fractions {MASS} R Documentation

## Rational Approximation

### Description

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

### Usage

```fractions(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.

### 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.

The numerators and denominators are then combined into a character vector that becomes the `"fracs"` attribute and used in printed representations.

Arithmetic operations on `"fractions"` objects have full floating point accuracy, but the character representation printed out may not.

### Value

An object of class `"fractions"`. A structure with `.Data` component the same as the input numeric `x`, but with the rational approximations held as a character vector attribute, `"fracs"`. Arithmetic operations on `"fractions"` objects are possible.

### References

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth Edition. Springer.

`rational`

### 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

fractions(solve(X, X/5))
##      [,1] [,2] [,3] [,4] [,5]
## [1,] 1/5    0    0    0    0
## [2,]   0  1/5    0    0    0
## [3,]   0    0  1/5    0    0
## [4,]   0    0    0  1/5    0
## [5,]   0    0    0    0  1/5

fractions(solve(X, X/5)) + 1
##      [,1] [,2] [,3] [,4] [,5]
## [1,] 6/5    1    1    1    1
## [2,]   1  6/5    1    1    1
## [3,]   1    1  6/5    1    1
## [4,]   1    1    1  6/5    1
## [5,]   1    1    1    1  6/5
```

[Package MASS version 7.2-33 Index]