rat, rats

Purpose

Rational fraction approximation.

Synopsis

[N,D] = rat(X,tol)
[N,D] = rat(X)
rat(...)
s = rats(X,strlen)
s = rats(X)

Description

Even though all floating-point numbers are rational numbers, it is sometimes desirable to approximate them by simple rational numbers, which are fractions whose numerator and denominator are small integers. rat(X) and rats(X) attempt to do this. The rational approximations are generated by truncating continued fraction expansions.

[N,D] = rat(X,tol) returns two integer matrices so that N./D is close to X in the sense that abs(N./D - X) <= tol*abs(X).

[N,D] = rat(X) uses a default tolerance, 1.e-6*norm(X(:),1).

rat(...) displays the continued fraction.

s = rats(X,strlen) uses rat to display simple rational approximations to the elements of X. The string length for each element is strlen. The default is strlen = 13, which allows 6 elements in 78 spaces. Asterisks are used for elements which cannot be printed in the allotted space, but are not negligible compared to the other elements in X.

s = rats(X) is an M-file which produces the same results as those printed by MATLAB with

 format rat

Examples

Ordinarily, the statement

 s = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7
produces

 s =
    0.7595
However, with

 format rat
or with

 rats(s)
the printed result is

 s = 
    319/420
This is a simple rational number. Its denominator is 420, the least common multiple of the denominators of the terms involved in the original expression. Even though the quantity s is stored internally as a binary floating-point number, the desired rational form can be reconstructed.

To see how the rational approximation is generated, the statement rat(s)

produces

 1 + 1/(-4 + 1/(-6 + 1/(-3 + 1/(-5))))
And the statement

 [n,d] = rat(s)
produces 

n = 319, d = 420
The mathematical quantity

is certainly not a rational number, but the MATLAB quantity pi which approximates it is a rational number. With IEEE floating-point arithmetic, pi is the ratio of a large integer and 2^52:

 14148475504056880/4503599627370496
However, this is not a simple rational number. The value printed for pi with format rat, or with rats(pi), is

 355/113
This approximation was known in Euclid's time. Its decimal representation is

 3.14159292035398
and so it agrees with pi to seven significant figures. The statement

 rat(pi)
produces

 3 + 1/(7 + 1/(16))
This shows how the 355/113 was obtained. The less accurate, but more familiar approximation 22/7 is obtained from the first two terms of this continued fraction.

Algorithm

rat(X) approximates each element of X by a continued fraction of the form:

The d's are obtained by repeatedly picking off the integer part and then taking the reciprocal of the fractional part. The accuracy of the approximation increases exponentially with the number of terms and is worst when X = sqrt(2). For x = sqrt(2), the error with k terms is about 2.68*(.173)^k, so each additional term increases the accuracy by less than one decimal digit. It takes 21 terms to get full floating-point accuracy.

(c) Copyright 1994 by The MathWorks, Inc.