## 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 to a given floating point value is calculated iteratively. At successive stages in the expansion, the approximation error for the convergents alternates in sign but with diminishing magnitude. The expansion is terminated when either two conditions is satisfied.
1: The magnitude of either the numerator or the denominator of the approximation is larger than a given limit. If so, the secondary convergents are tested to see if one provides a smaller error than backing off to the convergent from the previous iteration.
2: The absolute value of the difference between the approximation (N/D) and the given number is less than a specified value (tol).
If the input value is exactly a ratio of integers, this routine will find those integers if tol is set to zero, and MaxN and MaxD are 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 / Revision 1.7 2003/05/09

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