Package 'fractional'

Representation of a numeric vector in vulgar fractional form

Description

The object is flagged so that if it is coerced to character, or printed, the numerical quantities are represented by a rational approximation. In other respects the numerical object behaves as normally.

Usage

fractional(x, eps = 1e-06, maxConv = 20, sync = FALSE)

## S3 method for class 'fractional'
as.character(x, eps = attr(x, "eps"), maxConv = attr(x,
  "maxConv"), ...)

## S3 method for class 'charFrac'
print(x, ...)

## S3 method for class 'fractional'
print(x, ...)

Arguments

x	A numeric object
eps	An absolute error tolerance
maxConv	An upper limit on the number of convergents to use in the continued fractions.
sync	A logical value. Should the numerical value be changed to match the rational approximation, as closely as possible with floating point, (TRUE)? Or, should it be left and used in its original state (FALSE)?
...	Currently ignored.

Value

A numeric object of class "fractional".

Methods (by generic)

• as.character: S3 method for coercion to character, producing an object inheriting from class "charFrac"

• print: Print method for class "charFrac" objects, unquoted.

• print: Print method for "fractional" objects

See Also

fractions for a similar functionality.

Examples

(M <- solve(cbind(1, contr.helmert(5))))
(Mf <- fractional(M))     ## print method right justifies
(Mc <- as.character(Mf))  ## print method left justifies
(Mn <- numerical(Mc))
set.seed(123)
u <- matrix(runif(10), 2, 5)
(uf <- fractional(u))
(us <- fractional(u, sync = TRUE))  ## may look different!
unfractional(uf) - unfractional(us)  ## rational approximation errors

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Method for the group generic function for the elementary mathematical functions

Description

Allows graceful operations with mathematical functions.

Usage

## S3 method for class 'fractional'
Math(x, ...)

Arguments

x	A numerical object flagged as fractional
...	Passed on to further methods (but usually not required)

Value

A numeric object with the results of the computations, but NOT flagged as of class "fractional".

Examples

(M <- fractional(solve(cbind(1, contr.helmert(5)))))
(M0 <- abs(M)*sign(M))  ## fractional flag lost

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Extract the parts of a fractional object

Description

Generic function for extracting numerators with methods for "fractional" or "charFrac" objects

Generic function for extracting denominators with methods for "fractional" or "charFrac" objects

Usage

numerators(x)

## S3 method for class 'charFrac'
numerators(x)

## S3 method for class 'fractional'
numerators(x)

## Default S3 method:
numerators(x)

denominators(x)

## S3 method for class 'charFrac'
denominators(x)

## S3 method for class 'fractional'
denominators(x)

## Default S3 method:
denominators(x)

Arguments

x	An object of class "fractional" or "charFrac"

Value

An integer vector of numerators

An integer vector of denominators

Methods (by class)

• charFrac: numerators method function for "charFrac" objects

• fractional: numerators method function for "fractional" objects

• default: Default numerators method for numeric objects

• charFrac: denominators method function for "charFrac" objects

• fractional: denominators method function for "fractional" objects

• default: Default denominators method for numeric objects

Examples

(pi_approx <- vfractional(base::pi, eps = 0, maxConv = 1:10))
numerators(pi_approx)
denominators(pi_approx)

---

Convert a fractional object to the equivalent numeric object

Description

Convert an object of class "fractional" or "charFrac" to a purely numeric object. This is effectively a method function for the .Primitive generic function as.numeric but written as a separate function for purely technical reasons.

Usage

numerical(vulgar)

## S3 method for class 'fractional'
numerical(vulgar)

## S3 method for class 'charFrac'
numerical(vulgar)

## Default S3 method:
numerical(vulgar)

Arguments

vulgar

character string form of a class 'fractional' object.

Value

A numeric object as represented by its (usually fractional) display.

Methods (by class)

• fractional: Method for "fractional" objects

• charFrac: Method for "charFrac" objects

• default: Default method for numerical generic

Examples

suppressPackageStartupMessages(library(dplyr))
m <- 2*diag(5)
m[abs(row(m) - col(m)) == 1] <- -1
m  ## How much roundoff error does inverting entail?
(mi  <- solve(m) %>% fractional) ## patterned inverse
mi * max(denominators(mi)) ## clearer pattern
m1 <- solve(mi)
range(m1 - m)           ## roundoff still present
m2 <- m1 %>% numerical  ## remove roundoff error - hopefully!
identical(m2, m)        ## no roundoff

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Method for the group generic function for the arithmetic operators

Description

Provides arithmetic operations for numeric objects or of class "fractional".

Usage

## S3 method for class 'fractional'
Ops(e1, e2)

Arguments

e1	A numeric objet, possibly of class "fractional"
e2	A numeric objet, possibly of class "fractional"

Value

The result of the arithmetic operation, flagged as class "fractional"

Examples

(M <- fractional(1:10/7))
M + 1
1 + M + M^2

---

Calculate Rational Approximation Using Continued Fraction Methods

Description

This is a behind-the-scenes function not likely to be used other than internally within the package. It computes the rational approximations for each value in the principal argument.

Usage

rat(x, eps = 1e-06, maxConv = 20L)

.ratr(x, eps = 1e-06, maxConv = 20)

ratr(x, eps = 1e-06, maxConv = 20)

Arguments

x	A numeric vector for which rational approximations are required.
eps	An absolute error tolerance on the approximation
maxConv	An upper limit on the number of convergents that the continued fraction expansion may employ. The fraction is terminated once the desired accuracy is met (or the upper limit is about to be exceeded).

Value

A 3 column matrix giving, respectively, the numerators, denominators and number of convergents needed to achieve the error tolerance, in the columns

Functions

• rat: C++ version of the same function used for speed

• .ratr: Workhorse function for a single value

See Also

rat which has the same functionality, but is coded in C++.

Examples

fractional(base::pi)
ratr(base::pi)

set.seed(123)
(u <- matrix(runif(10), 2, 5))
(ru <- ratr(u, eps = 1.0e-3, maxConv = 6))
(abs_error <- matrix(abs(u - ru[, 1]/ru[, 2]), 2, 5))

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Demote a fractional object back to a numeric one

Description

Given an object of class "fractional" this simple function removes the attributes that signal that it is to be treated as a fractional object, thus returning it to its original numeric status alone

Usage

unfractional(x)

Arguments

x	A "fractional" object

Value

A simple numeric object like x

Examples

(tst <- fractional(matrix(0:9/10, 2, 5)))
(tst <- unfractional(tst))

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Vectorized form for fractional

Description

A function which allows any or all of the first three arguments of fractional to be vectors, with short vectors recycled in the usual way. Note that the return value is a character string vector and may not be used in arithmetic operations

Usage

vfractional(x, eps = 1e-06, maxConv = 20)

Arguments

x	as for fractional
eps	as for fractional, but may be a vector
maxConv	as for fractional but may be a vector

Value

A character string vector of class "charFrac"

Examples

oldOpt <- options(scipen = 15)
pi_approx <- vfractional(base::pi, eps = 0, maxConv = 1:10)
within(data.frame(pi_approx, stringsAsFactors = FALSE), {
  value = numerical(pi_approx)
  error = signif(base::pi - value, 3)
  n = seq_along(value) - 1
})[, c("n", "pi_approx", "value", "error")]
options(oldOpt)

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