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

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

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

Value

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

Examples

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

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

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

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

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

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

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

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))

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

Value

A simple numeric object like x

Examples

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

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

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)