Armadillo: API Documentation

[top] API Documentation for Armadillo 14.2 Preamble
For converting Matlab/Octave programs, see the syntax conversion table First time users: please see the short example program If you discover any bugs or regressions, please report them History of API additions |  |  | Please cite the following papers if you use Armadillo in your research and/or software. Citations are useful for the continued development and maintenance of the library. Conrad Sanderson and Ryan Curtin. Armadillo: a template-based C++ library for linear algebra. Journal of Open Source Software, Vol. 1, No. 2, pp. 26, 2016. Conrad Sanderson and Ryan Curtin. Practical Sparse Matrices in C++ with Hybrid Storage and Template-Based Expression Optimisation. Mathematical and Computational Applications, Vol. 24, No. 3, 2019.
Overview matrix, vector, cube and field classes member functions & variables generated vectors / matrices / cubes functions of vectors / matrices / cubes decompositions, factorisations, inverses and equation solvers (dense matrices) decompositions, factorisations, and equation solvers (sparse matrices) signal & image processing statistics and clustering miscellaneous (constants, configuration options, ...) Matrix, Vector, Cube and Field Classes
Mat<type>, mat, cx_mat |  | dense matrix class
Col<type>, colvec, vec |  | dense column vector class
Row<type>, rowvec |  | dense row vector class
Cube<type>, cube, cx_cube |  | dense cube class ("3D matrix")
field<object type> |  | class for storing arbitrary objects in matrix-like or cube-like layouts
SpMat<type>, sp_mat, sp_cx_mat |  | sparse matrix class
operators |  | `+ − * % / == != <= >= < > && ||`
Member Functions & Variables
attributes |  | .n_rows, .n_cols, .n_elem, .n_slices, ...
element access |  | element/object access via (), [] and .at()
element initialisation |  | set elements via initialiser lists
.zeros |  | set all elements to zero
.ones |  | set all elements to one
.eye |  | set elements along main diagonal to one and off-diagonal elements to zero
.randu / .randn |  | set all elements to random values
.fill |  | set all elements to specified value
.imbue |  | imbue (fill) with values provided by functor or lambda function
.clean |  | replace elements below a threshold with zeros
.replace |  | replace specific elements with a new value
.clamp |  | clamp values to lower and upper limits
.transform |  | transform each element via functor or lambda function
.for_each |  | apply a functor or lambda function to each element
.set_size |  | change size without keeping elements (fast)
.reshape |  | change size while keeping elements
.resize |  | change size while keeping elements and preserving layout
.copy_size |  | change size to be same as given object
.reset |  | change size to empty
submatrix views |  | read/write access to contiguous and non-contiguous submatrices
subcube views |  | read/write access to contiguous and non-contiguous subcubes
subfield views |  | read/write access to contiguous subfields
.diag |  | read/write access to matrix diagonals
.each_col / .each_row |  | vector operations applied to each column/row of matrix (aka "broadcasting")
.each_slice |  | matrix operations applied to each slice of cube (aka "broadcasting")
.set_imag / .set_real |  | set imaginary/real part
.insert_rows / cols / slices |  | insert vector/matrix/cube at specified row/column/slice
.shed_rows / cols / slices |  | remove specified rows/columns/slices
.swap_rows / cols |  | swap specified rows or columns
.swap |  | swap contents with given object
.memptr |  | raw pointer to memory
.colptr |  | raw pointer to memory used by specified column
iterators (matrices) |  | iterators and associated member functions for dense matrices and vectors
iterators (cubes) |  | iterators and associated member functions for cubes
iterators (sparse matrices) |  | iterators and associated member functions for sparse matrices
iterators (submatrices) |  | iterators and associated member functions for submatrices & subcubes
compat. container functions |  | compatibility container functions
.as_col / .as_row |  | return flattened matrix as column or row vector
.col_as_mat / .row_as_mat |  | return matrix representation of cube column or cube row
.as_dense |  | return dense vector/matrix representation of sparse matrix expression
.t / .st |  | return matrix transpose
.i |  | return inverse of square matrix
.min / .max |  | return extremum value
.index_min / .index_max |  | return index of extremum value
.eval |  | force evaluation of delayed expression
.in_range |  | check whether given location or span is valid
.is_empty |  | check whether object is empty
.is_vec |  | check whether matrix is a vector
.is_sorted |  | check whether vector or matrix is sorted
.is_trimatu / .is_trimatl |  | check whether matrix is upper/lower triangular
.is_diagmat |  | check whether matrix is diagonal
.is_square |  | check whether matrix is square sized
.is_symmetric |  | check whether matrix is symmetric
.is_hermitian |  | check whether matrix is hermitian
.is_sympd |  | check whether matrix is symmetric/hermitian positive definite
.is_zero |  | check whether all elements are zero
.is_finite |  | check whether all elements are finite
.has_inf |  | check whether any element is ±infinity
.has_nan |  | check whether any element is NaN
.print |  | print object to std::cout or user specified stream
.raw_print |  | print object without formatting
.brief_print |  | print object in shortened/abridged form
.save/.load (matrices & cubes) |  | save/load matrices and cubes in files or streams
.save/.load (fields) |  | save/load fields in files or streams
Generated Vectors / Matrices / Cubes
linspace |  | generate vector with linearly spaced elements
logspace |  | generate vector with logarithmically spaced elements
regspace |  | generate vector with regularly spaced elements
randperm |  | generate vector with random permutation of a sequence of integers
eye |  | generate identity matrix
ones |  | generate object filled with ones
zeros |  | generate object filled with zeros
randu |  | generate object with random values (uniform distribution)
randn |  | generate object with random values (normal distribution)
randg |  | generate object with random values (gamma distribution)
randi |  | generate object with random integer values in specified interval
speye |  | generate sparse identity matrix
spones |  | generate sparse matrix with non-zero elements set to one
sprandu / sprandn |  | generate sparse matrix with non-zero elements set to random values
toeplitz |  | generate Toeplitz matrix
Functions of Vectors / Matrices / Cubes
abs |  | obtain magnitude of each element
accu |  | accumulate (sum) all elements
affmul |  | affine matrix multiplication
all |  | check whether all elements are non-zero, or satisfy a relational condition
any |  | check whether any element is non-zero, or satisfies a relational condition
approx_equal |  | approximate equality
arg |  | phase angle of each element
as_scalar |  | convert 1x1 matrix to pure scalar
clamp |  | obtain clamped elements according to given limits
cond |  | condition number of matrix
conj |  | obtain complex conjugate of each element
conv_to |  | convert/cast between matrix types
cross |  | cross product
cumsum |  | cumulative sum
cumprod |  | cumulative product
det |  | determinant
diagmat |  | generate diagonal matrix from given matrix or vector
diagvec |  | extract specified diagonal
diags / spdiags |  | generate band matrix from given set of vectors
diff |  | differences between adjacent elements
dot / cdot / norm_dot |  | dot product
eps |  | obtain distance of each element to next largest floating point representation
expmat |  | matrix exponential
expmat_sym |  | matrix exponential of symmetric matrix
find |  | find indices of non-zero elements, or elements satisfying a relational condition
find_finite |  | find indices of finite elements
find_nonfinite |  | find indices of non-finite elements
find_nan |  | find indices of NaN elements
find_unique |  | find indices of unique elements
fliplr / flipud |  | flip matrix left to right or upside down
imag / real |  | extract imaginary/real part
ind2sub |  | convert linear index to subscripts
index_min / index_max |  | indices of extremum values
inplace_trans |  | in-place transpose
intersect |  | find common elements in two vectors/matrices
join_rows / join_cols |  | concatenation of matrices
join_slices |  | concatenation of cubes
kron |  | Kronecker tensor product
log_det |  | log determinant
log_det_sympd |  | log determinant of symmetric positive definite matrix
logmat |  | matrix logarithm
logmat_sympd |  | matrix logarithm of symmetric matrix
min / max |  | return extremum values
nonzeros |  | return non-zero values
norm |  | various norms of vectors and matrices
norm2est |  | fast estimate of the matrix 2-norm
normalise |  | normalise vectors to unit p-norm
pow |  | element-wise power
powmat |  | matrix power
prod |  | product of elements
rank |  | rank of matrix
rcond |  | reciprocal condition number
repelem |  | replicate elements
repmat |  | replicate matrix in block-like fashion
reshape |  | change size while keeping elements
resize |  | change size while keeping elements and preserving layout
reverse |  | reverse order of elements
roots |  | roots of polynomial
shift |  | shift elements
shuffle |  | randomly shuffle elements
size |  | obtain dimensions of given object
sort |  | sort elements
sort_index |  | vector describing sorted order of elements
sqrtmat |  | square root of matrix
sqrtmat_sympd |  | square root of symmetric matrix
sum |  | sum of elements
sub2ind |  | convert subscripts to linear index
symmatu / symmatl |  | generate symmetric matrix from given matrix
trace |  | sum of diagonal elements
trans |  | transpose of matrix
trapz |  | trapezoidal numerical integration
trimatu / trimatl |  | copy upper/lower triangular part
trimatu_ind / trimatl_ind |  | obtain indices of upper/lower triangular part
unique |  | return unique elements
vecnorm |  | obtain vector norm of each row or column of a matrix
vectorise |  | flatten matrix into vector
misc functions |  | miscellaneous element-wise functions: exp, log, sqrt, round, sign, ...
trig functions |  | trigonometric element-wise functions: cos, sin, tan, ...
Decompositions, Factorisations, Inverses and Equation Solvers (Dense Matrices)
chol |  | Cholesky decomposition
eig_sym |  | eigen decomposition of dense symmetric/hermitian matrix
eig_gen |  | eigen decomposition of dense general square matrix
eig_pair |  | eigen decomposition for pair of general dense square matrices
hess |  | upper Hessenberg decomposition
inv |  | inverse of general square matrix
inv_sympd |  | inverse of symmetric positive definite matrix
lu |  | lower-upper decomposition
null |  | orthonormal basis of null space
orth |  | orthonormal basis of range space
pinv |  | pseudo-inverse / generalised inverse
qr |  | QR decomposition
qr_econ |  | economical QR decomposition
qz |  | generalised Schur decomposition
schur |  | Schur decomposition
solve |  | solve systems of linear equations
svd |  | singular value decomposition
svd_econ |  | economical singular value decomposition
syl |  | Sylvester equation solver
Decompositions, Factorisations and Equation Solvers (Sparse Matrices)
eigs_sym |  | limited number of eigenvalues & eigenvectors of sparse symmetric real matrix
eigs_gen |  | limited number of eigenvalues & eigenvectors of sparse general square matrix
svds |  | truncated svd: limited number of singular values & singular vectors of sparse matrix
spsolve |  | solve sparse systems of linear equations
spsolve_factoriser |  | factoriser for solving sparse systems of linear equations
Signal & Image Processing
conv |  | 1D convolution
conv2 |  | 2D convolution
fft / ifft |  | 1D fast Fourier transform and its inverse
fft2 / ifft2 |  | 2D fast Fourier transform and its inverse
interp1 |  | 1D interpolation
interp2 |  | 2D interpolation
polyfit |  | find polynomial coefficients for data fitting
polyval |  | evaluate polynomial
Statistics & Clustering
stats functions |  | mean, median, standard deviation, variance
cov |  | covariance
cor |  | correlation
hist |  | histogram of counts
histc |  | histogram of counts with user specified edges
quantile |  | quantiles of a dataset
princomp |  | principal component analysis (PCA)
normpdf |  | probability density function of normal distribution
log_normpdf |  | logarithm version of probability density function of normal distribution
normcdf |  | cumulative distribution function of normal distribution
mvnrnd |  | random vectors from multivariate normal distribution
chi2rnd |  | random numbers from chi-squared distribution
wishrnd |  | random matrix from Wishart distribution
iwishrnd |  | random matrix from inverse Wishart distribution
running_stat |  | running statistics of scalars (one dimensional process/signal)
running_stat_vec |  | running statistics of vectors (multi-dimensional process/signal)
kmeans |  | cluster data into disjoint sets
gmm_diag/gmm_full |  | probabilistic clustering and likelihood calculation via mixture of Gaussians
Miscellaneous
constants |  | pi, inf, NaN, eps, speed of light, ...
wall_clock |  | timer for measuring number of elapsed seconds
RNG seed setting |  | functions for changing RNG seeds
output streams |  | streams for printing warnings and errors
uword / sword |  | shorthand for unsigned and signed integers
cx_double / cx_float |  | shorthand for std::complex<double> and std::complex<float>
Matlab/Armadillo syntax differences |  | examples of Matlab syntax and conceptually corresponding Armadillo syntax
example program |  | short example program
config.hpp |  | configuration options
API additions |  | API stability and list of API additions
- Matrix, Vector, Cube and Field Classes Mat<type> mat cx_mat Classes for dense matrices, with elements stored in column-major ordering (ie. column by column) The root matrix class is Mat<type>, where type is one of: float, double, std::complex<float>, std::complex<double>, short, int, long, and unsigned versions of short, int, long For convenience the following typedefs have been defined:
  `mat` | = | `Mat<double>`
  `dmat` | = | `Mat<double>`
  `fmat` | = | `Mat<float>`
  `cx_mat` | = | `Mat<cx_double>`
  `cx_dmat` | = | `Mat<cx_double>`
  `cx_fmat` | = | `Mat<cx_float>`
  `umat` | = | `Mat<uword>`
  `imat` | = | `Mat<sword>`
- In this documentation the mat type is used for convenience; it is possible to use other matrix types instead, eg. fmat Matrix types with integer elements (such as umat and imat) cannot hold special values such as NaN and Inf Functions which use LAPACK (generally matrix decompositions) are only valid for the following matrix types: mat, dmat, fmat, cx_mat, cx_dmat, cx_fmat Constructors:
  `mat()` |  |
  `mat(n_rows, n_cols)` |  |
  `mat(n_rows, n_cols, fill_form)` |  | (elements are initialised according to fill_form)
  `mat(size(X))` |  |
  `mat(size(X), fill_form)` |  | (elements are initialised according to fill_form)
  `mat(mat)` |  |
  `mat(vec)` |  |
  `mat(rowvec)` |  |
  `mat(initializer_list)` |  |
  `mat(string)` |  |
  `mat(std::vector)` |  | (treated as a column vector)
  `mat(sp_mat)` |  | (for converting a sparse matrix to a dense matrix)
  `cx_mat(mat,mat)` |  | (for constructing a complex matrix out of two real matrices)
- The elements can be explicitly initialised during construction by specifying fill_form, which is one of:
  `fill::zeros` | ↦ | set all elements to 0 (default in Armadillo >= 10.5)
  `fill::ones` | ↦ | set all elements to 1
  `fill::eye` | ↦ | set the elements on the main diagonal to 1 and off-diagonal elements to 0
  `fill::randu` | ↦ | set all elements to random values from a uniform distribution in the [0,1] interval
  `fill::randn` | ↦ | set all elements to random values from a normal/Gaussian distribution with zero mean and unit variance
  `fill::value(scalar)` | ↦ | set all elements to specified scalar
  `fill::none` | ↦ | do not initialise the elements (matrix may have garbage values)
- Caveat: in Armadillo >= 10.5, the default initialisation is `fill::zeros` in Armadillo <= 10.4, the default initialisation is `fill::none` For the mat(string) constructor, the format is elements separated by spaces, and rows denoted by semicolons; for example, the 2x2 identity matrix can be created using `"1 0; 0 1"`; note that string based initialisation is slower than directly setting the elements or using element initialisation Each instance of mat automatically allocates and releases internal memory. All internally allocated memory used by an instance of mat is automatically released as soon as the instance goes out of scope. For example, if an instance of mat is declared inside a function, it will be automatically destroyed at the end of the function. To forcefully release memory at any point, use .reset(); note that in normal use this is not required. Advanced constructors: `mat(ptr_aux_mem, n_rows, n_cols, copy_aux_mem = true, strict = false)` Create a matrix using data from writable auxiliary (external) memory, where ptr_aux_mem is a pointer to the memory. By default the matrix allocates its own memory and copies data from the auxiliary memory (for safety). However, if copy_aux_mem is set to false, the matrix will instead directly use the auxiliary memory (ie. no copying); this is faster, but can be dangerous unless you know what you are doing! The strict parameter comes into effect only when copy_aux_mem is set to false (ie. the matrix is directly using auxiliary memory) when strict is set to false, the matrix will use the auxiliary memory until a size change or an aliasing event when strict is set to true, the matrix will be bound to the auxiliary memory for its lifetime; the number of elements in the matrix can't be changed `mat(const ptr_aux_mem, n_rows, n_cols)` Create a matrix by copying data from read-only auxiliary memory, where ptr_aux_mem is a pointer to the memory `mat::fixed<n_rows, n_cols>` Create a fixed size matrix, with the size specified via template arguments. Memory for the matrix is reserved at compile time. This is generally faster than dynamic memory allocation, but the size of the matrix can't be changed afterwards (directly or indirectly). For convenience, there are several pre-defined typedefs for each matrix type (where the types are: umat, imat, fmat, mat, cx_fmat, cx_mat). The typedefs specify a square matrix size, ranging from 2x2 to 9x9. The typedefs were defined by appending a two digit form of the size to the matrix type; examples: mat33 is equivalent to mat::fixed<3,3>, while cx_mat44 is equivalent to cx_mat::fixed<4,4>. `mat::fixed<n_rows, n_cols>(fill_form)` Create a fixed size matrix, with the elements explicitly initialised according to fill_form `mat::fixed<n_rows, n_cols>(const ptr_aux_mem)` Create a fixed size matrix, with the size specified via template arguments; data is copied from auxiliary memory, where ptr_aux_mem is a pointer to the memory Examples:
      mat A(5, 5, fill::randu);
      double x = A(1,2);

      mat B = A + A;
      mat C = A * B;
      mat D = A % B;

      cx_mat X(A,B);

      B.zeros();
      B.set_size(10,10);
      B.ones(5,6);

      B.print("B:");

      mat::fixed<5,6> F;

      double aux_mem[24];
      mat H(&aux_mem[0], 4, 6, false);  // use auxiliary memory
- See also: matrix attributes accessing elements initialising elements math & relational operators submatrix views saving & loading matrices printing matrices element iterators .eval() conv_to() (convert between matrix types) Col class Row class Cube class SpMat class (sparse matrix with compressed sparse column format) config.hpp explanation of typedef (cplusplus.com) C data types (Wikipedia) Col<type> vec cx_vec Classes for column vectors (dense matrices with one column) The Col<type> class is derived from the Mat<type> class and inherits most of the member functions For convenience the following typedefs have been defined:
  `vec` | = | `colvec` | = | `Col<double>`
  `dvec` | = | `dcolvec` | = | `Col<double>`
  `fvec` | = | `fcolvec` | = | `Col<float>`
  `cx_vec` | = | `cx_colvec` | = | `Col<cx_double>`
  `cx_dvec` | = | `cx_dcolvec` | = | `Col<cx_double>`
  `cx_fvec` | = | `cx_fcolvec` | = | `Col<cx_float>`
  `uvec` | = | `ucolvec` | = | `Col<uword>`
  `ivec` | = | `icolvec` | = | `Col<sword>`
- The vec and colvec types have the same meaning and are used interchangeably In this documentation, the types vec or colvec are used for convenience; it is possible to use other column vector types instead, eg. fvec, fcolvec Functions which take Mat as input can generally also take Col as input; main exceptions are functions which require square matrices Constructors:
  `vec()` |  |
  `vec(n_elem)` |  |
  `vec(n_elem, fill_form)` |  | (elements are initialised according to fill_form)
  `vec(size(X))` |  |
  `vec(size(X), fill_form)` |  | (elements are initialised according to fill_form)
  `vec(vec)` |  |
  `vec(mat)` |  | (std::logic_error exception is thrown if the given matrix has more than one column)
  `vec(initializer_list)` |  |
  `vec(string)` |  | (elements separated by spaces)
  `vec(std::vector)` |  |
  `cx_vec(vec,vec)` |  | (for constructing a complex vector out of two real vectors)
- Caveat: in Armadillo >= 10.5, the default initialisation is `fill::zeros` in Armadillo <= 10.4, the default initialisation is `fill::none` Advanced constructors: `vec(ptr_aux_mem, number_of_elements, copy_aux_mem = true, strict = false)` Create a column vector using data from writable auxiliary (external) memory, where ptr_aux_mem is a pointer to the memory. By default the vector allocates its own memory and copies data from the auxiliary memory (for safety). However, if copy_aux_mem is set to false, the vector will instead directly use the auxiliary memory (ie. no copying); this is faster, but can be dangerous unless you know what you are doing! The strict parameter comes into effect only when copy_aux_mem is set to false (ie. the vector is directly using auxiliary memory) when strict is set to false, the vector will use the auxiliary memory until a size change or an aliasing event when strict is set to true, the vector will be bound to the auxiliary memory for its lifetime; the number of elements in the vector can't be changed `vec(const ptr_aux_mem, number_of_elements)` Create a column vector by copying data from read-only auxiliary memory, where ptr_aux_mem is a pointer to the memory `vec::fixed<number_of_elements>` Create a fixed size column vector, with the size specified via the template argument. Memory for the vector is reserved at compile time. This is generally faster than dynamic memory allocation, but the size of the vector can't be changed afterwards (directly or indirectly). For convenience, there are several pre-defined typedefs for each vector type (where the types are: uvec, ivec, fvec, vec, cx_fvec, cx_vec as well as the corresponding colvec versions). The pre-defined typedefs specify vector sizes ranging from 2 to 9. The typedefs were defined by appending a single digit form of the size to the vector type; examples: vec3 is equivalent to vec::fixed<3>, while cx_vec4 is equivalent to cx_vec::fixed<4>. `vec::fixed<number_of_elements>(fill_form)` Create a fixed size column vector, with the elements explicitly initialised according to fill_form `vec::fixed<number_of_elements>(const ptr_aux_mem)` Create a fixed size column vector, with the size specified via the template argument; data is copied from auxiliary memory, where ptr_aux_mem is a pointer to the memory Examples:
      vec x(10);
      vec y(10, fill::ones);

      mat A(10, 10, fill::randu);
      vec z = A.col(5); // extract a column vector
- See also: element initialisation Mat class Row class Row<type> rowvec cx_rowvec Classes for row vectors (dense matrices with one row) The template Row<type> class is derived from the Mat<type> class and inherits most of the member functions For convenience the following typedefs have been defined:
  `rowvec` | = | `Row<double>`
  `drowvec` | = | `Row<double>`
  `frowvec` | = | `Row<float>`
  `cx_rowvec` | = | `Row<cx_double>`
  `cx_drowvec` | = | `Row<cx_double>`
  `cx_frowvec` | = | `Row<cx_float>`
  `urowvec` | = | `Row<uword>`
  `irowvec` | = | `Row<sword>`
- In this documentation, the rowvec type is used for convenience; it is possible to use other row vector types instead, eg. frowvec Functions which take Mat as input can generally also take Row as input; main exceptions are functions which require square matrices Constructors:
  `rowvec()` |  |
  `rowvec(n_elem)` |  |
  `rowvec(n_elem, fill_form)` |  | (elements are initialised according to fill_form)
  `rowvec(size(X))` |  |
  `rowvec(size(X), fill_form)` |  | (elements are initialised according to fill_form)
  `rowvec(rowvec)` |  |
  `rowvec(mat)` |  | (std::logic_error exception is thrown if the given matrix has more than one row)
  `rowvec(initializer_list)` |  |
  `rowvec(string)` |  | (elements separated by spaces)
  `rowvec(std::vector)` |  |
  `cx_rowvec(rowvec,rowvec)` |  | (for constructing a complex row vector out of two real row vectors)
- Caveat: in Armadillo >= 10.5, the default initialisation is `fill::zeros` in Armadillo <= 10.4, the default initialisation is `fill::none` Advanced constructors: `rowvec(ptr_aux_mem, number_of_elements, copy_aux_mem = true, strict = false)` Create a row vector using data from writable auxiliary (external) memory, where ptr_aux_mem is a pointer to the memory. By default the vector allocates its own memory and copies data from the auxiliary memory (for safety). However, if copy_aux_mem is set to false, the vector will instead directly use the auxiliary memory (ie. no copying); this is faster, but can be dangerous unless you know what you are doing! The strict parameter comes into effect only when copy_aux_mem is set to false (ie. the vector is directly using auxiliary memory) when strict is set to false, the vector will use the auxiliary memory until a size change or an aliasing event when strict is set to true, the vector will be bound to the auxiliary memory for its lifetime; the number of elements in the vector can't be changed `rowvec(const ptr_aux_mem, number_of_elements)` Create a row vector by copying data from read-only auxiliary memory, where ptr_aux_mem is a pointer to the memory `rowvec::fixed<number_of_elements>` Create a fixed size row vector, with the size specified via the template argument. Memory for the vector is reserved at compile time. This is generally faster than dynamic memory allocation, but the size of the vector can't be changed afterwards (directly or indirectly). For convenience, there are several pre-defined typedefs for each vector type (where the types are: urowvec, irowvec, frowvec, rowvec, cx_frowvec, cx_rowvec). The pre-defined typedefs specify vector sizes ranging from 2 to 9. The typedefs were defined by appending a single digit form of the size to the vector type; examples: rowvec3 is equivalent to rowvec::fixed<3>, while cx_rowvec4 is equivalent to cx_rowvec::fixed<4>. `rowvec::fixed<number_of_elements>(fill_form)` Create a fixed size row vector, with the elements explicitly initialised according to fill_form `rowvec::fixed<number_of_elements>(const ptr_aux_mem)` Create a fixed size row vector, with the size specified via the template argument; data is copied from auxiliary memory, where ptr_aux_mem is a pointer to the memory Examples:
      rowvec x(10);
      rowvec y(10, fill::ones);

      mat    A(10, 10, fill::randu);
      rowvec z = A.row(5); // extract a row vector
- See also: element initialisation Mat class Col class Cube<type> cube cx_cube Classes for cubes (quasi 3rd order tensors), also known as "3D matrices" Data is stored as a set of slices (matrices) stored contiguously within memory; within each slice, elements are stored with column-major ordering (ie. column by column) The root cube class is Cube<type>, where type is one of: float, double, std::complex<float>, std::complex<double>, short, int, long and unsigned versions of short, int, long For convenience the following typedefs have been defined:
  `cube` | = | `Cube<double>`
  `dcube` | = | `Cube<double>`
  `fcube` | = | `Cube<float>`
  `cx_cube` | = | `Cube<cx_double>`
  `cx_dcube` | = | `Cube<cx_double>`
  `cx_fcube` | = | `Cube<cx_float>`
  `ucube` | = | `Cube<uword>`
  `icube` | = | `Cube<sword>`
- In this documentation the cube type is used for convenience; it is possible to use other types instead, eg. fcube Constructors:
  `cube()`
  `cube(n_rows, n_cols, n_slices)` |  |
  `cube(n_rows, n_cols, n_slices, fill_form)` |  | (elements are initialised according to fill_form)
  `cube(size(X))` |  |
  `cube(size(X), fill_form)` |  | (elements are initialised according to fill_form)
  `cube(cube)` |  |
  `cx_cube(cube, cube)` |  | (for constructing a complex cube out of two real cubes)
- The elements can be explicitly initialised during construction by specifying fill_form, which is one of:
  `fill::zeros` | ↦ | set all elements to 0 (default in Armadillo >= 10.5)
  `fill::ones` | ↦ | set all elements to 1
  `fill::randu` | ↦ | set all elements to random values from a uniform distribution in the [0,1] interval
  `fill::randn` | ↦ | set all elements to random values from a normal/Gaussian distribution with zero mean and unit variance
  `fill::value(scalar)` | ↦ | set all elements to specified scalar
  `fill::none` | ↦ | do not initialise the elements (cube may have garbage values)
- Caveat: in Armadillo >= 10.5, the default initialisation is `fill::zeros` in Armadillo <= 10.4, the default initialisation is `fill::none` Each instance of cube automatically allocates and releases internal memory. All internally allocated memory used by an instance of cube is automatically released as soon as the instance goes out of scope. For example, if an instance of cube is declared inside a function, it will be automatically destroyed at the end of the function. To forcefully release memory at any point, use .reset(); note that in normal use this is not required. Advanced constructors: `cube::fixed<n_rows, n_cols, n_slices>` Create a fixed size cube, with the size specified via template arguments. Memory for the cube is reserved at compile time. This is generally faster than dynamic memory allocation, but the size of the cube can't be changed afterwards (directly or indirectly). `cube(ptr_aux_mem, n_rows, n_cols, n_slices, copy_aux_mem = true, strict = false)` Create a cube using data from writable auxiliary (external) memory, where ptr_aux_mem is a pointer to the memory. By default the cube allocates its own memory and copies data from the auxiliary memory (for safety). However, if copy_aux_mem is set to false, the cube will instead directly use the auxiliary memory (ie. no copying); this is faster, but can be dangerous unless you know what you are doing! The strict parameter comes into effect only when copy_aux_mem is set to false (ie. the cube is directly using auxiliary memory) when strict is set to false, the cube will use the auxiliary memory until a size change or an aliasing event when strict is set to true, the cube will be bound to the auxiliary memory for its lifetime; the number of elements in the cube can't be changed `cube(const ptr_aux_mem, n_rows, n_cols, n_slices)` Create a cube by copying data from read-only auxiliary memory, where ptr_aux_mem is a pointer to the memory Examples:
      cube x(1, 2, 3);
      cube y(4, 5, 6, fill::randu);

      mat A = y.slice(1);  // extract a slice from the cube
                           // (each slice is a matrix)

      mat B(4, 5, fill::randu);
      y.slice(2) = B;     // set a slice in the cube

      cube q = y + y;     // cube addition
      cube r = y % y;     // element-wise cube multiplication

      cube::fixed<4,5,6> f;
      f.ones();
- - Notes: Each cube slice can be interpreted as a matrix, hence functions which take Mat as input can generally also take cube slices as input The size of individual slices can't be changed. For example, the following will not work:
        cube c(5,6,7);
        c.slice(0) = randu<mat>(10,20); // wrong size
- See also: cube attributes accessing elements math & relational operators subcube views and slices saving & loading cubes element iterators field class Mat class field<object_type> Class for storing arbitrary objects in matrix-like or cube-like layouts Somewhat similar to a matrix or cube, but instead of each element being a scalar, each element can be a vector, or matrix, or cube Each element can have an arbitrary size (eg. in a field of matrices, each matrix can have a unique size) Constructors, where object_type is another class, eg. vec, mat, std::string, etc:
  `field<object_type>()`
  `field<object_type>(n_elem)`
  `field<object_type>(n_rows, n_cols)`
  `field<object_type>(n_rows, n_cols, n_slices)`
  `field<object_type>(size(X))`
  `field<object_type>(field<object_type>)`
- Caveat: to store a set of matrices of the same size, the Cube class is more efficient Examples:
      mat A = randn(2,3);
      mat B = randn(4,5);

      field<mat> F(2,1);
      F(0,0) = A;
      F(1,0) = B;

      F.print("F:");

      F.save("mat_field");
- See also: field attributes subfield views saving / loading fields Cube class SpMat<type> sp_mat sp_cx_mat Classes for sparse matrices; intended for storing large matrices, where most of the elements are zeros The root sparse matrix class is SpMat<type>, where type is one of: float, double, std::complex<float>, std::complex<double>, short, int, long and unsigned versions of short, int, long For convenience the following typedefs have been defined:
  `sp_mat` | = | `SpMat<double>`
  `sp_dmat` | = | `SpMat<double>`
  `sp_fmat` | = | `SpMat<float>`
  `sp_cx_mat` | = | `SpMat<cx_double>`
  `sp_cx_dmat` | = | `SpMat<cx_double>`
  `sp_cx_fmat` | = | `SpMat<cx_float>`
  `sp_umat` | = | `SpMat<uword>`
  `sp_imat` | = | `SpMat<sword>`
- In this documentation the sp_mat type is used for convenience; it is possible to use other types instead, eg. sp_fmat Constructors:
  `sp_mat()` |  |
  `sp_mat(n_rows, n_cols)` |  |
  `sp_mat(size(X))` |  |
  `sp_mat(sp_mat)` |  |
  `sp_mat(mat)` |  | (for converting a dense matrix to a sparse matrix)
  `sp_cx_mat(sp_mat,sp_mat)` |  | (for constructing a complex matrix out of two real matrices)
- All elements are treated as zero by default (ie. the matrix is initialised to contain zeros) Non-zero elements are stored in compressed sparse column (CSC) format (ie. column-major ordering); zero-valued elements are never stored This class behaves in a similar manner to the dense matrix Mat class; however, member functions which set all elements to non-zero values (and hence do not make sense for sparse matrices) have been deliberately omitted; examples of omitted functions: .fill(), .ones(), += scalar, etc. Caveat: the sparse matrix class is not intended for small matrices (eg. with size ≤ 100x100), due to the overhead of the compressed storage format; for small matrices use the Mat class, even if most of the elements are zeros Batch insertion constructors: form 1: `sp_mat(locations, values, sort_locations = true)` form 2: `sp_mat(locations, values, n_rows, n_cols, sort_locations = true, check_for_zeros = true)` form 3: `sp_mat(add_values, locations, values, n_rows, n_cols, sort_locations = true, check_for_zeros = true)` form 4: `sp_mat(rowind, colptr, values, n_rows, n_cols, check_for_zeros = true)` For forms 1, 2, 3, locations is a dense matrix of type umat, with a size of 2 x N, where N is the number of values to be inserted; the location of the i-th element is specified by the contents of the i-th column of the locations matrix, where the row is in locations(0,i), and the column is in locations(1,i) For form 4, rowind is a dense column vector of type uvec containing the row indices of the values to be inserted, and colptr is a dense column vector of type uvec (with length n_cols + 1) containing indices of values corresponding to the start of new columns; the vectors correspond to the arrays used by the compressed sparse column format; this form is useful for copying data from other CSC sparse matrix containers For all forms, values is a dense column vector containing the values to be inserted; it must have the same element type as the sparse matrix. For forms 1 and 2, the value in values[i] will be inserted at the location specified by the i-th column of the locations matrix. For form 3, add_values is either true or false; when set to true, identical locations are allowed, and the values at identical locations are added The size of the constructed matrix is either automatically determined from the maximal locations in the locations matrix (form 1), or manually specified via n_rows and n_cols (forms 2, 3, 4) If sort_locations is set to false, the locations matrix is assumed to contain locations that are already sorted according to column-major ordering; do not set this to false unless you know what you are doing! If check_for_zeros is set to false, the values vector is assumed to contain no zero values; do not set this to false unless you know what you are doing! The following subset of operations & functions is available for sparse matrices: fundamental arithmetic operations (such as addition and multiplication) submatrix views: most contiguous forms and the non-contiguous form of X.cols(vector_of_column_indices) diagonal views saving and loading (using arma_binary, coord_ascii, and csv_ascii formats) element-wise functions: abs(), cbrt(), ceil(), conj(), floor(), imag(), real(), round(), sign(), sqrt(), square(), trunc() scalar functions of matrices: accu(), as_scalar(), dot(), norm(), norm2est(), trace() vector valued functions of matrices: diagvec(), min(), max(), nonzeros(), sum(), mean(), stddev(), var(), vecnorm(), vectorise() matrix valued functions of matrices: clamp(), diagmat(), spdiags(), flipud()/fliplr(), join_rows(), join_cols(), kron(), normalise(), repelem(), repmat(), reshape(), resize(), reverse(), shift(), symmatu()/symmatl(), trimatu()/trimatl(), .t(), trans() generated matrices: speye(), spones(), sprandu(), sprandn(), zeros() eigen decompositions and SVD: eigs_sym(), eigs_gen(), svds() solution of sparse linear systems: spsolve() miscellaneous: approx_equal(), element access, element iterators, .as_col() / .as_row(), .as_dense(), .for_each(), .print(), .clean(), .replace(), .transform(), .is_finite(), .is_symmetric(), .is_hermitian(), .is_trimatu(), .is_trimatl(), .is_diagmat() Examples:
      sp_mat A = sprandu(1000, 2000, 0.01);
      sp_mat B = sprandu(2000, 1000, 0.01);

      sp_mat C = 2*B;
      sp_mat D = A*C;

      sp_mat E(1000,1000);
      E(1,2) = 123;

      // batch insertion of 3 values at
      // locations (1, 2), (7, 8), (9, 9)

      umat locations = { { 1, 7, 9 },
                         { 2, 8, 9 } };

      vec values = { 1.0, 2.0, 3.0 };

      sp_mat X(locations, values);
- See also: element access element iterators (sparse matrices) printing matrices Sparse Matrix in Wikipedia Mat class (dense matrix) operators: `+ − * % / == != <= >= < > && ||` Overloaded operators for Mat, Col, Row and Cube classes Operations:
  `+` |  | addition of two objects
  `−` |  | subtraction of one object from another or negation of an object
  `*` |  | matrix multiplication of two objects; not applicable to the Cube class unless multiplying by a scalar
  `%` |  | element-wise multiplication of two objects (Schur product)
  `/` |  | element-wise division of an object by another object or a scalar
  `==` |  | element-wise equality evaluation of two objects; generates a matrix/cube of type umat/ucube
  `!=` |  | element-wise non-equality evaluation of two objects; generates a matrix/cube of type umat/ucube
  `>=` |  | element-wise "greater than or equal to" evaluation of two objects; generates a matrix/cube of type umat/ucube
  `<=` |  | element-wise "less than or equal to" evaluation of two objects; generates a matrix/cube of type umat/ucube
  `>` |  | element-wise "greater than" evaluation of two objects; generates a matrix/cube of type umat/ucube
  `<` |  | element-wise "less than" evaluation of two objects; generates a matrix/cube of type umat/ucube
  `&&` |  | element-wise logical AND evaluation of two objects; generates a matrix/cube of type umat/ucube
  `||` |  | element-wise logical OR evaluation of two objects; generates a matrix/cube of type umat/ucube
- For element-wise relational and logical operations (ie. `==`, `!=`, `>=`, `<=`, `>`, `<`, `&&`, `||`) each element in the generated object is either 0 or 1, depending on the result of the operation Caveat: operators involving equality comparison (ie. `==`, `!=`, `>=`, `<=`) are not recommended for matrices of type mat or fmat, due to the necessarily limited precision of floating-point element types; consider using approx_equal() instead If the `+`, `−` and `%` operators are chained, Armadillo aims to avoid the generation of temporaries; no temporaries are generated if all given objects are of the same type and size If the `*` operator is chained, Armadillo aims to find an efficient ordering of the matrix multiplications Broadcasting operations are available via .each_col(), .each_row(), .each_slice() If incompatible object sizes are used, a std::logic_error exception is thrown Examples:
      mat A(5, 10, fill::randu);
      mat B(5, 10, fill::randu);
      mat C(10, 5, fill::randu);

      mat P = A + B;
      mat Q = A - B;
      mat R = -B;
      mat S = A / 123.0;
      mat T = A % B;
      mat U = A * C;

      // V is constructed without temporaries
      mat V = A + B + A + B;

      imat AA = "1 2 3; 4 5 6; 7 8 9;";
      imat BB = "3 2 1; 6 5 4; 9 8 7;";

      // compare elements
      umat ZZ = (AA >= BB);
See also: approx_equal() pow() any() all() affmul() accu() as_scalar() find() .replace() .transform() .each_col() & .each_row() (vector operations applied to each column or row) miscellaneous element-wise functions (exp, log, sqrt, square, round, ...) floating point arithmetic in Wikipedia floating point representation in MathWorld Member Functions & Variables attributes
`.n_rows` |  | number of rows; present in Mat, Col, Row, Cube, field and SpMat
`.n_cols` |  | number of columns; present in Mat, Col, Row, Cube, field and SpMat
`.n_elem` |  | total number of elements; present in Mat, Col, Row, Cube, field and SpMat
`.n_slices` |  | number of slices; present in Cube and field
`.n_nonzero` |  | number of non-zero elements; present in SpMat
- The variables are of type uword The variables are read-only; to change the size, use .set_size(), .copy_size(), .zeros(), .ones(), or .reset() For the Col and Row classes, n_elem also indicates vector length Examples:
      mat X(4,5);
      cout << "X has " << X.n_cols << " columns" << endl;
- See also: .set_size() .copy_size() .zeros() .ones() .reset() size() element/object access via (), [] and .at() Provide access to individual elements or objects stored in a container object (ie. Mat, Col, Row, Cube, field)
  `(i)` |  | For vec and rowvec, access the element stored at index i. For mat, cube and field, access the element/object stored at index i under the assumption of a flat layout, with column-major ordering of data (ie. column by column). An exception is thrown if the requested element is out of bounds.
  `.at(i)` or `[i]` |  | As for `(i)`, but without a bounds check; not recommended; see the caveats below
  `(r,c)` |  | For mat and 2D field classes, access the element/object stored at row r and column c. An exception is thrown if the requested element is out of bounds.
  `.at(r,c)` |  | As for `(r,c)`, but without a bounds check; not recommended; see the caveats below |
  `(r,c,s)` |  | For cube and 3D field classes, access the element/object stored at row r, column c, and slice s. An exception is thrown if the requested element is out of bounds.
  `.at(r,c,s)` |  | As for `(r,c,s)`, but without a bounds check; not recommended; see the caveats below
- The indices of elements are specified via the uword type, which is a typedef for an unsigned integer type. When using loops to access elements, it is best to use uword instead of int. For example: `for(uword i=0; i<X.n_elem; ++i) { X(i) = ... }` Caveats: accessing elements without bounds checks is slightly faster, but is not recommended until your code has been thoroughly debugged first indexing in C++ starts at 0 accessing elements via `[r,c]` and `[r,c,s]` does not work correctly in C++; instead use `(r,c)` and `(r,c,s)` Examples:
      mat M(10, 10, fill::randu);
      M(9,9) = 123.0;
      double x = M(1,2);

      vec v(10, fill::randu);
      v(9) = 123.0;
      double y = v(0);
- See also: .in_range() element initialisation ind2sub() sub2ind() .index_min() / .index_max() submatrix views .memptr() .transform() .for_each() iterators (dense matrices) iterators (cubes) iterators (sparse matrices) config.hpp element initialisation Set elements in Mat, Col, Row via braced initialiser lists Examples:
      vec v = { 1, 2, 3 };

      mat A = { {1, 3, 5},
                {2, 4, 6} };
See also: element access .reshape() .print() saving & loading matrices advanced constructors (matrices)
.zeros() |  |  | (member function of Mat, Col, Row, SpMat, Cube)
.zeros( n_elem ) |  |  | (member function of Col and Row)
.zeros( n_rows, n_cols ) |  |  | (member function of Mat and SpMat)
.zeros( n_rows, n_cols, n_slices ) |  |  | (member function of Cube)
.zeros( size(X) ) |  |  | (member function of Mat, Col, Row, Cube, SpMat)
- Set the elements of an object to zero, optionally first changing the size to specified dimensions Examples:
      mat A;
      A.zeros(5, 10);   // or:  mat A(5, 10, fill::zeros);

      mat B;
      B.zeros( size(A) );

      mat C(5, 10, fill::randu);
      C.zeros();
See also: zeros() (standalone function) .ones() .clean() .is_zero() .randu() .fill() .imbue() .reset() .set_size() size()
.ones() |  |  | (member function of Mat, Col, Row, Cube)
.ones( n_elem ) |  |  | (member function of Col and Row)
.ones( n_rows, n_cols ) |  |  | (member function of Mat)
.ones( n_rows, n_cols, n_slices ) |  |  | (member function of Cube)
.ones( size(X) ) |  |  | (member function of Mat, Col, Row, Cube)
- Set all the elements of an object to one, optionally first changing the size to specified dimensions Examples:
      mat A;
      A.ones(5, 10);   // or:  mat A(5, 10, fill::ones);

      mat B;
      B.ones( size(A) );

      mat C(5, 10, fill::randu);
      C.ones();
- See also: ones() (standalone function) .eye() .zeros() .fill() .imbue() .randu() size() .eye() .eye( n_rows, n_cols ) .eye( size(X) ) Member functions of Mat and SpMat Set the elements along the main diagonal to one and off-diagonal elements to zero, optionally first changing the size to specified dimensions An identity matrix is generated when n_rows = n_cols Examples:
      mat A;
      A.eye(5, 5);  // or:  mat A(5, 5, fill::eye);

      mat B;
      B.eye( size(A) );

      mat C(5, 5, fill::randu);
      C.eye();
See also: .ones() .diag() diagmat() diagvec() eye() (standalone function) size()
.randu() |  |  | (member function of Mat, Col, Row, Cube)
.randu( n_elem ) |  |  | (member function of Col and Row)
.randu( n_rows, n_cols ) |  |  | (member function of Mat)
.randu( n_rows, n_cols, n_slices ) |  |  | (member function of Cube)
.randu( size(X) ) |  |  | (member function of Mat, Col, Row, Cube)
.randn() |  |  | (member function of Mat, Col, Row, Cube)
.randn( n_elem ) |  |  | (member function of Col and Row)
.randn( n_rows, n_cols ) |  |  | (member function of Mat)
.randn( n_rows, n_cols, n_slices ) |  |  | (member function of Cube)
.randn( size(X) ) |  |  | (member function of Mat, Col, Row, Cube)
- Set all the elements to random values, optionally first changing the size to specified dimensions .randu() uses a uniform distribution in the [0,1] interval .randn() uses a normal/Gaussian distribution with zero mean and unit variance For complex elements, the real and imaginary parts are treated separately Examples:
      mat A;
      A.randu(5, 10);   // or:  mat A(5, 10, fill::randu);

      mat B;
      B.randu( size(A) );

      mat C(5, 10, fill::zeros);
      C.randu();
- See also: randu() (standalone function with extended functionality) randn() (standalone function with extended functionality) .fill() .imbue() .ones() .zeros() size() RNG seed setting uniform distribution in Wikipedia normal distribution in Wikipedia .fill( value ) Member function of Mat, Col, Row, Cube, field Sets the elements to a specified value The type of value must match the type of elements used by the container object (eg. for mat the type is double) Examples:
      mat A(5, 6);

      A.fill(123.0);   // or:  mat A(5, 6, fill::value(123.0));
- Note: to set all elements to zero during matrix construction, use the following more compact form:
      mat A(5, 6, fill::zeros);
- See also: .imbue() .ones() .zeros() .randu() & .randn() .replace() constants (pi, nan, inf, ...) .imbue( functor ) .imbue( lambda_function ) Member functions of Mat, Col, Row and Cube Imbue (fill) with values provided by a functor or lambda function For matrices, filling is done column-by-column (ie. column 0 is filled, then column 1, ...) For cubes, filling is done slice-by-slice, with each slice treated as a matrix Examples:
      std::mt19937 engine;  // Mersenne twister random number engine

      std::uniform_real_distribution<double> distr(0.0, 1.0);

      mat A(4, 5, fill::none);

      A.imbue( [&]() { return distr(engine); } );
- See also: .fill() .transform() element access function object at Wikipedia C++11 lambda functions at Wikipedia lambda function at cprogramming.com .clean( threshold ) Member function of Mat, Col, Row, Cube and SpMat For objects with non-complex elements: each element with an absolute value ≤ threshold is replaced by zero For objects with complex elements: for each element, each component (real and imaginary) with an absolute value ≤ threshold is replaced by zero Can be used to sparsify a matrix, in the sense of zeroing values with small magnitudes Caveat: to explicitly convert from dense storage to sparse storage, use the SpMat class Examples:
      sp_mat A;

      A.sprandu(1000, 1000, 0.01);

      A(12,34) =  datum::eps;
      A(56,78) = -datum::eps;

      A.clean(datum::eps);
- See also: .replace() .clamp() .transform() .is_zero() .zeros() nonzeros() datum::eps .replace( old_value, new_value ) Member function of Mat, Col, Row, Cube and SpMat For all elements equal to old_value, set them to new_value The type of old_value and new_value must match the type of elements used by the container object (eg. for mat the type is double) Caveats: floating point numbers (float and double) are approximations due to their necessarily limited precision for sparse matrices (SpMat), replacement is not done when old_value = 0 Examples:
      mat A(5, 6, fill::randu);

      A.diag().fill(datum::nan);

      A.replace(datum::nan, 0);  // replace each NaN with 0
- See also: .transform() .for_each() .clean() .clamp() .fill() .has_nan() .has_inf() find() relational operators constants (pi, nan, inf, ...) .clamp( min_value, max_value ) Member function of Mat, Col, Row, Cube and SpMat Clamp each element to the [min_val,max_val] interval; any value lower than min_val will be set to min_val, and any value higher than max_val will be set to max_val For complex elements, the real and imaginary components are clamped separately For sparse matrices, clamping is applied only to the non-zero elements Examples:
      mat A(5, 6, fill::randu);

      A.clamp(0.2, 0.8);
- See also: .replace() .clean() .transform() .for_each() clamp() (standalone function) relational operators .transform( functor ) .transform( lambda_function ) Member functions of Mat, Col, Row, Cube and SpMat Transform each element using a functor or lambda function For dense matrices, transformation is done column-by-column for all elements For sparse matrices, transformation is done column-by-column for non-zero elements For cubes, transformation is done slice-by-slice, with each slice treated as a matrix Examples:
      mat A(4, 5, fill::ones);

      // add 123 to every element
      A.transform( [](double val) { return (val + 123.0); } );
- See also: .for_each() .replace() .imbue() .clean() .clamp() element access overloaded operators miscellaneous element-wise functions (exp, log, sqrt, square, round, ...) function object at Wikipedia C++11 lambda functions at Wikipedia lambda function at cprogramming.com .for_each( functor ) .for_each( lambda_function ) Member functions of Mat, Col, Row, Cube, SpMat and field For each element, pass its reference to a functor or lambda function For dense matrices and fields, the processing is done column-by-column for all elements For sparse matrices, the processing is done column-by-column for non-zero elements For cubes, processing is done slice-by-slice, with each slice treated as a matrix Examples:
      // add 123 to each element in a dense matrix

      mat A(4, 5, fill::ones);

      A.for_each( [](mat::elem_type& val) { val += 123.0; } );  // NOTE: the '&' is crucial!

      // add 123 to each non-zero element in a sparse matrix

      sp_mat S; S.sprandu(1000, 2000, 0.1);

      S.for_each( [](sp_mat::elem_type& val) { val += 123.0; } );  // NOTE: the '&' is crucial!

      // set the size of all matrices in field F

      field<mat> F(2,3);

      F.for_each( [](mat& X) { X.zeros(4,5); } );  // NOTE: the '&' is crucial!
See also: .transform() .replace() .each_col() & .each_row() .each_slice() element access miscellaneous element-wise functions (exp, log, sqrt, square, round, ...) function object at Wikipedia C++11 lambda functions at Wikipedia lambda function at cprogramming.com
.set_size( n_elem ) |  |  | (member function of Col, Row, field)
.set_size( n_rows, n_cols ) |  |  | (member function of Mat, SpMat, field)
.set_size( n_rows, n_cols, n_slices ) |  |  | (member function of Cube and field)
.set_size( size(X) ) |  |  | (member function of Mat, Col, Row, Cube, SpMat, field)
- Change the size of an object, without explicitly preserving data and without initialising the elements (ie. elements may contain garbage values, including NaN) To initialise the elements to zero while changing the size, use .zeros() instead To explicitly preserve data while changing the size, use .reshape() or .resize() instead; NOTE: .reshape() and .resize() are considerably slower than .set_size() Examples:
      mat A;
      A.set_size(5, 10);      // or:  mat A(5, 10, fill::none);

      mat B;
      B.set_size( size(A) );  // or:  mat B(size(A), fill::none);

      vec v;
      v.set_size(100);        // or:  vec v(100, fill::none);
See also: .reset() .copy_size() .reshape() .resize() .zeros() size()
.reshape( n_rows, n_cols ) |  |  | (member function of Mat and SpMat)
.reshape( n_rows, n_cols, n_slices ) |  |  | (member function of Cube)
.reshape( size(X) ) |  |  | (member function of Mat, Cube, SpMat)
- Recreate the object according to given size specifications, with the elements taken from the previous version of the object in a column-wise manner; the elements in the generated object are placed column-wise (ie. the first column is filled up before filling the second column) The layout of the elements in the recreated object will be different to the layout in the previous version of the object If the total number of elements in the previous version of the object is less than the specified size, the extra elements in the recreated object are set to zero If the total number of elements in the previous version of the object is greater than the specified size, only a subset of the elements is taken Caveats: to change the size without preserving data, use .set_size() instead, which is much faster to grow/shrink the object while preserving the elements as well as the layout of the elements, use .resize() instead to flatten a matrix into a vector, use vectorise() or .as_col() / .as_row() instead Examples:
      mat A(4, 5, fill::randu);

      A.reshape(5,4);
See also: .resize() .set_size() .copy_size() .zeros() .reset() .as_col() / .as_row() reshape() (standalone function) vectorise() size()
.resize( n_elem ) |  |  | (member function of Col, Row)
.resize( n_rows, n_cols ) |  |  | (member function of Mat and SpMat)
.resize( n_rows, n_cols, n_slices ) |  |  | (member function of Cube)
.resize( size(X) ) |  |  | (member function of Mat, Col, Row, Cube, SpMat)
- Recreate the object according to given size specifications, while preserving the elements as well as the layout of the elements Can be used for growing or shrinking an object (ie. adding/removing rows, and/or columns, and/or slices) Caveat: to change the size without preserving data, use .set_size() instead, which is much faster Examples:
      mat A(4, 5, fill::randu);

      A.resize(7,6);
- See also: .reshape() .set_size() .copy_size() .zeros() .reset() .insert_rows / cols / slices .shed_rows / cols / slices resize() (standalone function) vectorise() size() .copy_size( A ) Set the size to be the same as object A Object A must be of the same root type as the object being modified (eg. the size of a matrix can't be set by providing a cube) Examples:
      mat A(5, 6, fill::randu);

      mat B;
      B.copy_size(A);

      cout << B.n_rows << endl;
      cout << B.n_cols << endl;
- See also: .reset() .set_size() .reshape() .resize() .zeros() size() .reset() Reset the size to zero (the object will have no elements) Examples:
      mat A(5, 5, fill::randu);
      A.reset();
- See also: .set_size() .is_empty() .zeros() submatrix views A collection of member functions of Mat, Col and Row classes that provide read/write access to submatrix views
contiguous views for matrix X: X.col( col_number ) X.row( row_number ) X.cols( first_col, last_col ) X.rows( first_row, last_row ) X.submat( first_row, first_col, last_row, last_col ) X( span(first_row, last_row), span(first_col, last_col) ) X( first_row, first_col, size(n_rows, n_cols) ) X( first_row, first_col, size(Y) ) [ Y is a matrix ] X( span(first_row, last_row), col_number ) X( row_number, span(first_col, last_col) ) X.head_cols( number_of_cols ) X.head_rows( number_of_rows ) X.tail_cols( number_of_cols ) X.tail_rows( number_of_rows ) X.unsafe_col( col_number ) [ use with caution ] contiguous views for vector V: V( span(first_index, last_index) ) V.subvec( first_index, last_index ) V.subvec( first_index, size(W) ) [ W is a vector ] V.head( number_of_elements ) V.tail( number_of_elements )

non-contiguous views for matrix or vector X: X.elem( vector_of_indices ) X( vector_of_indices ) X.cols( vector_of_column_indices ) X.rows( vector_of_row_indices ) X.submat( vector_of_row_indices, vector_of_column_indices ) X( vector_of_row_indices, vector_of_column_indices ) related matrix views (documented separately) X.diag() X.each_row() X.each_col()

- Instances of span(start,end) can be replaced by span::all to indicate the entire range
- For functions requiring one or more vector of indices, eg. X.submat(vector_of_row_indices, vector_of_column_indices), each vector of indices must be of type uvec
- In the function X.elem(vector_of_indices), elements specified in vector_of_indices are accessed. X is interpreted as one long vector, with column-by-column ordering of the elements of X. The vector_of_indices must evaluate to a vector of type uvec (eg. generated by the find() function). The aggregate set of the specified elements is treated as a column vector (ie. the output of X.elem() is always a column vector).
- The function .unsafe_col() is provided for speed reasons and should be used only if you know what you are doing. It creates a seemingly independent Col vector object (eg. vec), but uses memory from the existing matrix object. As such, the created vector is not alias safe, and does not take into account that the underlying matrix memory could be freed (eg. due to any operation involving a size change of the matrix).
- Examples:

      mat A(5, 10, fill::zeros);

      A.submat( 0,1, 2,3 )      = randu<mat>(3,3);
      A( span(0,2), span(1,3) ) = randu<mat>(3,3);
      A( 0,1, size(3,3) )       = randu<mat>(3,3);

      mat B = A.submat( 0,1, 2,3 );
      mat C = A( span(0,2), span(1,3) );
      mat D = A( 0,1, size(3,3) );

      A.col(1)        = randu<mat>(5,1);
      A(span::all, 1) = randu<mat>(5,1);

      mat X(5, 5, fill::randu);

      // get all elements of X that are greater than 0.5
      vec q = X.elem( find(X > 0.5) );

      // add 123 to all elements of X greater than 0.5
      X.elem( find(X > 0.5) ) += 123.0;

      // set four specific elements of X to 1
      uvec indices = { 2, 3, 6, 8 };

      X.elem(indices) = ones<vec>(4);

      // add 123 to the last 5 elements of vector a
      vec a(10, fill::randu);
      a.tail(5) += 123.0;

      // add 123 to the first 3 elements of column 2 of X
      X.col(2).head(3) += 123;

- See also:
  - diagonal views
  - .each_col() & .each_row() (vector operations applied to each column or row)
  - .colptr()
  - .in_range()
  - find()
  - join_rows / cols / slices
  - .shed_rows / cols / slices
  - .insert_rows / cols / slices
  - size()
  - subcube views

subcube views and slices

- A collection of member functions of the Cube class that provide subcube views

contiguous views for cube Q: Q.slice( slice_number ) Q.slices( first_slice, last_slice ) Q.row( row_number ) Q.rows( first_row, last_row ) Q.col( col_number ) Q.cols( first_col, last_col ) Q.subcube( first_row, first_col, first_slice, last_row, last_col, last_slice ) Q( span(first_row, last_row), span(first_col, last_col), span(first_slice, last_slice) ) Q( first_row, first_col, first_slice, size(n_rows, n_cols, n_slices) ) Q( first_row, first_col, first_slice, size(R) ) [ R is a cube ] Q.head_slices( number_of_slices ) Q.tail_slices( number_of_slices ) Q.tube( row, col ) Q.tube( first_row, first_col, last_row, last_col ) Q.tube( span(first_row, last_row), span(first_col, last_col) ) Q.tube( first_row, first_col, size(n_rows, n_cols) ) |  |  |  | non-contiguous views for cube Q: Q.elem( vector_of_indices ) Q( vector_of_indices ) Q.slices( vector_of_slice_indices ) related cube views (documented separately) Q.each_slice() Q.col_as_mat() Q.row_as_mat()

- Instances of span(a,b) can be replaced by:
  - span() or span::all, to indicate the entire range
  - span(a), to indicate a particular row, column or slice
- An individual slice, accessed via .slice(), is an instance of the Mat class (a reference to a matrix is provided)
- All .tube() forms are variants of .subcube(), using first_slice = 0 and last_slice = Q.n_slices-1
- The .tube(row,col) form uses row = first_row = last_row, and col = first_col = last_col
- In the function Q.elem(vector_of_indices), elements specified in vector_of_indices are accessed. Q is interpreted as one long vector, with slice-by-slice and column-by-column ordering of the elements of Q. The vector_of_indices must evaluate to a vector of type uvec (eg. generated by the find() function). The aggregate set of the specified elements is treated as a column vector (ie. the output of Q.elem() is always a column vector).
- In the function Q.slices(vector_of_slice_indices), slices specified in vector_of_slice_indices are accessed. The vector_of_slice_indices must evaluate to a vector of type uvec.
- Examples:

      cube A(2, 3, 4, fill::randu);

      mat  B = A.slice(1); // each slice is a matrix

      A.slice(0) = randu<mat>(2,3);
      A.slice(0)(1,2) = 99.0;

      A.subcube(0,0,1,  1,1,2)             = randu<cube>(2,2,2);
      A( span(0,1), span(0,1), span(1,2) ) = randu<cube>(2,2,2);
      A( 0,0,1, size(2,2,2) )              = randu<cube>(2,2,2);

      // add 123 to all elements of A greater than 0.5
      A.elem( find(A > 0.5) ) += 123.0;

      cube C = A.head_slices(2);  // get first two slices

      A.head_slices(2) += 123.0;

- See also:
  - .in_range()
  - .each_slice()
  - .col_as_mat() / .row_as_mat()
  - join_slices()
  - shed_slices()
  - insert_slices()
  - size()
  - submatrix views

subfield views

- A collection of member functions of the field class that provide subfield views
- For a 2D field F, the subfields are accessed as:

F.row( row_number )
F.col( col_number )
F.rows( first_row, last_row )
F.cols( first_col, last_col )
F.subfield( first_row, first_col, last_row, last_col )
F( span(first_row, last_row), span(first_col, last_col) )
F( first_row, first_col, size(G) ) [ G is a 2D field ]
F( first_row, first_col, size(n_rows, n_cols) )

- For a 3D field F, the subfields are accessed as:

F.slice( slice_number )
F.slices( first_slice, last_slice )
F.subfield( first_row, first_col, first_slice, last_row, last_col, last_slice )
F( span(first_row, last_row), span(first_col, last_col), span(first_slice, last_slice) )
F( first_row, first_col, first_slice, size(G) ) [ G is a 3D field ]
F( first_row, first_col, first_slice, size(n_rows, n_cols, n_slices) )

- Instances of span(a,b) can be replaced by:
  - span() or span::all, to indicate the entire range
  - span(a), to indicate a particular row or column
- See also:
  - .in_range()
  - size()
  - submatrix views
  - subcube views

.diag()
.diag( k )

- Member function of Mat and SpMat
- Read/write access to a diagonal in a matrix
- The argument k is optional; by default k = 0
- The argument k specifies the diagonal to use:
  - k = 0 indicates the main diagonal (default setting)
  - k < 0 indicates the k-th sub-diagonal (below main diagonal, towards bottom-left corner)
  - k > 0 indicates the k-th super-diagonal (above main diagonal, towards top-right corner)
- The diagonal is interpreted as a column vector within expressions
- Note: to calculate only the diagonal elements of a compound expression, use diagvec() or diagmat()
- Examples:

      mat X(5, 5, fill::randu);

      vec a = X.diag();
      vec b = X.diag(1);
      vec c = X.diag(-2);

      X.diag() = randu<vec>(5);
      X.diag() += 6;
      X.diag().ones();

      sp_mat S = sprandu<sp_mat>(10,10,0.1);

      vec v(S.diag());  // copy sparse diagonal to dense vector

- See also:
  - .eye()
  - diagvec()
  - diagmat()
  - diags() / spdiags()
  - submatrix views
  - .each_col() & .each_row()
  - trace()

.each_col() |  | .each_row() |  | (form 1)
.each_col( vector_of_indices ) |  | .each_row( vector_of_indices ) |  | (form 2)
.each_col( lambda_function ) |  | .each_row( lambda_function ) |  | (form 3)

- Member functions of Mat
- Apply a vector operation to each column or row of a matrix
- Similar to "broadcasting" in Matlab / Octave
- Supported operations for forms 1 and 2:

  `+` |  | addition |  | `+=` |  | in-place addition
  `−` |  | subtraction |  | `−=` |  | in-place subtraction
  `%` |  | element-wise multiplication |  | `%=` |  | in-place element-wise multiplication
  `/` |  | element-wise division |  | `/=` |  | in-place element-wise division
  `=` |  | assignment (copy) |  |  |  |

- For form 2:
  - the argument vector_of_indices contains a list of indices of the columns/rows to be used; it must evaluate to a vector of type uvec
  - arithmetic operations as per form 1 are supported
- For form 3:
  - apply the given lambda_function to each column vector or row vector
  - the function must accept a reference to a Col or Row object with the same element type as the underlying matrix
- Examples:

      mat X(6, 5, fill::ones);
      vec v = linspace<vec>(10,15,6);

      X.each_col() += v;         // in-place addition of v to each column vector of X

      mat Y = X.each_col() + v;  // generate Y by adding v to each column vector of X

      // subtract v from columns 0 through to 3 in X
      X.cols(0,3).each_col() -= v;

      uvec indices(2);
      indices(0) = 2;
      indices(1) = 4;

      X.each_col(indices) = v;   // copy v to columns 2 and 4 in X

      X.each_col( [](vec& a){ a.print(); } );     // lambda function with non-const vector

      const mat& XX = X;
      XX.each_col( [](const vec& b){ b.print(); } );  // lambda function with const vector

- See also:
  - math & relational operators
  - submatrix views
  - diagonal views
  - repmat()
  - .for_each()
  - .each_slice()

.each_slice() |  | (form 1)
.each_slice( vector_of_indices ) |  | (form 2)
.each_slice( lambda_function ) |  | (form 3)
.each_slice( lambda_function, use_mp ) |  | (form 4)

- Member function of Cube
- Apply a matrix operation to each slice of a cube, with each slice treated as a matrix
- Similar to "broadcasting" in Matlab / Octave
- Supported operations for form 1:

  `+` |  | addition |  | `+=` |  | in-place addition
  `−` |  | subtraction |  | `−=` |  | in-place subtraction
  `%` |  | element-wise multiplication |  | `%=` |  | in-place element-wise multiplication
  `/` |  | element-wise division |  | `/=` |  | in-place element-wise division
  `*` |  | matrix multiplication |  | `*=` |  | in-place matrix multiplication
  `=` |  | assignment (copy) |  |  |  |

- For form 2:
  - the argument vector_of_indices contains a list of indices of the slices to be used; it must evaluate to a vector of type uvec
  - arithmetic operations as per form 1 are supported, except for `*` and `*=` (ie. matrix multiplication)
- For form 3:
  - apply the given lambda_function to each slice
  - the function must accept a reference to a Mat object with the same element type as the underlying cube
- For form 4:
  - apply the given lambda_function to each slice, as per form 3
  - the argument use_mp is a bool to enable the use of OpenMP for multi-threaded execution of lambda_function on multiple slices at the same time
  - the order of processing the slices is not deterministic (eg. slice 2 can be processed before slice 1)
  - lambda_function must be thread-safe, ie. it must not write to variables outside of its scope
- Examples:

      cube C(4, 5, 6, fill::randu);

      mat M = repmat(linspace<vec>(1,4,4), 1, 5);

      C.each_slice() += M;          // in-place addition of M to each slice of C

      cube D = C.each_slice() + M;  // generate D by adding M to each slice of C

      uvec indices(2);
      indices(0) = 2;
      indices(1) = 4;

      C.each_slice(indices) = M;    // copy M to slices 2 and 4 in C

      C.each_slice( [](mat& X){ X.print(); } );     // lambda function with non-const matrix

      const cube& CC = C;
      CC.each_slice( [](const mat& X){ X.print(); } );  // lambda function with const matrix

- See also:
  - math & relational operators
  - subcube views
  - .for_each()
  - .each_col() & .each_row()
  - lambda function at cprogramming.com

.set_imag( X )
.set_real( X )

- Set the imaginary/real part of an object
- X must have the same size as the recipient object
- Examples:

         mat A(4, 5, fill::randu);
         mat B(4, 5, fill::randu);

      cx_mat C(4, 5, fill::zeros);

      C.set_real(A);
      C.set_imag(B);

- Caveat: to directly construct a complex matrix out of two real matrices, the following code is faster:

         mat A(4, 5, fill::randu);
         mat B(4, 5, fill::randu);

      cx_mat C = cx_mat(A,B);

- See also:
  - matrix constructors
  - cube constructors
  - imag() / real()

.insert_rows( row_number, X ) .insert_rows( row_number, number_of_rows ) |  | (member functions of Mat, Col and Cube)
.insert_cols( col_number, X ) .insert_cols( col_number, number_of_cols ) |  | (member functions of Mat, Row and Cube)
.insert_slices( slice_number, X ) .insert_slices( slice_number, number_of_slices ) |  | (member functions of Cube)

- Functions with the X argument: insert a copy of X at the specified row/column/slice
  - if inserting rows, X must have the same number of columns (and slices) as the recipient object
  - if inserting columns, X must have the same number of rows (and slices) as the recipient object
  - if inserting slices, X must have the same number of rows and columns as the recipient object (ie. all slices must have the same size)
- Functions with the number_of_... argument:
  - expand the object by creating new rows/columns/slices
  - the elements in the new rows/columns/slices are set to zero
- Examples:

      mat A(5, 10, fill::randu);
      mat B(5,  2, fill::ones );

      // at column 2, insert a copy of B;
      // A will now have 12 columns
      A.insert_cols(2, B);

      // at column 1, insert 5 zeroed columns;
      // B will now have 7 columns
      B.insert_cols(1, 5);

- See also:
  - .shed_rows / cols / slices
  - join_rows / cols / slices
  - .resize()
  - submatrix views
  - subcube views

.shed_row( row_number ) .shed_rows( first_row, last_row ) .shed_rows( vector_of_indices ) |  | (member function of Mat, Col, SpMat, Cube) (member function of Mat, Col, SpMat, Cube) (member function of Mat, Col)
.shed_col( column_number ) .shed_cols( first_column, last_column ) .shed_cols( vector_of_indices ) |  | (member function of Mat, Row, SpMat, Cube) (member function of Mat, Row, SpMat, Cube) (member function of Mat, Row)
.shed_slice( slice_number ) .shed_slices( first_slice, last_slice ) .shed_slices( vector_of_indices ) |  | (member functions of Cube)

- Functions with single scalar argument: remove the specified row/column/slice
- Functions with two scalar arguments: remove the specified range of rows/columns/slices
- The vector_of_indices must evaluate to a vector of type uvec; it contains the indices of rows/columns/slices to remove
- Examples:

      mat A(5, 10, fill::randu);
      mat B(5, 10, fill::randu);

      A.shed_row(2);
      A.shed_cols(2,4);

      uvec indices = {4, 6, 8};
      B.shed_cols(indices);

- See also:
  - .insert_rows / cols / slices
  - join_rows / cols / slices
  - .resize()
  - submatrix views
  - subcube views

.swap_rows( row1, row2 )
.swap_cols( col1, col2 )

- Member functions of Mat, Col, Row and SpMat
- Swap the contents of specified rows or columns
- Examples:

      mat X(5, 5, fill::randu);
      X.swap_rows(0,4);

- See also:
  - reverse()
  - fliplr() & flipud()
  - .swap()

.swap( X )

- Member function of Mat, Col, Row and Cube
- Swap contents with object X
- Examples:

      mat A(4, 5, fill::zeros);
      mat B(6, 7, fill::ones );

      A.swap(B);

- See also:
  - .swap_rows() & .swap_cols()

.memptr()

- Member function of Mat, Col, Row and Cube
- Obtain a raw pointer to the memory used for storing elements
- Data for matrices is stored in a column-by-column order
- Data for cubes is stored in a slice-by-slice (matrix-by-matrix) order
- Caveat: the pointer becomes invalid after any operation involving a size change or aliasing
- Caveat: this function is not recommended for use unless you know what you are doing!
- Examples:

            mat A(5, 5, fill::randu);
      const mat B(5, 5, fill::randu);

            double* A_mem = A.memptr();
      const double* B_mem = B.memptr();

- See also:
  - .colptr()
  - submatrix views
  - element access
  - iterators (dense matrices)
  - iterators (cubes)
  - advanced constructors (matrices)
  - advanced constructors (cubes)

.colptr( col_number )

- Member function of Mat
- Obtain a raw pointer to the memory used by elements in the specified column
- Caveat: the pointer becomes invalid after any operation involving a size change or aliasing
- Caveat: this function is not recommended for use unless you know what you are doing -- it is safer to use submatrix views instead
- Examples:

      mat A(5, 5, fill::randu);

      double* mem = A.colptr(2);

- See also:
  - .memptr()
  - submatrix views
  - element access
  - iterators (dense matrices)
  - advanced constructors (matrices)

iterators (dense matrices & vectors)

- Iterators and associated member functions of Mat, Col, Row
- Iterators for dense matrices and vectors traverse over all elements within the specified range
- Member functions:

  .begin() |  | iterator referring to the first element
  .end() |  | iterator referring to the past-the-end element
  .begin_col( col_number ) |  | iterator referring to the first element of the specified column
  .end_col( col_number ) |  | iterator referring to the past-the-end element of the specified column
  .begin_row( row_number ) |  | iterator referring to the first element of the specified row
  .end_row( row_number ) |  | iterator referring to the past-the-end element of the specified row

- Iterator types:

  mat::iterator vec::iterator rowvec::iterator |  | random access iterators, for read/write access to elements (which are stored column by column)
  mat::const_iterator vec::const_iterator rowvec::const_iterator |  | random access iterators, for read-only access to elements (which are stored column by column)
  mat::col_iterator vec::col_iterator rowvec::col_iterator |  | random access iterators, for read/write access to the elements of specified columns
  mat::const_col_iterator vec::const_col_iterator rowvec::const_col_iterator |  | random access iterators, for read-only access to the elements of specified columns
  mat::row_iterator |  | bidirectional iterator, for read/write access to the elements of specified rows
  mat::const_row_iterator |  | bidirectional iterator, for read-only access to the elements of specified rows
  vec::row_iterator rowvec::row_iterator |  | random access iterators, for read/write access to the elements of specified rows
  vec::const_row_iterator rowvec::const_row_iterator |  | random access iterators, for read-only access to the elements of specified rows

- Examples:

      mat X(5, 6, fill::randu);

      mat::iterator it     = X.begin();
      mat::iterator it_end = X.end();

      for(; it != it_end; ++it)
        {
        cout << (*it) << endl;
        }

      mat::col_iterator col_it     = X.begin_col(1);  // start of column 1
      mat::col_iterator col_it_end = X.end_col(3);    //   end of column 3

      for(; col_it != col_it_end; ++col_it)
        {
        cout << (*col_it) << endl;
        (*col_it) = 123.0;
        }

- See also:
  - Mat class
  - element access
  - .for_each()
  - .memptr()
  - .colptr()
  - submatrix views
  - iterators (submatrices)
  - iterators (cubes)
  - iterators (sparse matrices)
  - iterator at cplusplus.com

iterators (cubes)

- Iterators and associated member functions of Cube
- Iterators for cubes traverse over all elements within the specified range
- Member functions:

  .begin() |  | iterator referring to the first element
  .end() |  | iterator referring to the past-the-end element
  .begin_slice( slice_number ) |  | iterator referring to the first element of the specified slice
  .end_slice( slice_number ) |  | iterator referring to the past-the-end element of the specified slice

- Iterator types:

  cube::iterator |  | random access iterator, for read/write access to elements; the elements are ordered slice by slice; the elements within each slice are ordered column by column
  cube::const_iterator |  | random access iterator, for read-only access to elements
  cube::slice_iterator |  | random access iterator, for read/write access to the elements of a particular slice; the elements are ordered column by column
  cube::const_slice_iterator |  | random access iterator, for read-only access to the elements of a particular slice

- Examples:

      cube X(2, 3, 4, fill::randu);

      cube::iterator it     = X.begin();
      cube::iterator it_end = X.end();

      for(; it != it_end; ++it)
        {
        cout << (*it) << endl;
        }

      cube::slice_iterator s_it     = X.begin_slice(1);  // start of slice 1
      cube::slice_iterator s_it_end = X.end_slice(2);    // end of slice 2

      for(; s_it != s_it_end; ++s_it)
        {
        cout << (*s_it) << endl;
        (*s_it) = 123.0;
        }

- See also:
  - Cube class
  - element access
  - .for_each()
  - .memptr()
  - subcube views
  - iterators (subcubes)
  - iterators (dense matrices)
  - iterator at cplusplus.com

iterators (sparse matrices)

- Iterators and associated member functions of SpMat
- Iterators for sparse matrices traverse over non-zero elements within the specified range
- Caveats:
  - writing a zero value into a sparse matrix through an iterator will invalidate all current iterators associated with the sparse matrix
  - to modify the non-zero elements in a safer manner, use .transform() or .for_each() instead of iterators
- Member functions:

  .begin() |  | iterator referring to the first element
  .end() |  | iterator referring to the past-the-end element
  .begin_col( col_number ) |  | iterator referring to the first element of the specified column
  .end_col( col_number ) |  | iterator referring to the past-the-end element of the specified column
  .begin_row( row_number ) |  | iterator referring to the first element of the specified row
  .end_row( row_number ) |  | iterator referring to the past-the-end element of the specified row

- Iterator types:

  sp_mat::iterator |  | bidirectional iterator, for read/write access to elements (which are stored column by column)
  sp_mat::const_iterator |  | bidirectional iterator, for read-only access to elements (which are stored column by column)
  sp_mat::col_iterator |  | bidirectional iterator, for read/write access to the elements of a specific column
  sp_mat::const_col_iterator |  | bidirectional iterator, for read-only access to the elements of a specific column
  sp_mat::row_iterator |  | bidirectional iterator, for read/write access to the elements of a specific row
  sp_mat::const_row_iterator |  | bidirectional iterator, for read-only access to the elements of a specific row

- The iterators have .row() and .col() functions which return the row and column of the current element; the returned values are of type uword
- Examples:

      sp_mat X = sprandu<sp_mat>(1000, 2000, 0.1);

      sp_mat::const_iterator it     = X.begin();
      sp_mat::const_iterator it_end = X.end();

      for(; it != it_end; ++it)
        {
        cout << "val: " << (*it)    << endl;
        cout << "row: " << it.row() << endl;
        cout << "col: " << it.col() << endl;
        }

- See also:
  - SpMat class
  - element access
  - .transform()
  - .for_each()
  - .replace()
  - submatrix views
  - iterators (dense matrices)
  - iterator at cplusplus.com

iterators (dense submatrices & subcubes)

- iterators for dense submatrix and subcube views, allowing range-based for loops
- Caveat: These iterators are intended only to be used with range-based for loops. Any other use is not supported. For example, the direct use of the begin() and end() functions, as well as the underlying iterators types is not supported. The implementation of submatrices and subcubes uses short-lived temporary objects that are subject to automatic deletion, and as such are error-prone to handle manually.
- Examples:

      mat X(100, 200, fill::randu);

      for( double& val : X(span(40,60), span(50,100)) )
        {
        cout << val << endl;
        val = 123.0;
        }

- See also:
  - submatrix views
  - .for_each()
  - iterators (dense matrices)
  - iterators (cubes)
  - range-based for (cppreference.com)

compatibility container functions

- Member functions to mimic the functionality of containers in the C++ standard library:

  .front() |  | access the first element in a vector
  .back() |  | access the last element in a vector
  .clear() |  | causes an object to have no elements
  .empty() |  | returns true if the object has no elements; returns false if the object has one or more elements
  .size() |  | returns the total number of elements

- Examples:

      mat A(5, 5, fill::randu);
      cout << A.size() << endl;

      A.clear();
      cout << A.empty() << endl;

- See also:
  - iterators (dense matrices)
  - iterators (cubes)
  - iterators (sparse matrices)
  - matrix and vector attributes
  - .is_empty()
  - .reset()

.as_col()
.as_row()

- Member functions of any matrix expression
- .as_col(): return a flattened version of the matrix as a column vector; flattening is done by concatenating all columns
- .as_row(): return a flattened version of the matrix as a row vector; flattening is done by concatenating all rows
- Caveat: concatenating columns is faster than concatenating rows
- Examples:

      mat X(4, 5, fill::randu);
      vec v = X.as_col();

- See also:
  - .reshape()
  - .t() / .st()
  - .as_dense()
  - as_scalar()
  - vectorise()

.col_as_mat(col_number)
.row_as_mat(row_number)

- Member functions of any cube expression
- .col_as_mat( col_number ):
  - return a matrix representation of the specified cube column
  - the number of rows is preserved
  - given a cube with size R x C x S, the resultant matrix size is R x S
- .row_as_mat( row_number ):
  - return a matrix representation of the specified cube row
  - the number of columns is preserved
  - given a cube with size R x C x S, the resultant matrix size is S x C
- Examples:

      cube Q(5, 4, 3, fill::randu);

      mat A = Q.col_as_mat(2);  // size of A: 5x3

      mat B = Q.row_as_mat(2);  // size of B: 3x4

- See also:
  - .slice()
  - vectorise()

.as_dense()

- Member function of any sparse matrix expression
- Returns a dense vector/matrix representation of the sparse matrix expression
- This function aims to avoid construction of the intermediate sparse matrix representation of the expression
  (applicable to some expressions)
- Examples:

      sp_mat A; A.sprandu(1000, 1000, 0.1);

      // store the sum of each column of A directly in dense row vector
      rowvec r = sum(A).as_dense();

      // extract column 123 of A directly into dense column vector
      colvec c = A.col(123).as_dense();

- See also:
  - as_scalar()
  - .as_col() / .as_row()

.t()
.st()

- Member functions of any matrix or vector expression
- For real (non-complex) matrix:
  - .t() provides a transposed copy of the matrix
  - .st() is not applicable
- For complex matrix:
  - .t() provides a Hermitian (conjugate) transposed copy (ie. signs of imaginary components are flipped)
  - .st() provides a simple transposed copy (ie. signs of imaginary components are not flipped)
- Examples:

      mat A(4, 5, fill::randu);
      mat B = A.t();

- See also:
  - trans()
  - reverse()
  - .as_col() / .as_row()
  - transpose in Wikipedia
  - transpose in MathWorld
  - conjugate transpose in Wikipedia
  - conjugate transpose in MathWorld

.i()

- Member function of any matrix expression
- Provides an inverse of the matrix expression
- If the matrix expression is not square sized, a std::logic_error exception is thrown
- If the matrix expression appears to be singular, the output matrix is reset and a std::runtime_error exception is thrown
- Caveats:
  - if matrix A is know to be symmetric positive definite, inv_sympd() is faster
  - to solve a system of linear equations, such as Z = inv(X)*Y, solve() can be faster and/or more accurate
- Examples:

      mat A(4, 4, fill::randu);

      mat X = A.i();

      mat Y = (A+A).i();

- See also:
  - inv()
  - rcond()
  - pinv()
  - solve()

.min()
.max()

- Return the extremum value of any matrix or cube expression
- For objects with complex numbers, absolute values are used for comparison
- Examples:

      mat A(5, 5, fill::randu);

      double max_val = A.max();

- See also:
  - .index_min() & .index_max()
  - min() & max() (standalone functions with extended functionality)
  - clamp()
  - running_stat
  - running_stat_vec

.index_min()
.index_max()

- Return the linear index of the extremum value of any matrix or cube expression
- For objects with complex numbers, absolute values are used for comparison
- The returned index is of type uword
- Examples:

      mat A(5, 5, fill::randu);

      uword i = A.index_max();

      double max_val = A(i);

- See also:
  - .min() & .max()
  - index_min() & index_max() (standalone functions with extended functionality)
  - ind2sub()
  - sort_index()
  - find()
  - element access

.eval()

- Member function of any matrix or vector expression
- Explicitly forces the evaluation of a delayed expression and outputs a matrix
- This function should be used sparingly and only in cases where it is absolutely necessary; indiscriminate use can degrade performance
- Examples:

      cx_mat A( randu<mat>(4,4), randu<mat>(4,4) );

      real(A).eval().save("A_real.dat", raw_ascii);
      imag(A).eval().save("A_imag.dat", raw_ascii);

- See also:
  - as_scalar()
  - Mat class

.in_range( i ) |  |  | (member of Mat, Col, Row, Cube, SpMat, field)
.in_range( span(start, end) ) |  |  | (member of Mat, Col, Row, Cube, SpMat, field)
.in_range( row, col ) |  |  | (member of Mat, Col, Row, SpMat, field)
.in_range( span(start_row, end_row), span(start_col, end_col) ) |  |  | (member of Mat, Col, Row, SpMat, field)
.in_range( row, col, slice ) |  |  | (member of Cube and field)
.in_range( span(start_row, end_row), span(start_col, end_col), span(start_slice, end_slice) ) |  |  | (member of Cube and field)
.in_range( first_row, first_col, size(X) ) (X is a matrix or field) |  |  | (member of Mat, Col, Row, SpMat, field)
.in_range( first_row, first_col, size(n_rows, n_cols) ) |  |  | (member of Mat, Col, Row, SpMat, field)
.in_range( first_row, first_col, first_slice, size(Q) ) (Q is a cube or field) |  |  | (member of Cube and field)
.in_range( first_row, first_col, first_slice, size(n_rows, n_cols n_slices) ) |  |  | (member of Cube and field)

- Returns true if the given location or span is currently valid
- Returns false if the object is empty, the location is out of bounds, or the span is out of bounds
- Instances of span(a,b) can be replaced by:
  - span() or span::all, to indicate the entire range
  - span(a), to indicate a particular row, column or slice
- Examples:

      mat A(4, 5, fill::randu);

      cout << A.in_range(0,0) << endl;  // true
      cout << A.in_range(3,4) << endl;  // true
      cout << A.in_range(4,5) << endl;  // false

- See also:
  - element access
  - submatrix views
  - subcube views
  - subfield views
  - .set_size()

.is_empty()

- Returns true if the object has no elements
- Returns false if the object has one or more elements
- Examples:

      mat A(5, 5, fill::randu);
      cout << A.is_empty() << endl;

      A.reset();
      cout << A.is_empty() << endl;

- See also:
  - .is_square()
  - .is_vec()
  - .is_finite()
  - .reset()

.is_vec()
.is_colvec()
.is_rowvec()

- Member functions of Mat and SpMat
- .is_vec():
  - returns true if the matrix can be interpreted as a vector (either column or row vector)
  - returns false if the matrix does not have exactly one column or one row
- .is_colvec():
  - returns true if the matrix can be interpreted as a column vector
  - returns false if the matrix does not have exactly one column
- .is_rowvec():
  - returns true if the matrix can be interpreted as a row vector
  - returns false if the matrix does not have exactly one row
- Caveat: do not assume that the vector has elements if these functions return true; it is possible to have an empty vector (eg. 0x1)
- Examples:

      mat A(1, 5, fill::randu);
      mat B(5, 1, fill::randu);
      mat C(5, 5, fill::randu);

      cout << A.is_vec() << endl;
      cout << B.is_vec() << endl;
      cout << C.is_vec() << endl;

- See also:
  - .is_empty()
  - .is_square()
  - .is_finite()

.is_sorted()
.is_sorted( sort_direction )
.is_sorted( sort_direction, dim )

- Member function of Mat, Row and Col
- If the object is a vector, return a bool indicating whether the elements are sorted
- If the object is a matrix, return a bool indicating whether the elements are sorted in each column (dim = 0), or each row (dim = 1)
- The sort_direction argument is optional; sort_direction is one of:

  `"ascend"` | ↦ | elements are ascending; consecutive elements can be equal; this is the default operation
  `"descend"` | ↦ | elements are descending; consecutive elements can be equal
  `"strictascend"` | ↦ | elements are strictly ascending; consecutive elements cannot be equal
  `"strictdescend"` | ↦ | elements are strictly descending; consecutive elements cannot be equal

- The dim argument is optional; by default dim = 0 is used
- For matrices and vectors with complex numbers, order is checked via absolute values
- Examples:

      vec a(10, fill::randu);
      vec b = sort(a);

      bool check1 = a.is_sorted();
      bool check2 = b.is_sorted();

      mat A(10, 10, fill::randu);

      // check whether each column is sorted in descending manner
      cout << A.is_sorted("descend") << endl;

      // check whether each row is sorted in ascending manner
      cout << A.is_sorted("ascend", 1) << endl;

- See also:
  - sort()
  - sort_index()

.is_trimatu()
.is_trimatl()

- Member functions of Mat and SpMat
- .is_trimatu():
  - return true if the matrix is upper triangular, ie. the matrix is square sized and all elements below the main diagonal are zero; return false otherwise
  - caveat: if this function returns true, do not assume that the matrix contains non-zero elements on or above the main diagonal
- .is_trimatl():
  - return true if the matrix is lower triangular, ie. the matrix is square sized and all elements above the main diagonal are zero; return false otherwise
  - caveat: if this function returns true, do not assume that the matrix contains non-zero elements on or below the main diagonal
- Examples:

      mat A(5, 5, fill::randu);
      mat B = trimatl(A);

      cout << A.is_trimatu() << endl;
      cout << B.is_trimatl() << endl;

- See also:
  - trimatu() / trimatl()
  - .is_symmetric()
  - .is_diagmat()
  - Triangular matrix in MathWorld
  - Triangular matrix in Wikipedia

.is_diagmat()

- Member function of Mat and SpMat
- Return true if the matrix is diagonal, ie. all elements outside of the main diagonal are zero
- Return false otherwise
- Caveat: if this function returns true, do not assume that the matrix contains non-zero elements on the main diagonal
- Examples:

      mat A(5, 5, fill::randu);
      mat B = diagmat(A);

      cout << A.is_diagmat() << endl;
      cout << B.is_diagmat() << endl;

- See also:
  - diagmat()
  - .is_trimatu() / .is_trimatl()
  - .is_symmetric()
  - Diagonal matrix in MathWorld
  - Diagonal matrix in Wikipedia

.is_square()

- Member function of Mat and SpMat
- Returns true if the matrix is square, ie. number of rows is equal to the number of columns
- Returns false if the matrix is not square
- Examples:

      mat A(5, 5, fill::randu);
      mat B(6, 7, fill::randu);

      cout << A.is_square() << endl;
      cout << B.is_square() << endl;

- See also:
  - .is_symmetric()
  - .is_hermitian()
  - .is_empty()
  - .is_vec()
  - .is_finite()

.is_symmetric()
.is_symmetric( tol )

- Member function of Mat and SpMat
- Returns true if the matrix is symmetric
- Returns false if the matrix is not symmetric
- The tol argument is optional; if tol is specified, the given matrix X is considered symmetric if norm(X - X.st(), "inf") / norm (X, "inf") ≤ tol
- Examples:

      mat A(5, 5, fill::randu);
      mat B = A.t() * A;

      cout << A.is_symmetric() << endl;
      cout << B.is_symmetric() << endl;

- See also:
  - .is_hermitian()
  - .is_sympd()
  - .is_square()
  - .is_trimatu() / .is_trimatl()
  - Symmetric matrix in Wikipedia
  - Symmetric matrix in MathWorld

.is_hermitian()
.is_hermitian( tol )

- Member function of Mat and SpMat
- Returns true if the matrix is hermitian (self-adjoint)
- Returns false if the matrix is not hermitian
- The tol argument is optional; if tol is specified, the given matrix X is considered hermitian if norm(X - X.t(), "inf") / norm (X, "inf") ≤ tol
- Examples:

      cx_mat A(5, 5, fill::randu);
      cx_mat B = A.t() * A;

      cout << A.is_hermitian() << endl;
      cout << B.is_hermitian() << endl;

- See also:
  - .is_symmetric()
  - .is_sympd()
  - .is_square()
  - Hermitian matrix in Wikipedia
  - Hermitian matrix in MathWorld

.is_sympd()
.is_sympd( tol )

- Member function of Mat and any dense matrix expression
- Returns true if the matrix is symmetric/hermitian positive definite within the tolerance given by tol
- Returns false otherwise
- The tol argument is optional; if tol is not specified, by default tol = 100 * datum::eps * norm(X, "fro")
- Examples:

      mat A(5, 5, fill::randu);

      mat B = A.t() * A;

      cout << A.is_sympd() << endl;
      cout << B.is_sympd() << endl;

- See also:
  - .is_symmetric()
  - .is_hermitian()
  - datum::eps

.is_zero()
.is_zero( tolerance )

- For objects with non-complex elements: return true if each element has an absolute value ≤ tolerance; return false otherwise
- For objects with complex elements: return true if for each element, each component (real and imaginary) has an absolute value ≤ tolerance; return false otherwise
- The argument tolerance is optional; by default tolerance = 0
- Examples:

      mat A(5, 5, fill::zeros);

      A(0,0) = datum::eps;

      cout << A.is_zero()           << endl;
      cout << A.is_zero(datum::eps) << endl;

- See also:
  - .clean()
  - all()
  - datum::eps
  - approx_equal()

.is_finite()

- Member function of Mat, Col, Row, Cube, SpMat
- Returns true if all elements of the object are finite
- Returns false if at least one of the elements of the object is non-finite (±infinity or NaN)
- Examples:

      mat A(5, 5, fill::randu);
      mat B(5, 5, fill::randu);

      B(1,1) = datum::inf;

      cout << A.is_finite() << endl;
      cout << B.is_finite() << endl;

- See also:
  - .has_inf()
  - .has_nan()
  - find_finite()
  - find_nonfinite()
  - constants (pi, nan, inf, ...)

.has_inf()

- Member function of Mat, Col, Row, Cube, SpMat
- Returns true if at least one of the elements of the object is ±infinity
- Returns false otherwise
- Examples:

      mat A(5, 5, fill::randu);
      mat B(5, 5, fill::randu);

      B(1,1) = datum::inf;

      cout << A.has_inf() << endl;
      cout << B.has_inf() << endl;

- See also:
  - .has_nan()
  - .replace()
  - .is_finite()
  - find_nonfinite()
  - constants (pi, nan, inf, ...)

.has_nan()

- Member function of Mat, Col, Row, Cube, SpMat
- Returns true if at least one of the elements of the object is NaN (not-a-number)
- Returns false otherwise
- Caveat: `NaN` is not equal to anything, even itself
- Examples:

      mat A(5, 5, fill::randu);
      mat B(5, 5, fill::randu);

      B(1,1) = datum::nan;

      cout << A.has_nan() << endl;
      cout << B.has_nan() << endl;

- See also:
  - .has_inf()
  - .replace()
  - .is_finite()
  - find_nonfinite()
  - constants (pi, nan, inf, ...)

.print()
.print( header )
.print( stream )
.print( stream, header )

- Member functions of Mat, Col, Row, SpMat, Cube and field
- Print the contents of an object to the std::cout stream (default), or a user specified stream, with an optional header string
- Objects can also be printed using the << stream operator
- Elements of a field can only be printed if there is an associated operator<< function defined
- Examples:

      mat A(5, 5, fill::randu);
      mat B(6, 6, fill::randu);

      A.print();

      // print a transposed version of A
      A.t().print();

      // "B:" is the optional header line
      B.print("B:");

      cout << A << endl;

      cout << "B:" << endl;
      cout << B << endl;

- See also:
  - .raw_print()
  - .brief_print()
  - saving & loading matrices
  - initialising elements
  - output streams

.raw_print()
.raw_print( header )
.raw_print( stream )
.raw_print( stream, header )

- Member functions of Mat, Col, Row, SpMat and Cube
- Similar to the .print() member function, with the difference that no formatting of the output is done; the stream's parameters such as precision, cell width, etc. can be set manually
- If the cell width is set to zero, a space is printed between the elements
- Examples:

      mat A(5, 5, fill::randu);

      cout.precision(11);
      cout.setf(ios::fixed);

      A.raw_print(cout, "A:");

- See also:
  - .print()
  - .brief_print()
  - std::ios_base::fmtflags (cppreference.com)
  - std::ios_base::fmtflags (cplusplus.com)

.brief_print()
.brief_print( header )
.brief_print( stream )
.brief_print( stream, header )

- Member functions of Mat, Col, Row, SpMat and Cube
- Print a shortened/abridged version of an object and its size to the std::cout stream (default), or a user specified stream, with an optional header string
- Can be useful for gaining a glimpse of large matrices
- Examples:

      mat A(123, 456, fill::randu);

      A.brief_print("A:");

      // possible output:
      //
      // A:
      // [matrix size: 123x456]
      //    0.8402   0.7605   0.6218      ...   0.9744
      //    0.3944   0.9848   0.0409      ...   0.7799
      //    0.7831   0.9350   0.4140      ...   0.8835
      //         :        :        :        :        :
      //    0.4954   0.1826   0.9848      ...   0.1918

- See also:
  - .print()
  - .raw_print()

saving / loading matrices & cubes

.save( filename ) .save( filename, file_type ) .save( stream ) .save( stream, file_type ) .save( hdf5_name(filename, dataset) ) .save( hdf5_name(filename, dataset, settings) ) .save( csv_name(filename, header) ) .save( csv_name(filename, header, settings) ) |  | .load( filename ) .load( filename, file_type ) .load( stream ) .load( stream, file_type ) .load( hdf5_name(filename, dataset) ) .load( hdf5_name(filename, dataset, settings) ) .load( csv_name(filename, header) ) .load( csv_name(filename, header, settings) )

- Member functions of Mat, Col, Row, Cube and SpMat
- Store/retrieve data in a file or stream (caveat: the stream must be opened in binary mode)
- On success, .save() and .load() return a bool set to true
- On failure, .save() and .load() return a bool set to false; additionally, .load() resets the object so that it has no elements
- file_type can be one of the following:

  auto_detect |  | Used only by .load() only: attempt to automatically detect the file type as one of the formats described below; [ default operation for .load() ]
  arma_binary |  | Numerical data stored in machine dependent binary format, with a simple header to speed up loading. The header indicates the type and size of matrix/cube. [ default operation for .save() ]
  arma_ascii |  | Numerical data stored in human readable text format, with a simple header to speed up loading. The header indicates the type and size of matrix/cube.
  raw_binary |  | Numerical data stored in machine dependent raw binary format, without a header. Matrices are loaded to have one column, while cubes are loaded to have one slice with one column. The .reshape() function can be used to alter the size of the loaded matrix/cube without losing data.
  raw_ascii |  | Numerical data stored in raw ASCII format, without a header. The numbers are separated by whitespace. The number of columns must be the same in each row. Cubes are loaded as one slice. Data which was saved in Matlab/Octave using the -ascii option can be read in Armadillo, except for complex numbers. Complex numbers are stored in standard C++ notation, which is a tuple surrounded by brackets: eg. (1.23,4.56) indicates 1.24 + 4.56i.
  csv_ascii |  | Numerical data stored in comma separated value (CSV) text format, without a header. To save/load with a header, use the csv_name(filename,header) specification instead (more details below). Handles complex numbers stored in the compound form of 1.24+4.56i. Applicable to Mat and SpMat.
  coord_ascii |  | Numerical data stored as a text file in coordinate list format, without a header. Only non-zero values are stored. For real matrices, each line contains information in the following format: `row column value` For complex matrices, each line contains information in the following format: `row column real_value imag_value` The rows and columns start at zero. Armadillo ≥ 10.3: applicable to Mat and SpMat; Armadillo ≤ 10.2: applicable to SpMat only. Caveat: not supported by auto_detect.
  pgm_binary |  | Image data stored in Portable Gray Map (PGM) format. Applicable to Mat only. Saving int, float or double matrices is a lossy operation, as each element is copied and converted to an 8 bit representation. As such the matrix should have values in the [0,255] interval, otherwise the resulting image may not display correctly.
  ppm_binary |  | Image data stored in Portable Pixel Map (PPM) format. Applicable to Cube only. Saving int, float or double matrices is a lossy operation, as each element is copied and converted to an 8 bit representation. As such the cube/field should have values in the [0,255] interval, otherwise the resulting image may not display correctly.
  hdf5_binary |  | Numerical data stored in portable HDF5 binary format. for saving, the default dataset name within the HDF5 file is "dataset" for loading, the order of operations is: (1) try loading a dataset named "dataset", (2) try loading a dataset named "value", (3) try loading the first available dataset to explicitly control the dataset name, specify it via the hdf5_name() argument (more details below)

- Caveat: for saving / loading HDF5 files, support for HDF5 must be enabled within Armadillo's configuration; the hdf5.h header file must be available on your system and you will need to link with the HDF5 library (eg. `-lhdf5`). HDF5 support can be enabled by defining ARMA_USE_HDF5 before including the armadillo header:
  `#define ARMA_USE_HDF5
  #include <armadillo>`
- By providing either hdf5_name(filename, dataset) or hdf5_name(filename, dataset, settings), the file_type type is assumed to be hdf5_binary
  - the dataset argument specifies an HDF5 dataset name (eg. "my_dataset") that can include a full path (eg. "/group_name/my_dataset"); if a blank dataset name is specified (ie. ""), it is assumed to be "dataset"
  - the settings argument is optional; it is one of the following, or a combination thereof:

    `hdf5_opts::trans` |  | save/load the data with columns transposed to rows (and vice versa)
    `hdf5_opts::append` |  | instead of overwriting the file, append the specified dataset to the file; the specified dataset must not already exist in the file
    `hdf5_opts::replace` |  | instead of overwriting the file, replace the specified dataset in the file caveat: HDF5 may not automatically reclaim deleted space; use h5repack to clean HDF5 files

    the above settings can be combined using the `+` operator; for example: `hdf5_opts::trans + hdf5_opts::append`
- By providing either csv_name(filename, header) or csv_name(filename, header, settings), the file is assumed to have data in comma separated value (CSV) text format
  - the header argument specifies the object which stores the separate elements of the header line; it must have the type `field<std::string>`
  - the optional settings argument is one of the following, or a combination thereof:

    `csv_opts::trans` |  | save/load the data with columns transposed to rows (and vice versa)
    `csv_opts::no_header` |  | assume there is no header line; the header argument is not referenced
    `csv_opts::semicolon` |  | use semicolon (;) instead of comma (,) as the separator character
    `csv_opts::strict` |  | interpret missing values as NaN (not applicable to sparse matrices)

    the above settings can be combined using the `+` operator; for example: `csv_opts::trans + csv_opts::no_header`
- Examples:

      mat A(5, 6, fill::randu);

      // default save format is arma_binary
      A.save("A.bin");

      // save in raw_ascii format
      A.save("A.txt", raw_ascii);

      // save in CSV format without a header
      A.save("A.csv", csv_ascii);

      // save in CSV format with a header
      field<std::string> header(A.n_cols);
      header(0) = "foo";
      header(1) = "bar";  // etc
      A.save( csv_name("A.csv", header) );

      // save in HDF5 format with internal dataset named as "my_data"
      A.save(hdf5_name("A.h5", "my_data"));

      // automatically detect format type while loading
      mat B;
      B.load("A.bin");

      // force loading in arma_ascii format
      mat C;
      C.load("A.txt", arma_ascii);

      // example of testing for success
      mat D;
      bool ok = D.load("A.bin");

      if(ok == false)
        {
        cout << "problem with loading" << endl;
        }

- See also:
  - HDF in Wikipedia
  - CSV in Wikipedia
  - saving / loading fields

saving / loading fields

.save( name ) .save( name, file_type ) .save( stream ) .save( stream, file_type ) |  | .load( name ) .load( name, file_type ) .load( stream ) .load( stream, file_type )

- Store/retrieve data in a file or stream (caveat: the stream must be opened in binary mode)
- On success, .save() and .load() return a bool set to true
- On failure, .save() and .load() return a bool set to false; additionally, .load() resets the object so that it has no elements
- Fields with objects of type std::string are saved and loaded as raw text files. The text files do not have a header. Each string is separated by a whitespace. load() will only accept text files that have the same number of strings on each line. The strings can have variable lengths.
- Other than storing string fields as text files, the following file formats are supported:

  auto_detect |  | .load(): attempt to automatically detect the field format type as one of the formats described below; this is the default operation
  arma_binary |  | objects are stored in machine dependent binary format default type for fields of type Mat, Col, Row or Cube only applicable to fields of type Mat, Col, Row or Cube
  ppm_binary |  | image data stored in Portable Pixmap Map (PPM) format only applicable to fields of type Mat, Col or Row .load(): loads the specified image and stores the red, green and blue components as three separate matrices; the resulting field is comprised of the three matrices, with the red, green and blue components in the first, second and third matrix, respectively .save(): saves a field with exactly three matrices of equal size as an image; it is assumed that the red, green and blue components are stored in the first, second and third matrix, respectively; saving int, float or double matrices is a lossy operation, as each matrix element is copied and converted to an 8 bit representation

- See also:
  - saving / loading matrices and cubes

Generated Vectors / Matrices / Cubes

linspace( start, end )
linspace( start, end, N )

- Generate a vector with N elements; the values of the elements are linearly spaced from start to (and including) end
- The argument N is optional; by default N = 100
- Usage:
  - vec v = linspace(start, end, N)
  - vector_type v = linspace<vector_type>(start, end, N)
- Caveat: for N = 1, the generated vector will have a single element equal to end
- Examples:

         vec a = linspace(0, 5, 6);

      rowvec b = linspace<rowvec>(5, 0, 6);

- See also:
  - regspace()
  - logspace()
  - randperm()
  - ones()
  - interp1()

logspace( A, B )
logspace( A, B, N )

- Generate a vector with N elements; the values of the elements are logarithmically spaced from 10A to (and including) 10B
- The argument N is optional; by default N = 50
- Usage:
  - vec v = logspace(A, B, N)
  - vector_type v = logspace<vector_type>(A, B, N)
- Examples:

         vec a = logspace(0, 5, 6);

      rowvec b = logspace<rowvec>(5, 0, 6);

- See also:
  - linspace()
  - regspace()

regspace( start, end )
regspace( start, delta, end )

- Generate a vector with regularly spaced elements:
  [ (start + 0*delta), (start + 1*delta), (start + 2*delta), ⋯, (start + M*delta) ]
  where M = floor((end-start) / delta), so that (start + M*delta) ≤ end
- Similar in operation to the Matlab/Octave colon operator, ie. start:end and start:delta:end
- If delta is not specified:
  - delta= +1, if start ≤ end
  - delta= −1, if start > end (caveat: this is different to Matlab/Octave)
- An empty vector is generated when one of the following conditions is true:
  - start < end, and delta < 0
  - start > end, and delta > 0
  - delta = 0
- Usage:
  - vec v = regspace(start, end)
  - vec v = regspace(start, delta, end)
  - vector_type v = regspace<vector_type>(start, end)
  - vector_type v = regspace<vector_type>(start, delta, end)
- Examples:

       vec a = regspace(0,  9);             // 0,  1, ...,   9

      uvec b = regspace<uvec>(2,  2,  10);  // 2,  4, ...,  10

      ivec c = regspace<ivec>(0, -1, -10);  // 0, -1, ..., -10

- Caveat: do not use regspace() to specify ranges for contiguous submatrix views; use span() instead
- See also:
  - linspace()
  - logspace()
  - randperm()

randperm( N )
randperm( N, M )

- Generate a vector with a random permutation of integers from 0 to N-1
- The optional argument M indicates the number of elements to return, sampled without replacement from 0 to N-1
- Examples:

      uvec X = randperm(10);
      uvec Y = randperm(10,2);

- See also:
  - randi()
  - shuffle()
  - linspace()
  - regspace()
  - RNG seed setting

eye( n_rows, n_cols )
eye( size(X) )

- Generate a matrix with the elements along the main diagonal set to one and off-diagonal elements set to zero
- An identity matrix is generated when n_rows = n_cols
- Usage:
  - mat X = eye( n_rows, n_cols )
  - matrix_type X = eye<matrix_type>( n_rows, n_cols )
  - matrix_type Y = eye<matrix_type>( size(X) )
- Examples:

         mat A = eye(5,5);  // or:  mat A(5,5,fill::eye);

        fmat B = 123.0 * eye<fmat>(5,5);

      cx_mat C = eye<cx_mat>( size(B) );

- See also:
  - .eye() (member function of Mat)
  - .diag()
  - ones()
  - diagmat()
  - diagvec()
  - speye()
  - size()

ones( n_elem )
ones( n_rows, n_cols )
ones( n_rows, n_cols, n_slices )
ones( size(X) )

- Generate a vector, matrix or cube with all elements set to one
- Usage:
  - vector_type v = ones<vector_type>( n_elem )
  - matrix_type X = ones<matrix_type>( n_rows, n_cols )
  - matrix_type Y = ones<matrix_type>( size(X) )
  - cube_type Q = ones<cube_type>( n_rows, n_cols, n_slices )
  - cube_type R = ones<cube_type>( size(Q) )
- Caveat: specifying fill::ones during object construction is more compact, eg. `mat A(5, 6, fill::ones)`
- Examples:

         vec v = ones(10);    // or: vec v(10, fill::ones);
        uvec u = ones<uvec>(10);
      rowvec r = ones<rowvec>(10);

       mat A = ones(5,6);     // or: mat A(5, 6, fill::ones);
      fmat B = ones<fmat>(5,6);
      umat C = ones<umat>(5,6);

       cube Q = ones(5,6,7);  // or: cube Q(5, 6, 7, fill::ones);
      fcube R = ones<fcube>(5,6,7);

- See also:
  - .ones() (member function of Mat, Col, Row and Cube)
  - .fill()
  - eye()
  - linspace()
  - regspace()
  - zeros()
  - randu()
  - spones()
  - size()

zeros( n_elem )
zeros( n_rows, n_cols )
zeros( n_rows, n_cols, n_slices )
zeros( size(X) )

- Generate a vector, matrix or cube with the elements set to zero
- Usage:
  - vector_type v = zeros<vector_type>( n_elem )
  - matrix_type X = zeros<matrix_type>( n_rows, n_cols )
  - matrix_type Y = zeros<matrix_type>( size(X) )
  - cube_type Q = zeros<cube_type>( n_rows, n_cols, n_slices )
  - cube_type R = zeros<cube_type>( size(Q) )
- Caveat: specifying fill::zeros during object construction is more compact, eg. `mat A(5, 6, fill::zeros)`
- Examples:

         vec v = zeros(10);    // or: vec v(10, fill::zeros);
        uvec u = zeros<uvec>(10);
      rowvec r = zeros<rowvec>(10);

       mat A = zeros(5,6);     // or: mat A(5, 6, fill::zeros);
      fmat B = zeros<fmat>(5,6);
      umat C = zeros<umat>(5,6);

       cube Q = zeros(5,6,7);  // or: cube Q(5, 6, 7, fill::zeros);
      fcube R = zeros<fcube>(5,6,7);

- See also:
  - .zeros() (member function of Mat, Col, Row, SpMat and Cube)
  - .fill()
  - ones()
  - randu()
  - size()

randu( )
randu( distr_param(a,b) )
randu( n_elem )
randu( n_elem, distr_param(a,b) )
randu( n_rows, n_cols )
randu( n_rows, n_cols, distr_param(a,b) )
randu( n_rows, n_cols, n_slices )
randu( n_rows, n_cols, n_slices, distr_param(a,b) )
randu( size(X) )
randu( size(X), distr_param(a,b) )

- Generate a scalar, vector, matrix or cube with the elements set to random floating point values uniformly distributed in the [a,b] interval
- The default distribution parameters are a = 0 and b = 1
- For complex elements, the real and imaginary parts are treated separately
- Usage:
  - scalar_type s = randu<scalar_type>( ), where scalar_type ∈ { float, double, cx_float, cx_double }
  - scalar_type s = randu<scalar_type>( distr_param(a,b) ), where scalar_type ∈ { float, double, cx_float, cx_double }
  - vector_type v = randu<vector_type>( n_elem )
  - vector_type v = randu<vector_type>( n_elem, distr_param(a,b) )
  - matrix_type X = randu<matrix_type>( n_rows, n_cols )
  - matrix_type X = randu<matrix_type>( n_rows, n_cols, distr_param(a,b) )
  - cube_type Q = randu<cube_type>( n_rows, n_cols, n_slices )
  - cube_type Q = randu<cube_type>( n_rows, n_cols, n_slices, distr_param(a,b) )
- Caveat: to generate a matrix with random integer values instead of floating point values, use randi() instead
- Examples:

      double a = randu();
      double b = randu(distr_param(10,20));

      vec v1 = randu(5);    // or: vec v1(5, fill::randu);
      vec v2 = randu(5, distr_param(10,20));

      rowvec r1 = randu<rowvec>(5);
      rowvec r2 = randu<rowvec>(5, distr_param(10,20));

      mat A1 = randu(5, 6);  // or: mat A1(5, 6, fill::randu);
      mat A2 = randu(5, 6, distr_param(10,20));

      fmat B1 = randu<fmat>(5, 6);
      fmat B2 = randu<fmat>(5, 6, distr_param(10,20));

- See also:
  - .randu() (member function)
  - randn()
  - randg()
  - randi()
  - .imbue()
  - ones()
  - zeros()
  - shuffle()
  - sprandu()
  - size()
  - RNG seed setting
  - uniform distribution in Wikipedia

randn( )
randn( distr_param(mu,sd) )
randn( n_elem )
randn( n_elem, distr_param(mu,sd) )
randn( n_rows, n_cols )
randn( n_rows, n_cols, distr_param(mu,sd) )
randn( n_rows, n_cols, n_slices )
randn( n_rows, n_cols, n_slices, distr_param(mu,sd) )
randn( size(X) )
randn( size(X), distr_param(mu,sd) )

- Generate a scalar, vector, matrix or cube with the elements set to random values with normal / Gaussian distribution, parameterised by mean mu and standard deviation sd
- The default distribution parameters are mu = 0 and sd = 1
- For complex elements, the real and imaginary parts are treated separately
- Usage:
  - scalar_type s = randn<scalar_type>( ), where scalar_type ∈ { float, double, cx_float, cx_double }
  - scalar_type s = randn<scalar_type>( distr_param(mu,sd) ), where scalar_type ∈ { float, double, cx_float, cx_double }
  - vector_type v = randn<vector_type>( n_elem )
  - vector_type v = randn<vector_type>( n_elem, distr_param(mu,sd) )
  - matrix_type X = randn<matrix_type>( n_rows, n_cols )
  - matrix_type X = randn<matrix_type>( n_rows, n_cols, distr_param(mu,sd) )
  - cube_type Q = randn<cube_type>( n_rows, n_cols, n_slices )
  - cube_type Q = randn<cube_type>( n_rows, n_cols, n_slices, distr_param(mu,sd) )
- Examples:

      double a = randn();
      double b = randn(distr_param(10,5));

      vec v1 = randn(5);    // or: vec v1(5, fill::randn);
      vec v2 = randn(5, distr_param(10,5));

      rowvec r1 = randn<rowvec>(5);
      rowvec r2 = randn<rowvec>(5, distr_param(10,5));

      mat A1 = randn(5, 6);  // or: mat A1(5, 6, fill::randn);
      mat A2 = randn(5, 6, distr_param(10,5));

      fmat B1 = randn<fmat>(5, 6);
      fmat B2 = randn<fmat>(5, 6, distr_param(10,5));

- See also:
  - .randn() (member function)
  - randu()
  - randg()
  - randi()
  - mvnrnd()
  - normpdf()
  - .imbue()
  - sprandn()
  - size()
  - RNG seed setting
  - normal distribution in Wikipedia

randg( )
randg( distr_param(a,b) )
randg( n_elem )
randg( n_elem, distr_param(a,b) )
randg( n_rows, n_cols )
randg( n_rows, n_cols, distr_param(a,b) )
randg( n_rows, n_cols, n_slices )
randg( n_rows, n_cols, n_slices, distr_param(a,b) )
randg( size(X) )
randg( size(X), distr_param(a,b) )

- Generate a scalar, vector, matrix or cube with the elements set to random values from a gamma distribution:

  |  | xa-1 exp( -x / b )
  p(x | a,b) | = |
  |  | ba Γ(a)

  where a is the shape parameter and b is the scale parameter, with constraints a > 0 and b > 0
- The default distribution parameters are a = 1 and b = 1
- Usage:
  - scalar_type s = randg<scalar_type>( ), where scalar_type is either float or double
  - scalar_type s = randg<scalar_type>( distr_param(a,b) ), where scalar_type is either float or double
  - vector_type v = randg<vector_type>( n_elem )
  - vector_type v = randg<vector_type>( n_elem, distr_param(a,b) )
  - matrix_type X = randg<matrix_type>( n_rows, n_cols )
  - matrix_type X = randg<matrix_type>( n_rows, n_cols, distr_param(a,b) )
  - cube_type Q = randg<cube_type>( n_rows, n_cols, n_slices )
  - cube_type Q = randg<cube_type>( n_rows, n_cols, n_slices, distr_param(a,b) )
- Examples:

      vec v1 = randg(100);
      vec v2 = randg(100, distr_param(2,1));

      rowvec r1 = randg<rowvec>(100);
      rowvec r2 = randg<rowvec>(100, distr_param(2,1));

      mat A1 = randg(10, 10);
      mat A2 = randg(10, 10, distr_param(2,1));

      fmat B1 = randg<fmat>(10, 10);
      fmat B2 = randg<fmat>(10, 10, distr_param(2,1));

- See also:
  - randu()
  - randn()
  - randi()
  - chi2rnd()
  - .imbue()
  - size()
  - RNG seed setting
  - gamma distribution in Wikipedia

randi( )
randi( distr_param(a,b) )
randi( n_elem )
randi( n_elem, distr_param(a,b) )
randi( n_rows, n_cols )
randi( n_rows, n_cols, distr_param(a,b) )
randi( n_rows, n_cols, n_slices )
randi( n_rows, n_cols, n_slices, distr_param(a,b) )
randi( size(X) )
randi( size(X), distr_param(a,b) )

- Generate a scalar, vector, matrix or cube with the elements set to random integer values uniformly distributed in the [a,b] interval
- The default distribution parameters are a = 0 and b = maximum_int
- Usage:
  - scalar_type v = randi<scalar_type>( )
  - scalar_type v = randi<scalar_type>( distr_param(a,b) )
  - vector_type v = randi<vector_type>( n_elem )
  - vector_type v = randi<vector_type>( n_elem, distr_param(a,b) )
  - matrix_type X = randi<matrix_type>( n_rows, n_cols )
  - matrix_type X = randi<matrix_type>( n_rows, n_cols, distr_param(a,b) )
  - cube_type Q = randi<cube_type>( n_rows, n_cols, n_slices )
  - cube_type Q = randi<cube_type>( n_rows, n_cols, n_slices, distr_param(a,b) )
- Caveat: to generate a matrix with random floating point values (ie. float or double) instead of integers, use randu() instead
- Examples:

      int a = randi();
      int b = randi(distr_param(-10, +20));

      imat A1 = randi(5, 6);
      imat A2 = randi(5, 6, distr_param(-10, +20));

      mat B1 = randi<mat>(5, 6);
      mat B2 = randi<mat>(5, 6, distr_param(-10, +20));

- See also:
  - randu()
  - randperm()
  - .imbue()
  - ones()
  - zeros()
  - shuffle()
  - size()
  - RNG seed setting

speye( n_rows, n_cols )
speye( size(X) )

- Generate a sparse matrix with the elements along the main diagonal set to one and off-diagonal elements set to zero
- An identity matrix is generated when n_rows = n_cols
- Usage:
  - sparse_matrix_type X = speye<sparse_matrix_type>( n_rows, n_cols )
  - sparse_matrix_type Y = speye<sparse_matrix_type>( size(X) )
- Examples:

      sp_mat A = speye<sp_mat>(5,5);

- See also:
  - spones()
  - sprandu() & sprandn()
  - eye()
  - size()

spones( A )

- Generate a sparse matrix with the same structure as sparse matrix A, but with the non-zero elements set to one
- Examples:

      sp_mat A = sprandu<sp_mat>(100, 200, 0.1);

      sp_mat B = spones(A);

- See also:
  - speye()
  - sprandu() & sprandn()
  - ones()

sprandu( n_rows, n_cols, density )
sprandn( n_rows, n_cols, density )
sprandu( size(X), density )
sprandn( size(X), density )

- Generate a sparse matrix with the non-zero elements set to random values
- The density argument specifies the percentage of non-zero elements; it must be in the [0,1] interval
- sprandu() uses a uniform distribution in the [0,1] interval
- sprandn() uses a normal/Gaussian distribution with zero mean and unit variance
- Usage:
  - sparse_matrix_type X = sprandu<sparse_matrix_type>( n_rows, n_cols, density )
  - sparse_matrix_type Y = sprandu<sparse_matrix_type>( size(X), density )
- Examples:

      sp_mat A = sprandu<sp_mat>(100, 200, 0.1);

- See also:
  - speye()
  - spones()
  - randu()
  - randn()
  - size()
  - RNG seed setting
  - uniform distribution in Wikipedia
  - normal distribution in Wikipedia

toeplitz( A )
toeplitz( A, B )

- Generate a Toeplitz matrix, with the first column specified by vector A, and (optionally) the first row specified by vector B
- If argument B is omitted, the first row is assumed to be the same as the first column
- Examples:

      vec A(5, fill::randu);

      mat X = toeplitz(A);

- See also:
  - Toeplitz matrix in MathWorld
  - Toeplitz matrix in Wikipedia

Functions of Vectors / Matrices / Cubes

abs( X )

- Obtain the magnitude of each element
- Usage for non-complex X:
  - Y = abs(X)
  - X and Y must have the same matrix type or cube type, such as mat or cube
- Usage for complex X:
  - real_object_type Y = abs(X)
  - The type of X must be a complex matrix or complex cube, such as cx_mat or cx_cube
  - The type of Y must be the real counterpart to the type of X; if X has the type cx_mat, then the type of Y must be mat
- Examples:

      mat A(5, 5, fill::randu);
      mat B = abs(A);

      cx_mat X(5, 5, fill::randu);
         mat Y = abs(X);

- See also:
  - arg()
  - conj()
  - imag() / real()
  - pow()
  - miscellaneous element-wise functions

accu( X )

- Accumulate (sum) all elements of a vector, matrix or cube
- Examples:

      mat A(5, 6, fill::randu);
      mat B(5, 6, fill::randu);

      double x = accu(A);

      double y = accu(A % B);

      // accu(A % B) is a "multiply-and-accumulate" operation
      // as operator % performs element-wise multiplication

- See also:
  - sum()
  - cumsum()
  - trace()
  - mean()
  - dot()
  - as_scalar()

affmul( A, B )

- Multiply matrix A by an automatically extended form of B
- A is typically an affine transformation matrix
- B can be a vector or matrix, and is treated as having an additional row of ones
- The number of columns in A must be equal to number of rows in the extended form of B (ie. A.n_cols = B.n_rows+1)
- If A is 3x3 and B is 2x1, the equivalent matrix multiplication is:

    ⎡ C0 ⎤   ⎡ A00 A01 A02 ⎤   ⎡ B0 ⎤
    ⎢ C1 ⎥ = ⎢ A10 A11 A12 ⎥ x ⎢ B1 ⎥
    ⎣ C2 ⎦   ⎣ A20 A21 A22 ⎦   ⎣ 1  ⎦

- If A is 2x3 and B is 2x1, the equivalent matrix multiplication is:

    ⎡ C0 ⎤   ⎡ A00 A01 A02 ⎤   ⎡ B0 ⎤
    ⎣ C1 ⎦ = ⎣ A10 A11 A12 ⎦ x ⎢ B1 ⎥
                              ⎣ 1  ⎦

- Examples:

      mat A(2, 3, fill::randu);
      vec B(2,    fill::randu);

      vec C = affmul(A,B);

- See also:
  - math operators
  - Affine transformation in Wikipedia
  - Transformation matrix in Wikipedia

all( V )
all( X )
all( X, dim )

- For vector V, return true if all elements of the vector are non-zero or satisfy a relational condition
- For matrix X and
  - dim = 0, return a row vector (of type urowvec or umat), with each element (0 or 1) indicating whether the corresponding column of X has all non-zero elements
  - dim = 1, return a column vector (of type ucolvec or umat), with each element (0 or 1) indicating whether the corresponding row of X has all non-zero elements
- The dim argument is optional; by default dim = 0 is used
- Relational operators can be used instead of V or X, eg. A > 0.5
- Examples:

      vec V(10,   fill::randu);
      mat X(5, 5, fill::randu);

      // status1 will be set to true if vector V has all non-zero elements
      bool status1 = all(V);

      // status2 will be set to true if vector V has all elements greater than 0.5
      bool status2 = all(V > 0.5);

      // status3 will be set to true if matrix X has all elements greater than 0.6;
      // note the use of vectorise()
      bool status3 = all(vectorise(X) > 0.6);

      // generate a row vector indicating which columns of X have all elements greater than 0.7
      umat A = all(X > 0.7);

- See also:
  - any()
  - approx_equal()
  - find()
  - .is_zero()
  - conv_to() (convert between matrix/vector types)
  - vectorise()

any( V )
any( X )
any( X, dim )

- For vector V, return true if any element of the vector is non-zero or satisfies a relational condition
- For matrix X and
  - dim = 0, return a row vector (of type urowvec or umat), with each element (0 or 1) indicating whether the corresponding column of X has any non-zero elements
  - dim = 1, return a column vector (of type ucolvec or umat), with each element (0 or 1) indicating whether the corresponding row of X has any non-zero elements
- The dim argument is optional; by default dim = 0 is used
- Relational operators can be used instead of V or X, eg. A > 0.9
- Examples:

      vec V(10,   fill::randu);
      mat X(5, 5, fill::randu);

      // status1 will be set to true if vector V has any non-zero elements
      bool status1 = any(V);

      // status2 will be set to true if vector V has any elements greater than 0.5
      bool status2 = any(V > 0.5);

      // status3 will be set to true if matrix X has any elements greater than 0.6;
      // note the use of vectorise()
      bool status3 = any(vectorise(X) > 0.6);

      // generate a row vector indicating which columns of X have elements greater than 0.7
      umat A = any(X > 0.7);

- See also:
  - all()
  - approx_equal()
  - find()
  - conv_to() (convert between matrix/vector types)
  - vectorise()

approx_equal( A, B, method, tol )
approx_equal( A, B, method, abs_tol, rel_tol )

- Return true if all corresponding elements in A and B are approximately equal
- Return false if any of the corresponding elements in A and B are not approximately equal, or if A and B have different dimensions
- The argument method controls how the approximate equality is determined; it is one of:

  `"absdiff"` | ↦ | scalars x and y are considered equal if |x − y|  ≤  tol
  `"reldiff"` | ↦ | scalars x and y are considered equal if |x − y| / max( |x|, |y| )  ≤  tol
  `"both"` | ↦ | scalars x and y are considered equal if |x − y|  ≤  abs_tol or |x − y| / max( |x|, |y| )  ≤  rel_tol

- Examples:

      mat A(5, 5, fill::randu);
      mat B = A + 0.001;

      bool same1 = approx_equal(A, B, "absdiff", 0.002);

      mat C = 1000 * randu<mat>(5,5);
      mat D = C + 1;

      bool same2 = approx_equal(C, D, "reldiff", 0.1);

      bool same3 = approx_equal(C, D, "both", 2, 0.1);

- See also:
  - all()
  - any()
  - find()
  - .is_zero()
  - relational operators

arg( X )

- Obtain the phase angle (in radians) of each element
- Usage for non-complex X:
  - Y = arg(X)
  - X and Y must have the same matrix type or cube type, such as mat or cube
  - non-complex elements are treated as complex elements with zero imaginary component
- Usage for complex X:
  - real_object_type Y = arg(X)
  - The type of X must be a complex matrix or complex cube, such as cx_mat or cx_cube
  - The type of Y must be the real counterpart to the type of X; if X has the type cx_mat, then the type of Y must be mat
- Examples:

      cx_mat A(5, 5, fill::randu);
         mat B = arg(A);

- See also:
  - abs()
  - atan2()
  - Argument (complex analysis) in Wikipedia
  - Complex Argument in MathWorld

as_scalar( expression )

- Evaluate an expression that results in a 1x1 matrix, followed by converting the 1x1 matrix to a pure scalar
- Optimised expression evaluations are automatically used when a binary or trinary expression is given (ie. 2 or 3 terms)
- Examples:

      rowvec r(5, fill::randu);
      colvec q(5, fill::randu);

      mat X(5, 5, fill::randu);

      // examples of expressions which have optimised implementations

      double a = as_scalar(r*q);
      double b = as_scalar(r*X*q);
      double c = as_scalar(r*diagmat(X)*q);
      double d = as_scalar(r*inv(diagmat(X))*q);

- See also:
  - .as_col() / .as_row()
  - .as_dense()
  - vectorise()
  - accu()
  - trace()
  - dot()
  - norm()
  - conv_to()

clamp( X, min_val, max_val )

- Create a copy of X with each element clamped to the [min_val,max_val] interval;
  any value lower than min_val will be set to min_val, and any value higher than max_val will be set to max_val
- For objects with complex elements, the real and imaginary components are clamped separately
- If X is a sparse matrix, clamping is applied only to the non-zero elements
- Examples:

      mat A(5, 5, fill::randu);

      mat B = clamp(A, 0.2,     0.8);

      mat C = clamp(A, A.min(), 0.8);

      mat D = clamp(A, 0.2, A.max());

- See also:
  - .clamp() (member function)
  - .min() & .max()
  - .clean()
  - .replace()
  - find()

cond( A )

- Return the condition number of matrix A (the ratio of the largest singular value to the smallest)
- Matrix A is well-conditioned when the condition number is close to 1
- Matrix A is ill-conditioned (poorly conditioned) when the condition number is large
- The computation is based on singular value decomposition
- Caveat: calculating the approximate reciprocal condition number via rcond() is considerably more efficient
- Examples:

      mat A(5, 5, fill::randu);

      double c = cond(A);

- See also:
  - rcond()
  - rank()
  - inv()
  - svd()
  - condition number in MathWorld
  - condition number in Wikipedia

conj( X )

- Obtain the complex conjugate of each element in a complex matrix or cube
- Examples:

      cx_mat X(5, 5, fill::randu);
      cx_mat Y = conj(X);

- See also:
  - abs()
  - imag() / real()
  - trans()

conv_to< type >::from( X )

- Convert (cast) from one matrix type to another (eg. mat to imat), or one cube type to another (eg. cube to icube)
- Conversion between std::vector and Armadillo matrices/vectors is also possible
- Conversion of a mat object into colvec, rowvec or std::vector is possible if the object can be interpreted as a vector
- Caveat: to convert an expression that results in a 1x1 matrix to a pure scalar value, use as_scalar()
- Examples:

       mat A(5, 5, fill::randu);
      fmat B = conv_to<fmat>::from(A);

      typedef std::vector<double> stdvec;

      stdvec x(3);
      x[0] = 0.0; x[1] = 1.0;  x[2] = 2.0;

      colvec y = conv_to< colvec >::from(x);
      stdvec z = conv_to< stdvec >::from(y);

- See also:
  - as_scalar()
  - .as_dense()
  - imag() / real()
  - advanced constructors (matrices)
  - advanced constructors (cubes)

cross( A, B )

- Calculate the cross product between A and B, under the assumption that A and B are 3 dimensional vectors
- Examples:

      vec a(3, fill::randu);
      vec b(3, fill::randu);

      vec c = cross(a,b);

- See also:
  - dot()
  - Cross product in Wikipedia
  - Cross product in MathWorld

cumsum( V )
cumsum( X )
cumsum( X, dim )

- For vector V, return a vector of the same orientation, containing the cumulative sum of elements
- For matrix X, return a matrix containing the cumulative sum of elements in each column (dim = 0), or each row (dim = 1)
- The dim argument is optional; by default dim = 0 is used
- Examples:

      mat A(5, 5, fill::randu);
      mat B = cumsum(A);
      mat C = cumsum(A, 1);

      vec x(10, fill::randu);
      vec y = cumsum(x);

- See also:
  - cumprod()
  - accu()
  - sum()
  - diff()

cumprod( V )
cumprod( X )
cumprod( X, dim )

- For vector V, return a vector of the same orientation, containing the cumulative product of elements
- For matrix X, return a matrix containing the cumulative product of elements in each column (dim = 0), or each row (dim = 1)
- The dim argument is optional; by default dim = 0 is used
- Examples:

      mat A(5, 5, fill::randu);
      mat B = cumprod(A);
      mat C = cumprod(A, 1);

      vec x(10, fill::randu);
      vec y = cumprod(x);

- See also:
  - cumsum()
  - prod()

val = det( A ) |  | (form 1)
det( val, A ) |  | (form 2)

- Calculate the determinant of square matrix A, based on LU decomposition
- form 1: return the determinant
- form 2: store the calculated determinant in val and return a bool indicating success
- If A is not square sized, a std::logic_error exception is thrown
- If the calculation fails:
  - val = det(A) throws a std::runtime_error exception
  - det(val,A) returns a bool set to false (exception is not thrown)
- Caveat: log_det() is preferred, as it's less likely to suffer from numerical underflows/overflows
- Examples:

      mat A(5, 5, fill::randu);

      double val1 = det(A);         // form 1

      double val2;
      bool success = det(val2, A);  // form 2

- See also:
  - log_det()
  - rcond()
  - determinant in MathWorld
  - determinant in Wikipedia

diagmat( V )
diagmat( V, k )
diagmat( X )
diagmat( X, k )

- Generate a diagonal matrix from vector V or matrix X
- Given vector V:
  - generate a square matrix with the k-th diagonal containing a copy of the vector; all other elements are set to zero
- Given matrix X:
  - generate a matrix with the k-th diagonal containing a copy of the k-th diagonal of X; all other elements are set to zero
  - if X is an expression, the evaluation of the expression aims to calculate only the diagonal elements
- The argument k is optional; by default k = 0
- The argument k specifies the diagonal to use:
  - k = 0 indicates the main diagonal (default setting)
  - k < 0 indicates the k-th sub-diagonal (below main diagonal, towards bottom-left corner)
  - k > 0 indicates the k-th super-diagonal (above main diagonal, towards top-right corner)
- Examples:

      mat A(5, 5, fill::randu);
      mat B = diagmat(A);
      mat C = diagmat(A,1);

      vec v(5, fill::randu);
      mat D = diagmat(v);
      mat E = diagmat(v,1);

- See also:
  - .diag()
  - .is_diagmat()
  - diagvec()
  - diags() / spdiags()
  - trace()
  - trimatu() / trimatl()
  - symmatu() / symmatl()
  - reshape()
  - vectorise()
  - diagonal matrix in Wikipedia

diagvec( X )
diagvec( X, k )

- Extract the k-th diagonal from matrix X
- The argument k is optional; by default k = 0
- The argument k specifies the diagonal to extract:
  - k = 0 indicates the main diagonal (default setting)
  - k < 0 indicates the k-th sub-diagonal (below main diagonal, towards bottom-left corner)
  - k > 0 indicates the k-th super-diagonal (above main diagonal, towards top-right corner)
- The extracted diagonal is interpreted as a column vector
- If X is an expression, the evaluation of the expression aims to calculate only the diagonal elements
- Examples:

      mat A(5, 5, fill::randu);

      vec d = diagvec(A);

- See also:
  - .diag()
  - diagmat()
  - trace()
  - vectorise()

diags( V, D, n_rows, n_cols )
spdiags( V, D, n_rows, n_cols )

- Generate a matrix with diagonals specified by vector D copied from corresponding column vectors in matrix V
- Function diags() generates a dense matrix, while function spdiags() generates a sparse matrix; in both cases the generated matrix has the same element type as matrix V
- Given vector D must be a vector of type ivec or irowvec
- The number of columns in given matrix V must match the number of elements in vector D
- Each element in vector D specifies diagonal k, where:
  - k = 0 indicates the main diagonal
  - k < 0 indicates the k-th sub-diagonal (below main diagonal, towards bottom-left corner)
  - k > 0 indicates the k-th super-diagonal (above main diagonal, towards top-right corner)
- For k < 0, vectors from matrix V are truncated at the end
- For k > 0, vectors from matrix V are truncated from the start
- Examples:

      mat  V(10, 3, fill::randu);

      ivec D = {-1, 0, +1};

      mat X = diags(V, D, 10, 10);

      sp_mat Y = spdiags(V, D, 10, 10);

- See also:
  - .diag()
  - diagmat()
  - band matrix in Wikipedia

diff( V )
diff( V, k )
diff( X )
diff( X, k )
diff( X, k, dim )

- For vector V, return a vector of the same orientation, containing the differences between consecutive elements
- For matrix X, return a matrix containing the differences between consecutive elements in each column (dim = 0), or each row (dim = 1)
- The optional argument k indicates that the differences are calculated recursively k times; by default k = 1 is used
- The resulting number of differences is n − k, where n is the number of elements; if n ≤ k, the number of differences is zero (ie. an empty vector/matrix is returned)
- The argument dim is optional; by default dim = 0
- Examples:

      vec a = linspace<vec>(1,10,10);

      vec b = diff(a);

- See also:
  - trapz()
  - numerical differentiation in Wikipedia
  - numerical differentiation in MathWorld

dot( A, B )
cdot( A, B )
norm_dot( A, B )

- dot(A,B): dot product of A and B, treating A and B as vectors
- cdot(A,B): as per dot(A,B), but the complex conjugate of A is used
- norm_dot(A,B): normalised dot product; equivalent to dot(A,B)  /  ( ∥A∥·∥B∥ )
- Caveat: norm() is more robust for calculating the norm, as it handles underflows and overflows
- Examples:

      vec a(10, fill::randu);
      vec b(10, fill::randu);

      double x = dot(a,b);

- See also:
  - norm()
  - as_scalar()
  - cross()
  - conj()

eps( X )

- Obtain the positive distance of the absolute value of each element of X to the next largest representable floating point number
- X can be a scalar (eg. double), vector or matrix
- Examples:

      mat A(4, 5, fill::randu);
      mat B = eps(A);

- See also:
  - datum::eps
  - Floating-Point Arithmetic in MathWorld
  - IEEE Standard for Floating-Point Arithmetic in Wikipedia

B = expmat( A )
expmat( B, A )

- Matrix exponential of general square matrix A
- If A is not square sized, a std::logic_error exception is thrown
- If the matrix exponential cannot be found:
  - B = expmat(A) resets B and throws a std::runtime_error exception
  - expmat(B,A) resets B and returns a bool set to false (exception is not thrown)
- Caveats:
  - the matrix exponential operation is generally not the same as applying the exp() function to each element
  - if matrix A is symmetric, expmat_sym() is faster
- Examples:

      mat A(5, 5, fill::randu);

      mat B = expmat(A);

- See also:
  - expmat_sym()
  - logmat()
  - sqrtmat()
  - miscellaneous element-wise functions
  - matrix exponential in Wikipedia
  - matrix exponential in MathWorld
  - Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later

B = expmat_sym( A )
expmat_sym( B, A )

- Matrix exponential of symmetric/hermitian matrix A
- The computation is based on eigen decomposition
- If A is not square sized, a std::logic_error exception is thrown
- If the matrix exponential cannot be found:
  - B = expmat_sym(A) resets B and throws a std::runtime_error exception
  - expmat_sym(B,A) resets B and returns a bool set to false (exception is not thrown)
- Caveat: the matrix exponential operation is generally not the same as applying the exp() function to each element
- Examples:

      mat A(5, 5, fill::randu);

      mat B = A*A.t();   // make symmetric matrix

      mat C = expmat_sym(B);

- See also:
  - expmat()
  - logmat_sympd()
  - sqrtmat_sympd()
  - .is_symmetric()
  - .is_hermitian()
  - miscellaneous element-wise functions
  - matrix exponential in Wikipedia
  - matrix exponential in MathWorld
  - Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later

find( X )
find( X, k )
find( X, k, s )

- Return a column vector containing the indices of elements of X that are non-zero or satisfy a relational condition
- The output vector must have the type uvec (ie. the indices are stored as unsigned integers of type uword)
- X is interpreted as a vector, with column-by-column ordering of the elements of X
- Relational operators can be used instead of X, eg. A > 0.5
- If k = 0 (default), return the indices of all non-zero elements, otherwise return at most k of their indices
- If s = "first" (default), return at most the first k indices of the non-zero elements
- If s = "last", return at most the last k indices of the non-zero elements
- Caveats:
  - to clamp values to an interval, clamp() is more efficient
  - to replace a specific value, .replace() is more efficient
- Examples:

      mat A(5, 5, fill::randu);
      mat B(5, 5, fill::randu);

      uvec q1 = find(A > B);
      uvec q2 = find(A > 0.5);
      uvec q3 = find(A > 0.5, 3, "last");

      // change elements of A greater than 0.5 to 1
      A.elem( find(A > 0.5) ).ones();

- See also:
  - all()
  - any()
  - clamp()
  - .transform()
  - find_finite()
  - find_nonfinite()
  - find_nan()
  - find_unique()
  - nonzeros()
  - unique()
  - sort_index()
  - ind2sub()
  - .index_min() & .index_max()
  - trimatu_ind() / trimatl_ind()
  - submatrix views
  - subcube views

find_finite( X )

- Return a column vector containing the indices of elements of X that are finite (ie. not ±Inf and not NaN)
- The output vector must have the type uvec (ie. the indices are stored as unsigned integers of type uword)
- X is interpreted as a vector, with column-by-column ordering of the elements of X
- Examples:

      mat A(5, 5, fill::randu);

      A(1,1) = datum::inf;

      // accumulate only finite elements
      double val = accu( A.elem( find_finite(A) ) );

- See also:
  - find()
  - find_nonfinite()
  - find_nan()
  - .is_finite()
  - .replace()
  - .has_inf()
  - .has_nan()
  - submatrix views
  - constants (pi, nan, inf, ...)

find_nonfinite( X )

- Return a column vector containing the indices of elements of X that are non-finite (ie. ±Inf or NaN)
- The output vector must have the type uvec (ie. the indices are stored as unsigned integers of type uword)
- X is interpreted as a vector, with column-by-column ordering of the elements of X
- Examples:

      mat A(5, 5, fill::randu);

      A(1,1) = datum::inf;
      A(2,2) = datum::nan;

      // change non-finite elements to zero
      A.elem( find_nonfinite(A) ).zeros();

- Caveat: to replace instances of a specific non-finite value (eg. NaN or Inf), it is more efficient to use .replace()
- See also:
  - find()
  - find_finite()
  - find_nan()
  - .is_finite()
  - .replace()
  - .has_inf()
  - .has_nan()
  - submatrix views
  - constants (pi, nan, inf, ...)

find_nan( X )

- Return a column vector containing the indices of elements of X that are NaN (not-a-number)
- The output vector must have the type uvec (ie. the indices are stored as unsigned integers of type uword)
- X is interpreted as a vector, with column-by-column ordering of the elements of X
- Examples:

      mat A(5, 5, fill::randu);

      A(2,3) = datum::nan;

      uvec indices = find_nan(A);

- Caveat: to replace instances of NaN values, it is more efficient to use .replace()
- See also:
  - find()
  - find_finite()
  - find_nonfinite()
  - .replace()
  - .has_nan()
  - constants (pi, nan, inf, ...)
  - NaN in Wikipedia

find_unique( X )
find_unique( X, ascending_indices )

- Return a column vector containing the indices of unique elements of X
- The output vector must have the type uvec (ie. the indices are stored as unsigned integers of type uword)
- X is interpreted as a vector, with column-by-column ordering of the elements of X
- The ascending_indices argument is optional; it is one of:

  `true` | = | the returned indices are sorted to be ascending (default setting)
  `false` | = | the returned indices are in arbitrary order (faster operation)

- Examples:

      mat A = { { 2, 2, 4 },
                { 4, 6, 6 } };

      uvec indices = find_unique(A);

- See also:
  - find()
  - unique()
  - intersect()
  - submatrix views

fliplr( X )
flipud( X )

- fliplr(): generate a copy of matrix X, with the order of the columns reversed
- flipud(): generate a copy of matrix X, with the order of the rows reversed
- Examples:

      mat A(5, 5, fill::randu);

      mat B = fliplr(A);
      mat C = flipud(A);

- See also:
  - reverse()
  - shift()
  - .swap_rows() & .swap_cols()
  - .t()

imag( X )
real( X )

- Extract the imaginary/real part of a complex matrix or cube
- Examples:

      cx_mat C(5, 5, fill::randu);

      mat    A = imag(C);
      mat    B = real(C);

- See also:
  - .set_imag() / .set_real()
  - abs()
  - conj()
  - conv_to()

uvec | sub = ind2sub( size(X), index ) |  | (form 1)
umat | sub = ind2sub( size(X), vector_of_indices ) |  | (form 2)

- Convert a linear index, or a vector of indices, to subscript notation
- The argument size(X) can be replaced with size(n_rows,n_cols) or size(n_rows,n_cols,n_slices)
- If an index is out of range, a std::logic_error exception is thrown
- When only one index is given (form 1), the subscripts are returned in a vector of type uvec
- When a vector of indices (of type uvec) is given (form 2), the corresponding subscripts are returned in each column of an m x n matrix of type umat; m=2 for matrix subscripts, while m=3 for cube subscripts
- Examples:

      mat M(4, 5, fill::randu);

      uvec s = ind2sub( size(M), 6 );

      cout << "row: " << s(0) << endl;
      cout << "col: " << s(1) << endl;

      uvec indices = find(M > 0.5);
      umat t       = ind2sub( size(M), indices );

      cube Q(2,3,4);

      uvec u = ind2sub( size(Q), 8 );

      cout << "row:   " << u(0) << endl;
      cout << "col:   " << u(1) << endl;
      cout << "slice: " << u(2) << endl;

- See also:
  - size()
  - sub2ind()
  - element access
  - find()

index_min( V ) index_min( M ) index_min( M, dim ) index_min( Q ) index_min( Q, dim ) |  | index_max( V ) index_max( M ) index_max( M, dim ) index_max( Q ) index_max( Q, dim )

- For vector V, return the linear index of the extremum value; the returned index is of type uword
- For matrix M and:
  - dim = 0, return a row vector (of type urowvec or umat), with each column containing the index of the extremum value in the corresponding column of M
  - dim = 1, return a column vector (of type uvec or umat), with each row containing the index of the extremum value in the corresponding row of M
- For cube Q, return a cube (of type ucube) containing the indices of extremum values of elements along dimension dim, where dim∈ { 0, 1, 2 }
- For each column, row, or slice, the index starts at zero
- The dim argument is optional; by default dim = 0 is used
- For objects with complex numbers, absolute values are used for comparison
- Examples:

      vec v(10, fill::randu);

      uword i = index_max(v);
      double max_val_in_v = v(i);

      mat M(5, 6, fill::randu);

      urowvec ii = index_max(M);
      ucolvec jj = index_max(M,1);

      double max_val_in_col_2 = M( ii(2), 2 );

      double max_val_in_row_4 = M( 4, jj(4) );

- See also:
  - min() & max()
  - .index_min() & .index_max() (member functions)
  - sort_index()
  - find()

inplace_trans( X )
inplace_trans( X, method )
inplace_strans( X )
inplace_strans( X, method )

- In-place / in-situ transpose of matrix X
- For real (non-complex) matrix:
  - inplace_trans() performs a normal transpose
  - inplace_strans() not applicable
- For complex matrix:
  - inplace_trans() performs a Hermitian transpose (ie. the conjugate of the elements is taken during the transpose)
  - inplace_strans() provides a transposed copy without taking the conjugate of the elements
- The argument method is optional
- By default, a greedy transposition algorithm is used; a low-memory algorithm can be used instead by explicitly setting method to `"lowmem"`
- The low-memory algorithm is considerably slower than the greedy algorithm; using the low-memory algorithm is only recommended for cases where X takes up more than half of available memory (ie. very large X)
- Examples:

      mat X(4,     5,     fill::randu);
      mat Y(20000, 30000, fill::randu);

      inplace_trans(X);            // use greedy algorithm by default

      inplace_trans(Y, "lowmem");  // use low-memory (and slow) algorithm

- See also:
  - .t()
  - trans()
  - inplace matrix transpose in Wikipedia

C = intersect( A, B ) |  | (form 1)
intersect( C, iA, iB, A, B ) |  | (form 2)

- For form 1:
  - return the unique elements common to both A and B, sorted in ascending order
- For form 2:
  - store in C the unique elements common to both A and B, sorted in ascending order
  - store in iA and iB the indices of the unique elements, such that C = A.elem(iA) and C = B.elem(iB)
  - iA and iB must have the type uvec (ie. the indices are stored as unsigned integers of type uword)
- C is a column vector if either A or B is a matrix or column vector; C is a row vector if both A and B are row vectors
- For matrices and vectors with complex numbers, ordering is via absolute values
- Examples:

      ivec A = regspace<ivec>(4, 1);  // 4, 3, 2, 1
      ivec B = regspace<ivec>(3, 6);  // 3, 4, 5, 6

      ivec C = intersect(A,B);       // 3, 4

      ivec CC;
      uvec iA;
      uvec iB;

      intersect(CC, iA, iB, A, B);

- See also:
  - unique()
  - find_unique()
  - submatrix views

join_rows( A, B ) join_rows( A, B, C ) join_rows( A, B, C, D ) join_cols( A, B ) join_cols( A, B, C ) join_cols( A, B, C, D ) |  | join_horiz( A, B ) join_horiz( A, B, C ) join_horiz( A, B, C, D ) join_vert( A, B ) join_vert( A, B, C ) join_vert( A, B, C, D )

- join_rows() and join_horiz(): horizontal concatenation; join the corresponding rows of the given matrices; the given matrices must have the same number of rows
- join_cols() and join_vert(): vertical concatenation; join the corresponding columns of the given matrices; the given matrices must have the same number of columns
- Examples:

      mat A(4, 5, fill::randu);
      mat B(4, 6, fill::randu);
      mat C(6, 5, fill::randu);

      mat AB = join_rows(A,B);
      mat AC = join_cols(A,C);

- See also:
  - .shed_rows / cols / slices
  - .insert_rows / cols / slices
  - join_slices()
  - submatrix views

join_slices( cube C, cube D )
join_slices( mat M, mat N )
join_slices( mat M, cube C )
join_slices( cube C, mat M )

- for two cubes C and D: join the slices of C with the slices of D; cubes C and D must have the same number of rows and columns (ie. all slices must have the same size)
- for two matrices M and N: treat M and N as cube slices and join them to form a cube with 2 slices; matrices M and N must have the same number of rows and columns
- for matrix M and cube C: treat M as a cube slice and join it with the slices of C; matrix M and cube C must have the same number of rows and columns
- Examples:

      cube C(5, 10, 3, fill::randu);
      cube D(5, 10, 4, fill::randu);

      cube E = join_slices(C,D);

      mat M(10, 20, fill::randu);
      mat N(10, 20, fill::randu);

      cube Q = join_slices(M,N);

      cube R = join_slices(Q,M);

      cube S = join_slices(M,Q);

- See also:
  - .shed_rows / cols / slices
  - .insert_rows / cols / slices
  - join_rows / cols
  - subcube views

kron( A, B )

- Kronecker tensor product
- Given matrix A (with n rows and p columns) and matrix B (with m rows and q columns), generate a matrix (with nm rows and pq columns) that denotes the tensor product of A and B
- Examples:

      mat A(4, 5, fill::randu);
      mat B(5, 4, fill::randu);

      mat K = kron(A,B);

- See also:
  - repmat()
  - repelem()
  - Kronecker product in MathWorld
  - Kronecker product in Wikipedia

complex result = log_det( A ) |  | (form 1)
log_det( val, sign, A ) |  | (form 2)

- Log determinant of square matrix A, based on LU decomposition
- form 1: return the complex log determinant
  - if matrix A is real and the determinant is positive:
    - the real part is the log determinant
    - the imaginary part is zero
  - if matrix A is real and the determinant is negative:
    - the real part is log abs(determinant)
    - the imaginary part is equal to datum::pi
- form 2: store the calculated log determinant in val and sign, and return a bool indicating success;
  the determinant is equal to exp(val) · sign
- If A is not square sized, a std::logic_error exception is thrown
- If the log determinant cannot be found:
  - result = log_det(A) throws a std::runtime_error exception
  - log_det(val, sign, A) returns a bool set to false (exception is not thrown)
- Examples:

      mat A(5, 5, fill::randu);

      cx_double result = log_det(A);    // form 1

      double val;
      double sign;

      bool ok = log_det(val, sign, A);  // form 2

- See also:
  - log_det_sympd()
  - det()
  - rcond()
  - cx_double
  - determinant in MathWorld
  - determinant in Wikipedia

result = log_det_sympd( A ) |  | (form 1)
log_det_sympd( result, A ) |  | (form 2)

- Log determinant of symmetric positive definite matrix A
- form 1: return the log determinant
- form 2: store the calculated log determinant in result and return a bool indicating success
- If A is not square sized, a std::logic_error exception is thrown
- If the log determinant cannot be found:
  - result = log_det_sympd(A) throws a std::runtime_error exception
  - log_det_sympd(result, A) returns a bool set to false (exception is not thrown)
- Examples:

      mat A(5, 5, fill::randu);

      mat B = A.t() * A;  // make symmetric matrix

      double result1 = log_det_sympd(B);           // form 1

      double result2;
      bool   success = log_det_sympd(result2, B);  // form 2

- See also:
  - log_det()
  - rcond()
  - determinant in MathWorld
  - determinant in Wikipedia

B = logmat( A )
logmat( B, A )

- Complex matrix logarithm of general square matrix A
- If A is not square sized, a std::logic_error exception is thrown
- If the matrix logarithm cannot be found:
  - B = logmat(A) resets B and throws a std::runtime_error exception
  - logmat(B,A) resets B and returns a bool set to false (exception is not thrown)
- Caveats:
  - the matrix logarithm operation is generally not the same as applying the log() function to each element
  - if matrix A is symmetric positive definite, logmat_sympd() is faster
- Examples:

         mat A(5, 5, fill::randu);

      cx_mat B = logmat(A);

- See also:
  - logmat_sympd()
  - expmat()
  - sqrtmat()
  - real()
  - miscellaneous element-wise functions
  - matrix logarithm in Wikipedia

B = logmat_sympd( A )
logmat_sympd( B, A )

- Matrix logarithm of symmetric/hermitian positive definite matrix A
- If A is not square sized, a std::logic_error exception is thrown
- If the matrix logarithm cannot be found:
  - B = logmat_sympd(A) resets B and throws a std::runtime_error exception
  - logmat_sympd(B,A) resets B and returns a bool set to false (exception is not thrown)
- Caveat: the matrix logarithm operation is generally not the same as applying the log() function to each element
- Examples:

      mat A(5, 5, fill::randu);

      mat B = A*A.t();   // make symmetric matrix

      mat C = logmat_sympd(B);

- See also:
  - logmat()
  - expmat_sym()
  - sqrtmat_sympd()
  - .is_sympd()
  - miscellaneous element-wise functions
  - matrix logarithm in Wikipedia
  - positive definite matrix in Wikipedia
  - positive definite matrix in MathWorld

min( V ) min( M ) min( M, dim ) min( Q ) min( Q, dim ) min( A, B ) |  | max( V ) max( M ) max( M, dim ) max( Q ) max( Q, dim ) max( A, B )

- For vector V, return the extremum value
- For matrix M, return the extremum value for each column (dim = 0), or each row (dim = 1)
- For cube Q, return the extremum values of elements along dimension dim, where dim∈ { 0, 1, 2 }
- The dim argument is optional; by default dim = 0 is used
- For two matrices/cubes A and B, return a matrix/cube containing element-wise extremum values
- For objects with complex numbers, absolute values are used for comparison
- Examples:

      colvec v(10, fill::randu);
      double x = max(v);

      mat M(10, 10, fill::randu);

      rowvec a = max(M);
      rowvec b = max(M,0);
      colvec c = max(M,1);

      // element-wise maximum
      mat X(5, 6, fill::randu);
      mat Y(5, 6, fill::randu);
      mat Z = arma::max(X,Y);  // use arma:: prefix to distinguish from std::max()

- See also:
  - .min() & .max() (member functions)
  - index_min() & index_max()
  - clamp()
  - statistics functions
  - running_stat - class for running statistics of scalars
  - running_stat_vec - class for running statistics of vectors

nonzeros( X )

- Return a column vector containing the non-zero values of X
- X can be a sparse or dense matrix
- Caveats:
  - for dense matrices/vectors, to obtain the number of non-zero elements, the expression `accu(X != 0)` is more efficient
  - for sparse matrices, to obtain the number of non-zero elements, the `.n_nonzero` attribute is more efficient, eg. `X.n_nonzero`
- Examples:

      sp_mat A = sprandu<sp_mat>(100, 100, 0.1);
         vec a = nonzeros(A);

      mat B(100, 100, fill::eye);
      vec b = nonzeros(B);

- See also:
  - find()
  - unique()
  - vectorise()
  - .clean()
  - .for_each()

norm( X )
norm( X, p )

- Compute the p-norm of X, where X is a vector or matrix
- For vectors, p is an integer ≥ 1, or one of: `"-inf"`, `"inf"`, `"fro"`
- For matrices, p is one of: 1, 2, `"inf"`, `"fro"`
- `"-inf"` is the minimum quasi-norm, `"inf"` is the maximum norm, `"fro"` is the Frobenius norm
- The argument p is optional; by default p = 2 is used
- For vector norm with p = 2 and matrix norm with p = "fro", a robust algorithm is used to reduce the likelihood of underflows and overflows
- Caveats:
  - matrix 2-norm (spectral norm) is based on SVD, which is computationally intensive; a faster alternative is norm2est()
  - to obtain the vector norm of each row or column of a matrix, use vecnorm()
  - to obtain the zero/Hamming pseudo-norm (number of non-zero elements), use this expression: `accu(X != 0)`
- Examples:

      vec q(5, fill::randu);

      double x = norm(q, 2);
      double y = norm(q, "inf");

- See also:
  - vecnorm()
  - norm2est()
  - normalise()
  - dot()
  - vector norm in Wikipedia
  - vector norm in MathWorld
  - matrix norm in Wikipedia
  - matrix norm in MathWorld

norm2est( X )
norm2est( X, tol )
norm2est( X, tol, max_iter )

- Fast estimate of the 2-norm (spectral norm) of X, where X is a dense or sparse matrix
- The estimate is found via an iterative algorithm which finishes when one of the following conditions is met:
  - the relative difference between two consecutive estimates is less than the specified tolerance, ie. |est1- est2| / max(est1, est2) < tol
  - the number of iterations is equal to max_iter
- The optional argument tol specifies the tolerance for the relative difference; by default tol = 1e-6 is used
- The optional argument max_iter specifies the maximum number of iterations; by default max_iter = 100 is used
- Examples:

      mat X(2000, 3000, fill::randu);

      double x = norm2est(X);
      double y = norm2est(X, 1e-5);
      double z = norm2est(X, 1e-4, 10);

- See also:
  - norm()
  - matrix norm in Wikipedia
  - matrix norm in MathWorld

normalise( V )
normalise( V, p )
normalise( X )
normalise( X, p )
normalise( X, p, dim )

- For vector V, return its normalised version (ie. having unit p-norm)
- For matrix X, return its normalised version, where each column (dim = 0) or row (dim = 1) has been normalised to have unit p-norm
- The p argument is optional; by default p = 2 is used
- The dim argument is optional; by default dim = 0 is used
- Examples:

      vec A(10, fill::randu);
      vec B = normalise(A);
      vec C = normalise(A, 1);

      mat X(5, 6, fill::randu);
      mat Y = normalise(X);
      mat Z = normalise(X, 2, 1);

- See also:
  - norm()
  - vecnorm()
  - norm_dot()
  - Normalised vector in MathWorld
  - Unit vector in Wikipedia

pow( A, scalar ) |  | (form 1)
pow( A, B ) |  | (form 2)

- Element-wise power operations
- form 1: raise all elements in A to the power denoted by the given scalar
- form 2: raise each element in A to the power denoted by the corresponding element in B; sizes of A and B must be the same
- Caveats:
  - to raise all elements to the power 2, use square() instead
  - for the matrix power operation, which takes into account matrix structure, use powmat()
- Examples:

      mat A(5, 6, fill::randu);
      mat B(5, 6, fill::randu);

      mat X = pow(A, 3.45);
      mat Y = pow(A, B);

- See also:
  - powmat()
  - miscellaneous element-wise functions

B = powmat( A, n )
powmat( B, A, n )

- Matrix power operation: raise square matrix A to the power of n, where n has the type int or double
- If n has the type double, the resultant matrix B always has complex elements
- For n = 0, an identity matrix is generated
- If A is not square sized, a std::logic_error exception is thrown
- If the matrix power cannot be found:
  - B = powmat(A) resets B and throws a std::runtime_error exception
  - powmat(B,A) resets B and returns a bool set to false (exception is not thrown)
- Caveats:
  - the matrix power operation is generally not the same as applying the pow() function to each element
  - to find the inverse of a matrix, use inv() instead
  - to solve a system of linear equations, use solve() instead
  - to find the matrix square root, use sqrtmat() instead
- Examples:

         mat A(5, 5, fill::randu);

         mat B = powmat(A, 4);     //     integer exponent

      cx_mat C = powmat(A, 4.56);  // non-integer exponent

- See also:
  - pow()
  - sqrtmat()
  - inv()
  - eye()
  - operators
  - miscellaneous element-wise functions

prod( V )
prod( M )
prod( M, dim )

- For vector V, return the product of all elements
- For matrix M, return the product of elements in each column (dim = 0), or each row (dim = 1)
- The dim argument is optional; by default dim = 0 is used
- Examples:

      colvec v(10, fill::randu);
      double x = prod(v);

      mat M(10, 10, fill::randu);

      rowvec a = prod(M);
      rowvec b = prod(M,0);
      colvec c = prod(M,1);

- See also:
  - cumprod()
  - Schur product in operators

r = rank( X ) |  | (form 1)
r = rank( X, tolerance ) |  |
rank( r, X ) |  | (form 2)
rank( r, X, tolerance ) |  |

- Calculate the rank of matrix X, based on singular value decomposition
- form 1: return the rank
- form 2: store the calculated rank in r and return a bool indicating success
- Any singular values less than tolerance are treated as zero
- The tolerance argument is optional; by default tolerance = max_rc · max_sv · epsilon, where:
  - max_rc = max(X.n_rows, X.n_cols)
  - max_sv = maximum singular value of X (ie. spectral norm)
  - epsilon = difference between 1 and the least value greater than 1 that is representable
- If the calculation fails:
  - r = rank(X) throws a std::runtime_error exception
  - rank(r,X) returns a bool set to false (exception is not thrown)
- Caveat: to distinguish from `std::rank`, use the `arma::` prefix, ie. `arma::rank(X)`
- Examples:

      mat A(4, 5, fill::randu);

      uword r1 = rank(A);           // form 1

      uword r2;
      bool success = rank(r2,  A);  // form 2

- See also:
  - rcond()
  - svd()
  - orth()
  - datum::eps
  - Rank in MathWorld
  - Rank in Wikipedia

rcond( A )

- Return the 1-norm estimate of the reciprocal condition number of square matrix A
- Values close to 1 suggest that A is well-conditioned
- Values close to 0 suggest that A is badly conditioned
- If A is not square sized, a std::logic_error exception is thrown
- Caveat: to efficiently calculate rcond and matrix inverse at the same time, use inv()
- Examples:

      mat A(5, 5, fill::randu);

      double r = rcond(A);

- See also:
  - cond()
  - rank()
  - det()
  - inv()
  - solve()

repelem( A, num_copies_per_row, num_copies_per_col )

- Generate a matrix by replicating each element of matrix A
- The generated matrix has the following size:

  n_rows | = | num_copies_per_row | * | A.n_rows
  n_cols | = | num_copies_per_col | * | A.n_cols

- Examples:

      mat A(2, 3, fill::randu);

      mat B = repelem(A, 4, 5);

- See also:
  - repmat()
  - kron()
  - reshape()
  - resize()

repmat( A, num_copies_per_row, num_copies_per_col )

- Generate a matrix by replicating matrix A in a block-like fashion
- The generated matrix has the following size:

  n_rows | = | num_copies_per_row | × | A.n_rows
  n_cols | = | num_copies_per_col | × | A.n_cols

- Caveat: to apply a vector operation on each row or column of a matrix, it is generally more efficient to use .each_row() or .each_col()
- Examples:

      mat A(2, 3, fill::randu);

      mat B = repmat(A, 4, 5);

- See also:
  - .each_col() & .each_row() (vector operations applied to each column or row)
  - repelem()
  - kron()
  - reshape()
  - resize()

reshape( X, n_rows, n_cols ) (X is a vector or matrix)
reshape( X, size(Y) )
reshape( Q, n_rows, n_cols, n_slices ) (Q is a cube)
reshape( Q, size(R) )

- Generate a vector/matrix/cube with given size specifications, whose elements are taken from the given object in a column-wise manner; the elements in the generated object are placed column-wise (ie. the first column is filled up before filling the second column)
- The layout of the elements in the generated object will be different to the layout in the given object
- If the total number of elements in the given object is less than the specified size, the remaining elements in the generated object are set to zero
- If the total number of elements in the given object is greater than the specified size, only a subset of elements is taken from the given object
- Caveats:
  - to change the size without preserving data, use .set_size() instead, which is much faster
  - to grow/shrink a matrix while preserving the elements as well as the layout of the elements, use resize() instead
  - to flatten a matrix into a vector, use vectorise() or .as_col() / .as_row() instead
- Examples:

      mat A(10, 5, fill::randu);

      mat B = reshape(A, 5, 10);

- See also:
  - .reshape() (member function)
  - .set_size()
  - resize()
  - vectorise()
  - as_scalar()
  - conv_to()
  - diagmat()
  - repmat()
  - repelem()
  - size()
  - interp2()

resize( X, n_rows, n_cols ) (X is a vector or matrix)
resize( X, size(Y) )
resize( Q, n_rows, n_cols, n_slices ) (Q is a cube)
resize( Q, size(R) )

- Generate a vector/matrix/cube with given size specifications, whose elements as well as the layout of the elements are taken from the given object
- Caveat: to change the size without preserving data, use .set_size() instead, which is much faster
- Examples:

      mat A(4, 5, fill::randu);

      mat B = resize(A, 7, 6);

- See also:
  - .resize() (member function of Mat and Cube)
  - .set_size() (member function of Mat and Cube)
  - reshape()
  - vectorise()
  - as_scalar()
  - conv_to()
  - repmat()
  - repelem()
  - size()

reverse( V )
reverse( X )
reverse( X, dim )

- For vector V, generate a copy of the vector with the order of elements reversed
- For matrix X, generate a copy of the matrix with the order of elements reversed in each column (dim = 0), or each row (dim = 1)
- The dim argument is optional; by default dim = 0 is used
- Examples:

      vec v(123, fill::randu);
      vec y = reverse(v);

      mat A(4, 5, fill::randu);
      mat B = reverse(A);
      mat C = reverse(A,1);

- See also:
  - fliplr() & flipud()
  - shift()
  - sort()
  - trans()
  - .t()
  - .swap_rows() & .swap_cols()

R = roots( P )
roots( R, P )

- Find the complex roots of a polynomial function represented via vector P and store them in column vector R
- The polynomial function is modelled as:
  y = p0xN + p1xN-1 + p2xN-2 + ... + pN-1x1 + pN
  where pi is the i-th polynomial coefficient in vector P
- The computation is based on eigen decomposition; if the decomposition fails:
  - R = roots(P) resets R and throws a std::runtime_error exception
  - roots(R,P) resets R and returns a bool set to false (exception is not thrown)
- Examples:

         vec P(5, fill::randu);

      cx_vec R = roots(P);

- See also:
  - polyval()
  - polyfit()
  - real()
  - solve()
  - zero of a function in Wikipedia

shift( V, N )
shift( X, N )
shift( X, N, dim )

- For vector V, generate a copy of the vector with the elements shifted by N positions in a circular manner
- For matrix X, generate a copy of the matrix with the elements shifted by N positions in each column (dim = 0), or each row (dim = 1)
- N can be positive or negative
- The dim argument is optional; by default dim = 0 is used
- Examples:

      mat A(4, 5, fill::randu);
      mat B = shift(A, -1);
      mat C = shift(A, +1);

- See also:
  - shuffle()
  - fliplr() & flipud()
  - reverse()

shuffle( V )
shuffle( X )
shuffle( X, dim )

- For vector V, generate a copy of the vector with the elements shuffled
- For matrix X, generate a copy of the matrix with the elements shuffled in each column (dim = 0), or each row (dim = 1)
- The dim argument is optional; by default dim = 0 is used
- Examples:

      mat A(4, 5, fill::randu);
      mat B = shuffle(A);

- See also:
  - shift()
  - sort()
  - unique()
  - randi()
  - randperm()
  - RNG seed setting

size( X )
size( n_rows, n_cols )
size( n_rows, n_cols, n_slices )

- Obtain the dimensions of object X, or explicitly specify the dimensions
- Dimensions provided via size(...) can be used in conjunction with matrix constructors, submatrix views, random matrix generation, ind2sub(), sub2ind(), etc.
- size(…) objects support simple arithmetic operations such as addition and multiplication
- Two size(…) objects can be compared for equality/inequality
- size(…) objects can be passed to an std::ostream (eg. cout) via <<
- Caveat: to prevent interference from std::size() in C++17, preface Armadillo's size(…) with the arma namespace qualification, eg. arma::size(…)
- Examples:

      mat A(5,6);

      mat B(size(A), fill::zeros);

      mat C; C.randu(size(A));

      mat D = ones<mat>(size(A));

      mat E(10,20, fill::ones);
      E(3,4,size(C)) = C;    // access submatrix of E

      mat F( size(A) + size(E) );

      mat G( size(A) * 2 );

      cout << "size of A: " << size(A) << endl;

      bool is_same_size = (size(A) == size(E));

- See also:
  - attributes

sort( V )
sort( V, sort_direction )
sort( X )
sort( X, sort_direction )
sort( X, sort_direction, dim )

- For vector V, return a vector which is a sorted version of the input vector
- For matrix X, return a matrix with the elements of the input matrix sorted in each column (dim = 0), or each row (dim = 1)
- The dim argument is optional; by default dim = 0 is used
- The sort_direction argument is optional; sort_direction is either `"ascend"` or `"descend"`; by default `"ascend"` is used
- For matrices and vectors with complex numbers, sorting is via absolute values
- Examples:

      mat A(10, 10, fill::randu);
      mat B = sort(A);

- See also:
  - sort_index()
  - .is_sorted()
  - shuffle()
  - unique()
  - reverse()
  - randi()

sort_index( X )
sort_index( X, sort_direction )
stable_sort_index( X )
stable_sort_index( X, sort_direction )

- Return a vector which describes the sorted order of the elements of X (ie. it contains the indices of the elements of X)
- The output vector must have the type uvec (ie. the indices are stored as unsigned integers of type uword)
- X is interpreted as a vector, with column-by-column ordering of the elements of X
- The sort_direction argument is optional; sort_direction is either `"ascend"` or `"descend"`; by default `"ascend"` is used
- The stable_sort_index() variant preserves the relative order of elements with equivalent values
- For matrices and vectors with complex numbers, sorting is via absolute values
- Examples:

      vec q(10, fill::randu);

      uvec indices = sort_index(q);

- See also:
  - sort()
  - find()
  - .is_sorted()
  - .index_min() & .index_max()
  - ind2sub()

B = sqrtmat( A )
sqrtmat( B, A )

- Complex square root of general square matrix A
- If A is not square sized, a std::logic_error exception is thrown
- If matrix A appears to be singular, an approximate square root is attempted; additionally, sqrtmat(B,A) returns a bool set to false
- Caveats:
  - the square root of a matrix is generally not the same as applying the sqrt() function to each element
  - if matrix A is symmetric positive definite, sqrtmat_sympd() is faster
- Examples:

         mat A(5, 5, fill::randu);

      cx_mat B = sqrtmat(A);

- See also:
  - sqrtmat_sympd()
  - powmat()
  - expmat()
  - logmat()
  - chol()
  - real()
  - miscellaneous element-wise functions
  - square root of a matrix in Wikipedia

B = sqrtmat_sympd( A )
sqrtmat_sympd( B, A )

- Square root of symmetric/hermitian positive definite matrix A
- If A is not square sized, a std::logic_error exception is thrown
- If the square root cannot be found:
  - B = sqrtmat_sympd(A) resets B and throws a std::runtime_error exception
  - sqrtmat_sympd(B,A) resets B and returns a bool set to false (exception is not thrown)
- Caveat: the matrix square root operation is generally not the same as applying the sqrt() function to each element
- Examples:

      mat A(5, 5, fill::randu);

      mat B = A*A.t();   // make symmetric matrix

      mat C = sqrtmat_sympd(B);

- See also:
  - sqrtmat()
  - expmat_sym()
  - logmat_sympd()
  - chol()
  - .is_sympd()
  - miscellaneous element-wise functions
  - square root of a matrix in Wikipedia
  - positive definite matrix in Wikipedia
  - positive definite matrix in MathWorld

sum( V )
sum( M )
sum( M, dim )
sum( Q )
sum( Q, dim )

- For vector V, return the sum of all elements
- For matrix M, return the sum of elements in each column (dim = 0), or each row (dim = 1)
- For cube Q, return the sums of elements along dimension dim, where dim ∈ { 0, 1, 2 }; for example, dim = 0 indicates the sum of elements in each column within each slice
- The dim argument is optional; by default dim = 0 is used
- Caveat: to get a sum of all the elements regardless of the object type (ie. vector, or matrix, or cube), use accu() instead
- Examples:

      colvec v(10, fill::randu);
      double x = sum(v);

      mat M(10, 10, fill::randu);

      rowvec a = sum(M);
      rowvec b = sum(M,0);
      colvec c = sum(M,1);

      double y = accu(M);   // find the overall sum regardless of object type

- See also:
  - accu()
  - cumsum()
  - trace()
  - trapz()
  - mean()
  - as_scalar()
  - .as_dense()

uword | index | = sub2ind( size(M), row, col ) | (M is a matrix)
uvec | indices | = sub2ind( size(M), matrix_of_subscripts ) |
uword | index | = sub2ind( size(Q), row, col, slice ) | (Q is a cube)
uvec | indices | = sub2ind( size(Q), matrix_of_subscripts ) |

- Convert subscripts to a linear index
- The argument size(X) can be replaced with size(n_rows,n_cols) or size(n_rows,n_cols,n_slices)
- For the matrix_of_subscripts argument, the subscripts must be stored in each column of an m x n matrix of type umat; m = 2 for matrix subscripts, while m = 3 for cube subscripts
- If a subscript is out of range, a std::logic_error exception is thrown
- Examples:

      mat  M(4,5);
      cube Q(4,5,6);

      uword i = sub2ind( size(M), 2, 3 );
      uword j = sub2ind( size(Q), 2, 3, 4 );

- See also:
  - size()
  - ind2sub()
  - element access

symmatu( A )
symmatu( A, do_conj )
symmatl( A )
symmatl( A, do_conj )

- symmatu(A): generate symmetric matrix from square matrix A, by reflecting the upper triangle to the lower triangle
- symmatl(A): generate symmetric matrix from square matrix A, by reflecting the lower triangle to the upper triangle
- If A is a complex matrix, the reflection uses the complex conjugate of the elements; to disable the complex conjugate, set do_conj to false
- If A is non-square, a std::logic_error exception is thrown
- Examples:

      mat A(5, 5, fill::randu);

      mat B = symmatu(A);
      mat C = symmatl(A);

- See also:
  - diagmat()
  - trimatu() / trimatl()
  - .is_symmetric()
  - .is_hermitian()
  - Symmetric matrix in Wikipedia

trace( X )

- Sum of the elements on the main diagonal of matrix X
- If X is an expression, the evaluation of the expression aims to calculate only the diagonal elements
- Examples:

      mat A(5, 5, fill::randu);

      double x = trace(A);

- See also:
  - accu()
  - as_scalar()
  - .diag()
  - diagvec()
  - diagmat()
  - sum()

trans( A )
strans( A )

- For real (non-complex) matrix:
  - trans() provides a transposed copy of the matrix
  - strans() is not applicable
- For complex matrix:
  - trans() provides a Hermitian (conjugate) transposed copy (ie. signs of imaginary components are flipped)
  - strans() provides a simple transposed copy (ie. signs of imaginary components are not flipped)
- Examples:

      mat A(5, 10, fill::randu);

      mat B = trans(A);
      mat C = A.t();    // equivalent to trans(A), but more compact

- See also:
  - .t()
  - inplace_trans()
  - reverse()
  - transpose in Wikipedia
  - transpose in MathWorld
  - conjugate transpose in Wikipedia
  - conjugate transpose in MathWorld

trapz( X, Y )
trapz( X, Y, dim )
trapz( Y )
trapz( Y, dim )

- Compute the trapezoidal integral of Y with respect to spacing in X, in each column (dim = 0) or each row (dim = 1) of Y
- X must be a vector; its length must equal either the number of rows in Y (when dim = 0), or the number of columns in Y (when dim = 1)
- If X is not specified, unit spacing is used
- The dim argument is optional; by default dim = 0
- Examples:

      vec X = linspace<vec>(0, datum::pi, 1000);
      vec Y = sin(X);

      mat Z = trapz(X,Y);

- See also:
  - diff()
  - sum()
  - linspace()
  - numerical integration in Wikipedia
  - numerical integration in MathWorld
  - trapezoidal rule in Wikipedia

trimatu( A )
trimatu( A, k )
trimatl( A )
trimatl( A, k )

- Create a new matrix by copying either the upper or lower triangular part from square matrix A, and setting the remaining elements to zero
  - trimatu() copies the upper triangular part
  - trimatl() copies the lower triangular part
- The argument k specifies the diagonal which inclusively delineates the boundary of the triangular part
  - for k > 0, the k-th super-diagonal is used (above main diagonal, towards top-right corner)
  - for k < 0, the k-th sub-diagonal is used (below main diagonal, towards bottom-left corner)
- The argument k is optional; by default the main diagonal is used (k = 0)
- If A is non-square, a std::logic_error exception is thrown
- Examples:

      mat A(5, 5, fill::randu);

      mat U  = trimatu(A);
      mat L  = trimatl(A);

      mat UU = trimatu(A,  1);  // omit the main diagonal
      mat LL = trimatl(A, -1);  // omit the main diagonal

- See also:
  - .is_trimatu() / .is_trimatl()
  - trimatu_ind() / trimatl_ind()
  - symmatu() / symmatl()
  - diagmat()
  - nonzeros()
  - Triangular matrix in MathWorld
  - Triangular matrix in Wikipedia

trimatu_ind( size(A) )
trimatu_ind( size(A), k )
trimatl_ind( size(A) )
trimatl_ind( size(A), k )

- Return a column vector containing the indices of elements that form the upper or lower triangle part of matrix A
  - trimatu_ind() refers to the upper triangular part
  - trimatl_ind() refers to the lower triangular part
- The output vector must have the type uvec (ie. the indices are stored as unsigned integers of type uword)
- The argument k specifies the diagonal which inclusively delineates the boundary of the triangular part
  - for k > 0, the k-th super-diagonal is used (above main diagonal, towards top-right corner)
  - for k < 0, the k-th sub-diagonal is used (below main diagonal, towards bottom-left corner)
- The argument k is optional; by default the main diagonal is used (k = 0)
- The argument size(A) can be replaced with size(n_rows, n_cols)
- Examples:

      mat A(5, 5, fill::randu);

      uvec upper_indices = trimatu_ind( size(A) );
      uvec lower_indices = trimatl_ind( size(A) );

      // extract upper/lower triangle into vector
      vec upper_part = A(upper_indices);
      vec lower_part = A(lower_indices);

      // obtain indices without the main diagonal
      uvec alt_upper_indices = trimatu_ind( size(A),  1);
      uvec alt_lower_indices = trimatl_ind( size(A), -1);

- See also:
  - trimatu() / trimatl()
  - find()
  - submatrix views

unique( A )

- Return the unique elements of A, sorted in ascending order
- If A is a vector, the output is also a vector with the same orientation (row or column) as A; if A is a matrix, the output is always a column vector
- Examples:

      mat X = { { 1, 2 }
                { 2, 3 } };

      mat Y = unique(X);

- See also:
  - find()
  - find_unique()
  - sort()
  - shuffle()
  - nonzeros()
  - intersect()

vecnorm( X )
vecnorm( X, p )
vecnorm( X, p, dim )

- Compute the p-norm of each column vector (dim = 0) or row vector (dim = 1) of matrix X
- p is an integer ≥ 1, or `"-inf"` (minimum quasi-norm), or `"inf"` (maximum norm)
- The arguments p and dim are optional; by default p = 2 and dim = 0 are used
- Caveat: to compute the matrix norm, use norm() instead
- Examples:

      mat X(4, 5, fill::randu);

      rowvec r = vecnorm(X, 2);

      colvec c = vecnorm(X, "inf", 1);

- See also:
  - norm()
  - normalise()
  - vector norm in Wikipedia
  - vector norm in MathWorld

vectorise( X )
vectorise( X, dim )
vectorise( Q )

- Generate a flattened version of matrix X or cube Q
- The argument dim is optional; by default dim = 0 is used
- For dim = 0, the elements are copied from X column-wise, resulting in a column vector; equivalent to concatenating all the columns of X
- For dim = 1, the elements are copied from X row-wise, resulting in a row vector; equivalent to concatenating all the rows of X
- Caveats:
  - column-wise vectorisation is faster than row-wise vectorisation
  - for sparse matrices, row-wise vectorisation is not recommended
- Examples:

      mat X(4, 5, fill::randu);

      vec v = vectorise(X);

- See also:
  - .as_col() / .as_row()
  - .col_as_mat() / .row_as_mat()
  - nonzeros()
  - reshape()
  - resize()
  - diagvec()
  - as_scalar()

miscellaneous element-wise functions:

exp |  | log |  | square |  | floor |  | erf |  | tgamma |  | sign
exp2 |  | log2 |  | sqrt |  | ceil |  | erfc |  | lgamma |  |
exp10 |  | log10 |  | cbrt |  | round |  |  |  |  |  |
expm1 |  | log1p |  |  |  | trunc |  |  |  |  |  |
trunc_exp |  | trunc_log |  |  |  |  |  |  |  |  |  |

- Apply a function to each element
- Usage:
  - B = fn(A), where fn(A) is one of the functions below
  - A and B must have the same matrix type or cube type, such as mat or cube

  `exp(A)` |  | base-e exponential: ex
  `exp2(A)` |  | base-2 exponential: 2x
  `exp10(A)` |  | base-10 exponential: 10x
  `expm1(A)` |  | compute `exp(A)-1` accurately for values of A close to zero (only for float and double elements)
  `trunc_exp(A)` |  | base-e exponential, truncated to avoid infinity (only for float and double elements)
  `log(A)` |  | natural log: logex
  `log2(A)` |  | base-2 log: log2x
  `log10(A)` |  | base-10 log: log10x
  `log1p(A)` |  | compute `log(1+A)` accurately for values of A close to zero (only for float and double elements)
  `trunc_log(A)` |  | natural log, truncated to avoid ±infinity (only for float and double elements)
  `square(A)` |  | square: x2
  `sqrt(A)` |  | square root: √x
  `cbrt(A)` |  | cube root: ∛x
  `floor(A)` |  | largest integral value that is not greater than the input value
  `ceil(A)` |  | smallest integral value that is not less than the input value
  `round(A)` |  | round to nearest integer, with halfway cases rounded away from zero
  `trunc(A)` |  | round to nearest integer, towards zero
  `erf(A)` |  | error function (only for float and double elements)
  `erfc(A)` |  | complementary error function (only for float and double elements)
  `tgamma(A)` |  | gamma function (only for float and double elements)
  `lgamma(A)` |  | natural log of the absolute value of gamma function (only for float and double elements)
  `sign(A)` |  | signum function; for each element a in A, the corresponding element b in B is:
  | ⎧ | −1 | if a < 0
  b = | ⎨ | 0 | if a = 0
  | ⎩ | +1 | if a > 0
  if a is complex and non-zero, then b = a / abs(a)

- Caveats:
  - all of the above functions are applied element-wise, where each element is treated independently
  - the element-wise functions exp(), log() and sqrt() have the corresponding functions expmat(), logmat() and sqrtmat() which take into account matrix structure
- Examples:

      mat A(5, 5, fill::randu);
      mat B = exp(A);

- See also:
  - abs()
  - pow()
  - clamp()
  - conj()
  - imag() / real()
  - .transform() (apply user-defined function to each element)
  - expmat()
  - logmat()
  - sqrtmat()
  - trigonometric functions
  - statistics functions
  - miscellaneous constants

trigonometric element-wise functions (cos, sin, tan, ...)

- For single argument functions, B = trig_fn(A), where trig_fn is applied to each element in A, with trig_fn as one of:
  - cos, acos, cosh, acosh
  - sin, asin, sinh, asinh
  - tan, atan, tanh, atanh
  - sinc, defined as sinc(x) = sin(π·x) / (π·x) for x ≠ 0, and sinc(x) = 1 for x = 0
- For dual argument functions, apply the function to each tuple of two corresponding elements in X and Y:
  - Z = atan2(Y, X)
  - Z = hypot(X, Y)
- Examples:

      mat X(5, 5, fill::randu);
      mat Y = cos(X);

- See also:
  - miscellaneous element-wise functions
  - trigonometric functions in Wikipedia
  - atan2 function in Wikipedia
  - hypot function in Wikipedia
  - sinc function in Wikipedia

Decompositions, Factorisations, Inverses and Equation Solvers (Dense Matrices)

R = chol( X ) |  | (form 1)
R = chol( X, layout ) |  | (form 2)
chol( R, X ) |  | (form 3)
chol( R, X, layout ) |  | (form 4)
chol( R, P, X, layout, "vector" ) |  | (form 5)
chol( R, P, X, layout, "matrix" ) |  | (form 6)

- Cholesky decomposition of symmetric/hermitian matrix X into triangular matrix R, with an optional permutation vector/matrix P
- By default, R is upper triangular
- The optional argument layout is either `"upper"` or `"lower"`, which specifies whether R is upper or lower triangular
- Forms 1 to 4 require X to be positive definite
- Forms 5 to 6 require X to be positive semi-definite; pivoted decomposition provides a permutation vector/matrix P with type uvec or umat
- The decomposition has the following form:
  - forms 1 and 3: X = R.t() * R
  - forms 2 and 4 with layout = "upper": X = R.t() * R
  - forms 2 and 4 with layout = "lower": X = R * R.t()
  - form 5 with layout = "upper": X(P,P) = R.t() * R, where X(P,P) is a non-contiguous view of X
  - form 5 with layout = "lower": X(P,P) = R * R.t(), where X(P,P) is a non-contiguous view of X
  - form 6 with layout = "upper": X = P * R.t() * R * P.t()
  - form 6 with layout = "lower": X = P * R * R.t() * P.t()
- If the decomposition fails:
  - the forms R = chol(X) and R = chol(X,layout) reset R and throw a std::runtime_error exception
  - the other forms reset R and P, and return a bool set to false (exception is not thrown)
- Caveat: to find the inverse of a positive definite matrix, use inv_sympd()
- Examples:

      mat A(5, 5, fill::randu);
      mat X = A.t()*A;

      mat R1 = chol(X);
      mat R2 = chol(X, "lower");

      mat R3;
      bool ok = chol(R3, X);

       mat R;
      uvec P_vec;
      umat P_mat;

      chol(R, P_vec, X, "upper", "vector");
      chol(R, P_mat, X, "lower", "matrix");

- See also:
  - sqrtmat()
  - lu()
  - qr()
  - inv_sympd()
  - .is_sympd()
  - Cholesky decomposition in MathWorld
  - Cholesky decomposition in Wikipedia
  - Definite matrix in Wikipedia

vec eigval = eig_sym( X )
eig_sym( eigval, X )
eig_sym( eigval, eigvec, X )
eig_sym( eigval, eigvec, X, method )

- Eigen decomposition of dense symmetric/hermitian matrix X
- The eigenvalues and corresponding eigenvectors are stored in eigval and eigvec, respectively
- The eigenvalues are in ascending order
- The eigenvectors are stored as column vectors
- If X is not square sized, a std::logic_error exception is thrown
- The method argument is optional; method is either `"dc"` or `"std"`
  - `"dc"` indicates divide-and-conquer method (default setting)
  - `"std"` indicates standard method
  - the divide-and-conquer method provides slightly different results than the standard method, but is considerably faster for large matrices
- If the decomposition fails:
  - eigval = eig_sym(X) resets eigval and throws a std::runtime_error exception
  - eig_sym(eigval,X) resets eigval and returns a bool set to false (exception is not thrown)
  - eig_sym(eigval,eigvec,X) resets eigval & eigvec and returns a bool set to false (exception is not thrown)
- Examples:

      // for matrices with real elements

      mat A(50, 50, fill::randu);
      mat B = A.t()*A;  // generate a symmetric matrix

      vec eigval;
      mat eigvec;

      eig_sym(eigval, eigvec, B);

      // for matrices with complex elements

      cx_mat C(50, 50, fill::randu);
      cx_mat D = C.t()*C;

         vec eigval2;
      cx_mat eigvec2;

      eig_sym(eigval2, eigvec2, D);

- See also:
  - eig_gen()
  - eig_pair()
  - svd()
  - svd_econ()
  - princomp()
  - eigs_sym()
  - .is_symmetric()
  - .is_hermitian()
  - eigen decomposition in MathWorld
  - eigenvalues & eigenvectors in Wikipedia
  - divide & conquer eigenvalue algorithm in Wikipedia

cx_vec eigval = eig_gen( X )
cx_vec eigval = eig_gen( X, bal )
eig_gen( eigval, X )
eig_gen( eigval, X, bal )
eig_gen( eigval, eigvec, X )
eig_gen( eigval, eigvec, X, bal )
eig_gen( eigval, leigvec, reigvec, X )
eig_gen( eigval, leigvec, reigvec, X, bal )

- Eigen decomposition of dense general (non-symmetric/non-hermitian) square matrix X
- The eigenvalues and corresponding right eigenvectors are stored in eigval and eigvec, respectively
- If both left and right eigenvectors are requested they are stored in leigvec and reigvec, respectively
- The eigenvectors are stored as column vectors
- The bal argument is optional; bal is one of:

  `"balance"` | ↦ | diagonally scale and permute X to improve conditioning of the eigenvalues
  `"nobalance"` | ↦ | do not balance X; this is the default operation

- If X is not square sized, a std::logic_error exception is thrown
- If the decomposition fails:
  - eigval = eig_gen(X) resets eigval and throws a std::runtime_error exception
  - eig_gen(eigval,X) resets eigval and returns a bool set to false (exception is not thrown)
  - eig_gen(eigval,eigvec,X) resets eigval & eigvec and returns a bool set to false (exception is not thrown)
  - eig_gen(eigval,leigvec,reigvec,X) resets eigval, leigvec & reigvec and returns a bool set to false (exception is not thrown)
- Examples:

      mat A(10, 10, fill::randu);

      cx_vec eigval;
      cx_mat eigvec;

      eig_gen(eigval, eigvec, A);
      eig_gen(eigval, eigvec, A, "balance");

- See also:
  - eig_pair()
  - eig_sym()
  - svd()
  - svd_econ()
  - schur()
  - eigs_gen()
  - eigen decomposition in MathWorld
  - eigenvalues & eigenvectors in Wikipedia

cx_vec eigval = eig_pair( A, B )
eig_pair( eigval, A, B )
eig_pair( eigval, eigvec, A, B )
eig_pair( eigval, leigvec, reigvec, A, B )

- Eigen decomposition for pair of general dense square matrices A and B of the same size, such that A*eigvec = B*eigvec*diagmat(eigval)
- The eigenvalues and corresponding right eigenvectors are stored in eigval and eigvec, respectively
- If both left and right eigenvectors are requested they are stored in leigvec and reigvec, respectively
- The eigenvectors are stored as column vectors
- If A or B is not square sized, a std::logic_error exception is thrown
- If the decomposition fails:
  - eigval = eig_pair(A,B) resets eigval and throws a std::runtime_error exception
  - eig_pair(eigval,A,B) resets eigval and returns a bool set to false (exception is not thrown)
  - eig_pair(eigval,eigvec,A,B) resets eigval & eigvec and returns a bool set to false (exception is not thrown)
  - eig_pair(eigval,leigvec,reigvec,A,B) resets eigval, leigvec & reigvec and returns a bool set to false (exception is not thrown)
- Examples:

      mat A(10, 10, fill::randu);
      mat B(10, 10, fill::randu);

      cx_vec eigval;
      cx_mat eigvec;

      eig_pair(eigval, eigvec, A, B);

- See also:
  - eig_gen()
  - eig_sym()
  - qz()
  - eigen decomposition in MathWorld
  - eigenvalues & eigenvectors in Wikipedia

H = hess( X )
hess( H, X )
hess( U, H, X )

- Upper Hessenberg decomposition of square matrix X, such that X = U*H*U.t()
- U is a unitary matrix containing the Hessenberg vectors
- H is a square matrix known as the upper Hessenberg matrix, with elements below the first subdiagonal set to zero
- If X is not square sized, a std::logic_error exception is thrown
- If the decomposition fails:
  - H = hess(X) resets H and throws a std::runtime_error exception
  - hess(H,X) resets H and returns a bool set to false (exception is not thrown)
  - hess(U,H,X) resets U & H and returns a bool set to false (exception is not thrown)
- Caveat: in general, upper Hessenberg decomposition is not unique
- Examples:

      mat X(20,20, fill::randu);

      mat U;
      mat H;

      hess(U, H, X);

- See also:
  - qz()
  - schur()
  - Hessenberg decomposition in MathWorld
  - Hessenberg matrix in Wikipedia

B = inv( A )
B = inv( A, settings )
inv( B, A )
inv( B, A, settings )
inv( B, rcond, A )

- Inverse of general square matrix A
- The settings argument is optional; it is one of the following:

  `inv_opts::no_ugly` |  | do not provide inverses for poorly conditioned matrices (where rcond < A.n_rows ·datum::eps)
  `inv_opts::allow_approx` |  | allow approximate inverses for rank deficient or poorly conditioned matrices

- The reciprocal condition number is optionally calculated and stored in rcond
  - rcond close to 1 suggests that A is well-conditioned
  - rcond close to 0 suggests that A is badly conditioned
- If A is not square sized, a std::logic_error exception is thrown
- If A appears to be singular:
  - B = inv(A) resets B and throws a std::runtime_error exception
  - inv(B,A) resets B and returns a bool set to false (exception is not thrown)
  - inv(B,rcond,A) resets B, sets rcond to zero, and returns a bool set to false (exception is not thrown)
- Caveats:
  - if matrix A is know to be symmetric positive definite, inv_sympd() is faster
  - if matrix A is know to be diagonal, use inv( diagmat(A) )
  - if matrix A is know to be triangular, use inv( trimatu(A) ) or inv( trimatl(A) )
  - to solve a system of linear equations, such as Z = inv(X)*Y, solve() can be faster and/or more accurate
- Examples:

      mat A(5, 5, fill::randu);

      mat B = inv(A);

      mat C;
      bool success = inv(C, A);

      mat D;
      double rcond_val;
      inv(D, rcond_val, A);

      A.col(1).zeros();
      mat E;
      inv(E, A, inv_opts::allow_approx);

- See also:
  - .i()
  - inv_sympd()
  - rcond()
  - pinv()
  - solve()
  - diagmat()
  - trimatu() / trimatl()
  - powmat()
  - matrix inverse in MathWorld
  - invertible matrix in Wikipedia

B = inv_sympd( A )
B = inv_sympd( A, settings )
inv_sympd( B, A )
inv_sympd( B, A, settings )
inv_sympd( B, rcond, A )

- Inverse of symmetric/hermitian positive definite matrix A
- The settings argument is optional; it is one of the following:

  `inv_opts::no_ugly` |  | do not provide inverses for poorly conditioned matrices (where rcond < A.n_rows ·datum::eps)
  `inv_opts::allow_approx` |  | allow approximate inverses for rank deficient or poorly conditioned symmetric matrices

- The reciprocal condition number is optionally calculated and stored in rcond
  - rcond close to 1 suggests that A is well-conditioned
  - rcond close to 0 suggests that A is badly conditioned
- If A is not square sized, a std::logic_error exception is thrown
- If A appears to be singular or not positive definite:
  - B = inv_sympd(A) resets B and throws a std::runtime_error exception
  - inv_sympd(B,A) resets B and returns a bool set to false (exception is not thrown)
  - inv_sympd(B,rcond,A) resets B, sets rcond to zero, and returns a bool set to false (exception is not thrown)
- Caveat: to solve a system of linear equations, such as Z = inv(X)*Y, using solve() can be faster and/or more accurate
- Examples:

      mat A(5, 5, fill::randu);
      mat B = A.t() * A;

      mat C = inv_sympd(B);

      mat D;
      bool success = inv_sympd(D, B);

      mat E;
      double rcond_val;
      inv_sympd(E, rcond_val, B);

      B.col(1).zeros();
      B.row(1).zeros();
      mat F;
      inv_sympd(F, B, inv_opts::allow_approx);

- See also:
  - inv()
  - rcond()
  - pinv()
  - solve()
  - eig_sym()
  - .is_sympd()
  - matrix inverse in MathWorld
  - invertible matrix in Wikipedia
  - positive definite matrix in MathWorld
  - positive definite matrix in Wikipedia

lu( L, U, P, X )
lu( L, U, X )

- Lower-upper decomposition (with partial pivoting) of matrix X
- The first form provides a lower-triangular matrix L, an upper-triangular matrix U, and a permutation matrix P, such that P.t()*L*U = X
- The second form provides permuted L and U, such that L*U = X; note that in this case L is generally not lower-triangular
- If the decomposition fails:
  - lu(L,U,P,X) resets L, U, P and returns a bool set to false (exception is not thrown)
  - lu(L,U,X) resets L, U and returns a bool set to false (exception is not thrown)
- Examples:

      mat A(5, 5, fill::randu);

      mat L, U, P;

      lu(L, U, P, A);

      mat B = P.t()*L*U;

- See also:
  - chol()
  - LU decomposition in Wikipedia
  - LU decomposition in MathWorld

B = null( A )
B = null( A, tolerance )
null( B, A )
null( B, A, tolerance )

- Find the orthonormal basis of the null space of matrix A
- The dimension of the range space is the number of singular values of A not greater than tolerance
- The tolerance argument is optional; by default tolerance = max_rc · max_sv · epsilon, where:
  - max_rc = max(A.n_rows, A.n_cols)
  - max_sv = maximum singular value of A (ie. spectral norm)
  - epsilon = difference between 1 and the least value greater than 1 that is representable
- The computation is based on singular value decomposition; if the decomposition fails:
  - B = null(A) resets B and throws a std::runtime_error exception
  - null(B,A) resets B and returns a bool set to false (exception is not thrown)
- Examples:

      mat A(5, 6, fill::randu);

      A.row(0).zeros();
      A.col(0).zeros();

      mat B = null(A);

- See also:
  - orth()
  - qr()
  - svd()
  - rank()
  - datum::eps
  - Orthonormal basis in Wikipedia

B = orth( A )
B = orth( A, tolerance )
orth( B, A )
orth( B, A, tolerance )

- Find the orthonormal basis of the range space of matrix A, so that B.t()*B ≈ eye(r,r), where r = rank(A)
- The dimension of the range space is the number of singular values of A greater than tolerance
- The tolerance argument is optional; by default tolerance = max_rc · max_sv · epsilon, where:
  - max_rc = max(A.n_rows, A.n_cols)
  - max_sv = maximum singular value of A (ie. spectral norm)
  - epsilon = difference between 1 and the least value greater than 1 that is representable
- The computation is based on singular value decomposition; if the decomposition fails:
  - B = orth(A) resets B and throws a std::runtime_error exception
  - orth(B,A) resets B and returns a bool set to false (exception is not thrown)
- Examples:

      mat A(5, 6, fill::randu);

      mat B = orth(A);

- See also:
  - null()
  - qr()
  - svd()
  - rank()
  - datum::eps
  - Orthonormal basis in Wikipedia

B = pinv( A )
B = pinv( A, tolerance )
B = pinv( A, tolerance, method )
pinv( B, A )
pinv( B, A, tolerance )
pinv( B, A, tolerance, method )

- Moore-Penrose pseudo-inverse (generalised inverse) of matrix A
- The computation is based on singular value decomposition
- The tolerance argument is optional; by default tolerance = max_rc · max_sv · epsilon, where:
  - max_rc = max(A.n_rows, A.n_cols)
  - max_sv = maximum singular value of A (ie. spectral norm)
  - epsilon = difference between 1 and the least value greater than 1 that is representable
- Any singular values less than tolerance are treated as zero
- The method argument is optional; method is either `"dc"` or `"std"`
  - `"dc"` indicates divide-and-conquer method (default setting)
  - `"std"` indicates standard method
  - the divide-and-conquer method provides slightly different results than the standard method, but is considerably faster for large matrices
- If the decomposition fails:
  - B = pinv(A) resets B and throws a std::runtime_error exception
  - pinv(B,A) resets B and returns a bool set to false (exception is not thrown)
- Caveats:
  - to find approximate solutions to under-/over-determined or rank deficient systems of linear equations, solve() can be considerably faster and/or more accurate
  - if the given matrix A is square-sized and only occasionally rank deficient, using inv() or inv_sympd() with the `inv_opts::allow_approx` option is faster
- Examples:

      mat A(4, 5, fill::randu);

      mat B = pinv(A);        // use default tolerance

      mat C = pinv(A, 0.01);  // set tolerance to 0.01

- See also:
  - .i()
  - inv()
  - solve()
  - svd()
  - datum::eps
  - Pseudoinverse in MathWorld
  - Moore-Penrose Matrix Inverse in MathWorld
  - Moore-Penrose inverse in Wikipedia

qr( Q, R, X ) |  | (form 1)
qr( Q, R, P, X, "vector" ) |  | (form 2)
qr( Q, R, P, X, "matrix" ) |  | (form 3)

- Decomposition of X into an orthogonal matrix Q and a right triangular matrix R, with an optional permutation matrix/vector P
  - form 1: decomposition has the form Q*R = X
  - form 2: P is permutation vector with type uvec; decomposition has the form Q*R = X.cols(P)
  - form 3: P is permutation matrix with type umat; decomposition has the form Q*R = X*P
- If P is specified, a column pivoting decomposition is used; the diagonal entries of R are ordered from largest to smallest magnitude
- If the decomposition fails, Q, R and P are reset and the function returns a bool set to false (exception is not thrown)
- Examples:

      mat X(5, 5, fill::randu);

      mat Q;
      mat R;

      qr(Q, R, X);

      uvec P_vec;
      umat P_mat;

      qr(Q, R, P_vec, X, "vector");
      qr(Q, R, P_mat, X, "matrix");

- See also:
  - qr_econ()
  - chol()
  - orth()
  - orthogonal matrix in Wikipedia
  - QR decomposition in Wikipedia
  - QR decomposition in MathWorld

qr_econ( Q, R, X )

- Economical decomposition of X (with size m x n) into an orthogonal matrix Q and a right triangular matrix R, such that Q*R = X
- If m > n, only the first n rows of R and the first n columns of Q are calculated (ie. the zero rows of R and the corresponding columns of Q are omitted)
- If the decomposition fails, Q and R are reset and the function returns a bool set to false (exception is not thrown)
- Examples:

      mat X(6, 5, fill::randu);

      mat Q;
      mat R;

      qr_econ(Q, R, X);

- See also:
  - qr()
  - orthogonal matrix in Wikipedia
  - QR decomposition in Wikipedia
  - QR decomposition in Octave
  - QR decomposition in MathWorld

qz( AA, BB, Q, Z, A, B )
qz( AA, BB, Q, Z, A, B, select )

- Generalised Schur decomposition for pair of general square matrices A and B of the same size,
  such that A = Q.t()*AA*Z.t() and B = Q.t()*BB*Z.t()
- The select argument is optional and specifies the ordering of the top left of the Schur form; it is one of the following:

  `"none"` |  | no ordering (default operation)
  `"lhp"` |  | left-half-plane: eigenvalues with real part < 0
  `"rhp"` |  | right-half-plane: eigenvalues with real part > 0
  `"iuc"` |  | inside-unit-circle: eigenvalues with absolute value < 1
  `"ouc"` |  | outside-unit-circle: eigenvalues with absolute value > 1

- The left and right Schur vectors are stored in Q and Z, respectively
- In the complex-valued problem, the generalised eigenvalues are found in diagvec(AA) / diagvec(BB)
- If A or B is not square sized, a std::logic_error exception is thrown
- If the decomposition fails, AA, BB, Q and Z are reset, and the function returns a bool set to false (exception is not thrown)
- Examples:

      mat A(10, 10, fill::randu);
      mat B(10, 10, fill::randu);

      mat AA;
      mat BB;
      mat Q;
      mat Z;

      qz(AA, BB, Q, Z, A, B);

- See also:
  - hess()
  - schur()
  - eig_pair()
  - generalised Schur decomposition in Wikipedia

S = schur( X )
schur( S, X )
schur( U, S, X )

- Schur decomposition of square matrix X, such that X = U*S*U.t()
- U is a unitary matrix containing the Schur vectors
- S is an upper triangular matrix, called the Schur form of X
- If X is not square sized, a std::logic_error exception is thrown
- If the decomposition fails:
  - S = schur(X) resets S and throws a std::runtime_error exception
  - schur(S,X) resets S and returns a bool set to false (exception is not thrown)
  - schur(U,S,X) resets U & S and returns a bool set to false (exception is not thrown)
- Caveat: in general, Schur decomposition is not unique
- Examples:

      mat X(20,20, fill::randu);

      mat U;
      mat S;

      schur(U, S, X);

- See also:
  - hess()
  - qz()
  - eig_gen()
  - Schur decomposition in MathWorld
  - Schur decomposition in Wikipedia

X = solve( A, B )
X = solve( A, B, settings )
solve( X, A, B )
solve( X, A, B, settings )

- Solve a dense system of linear equations, A*X = B, where X is unknown; similar functionality to the \ operator in Matlab/Octave, ie. X = A \ B
- A can be square sized (critically determined system), or non-square (under/over-determined system); A can be rank deficient
- B can be a vector or matrix
- The number of rows in A and B must be the same
- By default, matrix A is analysed to automatically determine whether it is a general matrix, band matrix, diagonal matrix, or symmetric/hermitian positive definite (sympd) matrix; based on the detected matrix structure, a specialised solver is used for faster execution; if no solution is found, an approximate solver is automatically used as a fallback; see the associated paper for more details
- If A is known to be a triangular matrix, the solution can be computed faster by explicitly indicating that A is triangular through trimatu() or trimatl(); see examples below
- The settings argument is optional; it is one of the following, or a combination thereof:

  `solve_opts::fast` |  | fast mode: disable determining solution quality via rcond, disable iterative refinement, disable equilibration
  `solve_opts::refine` |  | apply iterative refinement to improve solution quality (matrix A must be square)
  `solve_opts::equilibrate` |  | equilibrate the system before solving (matrix A must be square)
  `solve_opts::likely_sympd` |  | indicate that matrix A is likely symmetric/hermitian positive definite (sympd)
  `solve_opts::allow_ugly` |  | keep solutions of systems that are singular to working precision
  `solve_opts::no_approx` |  | do not find approximate solutions for rank deficient systems
  `solve_opts::force_sym` |  | force use of the symmetric/hermitian solver (not limited to sympd matrices)
  `solve_opts::force_approx` |  | force use of the approximate solver

  the above settings can be combined using the `+` operator; for example: `solve_opts::fast + solve_opts::no_approx`
- If a rank deficient system is detected and the `solve_opts::no_approx` option is not enabled, a warning is emitted and an approximate solution is attempted;
  since Armadillo 10.4, this warning can be disabled by setting ARMA_WARN_LEVEL to 1 before including the armadillo header:
  `#define ARMA_WARN_LEVEL 1`
  `#include <armadillo>`
- Caveats:
  - using `solve_opts::fast` will speed up finding the solution, but for poorly conditioned systems the solution may have lower quality
  - not all sympd matrices are automatically detected; to directly indicate that matrix A is likely sympd, use `solve_opts::likely_sympd`
  - using `solve_opts::force_approx` is only advised if the system is known to be rank deficient; the approximate solver is considerably slower
- If no solution is found:
  - X = solve(A,B) resets X and throws a std::runtime_error exception
  - solve(X,A,B) resets X and returns a bool set to false (exception is not thrown)
- Implementation details are available in the following paper:
  Conrad Sanderson and Ryan Curtin.
  An Adaptive Solver for Systems of Linear Equations.
  International Conference on Signal Processing and Communication Systems, 2020.
- Examples:

      mat A(5, 5, fill::randu);
      vec b(5,    fill::randu);
      mat B(5, 5, fill::randu);

      vec x1 = solve(A, b);

      vec x2;
      bool status = solve(x2, A, b);

      mat X1 = solve(A, B);

      mat X2 = solve(A, B, solve_opts::fast);  // enable fast mode

      mat X3 = solve(trimatu(A), B);  // indicate that A is triangular

- See also:
  - inv()
  - pinv()
  - rcond()
  - roots()
  - syl()
  - spsolve() - solve sparse system of linear equations
  - linear system of equations in MathWorld
  - system of linear equations in Wikipedia
  - band matrix in Wikipedia
  - definiteness of a matrix in Wikipedia
  - positive definite matrix in MathWorld
  - iterative refinement
  - ensmallen - C++ library for solving numerical optimisation problems

vec s = svd( X )
svd( vec s, X )
svd( mat U, vec s, mat V, mat X )
svd( mat U, vec s, mat V, mat X, method )
svd( cx_mat U, vec s, cx_mat V, cx_mat X )
svd( cx_mat U, vec s, cx_mat V, cx_mat X, method )

- Singular value decomposition of dense matrix X
- If X is square, it can be reconstructed using X = U*diagmat(s)*V.t()
- The singular values are in descending order
- The method argument is optional; method is either `"dc"` or `"std"`
  - `"dc"` indicates divide-and-conquer method (default setting)
  - `"std"` indicates standard method
  - the divide-and-conquer method provides slightly different results than the standard method, but is considerably faster for large matrices
- If the decomposition fails, the output objects are reset and:
  - s = svd(X) resets s and throws a std::runtime_error exception
  - svd(s,X) resets s and returns a bool set to false (exception is not thrown)
  - svd(U,s,V,X) resets U, s, V and returns a bool set to false (exception is not thrown)
- Examples:

      mat X(5, 5, fill::randu);

      mat U;
      vec s;
      mat V;

      svd(U,s,V,X);

- See also:
  - svd_econ()
  - eig_gen()
  - eig_sym()
  - princomp()
  - pinv()
  - svds()
  - singular value decomposition in Wikipedia
  - singular value decomposition in MathWorld

svd_econ( mat U, vec s, mat V, mat X )
svd_econ( mat U, vec s, mat V, mat X, mode )
svd_econ( mat U, vec s, mat V, mat X, mode, method )
svd_econ( cx_mat U, vec s, cx_mat V, cx_mat X )
svd_econ( cx_mat U, vec s, cx_mat V, cx_mat X, mode )
svd_econ( cx_mat U, vec s, cx_mat V, cx_mat X, mode, method )

- Economical singular value decomposition of dense matrix X
- The singular values are in descending order
- The mode argument is optional; mode is one of:

  | `"both"` | = | compute both left and right singular vectors (default operation)
  | `"left"` | = | compute only left singular vectors
  | `"right"` | = | compute only right singular vectors

- The method argument is optional; method is either `"dc"` or `"std"`
  - `"dc"` indicates divide-and-conquer method (default setting)
  - `"std"` indicates standard method
  - the divide-and-conquer method provides slightly different results than the standard method, but is considerably faster for large matrices
- If the decomposition fails, U, s, V are reset and a bool set to false is returned (exception is not thrown)
- Examples:

      mat X(4, 5, fill::randu);

      mat U;
      vec s;
      mat V;

      svd_econ(U, s, V, X);

- See also:
  - svd()
  - eig_gen()
  - eig_sym()
  - princomp()
  - svds()
  - singular value decomposition in Wikipedia
  - singular value decomposition in MathWorld

X = syl( A, B, C )
syl( X, A, B, C )

- Solve the Sylvester equation, ie. AX + XB + C = 0, where X is unknown
- Matrices A, B and C must be square sized
- If no solution is found:
  - syl(A,B,C) resets X and throws a std::runtime_error exception
  - syl(X,A,B,C) resets X and returns a bool set to false (exception is not thrown)
- Examples:

      mat A(5, 5, fill::randu);
      mat B(5, 5, fill::randu);
      mat C(5, 5, fill::randu);

      mat X1 = syl(A, B, C);

      mat X2;
      syl(X2, A, B, C);

- See also:
  - solve()
  - Sylvester equation in Wikipedia

Decompositions, Factorisations and Equation Solvers (Sparse Matrices)

vec eigval = eigs_sym( X, k )
vec eigval = eigs_sym( X, k, form )
vec eigval = eigs_sym( X, k, form, opts )
vec eigval = eigs_sym( X, k, sigma )
vec eigval = eigs_sym( X, k, sigma, opts )
eigs_sym( eigval, X, k )
eigs_sym( eigval, X, k, form )
eigs_sym( eigval, X, k, form, opts )
eigs_sym( eigval, X, k, sigma )
eigs_sym( eigval, X, k, sigma, opts )
eigs_sym( eigval, eigvec, X, k )
eigs_sym( eigval, eigvec, X, k, form )
eigs_sym( eigval, eigvec, X, k, form, opts )
eigs_sym( eigval, eigvec, X, k, sigma )
eigs_sym( eigval, eigvec, X, k, sigma, opts )

- Obtain a limited number of eigenvalues and eigenvectors of sparse symmetric real matrix X
- k specifies the number of eigenvalues and eigenvectors
- The argument form is optional; form is one of:

  `"lm"` | = | obtain eigenvalues with largest magnitude (default operation)
  `"sm"` | = | obtain eigenvalues with smallest magnitude (see the caveats below)
  `"la"` | = | obtain eigenvalues with largest algebraic value
  `"sa"` | = | obtain eigenvalues with smallest algebraic value

- The argument sigma is optional; if sigma is given, eigenvalues closest to sigma are found via shift-invert mode
  NOTE: to use sigma, both ARMA_USE_ARPACK and ARMA_USE_SUPERLU must be enabled in config.hpp
- The opts argument is optional; opts is an instance of the eigs_opts structure:

      struct eigs_opts
        {
        double       tol;     // default: 0
        unsigned int maxiter; // default: 1000
        unsigned int subdim;  // default: max(2*k+1, 20)
        };

  - tol specifies the tolerance for convergence
  - maxiter specifies the maximum number of Arnoldi iterations
  - subdim specifies the dimension of the Krylov subspace, with the constraint `k < subdim ≤ X.n_rows`; recommended value is `subdim ≥ 2*k`
- The eigenvalues and corresponding eigenvectors are stored in eigval and eigvec, respectively
- If X is not square sized, a std::logic_error exception is thrown
- If the decomposition fails:
  - eigval = eigs_sym(X,k) resets eigval and throws a std::runtime_error exception
  - eigs_sym(eigval,X,k) resets eigval and returns a bool set to false (exception is not thrown)
  - eigs_sym(eigval,eigvec,X,k) resets eigval & eigvec and returns a bool set to false (exception is not thrown)
- Caveats:
  - the number of obtained eigenvalues/eigenvectors may be lower than requested, depending on the given data
  - if the decomposition fails, try first increasing opts.subdim (Krylov subspace dimension), and, as secondary options, try increasing opts.maxiter (maximum number of iterations), and/or opts.tol (tolerance for convergence), and/or k (number of eigenvalues)
  - for an alternative to the `"sm"` form, use the shift-invert mode with sigma set to 0.0
- NOTE: the implementation in Armadillo 12.6 is considerably faster than earlier versions; further speedups can be obtained by enabling OpenMP in your compiler (eg. -fopenmp in GCC and clang)
- Examples:

      // generate sparse matrix
      sp_mat A = sprandu<sp_mat>(1000, 1000, 0.1);
      sp_mat B = A.t()*A;

      vec eigval;
      mat eigvec;

      eigs_sym(eigval, eigvec, B, 5);  // find 5 eigenvalues/eigenvectors

      eigs_opts opts;
      opts.maxiter = 10000;            // increase max iterations to 10000

      eigs_sym(eigval, eigvec, B, 5, "lm", opts);

- See also:
  - eigs_gen()
  - eig_sym()
  - svds()
  - .is_symmetric()
  - shift-invert mode in ARPACK
  - eigen decomposition in MathWorld
  - eigenvalues & eigenvectors in Wikipedia

cx_vec eigval = eigs_gen( X, k )
cx_vec eigval = eigs_gen( X, k, form )
cx_vec eigval = eigs_gen( X, k, sigma )
cx_vec eigval = eigs_gen( X, k, form, opts )
cx_vec eigval = eigs_gen( X, k, sigma, opts )
eigs_gen( eigval, X, k )
eigs_gen( eigval, X, k, form )
eigs_gen( eigval, X, k, sigma )
eigs_gen( eigval, X, k, form, opts )
eigs_gen( eigval, X, k, sigma, opts )
eigs_gen( eigval, eigvec, X, k )
eigs_gen( eigval, eigvec, X, k, form )
eigs_gen( eigval, eigvec, X, k, sigma )
eigs_gen( eigval, eigvec, X, k, form, opts )
eigs_gen( eigval, eigvec, X, k, sigma, opts )

- Obtain a limited number of eigenvalues and eigenvectors of sparse general (non-symmetric/non-hermitian) square matrix X
- k specifies the number of eigenvalues and eigenvectors
- The argument form is optional; form is one of:

  `"lm"` | = | obtain eigenvalues with largest magnitude (default operation)
  `"sm"` | = | obtain eigenvalues with smallest magnitude (see the caveats below)
  `"lr"` | = | obtain eigenvalues with largest real part
  `"sr"` | = | obtain eigenvalues with smallest real part
  `"li"` | = | obtain eigenvalues with largest imaginary part
  `"si"` | = | obtain eigenvalues with smallest imaginary part

- The argument sigma is optional; if sigma is given, eigenvalues closest to sigma are found via shift-invert mode
  NOTE: to use sigma, both ARMA_USE_ARPACK and ARMA_USE_SUPERLU must be enabled in config.hpp
- The opts argument is optional; opts is an instance of the eigs_opts structure:

      struct eigs_opts
        {
        double       tol;     // default: 0
        unsigned int maxiter; // default: 1000
        unsigned int subdim;  // default: max(2*k+1, 20)
        };

  - tol specifies the tolerance for convergence
  - maxiter specifies the maximum number of Arnoldi iterations
  - subdim specifies the dimension of the Krylov subspace, with the constraint `k + 2 < subdim ≤ X.n_rows`; recommended value is `subdim ≥ 2*k + 1`
- The eigenvalues and corresponding eigenvectors are stored in eigval and eigvec, respectively
- If X is not square sized, a std::logic_error exception is thrown
- If the decomposition fails:
  - eigval = eigs_gen(X,k) resets eigval and throws a std::runtime_error exception
  - eigs_gen(eigval,X,k) resets eigval and returns a bool set to false (exception is not thrown)
  - eigs_gen(eigval,eigvec,X,k) resets eigval & eigvec and returns a bool set to false (exception is not thrown)
- Caveats:
  - the number of obtained eigenvalues/eigenvectors may be lower than requested, depending on the given data
  - if the decomposition fails, try first increasing opts.subdim (Krylov subspace dimension) and, as secondary options, try increasing opts.maxiter (maximum number of iterations), and/or opts.tol (tolerance for convergence), and/or k (number of eigenvalues)
  - for an alternative to the `"sm"` form, use the shift-invert mode with sigma set to 0.0
- NOTE: the implementation in Armadillo 12.6 is considerably faster than earlier versions; further speedups can be obtained by enabling OpenMP in your compiler (eg. -fopenmp in GCC and clang)
- Examples:

      // generate sparse matrix
      sp_mat A = sprandu<sp_mat>(1000, 1000, 0.1);

      cx_vec eigval;
      cx_mat eigvec;

      eigs_gen(eigval, eigvec, A, 5);  // find 5 eigenvalues/eigenvectors

      eigs_opts opts;
      opts.maxiter = 10000;            // increase max iterations to 10000

      eigs_gen(eigval, eigvec, A, 5, "lm", opts);

- See also:
  - eigs_sym()
  - eig_gen()
  - svds()
  - shift-invert mode in ARPACK
  - eigen decomposition in MathWorld
  - eigenvalues & eigenvectors in Wikipedia

vec s = svds( X, k )
vec s = svds( X, k, tol )
svds( vec s, X, k )
svds( vec s, X, k, tol )
svds( mat U, vec s, mat V, sp_mat X, k )
svds( mat U, vec s, mat V, sp_mat X, k, tol )
svds( cx_mat U, vec s, cx_mat V, sp_cx_mat X, k )
svds( cx_mat U, vec s, cx_mat V, sp_cx_mat X, k, tol )

- Obtain a limited number of singular values and singular vectors (truncated SVD) of sparse matrix X
- The singular values and vectors are calculated via sparse eigen decomposition of: `

  ⎡ | zeros(X.n_rows, X.n_rows) |  | X | ⎤
  ⎣ | X.t() |  | zeros(X.n_cols, X.n_cols) | ⎦

  `
- k specifies the number of singular values and singular vectors
- The singular values are in descending order
- The argument tol is optional; it specifies the tolerance for convergence; it is passed as (tol ÷ √2) to eigs_sym()
- If the decomposition fails, the output objects are reset and:
  - s = svds(X,k) resets s and throws a std::runtime_error exception
  - svds(s,X,k) resets s and returns a bool set to false (exception is not thrown)
  - svds(U,s,V,X,k) resets U, s, V and returns a bool set to false (exception is not thrown)
- Caveats:
  - svds() is intended only for finding a few singular values from a large sparse matrix; to find all singular values, use svd() instead
  - depending on the given matrix, svds() may find fewer singular values than specified
- NOTE: the implementation in Armadillo 12.6 is considerably faster than earlier versions; further speedups can be obtained by enabling OpenMP in your compiler (eg. -fopenmp in GCC and clang)
- Examples:

      sp_mat X = sprandu<sp_mat>(100, 200, 0.1);

      mat U;
      vec s;
      mat V;

      svds(U, s, V, X, 10);

- See also:
  - eigs_gen()
  - eigs_sym()
  - svd()
  - singular value decomposition in Wikipedia
  - singular value decomposition in MathWorld

X = spsolve( A, B )
X = spsolve( A, B, solver )
X = spsolve( A, B, solver, opts )
spsolve( X, A, B )
spsolve( X, A, B, solver )
spsolve( X, A, B, solver, opts )

- Solve a sparse system of linear equations, A*X = B, where A is a sparse matrix, B is a dense matrix or vector, and X is unknown
- The number of rows in A and B must be the same
- If no solution is found:
  - X = spsolve(A, B) resets X and throws a std::runtime_error exception
  - spsolve(X, A, B) resets X and returns a bool set to false (no exception is thrown)
- The solver argument is optional; solver is either `"superlu"` or `"lapack"`; by default `"superlu"` is used
  - for `"superlu"`, ARMA_USE_SUPERLU must be enabled in config.hpp
  - for `"lapack"`, sparse matrix A is converted to a dense matrix before using the LAPACK solver; this considerably increases memory usage
- Notes:
  - the SuperLU solver is mainly useful for very large and/or very sparse matrices
  - to reuse the SuperLU factorisation of A for finding solutions where B is iteratively changed, see the spsolve_factoriser class
  - if there is sufficient amount of memory to store a dense version of matrix A, the LAPACK solver can be faster
- The opts argument is optional and applicable to the SuperLU solver;
  opts is an instance of the superlu_opts structure:

      struct superlu_opts
        {
        bool             allow_ugly;   // default: false
        bool             equilibrate;  // default: false
        bool             symmetric;    // default: false
        double           pivot_thresh; // default: 1.0
        permutation_type permutation;  // default: superlu_opts::COLAMD
        refine_type      refine;       // default: superlu_opts::REF_NONE
        };

  - allow_ugly is either true or false; indicates whether to keep solutions of systems singular to working precision
  - equilibrate is either true or false; indicates whether to equilibrate the system (scale the rows and columns of A to have unit norm)
  - symmetric is either true or false; indicates whether to use SuperLU symmetric mode, which gives preference to diagonal pivots
  - pivot_threshold is in the range [0.0, 1.0], used for determining whether a diagonal entry is an acceptable pivot (details in SuperLU documentation)
  - permutation specifies the type of column permutation; it is one of:

    `superlu_opts::NATURAL` |  | natural ordering
    `superlu_opts::MMD_ATA` |  | minimum degree ordering on structure of `A.t() * A`
    `superlu_opts::MMD_AT_PLUS_A` |  | minimum degree ordering on structure of `A.t() + A`
    `superlu_opts::COLAMD` |  | approximate minimum degree column ordering

  - refine specifies the type of iterative refinement; it is one of:

    `superlu_opts::REF_NONE` |  | no refinement
    `superlu_opts::REF_SINGLE` |  | iterative refinement in single precision
    `superlu_opts::REF_DOUBLE` |  | iterative refinement in double precision
    `superlu_opts::REF_EXTRA` |  | iterative refinement in extra precision

- Examples:

      sp_mat A = sprandu<sp_mat>(1000, 1000, 0.1);

      vec b(1000,    fill::randu);
      mat B(1000, 5, fill::randu);

      vec x = spsolve(A, b);  // solve one system
      mat X = spsolve(A, B);  // solve several systems

      bool status = spsolve(x, A, b);  // use default solver
      if(status == false)  { cout << "no solution" << endl; }

      spsolve(x, A, b, "lapack" );  // use LAPACK  solver
      spsolve(x, A, b, "superlu");  // use SuperLU solver

      superlu_opts opts;

      opts.allow_ugly  = true;
      opts.equilibrate = true;

      spsolve(x, A, b, "superlu", opts);

- See also:
  - spsolve_factoriser
  - solve() - solve dense system of linear equations
  - SuperLU home page
  - linear system of equations in MathWorld
  - system of linear equations in Wikipedia

spsolve_factoriser

- Class for factorisation of sparse matrix A for solving systems of linear equations in the form A*X = B
- Allows the SuperLU factorisation of A to be reused for finding solutions in cases where B is iteratively changed
- For an instance of spsolve_factoriser named as SF, the member functions are:
  SF.factorise(A)
  SF.factorise(A, opts)
  - factorise square-sized sparse matrix A
  - optional settings are given in the opts argument as per the spsolve() function
  - if the factorisation fails, a bool set to false is returned
  SF.solve(X, B)
  - using the given dense matrix B and the computed factorisation, store in X the solution to A*X = B
  - if computing the solution fails, X is reset and a bool set to false is returned
  SF.rcond()
  - return the 1-norm estimate of the reciprocal condition number computed during the factorisation
    - values close to 1 suggest that the factorised matrix is well-conditioned
    - values close to 0 suggest that the factorised matrix is badly conditioned
  SF.reset()
  - reset the instance and release all memory related to the stored factorisation; this is automatically done when the instance goes out of scope
- Notes:
  - if the factorisation of A does not need to be reused, use spsolve() instead
  - this class internally uses the SuperLU solver; ARMA_USE_SUPERLU must be enabled in config.hpp
- Examples:

      sp_mat A = sprandu<sp_mat>(1000, 1000, 0.1);

      spsolve_factoriser SF;

      bool status = SF.factorise(A);

      if(status == false) { cout << "factorisation failed" << endl; }

      double rcond_value = SF.rcond();

      vec B1(1000, fill::randu);
      vec B2(1000, fill::randu);

      vec X1;
      vec X2;

      bool solution1_ok = SF.solve(X1,B1);
      bool solution2_ok = SF.solve(X2,B2);

      if(solution1_ok == false) { cout << "couldn't find X1" << endl; }
      if(solution2_ok == false) { cout << "couldn't find X2" << endl; }

- See also:
  - spsolve()
  - linear system of equations in MathWorld
  - system of linear equations in Wikipedia

Signal & Image Processing

conv( A, B )
conv( A, B, shape )

- 1D convolution of vectors A and B
- The orientation of the result vector is the same as the orientation of A (ie. either column or row vector)
- The shape argument is optional; it is one of:

  `"full"` | = | return the full convolution (default setting), with the size equal to A.n_elem + B.n_elem - 1
  `"same"` | = | return the central part of the convolution, with the same size as vector A

- The convolution operation is also equivalent to FIR filtering
- Examples:

      vec A(256, fill::randu);

      vec B(16, fill::randu);

      vec C = conv(A, B);

      vec D = conv(A, B, "same");

- See also:
  - conv2()
  - fft()
  - cor()
  - interp1()
  - Convolution in MathWorld
  - Convolution in Wikipedia
  - FIR filter in Wikipedia

conv2( A, B )
conv2( A, B, shape )

- 2D convolution of matrices A and B
- The shape argument is optional; it is one of:

  `"full"` | = | return the full convolution (default setting), with the size equal to size(A) + size(B) - 1
  `"same"` | = | return the central part of the convolution, with the same size as matrix A

- The implementation of 2D convolution in this version is preliminary
- Examples:

      mat A(256, 256, fill::randu);

      mat B(16, 16, fill::randu);

      mat C = conv2(A, B);

      mat D = conv2(A, B, "same");

- See also:
  - conv()
  - fft2()
  - interp2()
  - Convolution in MathWorld
  - Convolution in Wikipedia
  - Kernel (image processing) in Wikipedia

cx_mat Y = fft( X )
cx_mat Y = fft( X, n )
cx_mat Z = ifft( cx_mat Y )
cx_mat Z = ifft( cx_mat Y, n )

- fft(): fast Fourier transform of a vector or matrix (real or complex)
- ifft(): inverse fast Fourier transform of a vector or matrix (complex only)
- If given a matrix, the transform is done on each column vector of the matrix
- The optional n argument specifies the transform length:
  - if n is larger than the length of the input vector, a zero-padded version of the vector is used
  - if n is smaller than the length of the input vector, only the first n elements of the vector are used
- If n is not specified, the transform length is the same as the length of the input vector
- Caveat: the transform is fastest when the transform length is a power of 2, eg. 64, 128, 256, 512, 1024, ...
- By default, an internal FFT algorithm based on KISS FFT is used
- Since Armadillo 12.0, the FFTW3 library can be optionally used for faster execution, as follows:
  - ARMA_USE_FFTW3 must be defined before including the armadillo header:
    `#define ARMA_USE_FFTW3
    #include <armadillo>`
  - you will also need to link with the FFTW3 runtime library (eg. `-lfftw3`)
- Examples:

         vec X(100, fill::randu);

      cx_vec Y = fft(X, 128);

- See also:
  - fft2()
  - conv()
  - real()
  - fast Fourier transform in MathWorld
  - fast Fourier transform in Wikipedia

cx_mat Y = fft2( X )
cx_mat Y = fft2( X, n_rows, n_cols )
cx_mat Z = ifft2( cx_mat Y )
cx_mat Z = ifft2( cx_mat Y, n_rows, n_cols )

- fft2(): 2D fast Fourier transform of a matrix (real or complex)
- ifft2(): 2D inverse fast Fourier transform of a matrix (complex only)
- The optional arguments n_rows and n_cols specify the size of the transform; a truncated and/or zero-padded version of the input matrix is used
- Caveat: the transform is fastest when both n_rows and n_cols are a power of 2, eg. 64, 128, 256, 512, 1024, ...
- The implementation of the 2D transform in this version is preliminary
- Examples:

         mat A(100, 100, fill::randu);

      cx_mat B = fft2(A);
      cx_mat C = fft2(A, 128, 128);

- See also:
  - fft()
  - conv2()
  - real()
  - fast Fourier transform in MathWorld
  - fast Fourier transform in Wikipedia

interp1( X, Y, XI, YI )
interp1( X, Y, XI, YI, method )
interp1( X, Y, XI, YI, method, extrapolation_value )

- 1D data interpolation
- Given a 1D function specified in vectors X and Y (where X specifies locations and Y specifies the corresponding values),
  generate vector YI which contains interpolated values at locations XI
- The method argument is optional; it is one of:

  `"nearest"` | = | interpolate using single nearest neighbour
  `"linear"` | = | linear interpolation between two nearest neighbours (default setting)
  `"*nearest"` | = | as per `"nearest"`, but faster by assuming that X and XI are monotonically increasing
  `"*linear"` | = | as per `"linear"`, but faster by assuming that X and XI are monotonically increasing

- If a location in XI is outside the domain of X, the corresponding value in YI is set to extrapolation_value
- The extrapolation_value argument is optional; by default it is datum::nan (not-a-number)
- Examples:

      vec x = linspace<vec>(0, 3, 20);
      vec y = square(x);

      vec xx = linspace<vec>(0, 3, 100);

      vec yy;

      interp1(x, y, xx, yy);  // use linear interpolation by default

      interp1(x, y, xx, yy, "*linear");  // faster than "linear"

      interp1(x, y, xx, yy, "nearest");

- See also:
  - interp2()
  - polyval()
  - linspace()
  - regspace()
  - conv()
  - interpolation in Wikipedia

interp2( X, Y, Z, XI, YI, ZI )
interp2( X, Y, Z, XI, YI, ZI, method )
interp2( X, Y, Z, XI, YI, ZI, method, extrapolation_value )

- 2D data interpolation
- Given a 2D function specified by matrix Z with coordinates given by vectors X and Y,
  generate matrix ZI which contains interpolated values at the coordinates given by vectors XI and YI
- The vector pairs (X, Y) and (XI, YI) define 2D coordinates in a grid;
  for example, X defines the horizontal coordinates and Y defines the corresponding vertical coordinates,
  so that (X(m),Y(n) ) is the 2D coordinate of element Z(n,m)
- The length of vector X must be equal to the number of columns in matrix Z
- The length of vector Y must be equal to the number of rows in matrix Z
- Vectors X, Y, XI, YI must contain monotonically increasing values (eg. 0.1, 0.2, 0.3, ...)
- The method argument is optional; it is one of:

  `"nearest"` | = | interpolate using nearest neighbours
  `"linear"` | = | linear interpolation between nearest neighbours (default setting)

- If a coordinate in the 2D grid specified by (XI, YI) is outside the domain of the 2D grid specified by (X, Y),
  the corresponding value in ZI is set to extrapolation_value
- The extrapolation_value argument is optional; by default it is datum::nan (not-a-number)
- Examples:

      mat Z;

      Z.load("input_image.pgm", pgm_binary);  // load an image in pgm format

      vec X = regspace(1, Z.n_cols);  // X = horizontal spacing
      vec Y = regspace(1, Z.n_rows);  // Y = vertical spacing

      vec XI = regspace(X.min(), 1.0/2.0, X.max()); // magnify by approx 2
      vec YI = regspace(Y.min(), 1.0/3.0, Y.max()); // magnify by approx 3

      mat ZI;

      interp2(X, Y, Z, XI, YI, ZI);  // use linear interpolation by default

      ZI.save("output_image.pgm", pgm_binary);

- See also:
  - interp1()
  - regspace()
  - conv2()
  - reshape()
  - bilinear interpolation in Wikipedia

P = polyfit( X, Y, N )
polyfit( P, X, Y, N )

- Given a 1D function specified in vectors X and Y (where X holds independent values and Y holds the corresponding dependent values),
  model the function as a polynomial of order N and store the polynomial coefficients in column vector P
- The given function is modelled as:
  y = p0xN + p1xN-1 + p2xN-2 + ... + pN-1x1 + pN
  where pi is the i-th polynomial coefficient; the coefficients are selected to minimise the overall error of the fit (least squares)
- The column vector P has N+1 coefficients
- N must be smaller than the number of elements in X
- If the polynomial coefficients cannot be found:
  - P = polyfit( X, Y, N ) resets P and throws a std::runtime_error exception
  - polyfit( P, X, Y, N ) resets P and returns a bool set to false (exception is not thrown)
- Examples:

      vec x = linspace<vec>(0,4*datum::pi,100);
      vec y = cos(x);

      vec p = polyfit(x,y,10);

- See also:
  - polyval()
  - roots()
  - interp1()
  - polynomial in Wikipedia
  - least squares in Wikipedia
  - curve fitting in Wikipedia
  - least squares fitting in MathWorld

Y = polyval( P, X )

- Given vector P of polynomial coefficients and vector X containing the independent values of a 1D function,
  generate vector Y which contains the corresponding dependent values
- For each x value in vector X, the corresponding y value in vector Y is generated using:
  y = p0xN + p1xN-1 + p2xN-2 + ... + pN-1x1 + pN
  where pi is the i-th polynomial coefficient in vector P
- P must contain polynomial coefficients in descending powers (eg. generated by the polyfit() function)
- Examples:

      vec x1 = linspace<vec>(0,4*datum::pi,100);
      vec y1 = cos(x1);
      vec p1 = polyfit(x1,y1,10);

      vec y2 = polyval(p1,x1);

- See also:
  - polyfit()
  - roots()
  - interp1()
  - polynomial in Wikipedia

Statistics & Clustering

mean, median, stddev, var, range

mean( V ) mean( M ) mean( M, dim ) mean( Q ) mean( Q, dim ) |  | ⎫ ⎪ ⎬ mean (average value) ⎪ ⎭
median( V ) median( M ) median( M, dim ) |  | ⎫ ⎬ median ⎭
stddev( V ) stddev( V, norm_type ) stddev( M ) stddev( M, norm_type ) stddev( M, norm_type, dim ) |  | ⎫ ⎪ ⎬ standard deviation ⎪ ⎭
var( V ) var( V, norm_type ) var( M ) var( M, norm_type ) var( M, norm_type, dim ) |  | ⎫ ⎪ ⎬ variance ⎪ ⎭
range( V ) range( M ) range( M, dim ) |  | ⎫ ⎬ range (difference between max and min) ⎭

- For vector V, return the statistic calculated using all the elements of the vector
- For matrix M, find the statistic for each column (dim = 0), or each row (dim = 1)
- For cube Q, find the statistics of elements along dimension dim, where dim∈ { 0, 1, 2 }
- The dim argument is optional; by default dim = 0 is used
- The norm_type argument is optional; by default norm_type = 0 is used
- For the var() and stddev() functions:
  - the default norm_type = 0 performs normalisation using N-1 (where N is the number of samples), providing the best unbiased estimator
  - using norm_type = 1 performs normalisation using N, which provides the second moment around the mean
- Caveat: to obtain statistics for an integer vector or matrix (eg. uvec, ivec), first use the conv_to() function to obtain a floating point representation (eg. vec)
- Examples:

      mat A(5, 5, fill::randu);

      mat B    = mean(A);
      mat C    = var(A);
      double m = mean(mean(A));

      vec v(5, fill::randu);
      double x = var(v);

      ivec w = {1, 2, 3, 4};  // integer vector
      double y = mean( conv_to<vec>::from(w) );

- See also:
  - cov()
  - cor()
  - hist()
  - histc()
  - quantile()
  - normpdf()
  - min() & max()
  - running_stat - class for running statistics of scalars
  - running_stat_vec - class for running statistics of vectors
  - gmm_diag / gmm_full - probabilistic clustering and likelihood calculation via Gaussian mixture models
  - kmeans()

cov( X, Y )
cov( X, Y, norm_type )
cov( X )
cov( X, norm_type )

- For two matrix arguments X and Y, if each row of X and Y is an observation and each column is a variable, the (i,j)-th entry of cov(X,Y) is the covariance between the i-th variable in X and the j-th variable in Y
- For vector arguments, the type of vector is ignored and each element in the vector is treated as an observation
- For matrices, X and Y must have the same dimensions
- For vectors, X and Y must have the same number of elements
- cov(X) is equivalent to cov(X, X)
- The norm_type argument is optional; by default norm_type = 0 is used
- the norm_type argument controls the type of normalisation used, with N denoting the number of observations:
  - for norm_type = 0, normalisation is done using N-1, providing the best unbiased estimation of the covariance matrix (if the observations are from a normal distribution)
  - for norm_type = 1, normalisation is done using N, which provides the second moment matrix of the observations about their mean
- Examples:

      mat X(4, 5, fill::randu);
      mat Y(4, 5, fill::randu);

      mat C = cov(X,Y);
      mat D = cov(X,Y, 1);

- See also:
  - cor()
  - statistics functions
  - running_stat_vec
  - Covariance in MathWorld

cor( X, Y )
cor( X, Y, norm_type )
cor( X )
cor( X, norm_type )

- For two matrix arguments X and Y, if each row of X and Y is an observation and each column is a variable, the (i,j)-th entry of cor(X,Y) is the correlation coefficient between the i-th variable in X and the j-th variable in Y
- For vector arguments, the type of vector is ignored and each element in the vector is treated as an observation
- For matrices, X and Y must have the same dimensions
- For vectors, X and Y must have the same number of elements
- cor(X) is equivalent to cor(X, X)
- The norm_type argument is optional; by default norm_type = 0 is used
- the norm_type argument controls the type of normalisation used, with N denoting the number of observations:
  - for norm_type = 0, normalisation is done using N-1
  - for norm_type = 1, normalisation is done using N
- Examples:

      mat X(4, 5, fill::randu);
      mat Y(4, 5, fill::randu);

      mat R = cor(X,Y);
      mat S = cor(X,Y, 1);

- See also:
  - cov()
  - conv()
  - statistics functions
  - running_stat_vec
  - Correlation in MathWorld
  - Autocorrelation in MathWorld

hist( V )
hist( V, n_bins )
hist( V, centers )
hist( X, centers )
hist( X, centers, dim )

- For vector V, produce an unsigned vector of the same orientation as V (ie. either uvec or urowvec) that represents a histogram of counts
- For matrix X, produce a umat matrix containing either column histogram counts (for dim = 0, default), or row histogram counts (for dim = 1)
- The bin centers can be automatically determined from the data, with the number of bins specified via n_bins (default is 10); the range of the bins is determined by the range of the data
- The bin centers can also be explicitly specified via the centers vector; the vector must contain monotonically increasing values (eg. 0.1, 0.2, 0.3, ...)
- Examples:

       vec v(1000, fill::randn); // Gaussian distribution

      uvec h1 = hist(v, 11);
      uvec h2 = hist(v, linspace<vec>(-2,2,11));

- See also:
  - histc()
  - quantile()
  - statistics functions
  - conv_to()

histc( V, edges )
histc( X, edges )
histc( X, edges, dim )

- For vector V, produce an unsigned vector of the same orientation as V (ie. either uvec or urowvec) that contains the counts of the number of values that fall between the elements in the edges vector
- For matrix X, produce a umat matrix containing either column histogram counts (for dim = 0, default), or row histogram counts (for dim = 1)
- The edges vector must contain monotonically increasing values (eg. 0.1, 0.2, 0.3, ...)
- Examples:

       vec v(1000, fill::randn);  // Gaussian distribution

      uvec h = histc(v, linspace<vec>(-2,2,11));

- See also:
  - hist()
  - statistics functions
  - conv_to()

quantile( V, P )
quantile( X, P )
quantile( X, P, dim )

- For a dataset stored in vector V or matrix X, calculate the quantiles corresponding to the cumulative probability values in the given P vector
- For vector V, produce a vector with the same orientation as V and the same length as P
- For matrix X, produce a matrix with the quantiles for each column vector (dim = 0) or each row vector (dim = 1)
- The dim argument is optional; by default dim = 0
- The P vector must contain values in the [0,1] interval (eg. 0.00, 0.25, 0.50, 0.75, 1.00)
- The algorithm for calculating the quantiles is based on Definition 5 in:
  Rob J. Hyndman and Yanan Fan. Sample Quantiles in Statistical Packages. The American Statistician, 50(4), 361-365, 1996. DOI: 10.2307/2684934
- Examples:

      vec V(1000, fill::randn);  // Gaussian distribution

      vec P = { 0.25, 0.50, 0.75 };

      vec Q = quantile(V, P);

- See also:
  - hist()
  - median()
  - normcdf()
  - quantile in Wikipedia

mat coeff = princomp( mat X )
cx_mat coeff = princomp( cx_mat X )
princomp( mat coeff, mat X )
princomp( cx_mat coeff, cx_mat X )
princomp( mat coeff, mat score, mat X )
princomp( cx_mat coeff, cx_mat score, cx_mat X )
princomp( mat coeff, mat score, vec latent, mat X )
princomp( cx_mat coeff, cx_mat score, vec latent, cx_mat X )
princomp( mat coeff, mat score, vec latent, vec tsquared, mat X )
princomp( cx_mat coeff, cx_mat score, vec latent, cx_vec tsquared, cx_mat X )

- Principal component analysis of matrix X
- Each row of X is an observation and each column is a variable
- output objects:
  - coeff: principal component coefficients
  - score: projected data
  - latent: eigenvalues of the covariance matrix of X
  - tsquared: Hotteling's statistic for each sample
- The computation is based on singular value decomposition
- If the decomposition fails:
  - coeff = princomp(X) resets coeff and throws a std::runtime_error exception
  - remaining forms of princomp() reset all output matrices and return a bool set to false (exception is not thrown)
- Examples:

      mat A(5, 4, fill::randu);

      mat coeff;
      mat score;
      vec latent;
      vec tsquared;

      princomp(coeff, score, latent, tsquared, A);

- See also:
  - eig_sym()
  - svd()
  - svd_econ()
  - principal components analysis in Wikipedia

normpdf( X )
normpdf( X, M, S )

- For each scalar x in X, compute its probability density function according to a Gaussian (normal) distribution using the corresponding mean value m in M and the corresponding standard deviation value s in S:
  `

  | 1 |  | ⎧ |  | (x-m)2 | ⎫
  y = | ‒‒‒‒ | exp | ⎪ | −0.5 | ‒‒‒‒‒‒ | ⎪
  | s·√τ |  | ⎩ |  | s2 | ⎭

  `
- X can be a scalar, vector, or matrix
- M and S can jointly be either scalars, vectors, or matrices
- If M and S are omitted, their values are assumed to be 0 and 1, respectively
- Caveat: to reduce the incidence of numerical underflows, consider using log_normpdf()
- Examples:

      vec X(10, fill::randu);
      vec M(10, fill::randu);
      vec S(10, fill::randu);

         vec P1 = normpdf(X);
         vec P2 = normpdf(X,    M,    S   );
         vec P3 = normpdf(1.23, M,    S   );
         vec P4 = normpdf(X,    4.56, 7.89);
      double P5 = normpdf(1.23, 4.56, 7.89);

- See also:
  - log_normpdf()
  - normcdf()
  - randn()
  - gmm_diag / gmm_full - probabilistic clustering and likelihood calculation via Gaussian mixture models
  - normal distribution in Wikipedia

log_normpdf( X )
log_normpdf( X, M, S )

- For each scalar x in X, compute the logarithm version of probability density function according to a Gaussian (normal) distribution using the corresponding mean value m in M and the corresponding standard deviation value s in S:
  `

  |  | ⎧ | 1 |  | ⎧ |  | (x-m)2 | ⎫ | ⎫
  y = | log | ⎪ | ‒‒‒‒ | exp | ⎪ | −0.5 | ‒‒‒‒‒‒ | ⎪ | ⎪
  |  | ⎩ | s·√τ |  | ⎩ |  | s2 | ⎭ | ⎭

  |  |  | ⎧ |  | (x-m)2 | ⎫
  = | −log(s·√τ) | + | ⎪ | −0.5 | ‒‒‒‒‒‒ | ⎪
  |  |  | ⎩ |  | s2 | ⎭

  `
- X can be a scalar, vector, or matrix
- M and S can jointly be either scalars, vectors, or matrices
- If M and S are omitted, their values are assumed to be 0 and 1, respectively
- Examples:

      vec X(10, fill::randu);
      vec M(10, fill::randu);
      vec S(10, fill::randu);

         vec P1 = log_normpdf(X);
         vec P2 = log_normpdf(X,    M,    S   );
         vec P3 = log_normpdf(1.23, M,    S   );
         vec P4 = log_normpdf(X,    4.56, 7.89);
      double P5 = log_normpdf(1.23, 4.56, 7.89);

- See also:
  - normpdf()
  - gmm_diag / gmm_full - probabilistic clustering and likelihood calculation via Gaussian mixture models
  - normal distribution in Wikipedia

normcdf( X )
normcdf( X, M, S )

- For each scalar x in X, compute its cumulative distribution function according to a Gaussian (normal) distribution using the corresponding mean value m in M and the corresponding standard deviation value s in S
- X can be a scalar, vector, or matrix
- M and S can jointly be either scalars, vectors, or matrices
- If M and S are omitted, their values are assumed to be 0 and 1, respectively
- Examples:

      vec X(10, fill::randu);
      vec M(10, fill::randu);
      vec S(10, fill::randu);

         vec P1 = normcdf(X);
         vec P2 = normcdf(X,    M,    S   );
         vec P3 = normcdf(1.23, M,    S   );
         vec P4 = normcdf(X,    4.56, 7.89);
      double P5 = normcdf(1.23, 4.56, 7.89);

- See also:
  - normpdf()
  - quantile()
  - randn()
  - normal distribution in Wikipedia
  - cumulative distribution function in Wikipedia

X = mvnrnd( M, C )
X = mvnrnd( M, C, N )
mvnrnd( X, M, C )
mvnrnd( X, M, C, N )

- Generate a matrix with random column vectors from a multivariate Gaussian (normal) distribution with parameters M and C:
  - M is the mean; must be a column vector
  - C is the covariance matrix; must be symmetric positive semi-definite (preferably positive definite)
- N is the number of column vectors to generate; if N is omitted, it is assumed to be 1
- Caveat: repeated generation of one vector (or a small number of vectors) using the same M and C parameters can be inefficient;
  for repeated generation consider using the generate() function in the gmm_diag and gmm_full classes
- If generating the random vectors fails:
  - X = mvnrnd(M, C) and X = mvnrnd(M, C, N) reset X and throw a std::runtime_error exception
  - mvnrnd(X, M, C) and mvnrnd(X, M, C, N) reset X and return a bool set to false (exception is not thrown)
- Examples:

      vec M(5, fill::randu);

      mat B(5, 5, fill::randu);
      mat C = B.t() * B;

      mat X = mvnrnd(M, C, 100);

- See also:
  - randn()
  - chi2rnd()
  - wishrnd()
  - cov()
  - .is_sympd()
  - gmm_diag / gmm_full - probabilistic clustering and likelihood calculation via Gaussian mixture models
  - multivariate normal distribution in Wikipedia

chi2rnd( DF )
chi2rnd( DF_scalar )
chi2rnd( DF_scalar, n_elem )
chi2rnd( DF_scalar, n_rows, n_cols )
chi2rnd( DF_scalar, size(X) )

- Generate a random scalar, vector or matrix with elements sampled from a chi-squared distribution with the degrees of freedom specified by parameter DF or DF_scalar
- DF is a vector or matrix, while DF_scalar is a scalar
- Each value in DF and DF_scalar must be greater than zero
- For the chi2rnd(DF) form, the output vector/matrix has the same size and type as DF; each element in DF specifies a separate degree of freedom
- Usage:
  - vector_type v = chi2rnd( DF ), where the type of DF is a real vector_type
  - matrix_type X = chi2rnd( DF ), where the type of DF is a real matrix_type
  - scalar_type s = chi2rnd<scalar_type>( DF_scalar ), where scalar_type is either float or double
  - vector_type v = chi2rnd<vector_type>( DF_scalar, n_elem )
  - matrix_type X = chi2rnd<matrix_type>( DF_scalar, n_rows, n_cols )
  - matrix_type Y = chi2rnd<matrix_type>( DF_scalar, size(X) )
- Examples:

      mat X = chi2rnd(2, 5, 6);

      mat A = randi<mat>(5, 6, distr_param(1, 10));
      mat B = chi2rnd(A);

- See also:
  - randg()
  - randn()
  - mvnrnd()
  - wishrnd()
  - size()
  - chi-squared distribution in Wikipedia

W = wishrnd( S, df )
W = wishrnd( S, df, D )
wishrnd( W, S, df )
wishrnd( W, S, df, D )

- Generate a random matrix sampled from the Wishart distribution with parameters S and df:
  - S is a symmetric positive definite matrix (eg. a covariance matrix)
  - df is a scalar specifying the degrees of freedom; it can be a non-integer value
- D is an optional argument; it specifies the Cholesky decomposition of S; if D is provided, S is ignored;
  using D is more efficient if wishrnd() needs to be used many times for the same S matrix
- If generating the random matrix fails:
  - W = wishrnd(S, df) and W = wishrnd(S, df, D) reset W and throw a std::runtime_error exception
  - wishrnd(W, S, df) and wishrnd(W, S, df, D) reset W and return a bool set to false (exception is not thrown)
- Caveat: for the inverse (inverted) Wishart distribution, use iwishrnd()
- Examples:

      mat X(5, 5, fill::randu);

      mat S = X.t() * X;

      mat W = wishrnd(S, 6.7);

- See also:
  - iwishrnd()
  - chi2rnd()
  - mvnrnd()
  - chol()
  - cov()
  - randn()
  - .is_sympd()
  - Wishart distribution in Wikipedia

W = iwishrnd( T, df )
W = iwishrnd( T, df, Dinv )
iwishrnd( W, T, df )
iwishrnd( W, T, df, Dinv )

- Generate a random matrix sampled from the inverse Wishart distribution with parameters T and df:
  - T is a symmetric positive definite matrix
  - df is a scalar specifying the degrees of freedom; it can be a non-integer value
- Dinv is an optional argument; it specifies the Cholesky decomposition of the inverse of T; if Dinv is provided, T is ignored
  using Dinv is more efficient if iwishrnd() needs to be used many times for the same T matrix
- If generating the random matrix fails:
  - W = iwishrnd(T, df) and W = iwishrnd(T, df, Dinv) reset W and throw a std::runtime_error exception
  - iwishrnd(W, T, df) and iwishrnd(W, T, df, Dinv) reset W and return a bool set to false (exception is not thrown)
- Caveat: for the plain (non-inverse) Wishart distribution, use wishrnd()
- Examples:

      mat X(5, 5, fill::randu);

      mat T = X.t() * X;

      mat W = iwishrnd(T, 6.7);

- See also:
  - wishrnd()
  - chol()
  - inv_sympd()
  - .is_sympd()
  - inverse Wishart distribution in Wikipedia

running_stat<type>

- Class for running statistics (online statistics) of scalars (one dimensional process/signal)
- Useful if the storage of all samples (scalars) is impractical, or if the number of samples is not known in advance
- type is one of: float, double, cx_float, cx_double
- For an instance of running_stat named as X, the member functions are:

  X(scalar) |  | update the statistics using the given scalar
  X.min() |  | current minimum value
  X.max() |  | current maximum value
  X.range() |  | current range
  X.mean() |  | current mean or average value
  X.var() and X.var(norm_type) |  | current variance
  X.stddev() and X.stddev(norm_type) |  | current standard deviation
  X.reset() |  | reset all statistics and set the number of samples to zero
  X.count() |  | current number of samples

- The norm_type argument is optional; by default norm_type = 0 is used
- For the .var() and .stddev() functions, the default norm_type = 0 performs normalisation using N-1 (where N is the number of samples so far), providing the best unbiased estimator; using norm_type = 1 causes normalisation to be done using N, which provides the second moment around the mean
- The return type of .count() depends on the underlying form of type: it is either float or double
- Examples:

      running_stat<double> stats;

      for(uword i=0; i<10000; ++i)
        {
        double sample = randn();
        stats(sample);
        }

      cout << "mean = " << stats.mean() << endl;
      cout << "var  = " << stats.var()  << endl;
      cout << "min  = " << stats.min()  << endl;
      cout << "max  = " << stats.max()  << endl;

- See also:
  - running_stat_vec (running statistics of vectors)
  - statistics functions
  - gmm_diag / gmm_full - probabilistic clustering via Gaussian mixture models

running_stat_vec<vec_type>
running_stat_vec<vec_type>(calc_cov)

- Class for running statistics (online statistics) of vectors (multi-dimensional process/signal)
- Useful if the storage of all samples (vectors) is impractical, or if the number of samples is not known in advance
- This class is similar to running_stat, with the difference that vectors are processed instead of scalars
- vec_type is the vector type of the samples; for example: vec, cx_vec, rowvec, ...
- For an instance of running_stat_vec named as X, the member functions are:

  X(vector) |  | update the statistics using the given vector
  X.min() |  | vector of current minimum values
  X.max() |  | vector of current maximum values
  X.range() |  | vector of current ranges
  X.mean() |  | vector of current means
  X.var() and X.var(norm_type) |  | vector of current variances
  X.stddev() and X.stddev(norm_type) |  | vector of current standard deviations
  X.cov() and X.cov(norm_type) |  | matrix of current covariances; valid if calc_cov=true during construction of running_stat_vec
  X.reset() |  | reset all statistics and set the number of samples to zero
  X.count() |  | current number of samples

- The calc_cov argument is optional; by default calc_cov=false, indicating that the covariance matrix will not be calculated; to enable the covariance matrix, use calc_cov=true during construction; for example: running_stat_vec<vec> X(true);
- The norm_type argument is optional; by default norm_type = 0 is used
- For the .var() and .stddev() functions, the default norm_type = 0 performs normalisation using N-1 (where N is the number of samples so far), providing the best unbiased estimator; using norm_type = 1 causes normalisation to be done using N, which provides the second moment around the mean
- The return type of .count() depends on the underlying form of vec_type: it is either float or double
- Examples:

      running_stat_vec<vec> stats;

      vec sample;

      for(uword i=0; i<10000; ++i)
        {
        sample = randu<vec>(5);
        stats(sample);
        }

      cout << "mean = " << endl << stats.mean() << endl;
      cout << "var  = " << endl << stats.var()  << endl;
      cout << "max  = " << endl << stats.max()  << endl;

      //
      //

      running_stat_vec<rowvec> more_stats(true);

      for(uword i=0; i<20; ++i)
        {
        sample = randu<rowvec>(3);

        sample(1) -= sample(0);
        sample(2) += sample(1);

        more_stats(sample);
        }

      cout << "covariance matrix = " << endl;
      cout << more_stats.cov() << endl;

      rowvec sd = more_stats.stddev();

      cout << "correlations = " << endl;
      cout << more_stats.cov() / (sd.t() * sd);

- See also:
  - running_stat (running statistics of scalars)
  - statistics functions
  - cov()
  - cor()
  - gmm_diag / gmm_full - probabilistic clustering via Gaussian mixture models

kmeans( means, data, k, seed_mode, n_iter, print_mode )

- Cluster given data into k disjoint sets
- The means parameter is the output matrix for storing the resulting centroids of the sets, with each centroid stored as a column vector
- The data parameter is the input data matrix, with each sample stored as a column vector
- The k parameter indicates the number of centroids; the number of samples in the data matrix should be much larger than k
- The seed_mode parameter specifies how the initial centroids are seeded; it is one of:

  `keep_existing` |  | use the centroids specified in the means matrix as the starting point
  `static_subset` |  | use a subset of the data vectors (repeatable)
  `random_subset` |  | use a subset of the data vectors (random)
  `static_spread` |  | use a maximally spread subset of data vectors (repeatable)
  `random_spread` |  | use a maximally spread subset of data vectors (random start)

  Caveat: seeding the initial centroids with `static_spread` and `random_spread` can be much more time consuming than with `static_subset` and `random_subset`
- The n_iter parameter specifies the number of clustering iterations; this is data dependent, but about 10 is typically sufficient
- The print_mode parameter is either true or false, indicating whether progress is printed during clustering
- If the clustering fails, the means matrix is reset and a bool set to false is returned
- For faster execution on multi-core machines, enable OpenMP in your compiler (eg. -fopenmp in GCC and clang)
- Caveat: for probabilistic clustering, use the gmm_diag or gmm_full classes instead
- Examples:

      uword d = 5;       // dimensionality
      uword N = 10000;   // number of vectors

      mat data(d, N, fill::randu);

      mat means;

      bool status = kmeans(means, data, 2, random_subset, 10, true);

      if(status == false)
        {
        cout << "clustering failed" << endl;
        }

      means.print("means:");

- See also:
  - gmm_diag / gmm_full
  - statistics functions
  - running_stat_vec
  - k-means clustering in Wikipedia
  - k-means clustering in MathWorld
  - OpenMP in Wikipedia

gmm_diag
gmm_full

- Classes for multi-variate probabilistic clustering and likelihood calculation via Gaussian Mixture Models (GMMs)
- Distribution of data is modelled as:

  |  | n_gaus-1 |
  p(x) | = | ∑ | hg N( x | mg ,Cg )
  |  | g=0 |

  where:
  - x is a column vector
  - n_gaus is the number of Gaussians; n_gaus≥ 1
  - N( x | mg ,Cg ) represents a Gaussian (normal) distribution
  - each Gaussian g has the following parameters:
    - hg is the heft (weight), with constraints hg≥ 0 and ∑hg= 1
    - mg is the mean vector (centroid) with dimensionality n_dims
    - Cg is the covariance matrix (either diagonal or full)
- gmm_diag is tailored for diagonal covariance matrices
- gmm_full is tailored for full covariance matrices
- gmm_diag is considerably faster than gmm_full, at the cost of some reduction in modelling accuracy
- Both gmm_diag and gmm_full include tailored k-means and Expectation Maximisation algorithms for learning model parameters from training data
- For faster execution on multi-core machines, enable OpenMP in your compiler (eg. -fopenmp in GCC and clang)
- Implementation details are available in the following paper:
  Conrad Sanderson and Ryan Curtin.
  An Open Source C++ Implementation of Multi-Threaded Gaussian Mixture Models, k-Means and Expectation Maximisation.
  International Conference on Signal Processing and Communication Systems, 2017.
- For an instance of gmm_diag or gmm_full named as M, the member functions and variables are:

  M.log_p(V) |  | return a scalar representing the log-likelihood of vector V (of type vec)
  M.log_p(V, g) |  | return a scalar representing the log-likelihood of vector V (of type vec), according to Gaussian with index g
  M.log_p(X) |  | return a row vector (of type `rowvec`) containing log-likelihoods of each column vector in matrix X (of type mat)
  M.log_p(X, g) |  | return a row vector (of type `rowvec`) containing log-likelihoods of each column vector in matrix X (of type mat), according to Gaussian with index g
  M.sum_log_p(X) |  | return a scalar representing the sum of log-likelihoods for all column vectors in matrix X (of type mat)
  M.sum_log_p(X, g) |  | return a scalar representing the sum of log-likelihoods for all column vectors in matrix X (of type mat), according to Gaussian with index g
  M.avg_log_p(X) |  | return a scalar representing the average log-likelihood of all column vectors in matrix X (of type mat)
  M.avg_log_p(X, g) |  | return a scalar representing the average log-likelihood of all column vectors in matrix X (of type mat), according to Gaussian with index g
  M.assign(V, dist_mode) |  | return the index of the closest mean (or Gaussian) to vector V (of type vec); parameter dist_mode is one of:
  `eucl_dist` |  | Euclidean distance (takes only means into account)
  `prob_dist` |  | probabilistic "distance", defined as the inverse likelihood (takes into account means, covariances and hefts)
  M.assign(X, dist_mode) |  | return a row vector (of type urowvec) containing the indices of the closest means (or Gaussians) to each column vector in matrix X (of type mat); parameter dist_mode is either `eucl_dist` or `prob_dist` (as per the .assign() function above)
  M.raw_hist(X, dist_mode) |  | return a row vector (of type `urowvec`) representing the raw histogram of counts; each entry is the number of counts corresponding to a Gaussian; each count is the number times the corresponding Gaussian was the closest to each column vector in matrix X; parameter dist_mode is either `eucl_dist` or `prob_dist` (as per the .assign() function above)
  M.norm_hist(X, dist_mode) |  | similar to the .raw_hist() function above; return a row vector (of type `rowvec`) containing normalised counts; the vector sums to one; parameter dist_mode is either `eucl_dist` or `prob_dist` (as per the .assign() function above)
  M.generate() |  | return a column vector (of type vec) representing a random sample generated according to the model's parameters
  M.generate(N) |  | return a matrix (of type mat) containing N column vectors, with each vector representing a random sample generated according to the model's parameters
  M.save(filename) |  | save the model to a file and return a bool indicating either success (true) or failure (false)
  M.load(filename) |  | load the model from a file and return a bool indicating either success (true) or failure (false)
  M.n_gaus() |  | return the number of means/Gaussians in the model
  M.n_dims() |  | return the dimensionality of the means/Gaussians in the model
  M.reset(n_dims, n_gaus) |  | set the model to have dimensionality n_dims, with n_gaus number of Gaussians; all the means are set to zero, all covariance matrix representations are equivalent to the identity matrix, and all the hefts (weights) are set to be uniform
  M.hefts |  | read-only row vector (of type rowvec) containing the hefts (weights)
  M.means |  | read-only matrix (of type mat) containing the means (centroids), stored as column vectors
  M.dcovs [only in gmm_diag] |  | read-only matrix (of type mat) containing the representation of diagonal covariance matrices, with the set of diagonal covariances for each Gaussian stored as a column vector; applicable only to the gmm_diag class
  M.fcovs [only in gmm_full] |  | read-only cube containing the full covariance matrices, with each covariance matrix stored as a slice within the cube; applicable only to the gmm_full class
  M.set_hefts(V) |  | set the hefts (weights) of the model to be as specified in row vector V (of type rowvec); the number of hefts must match the existing model
  M.set_means(X) |  | set the means to be as specified in matrix X (of type mat); the number of means and their dimensionality must match the existing model
  M.set_dcovs(X) [only in gmm_diag] |  | set the diagonal covariances matrices to be as specified in matrix X (of type mat), with the set of diagonal covariances for each Gaussian stored as a column vector; the number of covariance matrices and their dimensionality must match the existing model; applicable only to the gmm_diag class
  M.set_fcovs(X) [only in gmm_full] |  | set the full covariances matrices to be as specified in cube X, with each covariance matrix stored as a slice within the cube; the number of covariance matrices and their dimensionality must match the existing model; applicable only to the gmm_full class
  M.set_params(means, covs, hefts) |  | set all the parameters at the same time; the type and layout of the parameters is as per the .set_hefts(), .set_means(), .set_dcovs() and .set_fcovs() functions above; the number of Gaussians and dimensionality can be different from the existing model
  M.learn(data, n_gaus, dist_mode, seed_mode, km_iter, em_iter, var_floor, print_mode) learn the model parameters via multi-threaded k-means and/or EM algorithms; return a `bool` value, with true indicating success, and false indicating failure; the parameters have the following meanings:
  data |  | matrix (of type mat) containing training samples; each sample is stored as a column vector
  n_gaus |  | set the number of Gaussians to n_gaus; to help convergence, it is recommended that the given data matrix (above) contains at least 10 samples for each Gaussian
  dist_mode |  | specifies the distance used during the seeding of initial means and k-means clustering:
  `eucl_dist` |  | Euclidean distance
  `maha_dist` |  | Mahalanobis distance, which uses a global diagonal covariance matrix estimated from the training samples; this is recommended for probabilistic applications
  seed_mode |  | specifies how the initial means are seeded prior to running k-means and/or EM algorithms:
  `keep_existing` |  | keep the existing model (do not modify the means, covariances and hefts)
  `static_subset` |  | a subset of the training samples (repeatable)
  `random_subset` |  | a subset of the training samples (random)
  `static_spread` |  | a maximally spread subset of training samples (repeatable)
  `random_spread` |  | a maximally spread subset of training samples (random start)
  caveat: seeding the initial means with `static_spread` and `random_spread` can be much more time consuming than with `static_subset` and `random_subset`
  km_iter |  | the number of iterations of the k-means algorithm; this is data dependent, but typically 10 iterations are sufficient
  em_iter |  | the number of iterations of the EM algorithm; this is data dependent, but typically 5 to 10 iterations are sufficient
  var_floor |  | the variance floor (smallest allowed value) for the diagonal covariances; setting this to a small non-zero value can help with convergence and/or better quality parameter estimates
  print_mode |  | either `true` or `false`; enable or disable printing of progress during the k-means and EM algorithms

- Examples:

      // create synthetic data with 2 Gaussians

      uword d = 5;       // dimensionality
      uword N = 10000;   // number of vectors

      mat data(d, N, fill::zeros);

      vec mean0 = linspace<vec>(1,d,d);
      vec mean1 = mean0 + 2;

      uword i = 0;

      while(i < N)
        {
        if(i < N)  { data.col(i) = mean0 + randn<vec>(d); ++i; }
        if(i < N)  { data.col(i) = mean0 + randn<vec>(d); ++i; }
        if(i < N)  { data.col(i) = mean1 + randn<vec>(d); ++i; }
        }

      // model the data as a diagonal GMM with 2 Gaussians

      gmm_diag model;

      bool status = model.learn(data, 2, maha_dist, random_subset, 10, 5, 1e-10, true);

      if(status == false)
        {
        cout << "learning failed" << endl;
        }

      model.means.print("means:");

      double  scalar_likelihood = model.log_p( data.col(0)    );
      rowvec     set_likelihood = model.log_p( data.cols(0,9) );

      double overall_likelihood = model.avg_log_p(data);

      uword   gaus_id  = model.assign( data.col(0),    eucl_dist );
      urowvec gaus_ids = model.assign( data.cols(0,9), prob_dist );

      urowvec hist1 = model.raw_hist (data, prob_dist);
       rowvec hist2 = model.norm_hist(data, eucl_dist);

      model.save("my_model.gmm");

- See also:
  - normpdf()
  - mvnrnd()
  - statistics functions
  - running_stat_vec
  - kmeans()
  - covariance matrix in Wikipedia
  - covariance matrix in MathWorld
  - Mahalanobis distance in Wikipedia
  - multivariate normal distribution in Wikipedia
  - mixture model in Wikipedia
  - k-means clustering in Wikipedia
  - k-means clustering in MathWorld
  - Expectation-Maximisation algorithm in Wikipedia
  - maximum likelihood in MathWorld
  - OpenMP in Wikipedia

Miscellaneous

constants (pi, inf, eps, ...)

`datum::tau` |  | τ, the ratio of any circle's circumference to its radius; equivalent to 2π
`datum::pi` |  | π, the ratio of any circle's circumference to its diameter
`datum::inf` |  | ∞, infinity
`datum::nan` |  | “not a number” (NaN); caveat: NaN is not equal to anything, even itself
`datum::eps` |  | machine epsilon; approximately 2.2204e-16; difference between 1 and the next representable value
`datum::e` |  | base of the natural logarithm
`datum::sqrt2` |  | square root of 2
`datum::log_min` |  | log of minimum non-zero value (type and machine dependent)
`datum::log_max` |  | log of maximum value (type and machine dependent)
`datum::euler` |  | Euler's constant, aka Euler-Mascheroni constant
`datum::gratio` |  | golden ratio
`datum::m_u` |  | atomic mass constant (in kg)
`datum::N_A` |  | Avogadro constant
`datum::k` |  | Boltzmann constant (in joules per kelvin)
`datum::k_evk` |  | Boltzmann constant (in eV/K)
`datum::a_0` |  | Bohr radius (in meters)
`datum::mu_B` |  | Bohr magneton
`datum::Z_0` |  | characteristic impedance of vacuum (in ohms)
`datum::G_0` |  | conductance quantum (in siemens)
`datum::k_e` |  | Coulomb's constant (in meters per farad)
`datum::eps_0` |  | electric constant (in farads per meter)
`datum::m_e` |  | electron mass (in kg)
`datum::eV` |  | electron volt (in joules)
`datum::ec` |  | elementary charge (in coulombs)
`datum::F` |  | Faraday constant (in coulombs)
`datum::alpha` |  | fine-structure constant
`datum::alpha_inv` |  | inverse fine-structure constant
`datum::K_J` |  | Josephson constant
`datum::mu_0` |  | magnetic constant (in henries per meter)
`datum::phi_0` |  | magnetic flux quantum (in webers)
`datum::R` |  | molar gas constant (in joules per mole kelvin)
`datum::G` |  | Newtonian constant of gravitation (in newton square meters per kilogram squared)
`datum::h` |  | Planck constant (in joule seconds)
`datum::h_bar` |  | Planck constant over 2 pi, aka reduced Planck constant (in joule seconds)
`datum::m_p` |  | proton mass (in kg)
`datum::R_inf` |  | Rydberg constant (in reciprocal meters)
`datum::c_0` |  | speed of light in vacuum (in meters per second)
`datum::sigma` |  | Stefan-Boltzmann constant
`datum::R_k` |  | von Klitzing constant (in ohms)
`datum::b` |  | Wien wavelength displacement law constant

- The constants are stored in the Datum<type> class, where type is either float or double;
  for convenience, Datum<double> is typedefed as datum, and Datum<float> is typedefed as fdatum
- Caveat: `datum::nan` is not equal to anything, even itself; to check whether a scalar x is finite, use std::isfinite(x)
- The physical constants were mainly taken from NIST 2022 CODATA values, and some from WolframAlpha (as of 2009-06-23)
- Examples:

      cout << "speed of light = " << datum::c_0 << endl;

      cout << "log_max for floats = ";
      cout << fdatum::log_max << endl;

      cout << "log_max for doubles = ";
      cout << datum::log_max << endl;

- See also:
  - .is_finite()
  - .fill()
  - NaN in Wikipedia
  - physical constant in Wikipedia
  - replacement of 2π with τ in Wikipedia
  - The Tau Manifesto by Michael Hartl
  - std::numeric_limits in cplusplus.com
  - std::numeric_limits in cppreference.com

wall_clock

- Simple timer class for measuring the number of elapsed seconds
- An instance of the class has two member functions:

  `.tic()` |  | start the timer
  `.toc()` |  | return the number of seconds since the last call to `.tic()`

- Examples:

      wall_clock timer;

      timer.tic();

      // ... do something ...

      double n = timer.toc();

      cout << "number of seconds: " << n << endl;

- See also:
  - elapsed real time in Wikipedia

RNG seed setting

- There are two functions to change the seed used by the random number generator (RNG):

  `arma_rng::set_seed( value )` |  | set the RNG seed to the specified value
  `arma_rng::set_seed_random()` |  | set the RNG seed to a value drawn from std::random_device (if the reported entropy is not zero), or /dev/urandom (on Linux and macOS), or based on the current time (on systems without /dev/urandom)

- Since Armadillo 12.6.2, the default RNG on all systems is the 64-bit version of the Mersenne Twister (aka MT19937-64)
- Caveat: when using a multi-threading framework (such as OpenMP) and the underlying system supports the thread_local storage specifier, the above functions change the seed only within the thread they are running on
- To change the seeds on all OpenMP threads to the same value, adapt the following code:

      #pragma omp parallel
        {
        arma_rng::set_seed(123);
        }

- To change the seeds on all OpenMP threads to unique values, adapt the following code:

      std::atomic<std::size_t> counter(0);

      #pragma omp parallel
        {
        arma_rng::set_seed(123 + counter++);
        }

- See also:
  - randu(), randn(), randg(), randi(), sprandu(), sprandn(), shuffle(), randperm()
  - Pseudo-random number generator in Wikipedia
  - Mersenne Twister in Wikipedia
  - /dev/random in Wikipedia
  - std::mersenne_twister_engine in cppreference.com
  - std::random_device in cppreference.com

output streams

- The default stream for printing matrices and cubes is `std::cout`
  the stream can be changed via the `ARMA_COUT_STREAM` define; see config.hpp
- The default stream for printing warnings and errors is `std::cerr`
  the stream can be changed via the `ARMA_CERR_STREAM` define; see config.hpp
- The degree of printed warnings is controlled by the `ARMA_WARN_LEVEL` define; see config.hpp
- Example of changing the warning level:

      #define ARMA_WARN_LEVEL 1
      #include <armadillo>

- See also:
  - config.hpp
  - .print()
  - std::cout
  - std::cerr
  - std::ostream

uword, sword

- uword is a typedef for an unsigned integer type; it is used for matrix indices as well as all internal counters and loops
- sword is a typedef for a signed integer type
- The minimum width of both uword and sword is either 32 or 64 bits:
  - the default width is 32 bits on 32-bit platforms
  - the default width is 64 bits on 64-bit platforms
  - the default width is 32 bits when using Armadillo in the R environment (via RcppArmadillo) on either 32-bit or 64-bit platforms
- The width can also be forcefully set to 64 bits by enabling ARMA_64BIT_WORD via editing include/armadillo_bits/config.hpp
- See also:
  - C++ variable types
  - explanation of typedef
  - imat & umat matrix types
  - ivec & uvec vector types

cx_double, cx_float

- Convenience short forms (typedefs) for complex element types:

  `cx_double` |  | equivalent to `std::complex<double>`
  `cx_float` |  | equivalent to `std::complex<float>`

- Example:

      cx_mat X(5, 5, fill::randu);

      X(1,2) = cx_double(3.4, 5.6);

      cx_double val = X(2,3);

- See also:
  - complex numbers in the standard C++ library
  - explanation of typedef
  - cx_mat matrix type
  - cx_vec vector type

Examples of Matlab/Octave syntax and conceptually corresponding Armadillo syntax

Matlab/Octave |  | Armadillo |  | Notes
`A(1, 1)` |  | `A(0, 0)` |  | indexing in Armadillo starts at 0
`A(k, k)` |  | `A(k-1, k-1)` |  |
`size(A,1)` |  | `A.n_rows` |  | read only
`size(A,2)` |  | `A.n_cols` |  |
`size(Q,3)` |  | `Q.n_slices` |  | Q is a cube (3D array)
`numel(A)` |  | `A.n_elem` |  |
`A(:, k)` |  | `A.col(k)` |  | this is a conceptual example only; exact conversion from Matlab/Octave to Armadillo syntax will require taking into account that indexing starts at 0
`A(k, :)` |  | `A.row(k)` |  |
`A(:, p:q)` |  | `A.cols(p, q)` |  |
`A(p:q, :)` |  | `A.rows(p, q)` |  |
`A(p:q, r:s)` |  | `A( span(p,q), span(r,s) )` |  | A( span(first_row, last_row), span(first_col, last_col) )
`Q(:, :, k)` |  | `Q.slice(k)` |  | Q is a cube (3D array)
`Q(:, :, t:u)` |  | `Q.slices(t, u)` |  |
`Q(p:q, r:s, t:u)` |  | `Q( span(p,q), span(r,s), span(t,u) )` |  |
`A'` |  | `A.t() or trans(A)` |  | matrix transpose / Hermitian transpose (for complex matrices, the conjugate of each element is taken)
`A = zeros(size(A))` |  | `A.zeros()` |  |
`A = ones(size(A))` |  | `A.ones()` |  |
`A = zeros(k)` |  | `A = zeros<mat>(k,k)` |  |
`A = ones(k)` |  | `A = ones<mat>(k,k)` |  |
`C = complex(A,B)` |  | `cx_mat C = cx_mat(A,B)` |  |
`A .* B` |  | `A % B` |  | element-wise multiplication
`A ./ B` |  | `A / B` |  | element-wise division
`A \ B` |  | `solve(A,B)` |  | conceptually similar to inv(A)*B, but more efficient
`A = A + 1;` |  | `A++` |  |
`A = A - 1;` |  | `A--` |  |
`A = [ 1 2; 3 4; ]` |  | `A = { { 1, 2 }, { 3, 4 } };` |  | element initialisation
`X = A(:)` |  | `X = vectorise(A)` |  |
`X = [ A B ]` |  | `X = join_horiz(A,B)` |  |
`X = [ A; B ]` |  | `X = join_vert(A,B)` |  |
`A` |  | `cout << A << endl;` or `A.print("A =");` |  |
`save ‑ascii 'A.txt' A` |  | `A.save("A.txt", raw_ascii);` |  | Matlab/Octave matrices saved as ascii are readable by Armadillo (and vice-versa)
`load ‑ascii 'A.txt'` |  | `A.load("A.txt", raw_ascii);` |  |
`A = randn(2,3); B = randn(4,5); F = { A; B }` |  | `mat A = randn(2,3); mat B = randn(4,5); field<mat> F(2,1); F(0,0) = A; F(1,0) = B;` |  | fields store arbitrary objects, such as matrices

example program

    #include <iostream>
    #include <armadillo>

    using namespace std;
    using namespace arma;

    int main()
      {
      mat A(4, 5, fill::randu);
      mat B(4, 5, fill::randu);

      cout << A*B.t() << endl;

      return 0;
      }

- If the above program is stored as example.cpp, under Linux and macOS it can be compiled using:
  `g++ example.cpp -o example -std=c++11 -O2 -larmadillo`
- Armadillo extensively uses template meta-programming, so it's recommended to enable optimisation when compiling programs (eg. use the -O2 or -O3 options for GCC or clang)
- See the Questions page for more info on compiling and linking
- See also the example program that comes with the Armadillo archive

config.hpp

- Armadillo can be configured via editing the file include/armadillo_bits/config.hpp
- Specific functionality can be enabled or disabled by uncommenting or commenting out a particular #define, listed below.
- Some options can also be specified by explicitly defining them before including the armadillo header.

`ARMA_DONT_USE_WRAPPER` |  | Disable going through the run-time Armadillo wrapper library (libarmadillo.so) when calling LAPACK, BLAS, ARPACK, and SuperLU functions. You will need to directly link with BLAS, LAPACK, etc (eg. `-lblas -llapack`)
`ARMA_USE_LAPACK` |  | Enable use of LAPACK, or a high-speed replacement for LAPACK (eg. OpenBLAS, Intel MKL, or the Accelerate framework). Armadillo requires LAPACK for functions such as svd(), inv(), eig_sym(), solve(), etc.
`ARMA_DONT_USE_LAPACK` |  | Disable use of LAPACK; overrides ARMA_USE_LAPACK
`ARMA_USE_BLAS` |  | Enable use of BLAS, or a high-speed replacement for BLAS (eg. OpenBLAS, Intel MKL, or the Accelerate framework). BLAS is used for matrix multiplication. Without BLAS, Armadillo will use a built-in matrix multiplication routine, which might be slower for large matrices.
`ARMA_DONT_USE_BLAS` |  | Disable use of BLAS; overrides ARMA_USE_BLAS
`ARMA_USE_NEWARP` |  | Enable use of NEWARP (built-in alternative to ARPACK). This is used for the eigen decomposition of real (non-complex) sparse matrices, ie. eigs_gen(), eigs_sym() and svds(). Requires ARMA_USE_LAPACK to be enabled. If use of both NEWARP and ARPACK is enabled, NEWARP will be preferred.
`ARMA_DONT_USE_NEWARP` |  | Disable use of NEWARP (built-in alternative to ARPACK); overrides ARMA_USE_NEWARP
`ARMA_USE_ARPACK` |  | Enable use of ARPACK, or a high-speed replacement for ARPACK. Armadillo requires ARPACK for the eigen decomposition of complex sparse matrices, ie. eigs_gen(), eigs_sym() and svds(). If use of NEWARP is disabled, ARPACK will also be used for the eigen decomposition of real sparse matrices.
`ARMA_DONT_USE_ARPACK` |  | Disable use of ARPACK; overrides ARMA_USE_ARPACK
`ARMA_USE_SUPERLU` |  | Enable use of SuperLU, which is used by spsolve() for finding the solutions of sparse systems, as well as eigs_sym() and eigs_gen() in shift-invert mode. You will need to link with the superlu library, for example `-lsuperlu`
`ARMA_DONT_USE_SUPERLU` |  | Disable use of SuperLU; overrides ARMA_USE_SUPERLU
`ARMA_USE_HDF5` |  | Enable the ability to save and load matrices stored in the HDF5 format; the hdf5.h header file must be available on your system and you will need to link with the hdf5 library (eg. `-lhdf5`)
`ARMA_DONT_USE_HDF5` |  | Disable the use of the HDF5 library; overrides ARMA_USE_HDF5
`ARMA_USE_FFTW3` |  | Enable use of the FFTW3 library by fft() and ifft(); you will need to link with the FFTW3 library (eg. `-lfftw3`)
`ARMA_DONT_USE_FFTW3` |  | Disable the use of the FFTW3 library; overrides ARMA_USE_FFTW3
`ARMA_DONT_USE_STD_MUTEX` |  | Disable use of std::mutex; applicable if your compiler and/or environment doesn't support std::mutex
`ARMA_DONT_OPTIMISE_BAND` |  | Disable automatically optimised handling of band matrices by solve() and chol()
`ARMA_DONT_OPTIMISE_SYMPD` |  | Disable automatically optimised handling of symmetric/hermitian positive definite matrices by solve(), inv(), pinv(), expmat(), logmat(), sqrtmat(), powmat(), rcond()
`ARMA_USE_OPENMP` |  | Use OpenMP for parallelisation of computationally expensive element-wise operations (such as exp(), log(), cos(), etc). Automatically enabled when using a compiler which has OpenMP 3.1+ active (eg. the `-fopenmp` option for gcc and clang).
`ARMA_DONT_USE_OPENMP` |  | Disable use of OpenMP for parallelisation of element-wise operations; overrides ARMA_USE_OPENMP
`ARMA_OPENMP_THRESHOLD` |  | The minimum number of elements in a matrix to enable OpenMP based parallelisation of computationally expensive element-wise functions; default value is 320
`ARMA_OPENMP_THREADS` |  | The maximum number of threads for OpenMP based parallelisation of computationally expensive element-wise functions; default value is 8
`ARMA_BLAS_CAPITALS` |  | Use capitalised (uppercase) BLAS and LAPACK function names (eg. DGEMM vs dgemm)
`ARMA_BLAS_UNDERSCORE` |  | Append an underscore to BLAS and LAPACK function names (eg. dgemm_ vs dgemm). Enabled by default.
`ARMA_BLAS_LONG_LONG` |  | Use "long long" instead of "int" when calling BLAS and LAPACK functions; the "long long" type is a 64 bit integer type on all platforms
`ARMA_USE_FORTRAN_HIDDEN_ARGS` |  | Use so-called "hidden arguments" when calling BLAS and LAPACK functions. Enabled by default. See Fortran argument passing conventions for more details.
`ARMA_DONT_USE_FORTRAN_HIDDEN_ARGS` |  | Disable use of so-called "hidden arguments" when calling BLAS and LAPACK functions. May be necessary when using Armadillo in conjunction with broken MKL headers (eg. if you have `#include "mkl_lapack.h"` in your code).
`ARMA_USE_TBB_ALLOC` |  | Use Intel TBB scalable_malloc() and scalable_free() instead of standard malloc() and free() for managing matrix memory
`ARMA_USE_MKL_ALLOC` |  | Use Intel MKL mkl_malloc() and mkl_free() instead of standard malloc() and free() for managing matrix memory
`ARMA_USE_MKL_TYPES` |  | Use Intel MKL types for complex numbers. You will need to include appropriate MKL headers before the Armadillo header. You may also need to enable one or more of the following options: `ARMA_BLAS_LONG_LONG`, `ARMA_DONT_USE_FORTRAN_HIDDEN_ARGS`
`ARMA_64BIT_WORD` |  | Use 64 bit integers for matrix and vector sizes. Automatically enabled when using a 64-bit platform, except when using Armadillo in the R environment (via RcppArmadillo). Useful if matrices/vectors capable of holding more than 4 billion elements are required. This can also be enabled by adding `#define ARMA_64BIT_WORD` before each instance of `#include <armadillo>`. See also the `ARMA_BLAS_LONG_LONG` option.
`ARMA_MAT_PREALLOC` |  | The number of pre-allocated elements used by matrices and vectors. Must be always enabled and set to an integer that is at least 1. By default set to 16. If you mainly use lots of very small vectors (eg. ≤ 4 elements), change the number to the size of your vectors.
`ARMA_COUT_STREAM` |  | The default stream used for printing matrices and cubes by .print(). Must be always enabled. By default defined to std::cout
`ARMA_CERR_STREAM` |  | The default stream used for printing warnings and errors. Must be always enabled. By default defined to std::cerr
`ARMA_WARN_LEVEL` |  | The level of warning messages printed to ARMA_CERR_STREAM. Must be an integer ≥ 0. By default defined to 2.
0 | = | no warnings; generally not recommended
1 | = | only critical warnings about arguments and/or data which are likely to lead to incorrect results
2 | = | as per level 1, and warnings about poorly conditioned systems (low rcond) detected by solve(), spsolve(), etc
3 | = | as per level 2, and warnings about failed decompositions, failed saving / loading, etc
Example usage:
    #define ARMA_WARN_LEVEL 1
    #include <armadillo>

- See also:
  - element access
  - element initialisation
  - uword/sword
  - output streams

History of API Additions and Changes

- API Stability and Version Policy:
  - Each release of Armadillo has its public API (functions, classes, constants) described in the accompanying API documentation specific to that release.
  - Each release of Armadillo has its full version specified as A.B.C, where A is a major version number, B is a minor version number, and C is a patch level. The version specification has explicit meaning (similar to Semantic Versioning) as follows:
  - Within a major version (eg. 10), each minor version (eg. 10.2) has a public API that strongly strives to be backwards compatible (at the source level) with the public API of preceding minor versions. For example, user code written for version 10.0 should work with version 10.1, 10.2, etc. However, subsequent minor versions may have more features (API additions and extensions) than preceding minor versions. As such, user code specifically written for version 10.2 may not work with 10.1.
  - An increase in the patch level, while the major and minor versions are retained, indicates modifications to the code and/or documentation which aim to fix bugs without altering the public API.
  - We don't like changes to existing public API and strongly prefer not to break any user software. However, to allow evolution, the public API may be altered in future major versions while remaining backwards compatible in as many cases as possible (eg. major version 11 may have slightly different public API than major version 10).
  - Caveat: the above policy applies only to the public API described in the documentation. Any functionality within Armadillo which is not explicitly described in the public API documentation is considered as internal implementation detail, and may be changed or removed without notice.
- List of additions and changes for each version:
  - Version 14.2:
    - faster handling of symmetric matrices by inv(), rcond(), powmat()
    - faster handling of hermitian matrices by inv(), rcond(), powmat(), cond(), pinv(), rank()
    - added `solve_opts::force_sym` option to solve() to force use of the symmetric/hermitian solver (not limited to sympd matrices)
    - more efficient handling of compound expressions by solve()
  - Version 14.0
    - C++14 is now the minimum recommended C++ standard
    - faster handling of compound expressions by as_scalar(), accu(), dot()
    - faster interactions between sparse and dense matrices
    - expanded stddev() to handle sparse matrices
    - expanded relational operators to handle expressions between sparse matrices and scalars
    - added .as_dense() to obtain dense vector/matrix representation of any sparse matrix expression
    - updated physical constants to NIST 2022 CODATA values
  - Version 12.8:
    - faster detection of symmetric expressions by pinv() and rank()
    - faster handling of sparse submatrix views
    - expanded shift() to handle sparse matrices
    - expanded conv_to for more flexible conversions between sparse and dense matrices
    - added cbrt()
    - more compact representation of integers when saving matrices in CSV format
  - Version 12.6:
    - faster multiplication of dense vectors by sparse matrices (and vice versa)
    - faster eigs_sym(), eigs_gen(), svds()
    - faster conv() and conv2() when using OpenMP
    - added diags() and spdiags() for generating band matrices from set of vectors
  - Version 12.4:
    - added norm2est() for finding fast estimates of matrix 2-norm (spectral norm)
    - added vecnorm() for obtaining the vector norm of each row or column of a matrix
  - Version 12.2:
    - more efficient use of FFTW3 by fft() and ifft()
    - faster in-place element-wise multiplication of sparse matrices by dense matrices
    - added spsolve_factoriser class to allow reuse of sparse matrix factorisation for solving systems of linear equations
  - Version 12.0:
    - faster fft() and ifft() via optional use of FFTW3
    - faster min() and max()
    - faster index_min() and index_max()
    - added .col_as_mat() and .row_as_mat() which return matrix representation of cube column and cube row
    - added `csv_opts::strict` option to loading CSV files to interpret missing values as NaN
    - added `check_for_zeros` option to form 4 of sparse matrix batch constructors
    - inv() and inv_sympd() with options `inv_opts::no_ugly` or `inv_opts::allow_approx` now use a scaled threshold similar to pinv()
    - `set_cout_stream()` and `set_cerr_stream()` are now no-ops; instead use the options ARMA_WARN_LEVEL, or ARMA_COUT_STREAM, or ARMA_CERR_STREAM
  - Version 11.4:
    - extended pow() with secondary element-wise form
    - added find_nan() to find indices of NaN elements
    - faster handling of compound expressions by sum()
  - Version 11.2:
    - extended randu() and randn() to allow specification of distribution parameters
    - added `inv_opts::no_ugly` option to inv() and inv_sympd() to disallow inverses of poorly conditioned matrices
    - more efficient handling of rank-deficient matrices via `inv_opts::allow_approx` option in inv() and inv_sympd()
    - faster handling of sparse submatrix column views by norm(), accu(), nonzeros()
    - faster handling of symmetric and diagonal matrices by cond()
    - better detection of rank deficient matrices by solve()
  - Version 11.0:
    - added variants of inv() and inv_sympd() that provide rcond (reciprocal condition number)
    - added `inv_opts::allow_approx` option to inv() and inv_sympd() to allow approximate inverses of poorly conditioned matrices
    - stricter handling of singular matrices by inv() and inv_sympd()
    - stricter handling of non-sympd matrices by inv_sympd()
    - stricter handling of non-finite matrices by pinv()
    - more robust handling of rank deficient matrices by solve()
    - faster handling of diagonal matrices by rcond()
    - changed eigs_sym() and eigs_gen() to use higher quality RNG
    - quantile() and median() will now throw an exception if given matrices/vectors have NaN elements
  - Version 10.8:
    - faster handling of symmetric matrices by pinv() and rank()
    - faster handling of diagonal matrices by inv_sympd(), pinv(), rank()
    - expanded norm() to handle integer vectors and matrices
    - added datum::tau to replace 2π
  - Version 10.7:
    - faster handling of submatrix views accessed by `X.cols(first_col,last_col)`
    - faster handling of element-wise min() and max() in compound expressions
    - expanded solve() with `solve_opts::force_approx` option to force use of the approximate solver
  - Version 10.6:
    - expanded chol() to optionally use pivoted decomposition
    - expanded vector, matrix and cube constructors to allow element initialisation via `fill::value(scalar)`, eg. `mat X(4,5,fill::value(123))`
    - faster loading of CSV files when using OpenMP
    - added csv_opts::semicolon option to allow saving / loading of CSV files with semicolon (;) instead of comma (,) as the separator
  - Version 10.5:
    - added .clamp() member function
    - expanded the standalone clamp() function to handle complex values
    - more efficient use of OpenMP
    - vector, matrix and cube constructors now initialise elements to zero by default;
      element initialisation can be disabled via the `fill::none` specifier, eg. `mat X(4,5,fill::none)`
  - Version 10.4:
    - faster handling of triangular matrices by log_det()
    - added log_det_sympd() for log determinant of symmetric positive matrices
    - added `ARMA_WARN_LEVEL` configuration option, to control the degree of emitted warning messages
    - reduced the default degree of warning messages, so that failed decompositions, failed saving / loading, etc, no longer emit warnings
  - Version 10.3:
    - faster handling of symmetric positive definite matrices by pinv()
    - expanded .save() / .load() for dense matrices to handle coord_ascii format
    - for out of bounds access, element accessors now throw the more nuanced std::out_of_range exception, instead of only std::logic_error
    - improved quality of random numbers
  - Version 10.2:
    - faster handling of subcubes
    - added tgamma()
    - added .brief_print() for abridged printing of matrices & cubes
    - expanded sparse matrix forms of trimatu() and trimatl() to allow specifying the diagonal delimiter
    - expanded eigs_sym() and eigs_gen() with optional shift-invert mode
  - Version 10.1:
    - C++11 is now the minimum required C++ standard
    - faster handling of compound expressions by trimatu() and trimatl()
    - faster sparse matrix addition, subtraction and element-wise multiplication
    - expanded sparse submatrix views to handle the non-contiguous form of X.cols(vector_of_column_indices)
    - expanded eigs_sym() and eigs_gen() with optional fine-grained parameters
  - Version 9.900:
    - faster solve() for under/over-determined systems
    - faster eig_gen() and eig_pair() for large matrices
    - faster handling of matrix multiplication expressions by diagvec() and diagmat()
    - faster handling of relational expressions by accu()
    - faster handling of sympd matrices by expmat(), logmat(), sqrtmat()
    - faster access to columns in sparse submatrix views
    - added quantile()
    - added powmat()
    - added trimatu_ind() and trimatl_ind()
    - added log_normpdf()
    - added .is_zero()
    - added `ARMA_DONT_USE_CXX11_MUTEX` configuration option to disable use of std::mutex
    - expanded eig_gen() and eig_pair() to optionally provide left and right eigenvectors
    - expanded qr() to optionally use pivoted decomposition
    - expanded .save() and .load() to handle CSV files with headers via csv_name(filename,header) specification
    - more consistent detection of sparse vector expressions
    - updated physical constants to NIST 2018 CODATA values
    - workaround for save/load issues with HDF5 v1.12
  - Version 9.800:
    - faster solve() in default operation; iterative refinement is no longer applied by default; use `solve_opts::refine` to explicitly enable refinement
    - faster expmat()
    - faster handling of triangular matrices by rcond()
    - added .front() and .back()
    - added .is_trimatu() and .is_trimatl()
    - added .is_diagmat()
  - Version 9.700:
    - faster handling of cubes by vectorise()
    - faster handling of sparse matrices by nonzeros()
    - faster row-wise index_min() and index_max()
    - expanded join_rows() and join_cols() to handle joining up to 4 matrices
    - expanded .save() and .load() to allow storing sparse matrices in CSV format
    - added randperm() to generate a vector with random permutation of a sequence of integers
  - Version 9.600:
    - faster handling of sparse submatrices
    - faster handling of sparse diagonal views
    - faster handling of sparse matrices by symmatu() and symmatl()
    - faster handling of sparse matrices by join_cols()
    - expanded clamp() to handle sparse matrices
    - added .clean() to replace elements below a threshold with zeros
  - Version 9.500:
    - expanded solve() with `solve_opts::likely_sympd` to indicate that the given matrix is likely positive definite
    - more robust automatic detection of positive definite matrices by solve() and inv()
    - faster handling of sparse submatrices
    - expanded eigs_sym() to print a warning if the given matrix is not symmetric
    - extended LAPACK function prototypes to follow Fortran passing conventions for so-called "hidden arguments", in order to address GCC Bug 90329;
      to use previous LAPACK function prototypes without the "hidden arguments", `#define ARMA_DONT_USE_FORTRAN_HIDDEN_ARGS` before `#include <armadillo>`
  - Version 9.400:
    - faster cov() and cor()
    - added .as_col() and .as_row()
    - expanded .shed_rows() / .shed_cols() / .shed_slices() to remove rows/columns/slices specified in a vector
    - expanded vectorise() to handle sparse matrices
    - expanded element-wise versions of max() and min() to handle sparse matrices
    - optimised handling of sparse matrix expressions: sparse % (sparse +- scalar) and sparse / (sparse +- scalar)
    - expanded eig_sym(), chol(), expmat_sym(), logmat_sympd(), sqrtmat_sympd(), inv_sympd() to print a warning if the given matrix is not symmetric
    - more consistent detection of vector expressions
  - Version 9.300:
    - faster handling of compound complex matrix expressions by trace()
    - more efficient handling of element access for inplace modifications in sparse matrices
    - added .is_sympd() to check whether a matrix is symmetric/hermitian positive definite
    - added interp2() for 2D data interpolation
    - added expm1() and log1p()
    - expanded .is_sorted() with options `"strictascend"` and `"strictdescend"`
    - expanded eig_gen() to optionally perform balancing prior to decomposition
  - Version 9.200:
    - faster handling of symmetric positive definite matrices by rcond()
    - faster transpose of matrices with size ≥ 512x512
    - faster handling of compound sparse matrix expressions by accu(), diagmat(), trace()
    - faster handling of sparse matrices by join_rows()
    - added sinc()
    - expanded sign() to handle scalar arguments
    - expanded operators (*, %, +, −) to handle sparse matrices with differing element types (eg. multiplication of complex matrix by real matrix)
    - expanded conv_to() to allow conversion between sparse matrices with differing element types
    - expanded solve() to optionally allow keeping solutions of systems singular to working precision
  - Version 9.100:
    - faster handling of symmetric/hermitian positive definite matrices by solve()
    - faster handling of inv_sympd() in compound expressions
    - added .is_symmetric()
    - added .is_hermitian()
    - expanded spsolve() to optionally allow keeping solutions of systems singular to working precision
    - new configuration options ARMA_OPTIMISE_SOLVE_BAND and ARMA_OPTIMISE_SOLVE_SYMPD
    - smarter use of the element cache in sparse matrices
  - Version 8.600:
    - added hess() for Hessenberg decomposition
    - added .row(), .rows(), .col(), .cols() to subcube views
    - expanded .shed_rows() and .shed_cols() to handle cubes
    - expanded .insert_rows() and .insert_cols() to handle cubes
    - expanded subcube views to allow non-contiguous access to slices
    - improved tuning of sparse matrix element access operators
    - faster handling of tridiagonal matrices by solve()
    - faster multiplication of matrices with differing element types when using OpenMP
  - Version 8.500:
    - faster handling of sparse matrices by kron() and repmat()
    - faster transpose of sparse matrices
    - faster element access in sparse matrices
    - faster row iterators for sparse matrices
    - faster handling of compound expressions by trace()
    - more efficient handling of aliasing in submatrix views
    - expanded normalise() to handle sparse matrices
    - expanded .transform() and .for_each() to handle sparse matrices
    - added reverse() for reversing order of elements
    - added repelem() for replicating elements
    - added roots() for finding the roots of a polynomial
  - Version 8.400:
    - faster handling of sparse matrices by repmat()
    - faster loading of CSV files
    - expanded kron() to handle sparse matrices
    - expanded index_min() and index_max() to handle cubes
    - expanded randi(), randu(), randn(), randg() to output single scalars
    - added submatrix & subcube iterators
    - added normcdf()
    - added mvnrnd()
    - added chi2rnd()
    - added wishrnd() and iwishrnd()
  - Version 8.300:
    - faster handling of band matrices by solve()
    - faster handling of band matrices by chol()
    - faster randg() when using OpenMP
    - added normpdf()
    - expanded .save() to allow appending new datasets to existing HDF5 files
  - Version 8.200:
    - added intersect() for finding common elements in two vectors/matrices
    - expanded affmul() to handle non-square matrices
  - Version 8.100:
    - faster incremental construction of sparse matrices via element access operators
    - faster diagonal views in sparse matrices
    - expanded SpMat to save/load sparse matrices in coord format
    - expanded .save()/.load() to allow specification of datasets within HDF5 files
    - added affmul() to simplify application of affine transformations
    - warnings and errors are now printed by default to the std::cerr stream
    - added set_cerr_stream() and get_cerr_stream() to replace set_stream_err1(), set_stream_err2(), get_stream_err1(), get_stream_err2()
    - new configuration options ARMA_COUT_STREAM and ARMA_CERR_STREAM
  - Version 7.960:
    - faster randn() when using OpenMP
    - faster gmm_diag class, for Gaussian mixture models with diagonal covariance matrices
    - added .sum_log_p() to the gmm_diag class
    - added gmm_full class, for Gaussian mixture models with full covariance matrices
    - expanded .each_slice() to optionally use OpenMP for multi-threaded execution
  - Version 7.950:
    - expanded accu() and sum() to use OpenMP for processing expressions with computationally expensive element-wise functions
    - expanded trimatu() and trimatl() to allow specification of the diagonal which delineates the boundary of the triangular part
  - Version 7.900:
    - expanded clamp() to handle cubes
    - computationally expensive element-wise functions (such as exp(), log(), cos(), etc) can now be automatically sped up via OpenMP; this requires a C++11/C++14 compiler with OpenMP 3.1+ support
      - for GCC and clang compilers use the following options to enable both C++11 and OpenMP: `-std=c++11 -fopenmp`
      - Caveat: when using GCC, use of `-march=native` in conjunction with `-fopenmp` may lead to speed regressions on recent processors
  - Version 7.800:
    - changed license to the permissive Apache License 2.0; see the Questions page for more info
  - Version 7.700:
    - added polyfit() and polyval()
    - added second form of log_det() to directly return the result as a complex number
    - added range() to statistics functions
    - expanded trimatu()/trimatl() and symmatu()/symmatl() to handle sparse matrices
  - Version 7.600:
    - more accurate eigs_sym() and eigs_gen()
    - expanded floor(), ceil(), round(), trunc(), sign() to handle sparse matrices
    - added arg(), atan2(), hypot()
  - Version 7.500:
    - expanded qz() to optionally specify ordering of the Schur form
    - expanded .each_slice() to support matrix multiplication
  - Version 7.400:
    - added expmat_sym(), logmat_sympd(), sqrtmat_sympd()
    - added .replace()
  - Version 7.300:
    - added index_min() and index_max() standalone functions
    - expanded .subvec() to accept size() arguments
    - more robust handling of non-square matrices by lu()
  - Version 7.200:
    - added .index_min() and .index_max() member functions
    - expanded ind2sub() to handle vectors of indices
    - expanded sub2ind() to handle matrix of subscripts
    - expanded expmat(), logmat() and sqrtmat() to optionally return a bool indicating success
    - faster handling of compound expressions by vectorise()
  - Version 7.100:
    - added erf(), erfc(), lgamma()
    - added .head_slices() and .tail_slices() to subcube views
    - spsolve() now requires SuperLU 5.2
    - eigs_sym(), eigs_gen() and svds() now use a built-in reimplementation of ARPACK for real (non-complex) matrices; contributed by Yixuan Qiu
  - Version 6.700:
    - added trapz() for numerical integration
    - added logmat() for calculating the matrix logarithm
    - added regspace() for generating vectors with regularly spaced elements
    - added logspace() for generating vectors with logarithmically spaced elements
    - added approx_equal() for determining approximate equality
  - Version 6.600:
    - expanded sum(), mean(), min(), max() to handle cubes
    - expanded Cube class to handle arbitrarily sized empty cubes (eg. 0x5x2)
    - added shift() for circular shifts of elements
    - added sqrtmat() for finding the square root of a matrix
  - Version 6.500:
    - added conv2() for 2D convolution
    - added stand-alone kmeans() function for clustering data
    - added trunc()
    - extended conv() to optionally provide central convolution
    - faster handling of multiply-and-accumulate by accu() when using Intel MKL, ATLAS or OpenBLAS
  - Version 6.400:
    - expanded each_col(), each_row() and each_slice() to handle C++11 lambda functions
    - added ind2sub() and sub2ind()
  - Version 6.300:
    - expanded solve() to find approximate solutions for rank-deficient systems
    - faster handling of non-contiguous submatrix views in compound expressions
    - added .for_each() to Mat, Row, Col, Cube and field classes
    - added rcond() for estimating the reciprocal condition number
  - Version 6.200:
    - expanded diagmat() to handle non-square matrices and arbitrary diagonals
    - expanded trace() to handle non-square matrices
  - Version 6.100:
    - faster norm() and normalise() when using Intel MKL, ATLAS or OpenBLAS
    - added Schur decomposition: schur()
    - stricter handling of matrix objects by hist() and histc()
    - advanced constructors for using auxiliary memory by Mat, Col, Row and Cube now have the default of strict = false
    - Cube class now delays allocation of .slice() related structures until needed
    - expanded join_slices() to handle joining cubes with matrices
  - Version 5.600:
    - added .each_slice() for matrix operations applied to each slice of a cube
    - expanded .each_col() and .each_row() to handle out-of-place operations
  - Version 5.500:
    - expanded object constructors and generators to handle size() based specification of dimensions
  - Version 5.400:
    - added find_unique() for finding indices of unique values
    - added diff() for calculating differences between consecutive elements
    - added cumprod() for calculating cumulative product
    - added null() for finding the orthonormal basis of null space
    - expanded interp1() to handle repeated locations
    - expanded unique() to handle complex numbers
    - faster flipud()
    - faster row-wise cumsum()
  - Version 5.300:
    - added generalised Schur decomposition: qz()
    - added .has_inf() and .has_nan()
    - expanded interp1() to handle out-of-domain locations
    - expanded sparse matrix class with .set_imag() and .set_real()
    - expanded imag(), real() and conj() to handle sparse matrices
    - expanded diagmat(), reshape() and resize() to handle sparse matrices
    - faster sparse sum()
    - faster row-wise sum(), mean(), min(), max()
    - updated physical constants to NIST 2014 CODATA values
  - Version 5.200:
    - added orth() for finding the orthonormal basis of the range space of a matrix
    - expanded element initialisation to handle nested initialiser lists (C++11)
  - Version 5.100:
    - added interp1() for 1D interpolation
    - added .is_sorted() for checking whether a vector or matrix has sorted elements
    - updated physical constants to NIST 2010 CODATA values
  - Version 5.000:
    - added spsolve() for solving sparse systems of linear equations
    - added svds() for singular value decomposition of sparse matrices
    - added nonzeros() for extracting non-zero values from matrices
    - added handling of diagonal views by sparse matrices
    - expanded repmat() to handle sparse matrices
    - expanded join_rows() and join_cols() to handle sparse matrices
    - sort_index() and stable_sort_index() have been placed in the delayed operations framework for increased efficiency
    - use of 64 bit integers is automatically enabled when using a C++11 compiler
  - Version 4.650:
    - added randg() for generating random values from gamma distributions (C++11 only)
    - added .head_rows() and .tail_rows() to submatrix views
    - added .head_cols() and .tail_cols() to submatrix views
    - expanded eigs_sym() to optionally calculate eigenvalues with smallest/largest algebraic values
  - Version 4.600:
    - added .head() and .tail() to submatrix views
    - faster matrix transposes within compound expressions
    - faster in-place matrix multiplication
    - faster accu() and norm() when compiling with -O3 -ffast-math -march=native (gcc and clang)
  - Version 4.550:
    - added matrix exponential function: expmat()
    - faster .log_p() and .avg_log_p() functions in the gmm_diag class when compiling with OpenMP enabled
    - faster handling of in-place addition/subtraction of expressions with an outer product
  - Version 4.500:
    - faster handling of complex vectors by norm()
    - expanded chol() to optionally specify output matrix as upper or lower triangular
    - better handling of non-finite values when saving matrices as text files
  - Version 4.450:
    - faster handling of matrix transposes within compound expressions
    - expanded symmatu()/symmatl() to optionally disable taking the complex conjugate of elements
    - expanded sort_index() to handle complex vectors
    - expanded the gmm_diag class with functions to generate random samples
  - Version 4.400:
    - faster handling of subvectors by dot()
    - faster handling of aliasing by submatrix views
    - added clamp() for clamping values to be between lower and upper limits
    - added gmm_diag class for statistical modelling of data using Gaussian Mixture Models
    - expanded batch insertion constructors for sparse matrices to add values at repeated locations
  - Version 4.320:
    - expanded eigs_sym() and eigs_gen() to use an optional tolerance parameter
    - expanded eig_sym() to automatically fall back to standard decomposition method if divide-and-conquer fails
    - cmake-based installer enables use of C++11 random number generator when using gcc 4.8.3+ in C++11 mode
  - Version 4.300:
    - added find_finite() and find_nonfinite()
    - expressions X=inv(A)*B*C and X=A.i()*B*C are automatically converted to X=solve(A,B*C)
  - Version 4.200:
    - faster transpose of sparse matrices
    - more efficient handling of aliasing during matrix multiplication
    - faster inverse of matrices marked as diagonal
  - Version 4.100:
    - added normalise() for normalising vectors to unit p-norm
    - extended the field class to handle 3D layout
    - extended eigs_sym() and eigs_gen() to obtain eigenvalues of various forms (eg. largest or smallest magnitude)
    - automatic SIMD vectorisation of elementary expressions (eg. matrix addition) when using Clang 3.4+ with -O3 optimisation
    - faster handling of sparse submatrix views
  - Version 4.000:
    - added eigen decompositions of sparse matrices: eigs_sym() and eigs_gen()
    - added eigen decomposition for pair of matrices: eig_pair()
    - added simpler forms of eig_gen()
    - added condition number of matrices: cond()
    - expanded find() to handle cubes
    - expanded subcube views to access elements specified in a vector
    - template argument for running_stat_vec expanded to accept vector types
    - more robust fast inverse of 4x4 matrices
    - faster divide-and-conquer decompositions are now used by default for eig_sym(), pinv(), princomp(), rank(), svd(), svd_econ()
    - the form inv(sympd(X)) no longer assumes that X is positive definite; use inv_sympd() instead
  - Version 3.930:
    - added size() based specifications of submatrix view sizes
    - added element-wise variants of min() and max()
    - added divide-and-conquer variant of svd_econ()
    - added divide-and-conquer variant of pinv()
    - added randi() for generating matrices with random integer values
    - added inplace_trans() for memory efficient in-place transposes
    - added more intuitive specification of sort direction in sort() and sort_index()
    - added more intuitive specification of method in det(), .i(), inv() and solve()
    - more precise timer for the wall_clock class when using C++11
  - Version 3.920:
    - faster .zeros()
    - faster round(), exp2() and log2() when using C++11
    - added signum function: sign()
    - added move constructors when using C++11
    - added 2D fast Fourier transform: fft2()
    - added .tube() for easier extraction of vectors and subcubes from cubes
    - added specification of a fill type during construction of Mat, Col, Row and Cube classes, eg. mat X(4, 5, fill::zeros)
  - Version 3.910:
    - faster multiplication of a matrix with a transpose of itself, ie. X*X.t() and X.t()*X
    - added vectorise() for reshaping matrices into vectors
    - added all() and any() for indicating presence of elements satisfying a relational condition
  - Version 3.900:
    - added automatic SSE2 vectorisation of elementary expressions (eg. matrix addition) when using GCC 4.7+ with -O3 optimisation
    - faster median()
    - faster handling of compound expressions with transposes of submatrix rows
    - faster handling of compound expressions with transposes of complex vectors
    - added support for saving & loading of cubes in HDF5 format
  - Version 3.820:
    - faster as_scalar() for compound expressions
    - faster transpose of small vectors
    - faster matrix-vector product for small vectors
    - faster multiplication of small fixed size matrices
  - Version 3.810:
    - added fast Fourier transform: fft()
    - added handling of .imbue() and .transform() by submatrices and subcubes
    - added batch insertion constructors for sparse matrices
  - Version 3.800:
    - added .imbue() for filling a matrix/cube with values provided by a functor or lambda expression
    - added .swap() for swapping contents with another matrix
    - added .transform() for transforming a matrix/cube using a functor or lambda expression
    - added round() for rounding matrix elements towards nearest integer
    - faster find()
    - changed license to the Mozilla Public License 2.0
  - Version 3.6:
    - faster handling of compound expressions with submatrices and subcubes
    - faster trace()
    - added support for loading matrices as text files with NaN and Inf elements
    - added stable_sort_index(), which preserves the relative order of elements with equivalent values
    - added handling of sparse matrices by mean(), var(), norm(), abs(), square(), sqrt()
    - added saving and loading of sparse matrices in arma_binary format
  - Version 3.4:
    - added economical QR decomposition: qr_econ()
    - added .each_col() & .each_row() for vector operations applied to each column or row of a matrix
    - added preliminary support for sparse matrices
    - added ability to save and load matrices in HDF5 format
    - faster singular value decomposition via optional use of divide-and-conquer algorithm
    - faster .randn()
    - faster dot() and cdot() for complex numbers
  - Version 3.2:
    - added unique(), for finding unique elements of a matrix
    - added .eval(), for forcing the evaluation of delayed expressions
    - faster eigen decomposition via optional use of divide-and-conquer algorithm
    - faster transpose of vectors and compound expressions
    - faster handling of diagonal views
    - faster handling of tiny fixed size vectors (≤ 4 elements)
  - Version 3.0:
    - added shorthand for inverse: .i()
    - added datum class
    - added hist() and histc()
    - added non-contiguous submatrix views
    - faster handling of submatrix views with a single row or column
    - faster element access in fixed size matrices
    - faster repmat()
    - expressions X=inv(A)*B and X=A.i()*B are automatically converted to X=solve(A,B)
    - better detection of vector expressions by sum(), cumsum(), prod(), min(), max(), mean(), median(), stddev(), var()
    - faster generation of random numbers (eg. randu() and randn()), via an algorithm that produces slightly different numbers than in 2.x
    - support for tying writable auxiliary (external) memory to fixed size matrices has been removed; instead, you can use standard matrices with writable auxiliary memory, or initialise fixed size matrices by copying the memory; using auxiliary memory with standard matrices is unaffected
    - .print_trans() and .raw_print_trans() have been removed; instead, you can chain .t() and .print() to achieve a similar result: X.t().print()
  - Version 2.4:
    - added shorter forms of transposes: .t() and .st()
    - added .resize() and resize()
    - added optional use of 64 bit indices (allowing matrices to have more than 4 billion elements), enabled via ARMA_64BIT_WORD in include/armadillo_bits/config.hpp
    - added experimental support for C++11 initialiser lists, enabled via ARMA_USE_CXX11 in include/armadillo_bits/config.hpp
    - refactored code to eliminate warnings when using the Clang C++ compiler
    - umat, uvec, .min() and .max() have been changed to use the uword type instead of the u32 type; by default the uword and u32 types are equivalent (ie. unsigned integer type with a minimum width 32 bits); however, when the use of 64 bit indices is enabled via ARMA_64BIT_WORD in include/armadillo_bits/config.hpp, the uword type then has a minimum width of 64 bits
  - Version 2.2:
    - added svd_econ()
    - added .is_colvec() and .is_rowvec()
  - Version 2.0:
    - det(), inv() and solve() can be forced to use more precise algorithms for tiny matrices (≤ 4x4)
    - added syl(), for solving Sylvester's equation
    - added strans(), for transposing a complex matrix without taking the complex conjugate
    - added symmatu() and symmatl()
    - added submatrices of submatrices
    - faster inverse of symmetric positive definite matrices
    - faster element access for fixed size matrices
    - faster multiplication of tiny matrices (eg. 4x4)
    - faster compound expressions containing submatrices
    - added handling of arbitrarily sized empty matrices (eg. 5x0)
    - added .count() member function in running_stat and running_stat_vec
    - added loading & saving of matrices as CSV text files
    - trans() now takes the complex conjugate when transposing a complex matrix
    - forms of chol(), eig_sym(), eig_gen(), inv(), lu(), pinv(), princomp(), qr(), solve(), svd(), syl() that do not return a bool indicating success now throw std::runtime_error exceptions when failures are detected
    - princomp_cov() has been removed; eig_sym() in conjunction with cov() can be used instead
    - .is_vec() now outputs true for empty vectors (eg. 0x1)
    - set_log_stream() & get_log_stream() have been replaced by set_stream_err1() & get_stream_err1()
  - Version 1.2:
    - added .min() & .max() member functions of Mat and Cube
    - added floor() and ceil()
    - added representation of “not a number”: math::nan()
    - added representation of infinity: math::inf()
    - .in_range() expanded to use span() arguments
    - fixed size matrices and vectors can use auxiliary (external) memory
    - submatrices and subfields can be accessed via X( span(a,b), span(c,d) )
    - subcubes can be accessed via X( span(a,b), span(c,d), span(e,f) )
    - the two argument version of span can be replaced by span::all or span(), to indicate an entire range
    - for cubes, the two argument version of span can be replaced by a single argument version, span(a), to indicate a single column, row or slice
    - arbitrary "flat" subcubes can be interpreted as matrices; for example:

          cube Q = randu<cube>(5,3,4);
          mat  A = Q( span(1), span(1,2), span::all );
          // A has a size of 2x4

          vec v = ones<vec>(4);
          Q( span(1), span(1), span::all ) = v;

    - added interpretation of matrices as triangular through trimatu() / trimatl()
    - added explicit handling of triangular matrices by solve() and inv()
    - extended syntax for submatrices, including access to elements whose indices are specified in a vector
    - added ability to change the stream used for logging of errors and warnings
    - added ability to save/load matrices in raw binary format
    - added cumulative sum function: cumsum()
  - Changed in 1.0 (compared to earlier 0.x development versions):
    - the 3 argument version of lu(), eg. lu(L,U,X), provides L and U which should be the same as produced by Octave 3.2 (this was not the case in versions prior to 0.9.90)
    - rand() has been replaced by randu(); this has been done to avoid confusion with std::rand(), which generates random numbers in a different interval
    - In versions earlier than 0.9.0, some multiplication operations directly converted result matrices with a size of 1x1 into scalars. This is no longer the case. If you know the result of an expression will be a 1x1 matrix and wish to treat it as a pure scalar, use the as_scalar() wrapping function
    - Almost all functions have been placed in the delayed operations framework (for speed purposes). This may affect code which assumed that the output of some functions was a pure matrix. The solution is easy, as explained below.
      In general, Armadillo queues operations before executing them. As such, the direct output of an operation or function cannot be assumed to be a directly accessible matrix. The queued operations are executed when the output needs to be stored in a matrix, eg. mat B = trans(A) or mat B(trans(A)). If you need to force the execution of the delayed operations, place the operation or function inside the corresponding Mat constructor. For example, if your code assumed that the output of some functions was a pure matrix, eg. chol(m).diag(), change the code to mat(chol(m)).diag(). Similarly, if you need to pass the result of an operation such as A+B to one of your own functions, use my_function( mat(A+B) ).