rational {MASS} | R Documentation

## 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.

### 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.

### 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.

### 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