MASS (version 7.3-0)

rational: Rational Approximation

Description

Find rational approximations to the components of a real numeric object using a standard continued fraction method.

Usage

rational(x, cycles = 10, max.denominator = 2000, ...)

Arguments

x

Any object of mode numeric. Missing values are now allowed.

cycles

The maximum number of steps to be used in the continued fraction approximation process.

max.denominator

An early termination criterion. If any partial denominator exceeds max.denominator the continued fraction stops at that point.

...

arguments passed to or from other methods.

Value

• A numeric object with the same attributes as x but with entries rational approximations to the values. This effectively rounds relative to the size of the object and replaces very small entries by zero.

Details

Each component is first expanded in a continued fraction of the form

x = floor(x) + 1/(p1 + 1/(p2 + ...)))

where p1, p2, ...are positive integers, terminating either at cycles terms or when a pj > max.denominator. The continued fraction is then re-arranged to retrieve the numerator and denominator as integers and the ratio returned as the value.

See Also

fractions

Examples

X <- matrix(runif(25), 5, 5)
solve(X, X/5)
##             [,1]        [,2]       [,3]        [,4]        [,5]
## [1,]  2.0000e-01  3.7199e-17 1.2214e-16  5.7887e-17 -8.7841e-17
## [2,] -1.1473e-16  2.0000e-01 7.0955e-17  2.0300e-17 -1.0566e-16
## [3,]  2.7975e-16  1.3653e-17 2.0000e-01 -1.3397e-16  1.5577e-16
## [4,] -2.9196e-16  2.0412e-17 1.5618e-16  2.0000e-01 -2.1921e-16
## [5,] -3.6476e-17 -3.6430e-17 3.6432e-17  4.7690e-17  2.0000e-01

## rational(solve(X, X/5))
##      [,1] [,2] [,3] [,4] [,5]
## [1,]  0.2  0.0  0.0  0.0  0.0
## [2,]  0.0  0.2  0.0  0.0  0.0
## [3,]  0.0  0.0  0.2  0.0  0.0
## [4,]  0.0  0.0  0.0  0.2  0.0
## [5,]  0.0  0.0  0.0  0.0  0.2