TSP (libtsp/MS) - MSratio

## MSratio

#### Routine

void MSratio (double Val, long int *N, long int *D, double tol, long int MaxN, long int MaxD)

#### Purpose

Find a ratio of integers approximation to a floating point value

#### Description

A continued fraction expansion approximation for a given floating point value is calculated iteratively. The continued fraction expansion gives a fraction (ratio of integers) which approximates to the real number. At successive stages in the expansion, the approximation error alternates in sign but with diminishing magnitude. The expansion is terminated when either of the two following conditions is satisfied.

1: The absolute value of the difference between the approximation (N/D) and the given number is less than a specified value (tol).
2: The next iteration results in the magnitude of either the numerator or the denominator of the approximation being larger than given limits. At this point, the secondary convergents are tested to see if one provides a smaller error than backing off to the result of the previous iteration.

If the input value is exactly a ratio of integers, this routine will find those integers (with tol set to zero, and MaxN and MaxD appropriately large).

Consider that case that the iteration is stopped by either the numerator exceeding MaxN or the denominator exceeding MaxD. The resulting fraction N/D is the fraction nearest Val amongst those fractions with numerator and denominator not exceeding the given limits.

References:

R. B. J. T. Allenby and E. J. Redfern, "Introduction to Number Theory with Computing", Edward Arnold, 1989.

I. Niven and H. S. Zuckerman, "An Introduction to the Theory of Numbers", 4th ed., John Wiley & Sons, 1980.

#### Parameters

-> double Val
  Value to be approximated. This value should have a magnitude between 1/MaxD and MaxN.
<- long int N
  Numerator in the rational approximation to Val. N is negative if Val is negative.
<- long int D
  Denominator in the rational approximation to Val.
-> double tol
  Error tolerance used to terminate the approximation
-> long int MaxN
  Maximum value for N. This value should be at least floor(|Val|).
-> long int MaxD
  Maximum value for D. This value should be at least floor(1/|Val|).

#### Author / revision

P. Kabal Copyright (C) 1998 / Revision 1.2 1998/04/24

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