|MATLAB Function Reference|
Rational fraction approximation
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. The
rat function attempts to do this. Rational approximations are generated by truncating continued fraction expansions. The
rats function calls
rat, and returns strings.
[N,D] = rat(X)
D so that
X to within the default tolerance,
[N,D] = rat(X,tol
X to within
with no output arguments, simply displays the continued fraction.
returns a string containing simple rational approximations to the elements of
S = rats(X,strlen
X. Asterisks are used for elements that cannot be printed in the allotted space, but are not negligible compared to the other elements in
strlen is the length of each string element returned by the
rats function. The default is
13, which allows 6 elements in 78 spaces.
returns the same results as those printed by MATLAB with
S = rats(X)
Ordinarily, the statement
the printed result is
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
And the statement
The mathematical quantity is certainly not a rational number, but the MATLAB quantity
pi that approximates it is a rational number.
pi is the ratio of a large integer and
However, this is not a simple rational number. The value printed for
rat, or with
This approximation was known in Euclid's time. Its decimal representation is
and so it agrees with
pi to seven significant figures. The statement
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.
rat(X) function approximates each element of
X by a continued fraction of the form
The 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
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.