Operators for rational functions

Description

Basic arithmetic operators for rational functions.

Usage

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

Arguments

Value

A new object of "rationalfun" class.

Examples

r1 <- rationalfun(c(1, 2), c(1, 2, 1))
r2 <- rationalfun(c(1, 1), c(1, -2, 1))
r1 + r2
r1 * r2
r1^2

Convert object to character

Description

This function converts an object of class "rationalfun" to a character string.

Usage

## S3 method for class 'rationalfun'
as.character(x, ...)

Arguments

Value

A character string representing the rational function.

See Also

as.character.polynomial

Examples

r <- rationalfun(c(1, 1), c(3, 2, 1))
as.character(r)

Convert object to function

Description

This function converts an object of class "rationalfun" to a function.

Usage

## S3 method for class 'rationalfun'
as.function(x, ...)

Arguments

Value

A function with one argument which could be a real or complex vector.

See Also

as.function.polynomial

Examples

r <- rationalfun(c(1, 1), c(3, 2, 1))
r
f <- as.function(r)
f
f(1:10)
f(1:10 + (0+2i))

Differentiate a rational function

Description

Calculate the derivative of a rational function. The returned value result is still an object of class "rationalfun".

Usage

## S3 method for class 'rationalfun'
deriv(expr, ...)

Arguments

Value

An object of class "rationalfun" representing the derivative of the original rational function.

See Also

deriv.polynomial, deriv

Examples

# (x + 1) / (x^2 + x + 1)
r <- rationalfun(c(1, 1), c(1, 1, 1))
deriv(r)

Convert a call to a function

Description

Convert a function call to a function in R. In this package, the function is typically used to convert the result of integral.rationalfun() to a function with one argument.

Usage

int2fun(expr)

Arguments

Value

A function with one argument which could be a real or complex vector.

See Also

integral.polynomial

Examples

x <- rationalfun(c(-6, -1, -8, 15, -1, 8, -9, 2), 
    c(8, 12, 16, 4, 4))
int <- integral(x)
fun <- int2fun(int)
fun(c(0, 1))

Integrate a rational function

Description

Calculate the integral of a rational function. See "Details".

Usage

## S3 method for class 'rationalfun'
integral(expr, ...)

Arguments

Details

The returned value is a function call with argument named "x". That is, the integral is an expression in R with an explicit form, which could be evaluated directly by calling eval(), or indirectly using the int2fun() function.

The algorithm is based on the Hermite-Ostrogradski formula which is discussed in the reference. See the article for more details.

Value

A function call representing the explicit form of the integral.

References

T. N. Subramaniam, and Donald E. G. Malm, How to Integrate Rational Functions, The American Mathematical Monthly, Vol. 99, No.8 (1992), 762-772.

See Also

integral.polynomial

Examples

# (x + 1) / (x^2 + x + 1)
r <- rationalfun(c(1, 1), c(1, 1, 1))
expr <- integral(r)
# Evaluate the call directly
eval(expr, list(x = 2))
# Use int2fun()
f <- int2fun(expr)
f(2)

Evaluate a rational function

Description

Evaluate a rational function at a real or complex vector.

Usage

## S3 method for class 'rationalfun'
predict(object, newdata, ...)

Arguments

Value

A vector of evaluated results.

See Also

predict.polynomial

Examples

r <- rationalfun(c(1, 1), c(3, 2, 1))
predict(r, 1:10)

Print a rational function

Description

Print a rational function in a fraction form.

Usage

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

Arguments

Value

Invisible, the object itself.

See Also

print.polynomial

Examples

r <- rationalfun(c(1, 1), c(3, 2, 1))
print(r)

Construction of rational functions

Description

Construction of rational functions.

Usage

rationalfun(numer = c(0, 1), denom = c(1, 1, 1))

rfun(numer = c(0, 1), denom = c(1, 1, 1))

rationalfun.poly(numer = polynomial(c(0, 1)), denom = polynomial(c(1, 
    1, 1)))

rfun.poly(numer = polynomial(c(0, 1)), denom = polynomial(c(1, 
    1, 1)))

Arguments

Details

A rational function object could be constructed either by calling rationalfun() or by calling rationalfun.poly().

rationalfun() constructs a rational function from the coefficient vectors of the numerator and the denominator. For example, consider a rational function R(x) = P(x) / Q(x) where

P(x) = p_1 + p_2 x + p_3 x^2 + \dots + p_k x^{k-1}

and

Q(x) = q_1 + q_2 x + q_3 x^2 + \dots + q_m x^{m-1}

, you may call rationalfun(p[1:k], q[1:m]) to build the object.

For rationalfun.poly(), it receives two objects of class "polynomial" from the polynom package, representing the polynomials of the numerator and the denominator respectively. Use this function if you already have objects of "polynomial" class, typically by calling polynomial(), poly.calc() or poly.orth().

rfun() and rfun.poly() are aliases of rationalfun() and rationalfun.poly() in order to type fewer letters.

The value returned by rationalfun() and rationalfun.poly() is an object of class "rationalfun". You can coerce the object to a function, by calling as.function.rationalfun(), or to a character string, by calling as.character.rationalfun().

Objects of "ratioanlfun" class support basic operators including "+", "-", "*", "/" and "^". To evaluate a rational function at a given vector, use predict.rationalfun(). To compute the derivative and integral in explicit form, call deriv.rationalfun() and integral.rationalfun() respectively.

Value

An object of class "rationalfun".

See Also

polynomial, poly.calc, poly.orth

Examples

# (x + 1) / (x^2 + 2 * x + 3)
r1 <- rationalfun(c(1, 1), c(3, 2, 1))
print(r1)
# Construct from objects of 'polynomial' class
if (require(polynom)) {
    p1 <- poly.calc(c(1, 2))
    p2 <- polynomial(rep(1, 5))
    r2 <- rfun.poly(p1, p2)
    print(r2)
}

Simplify a rational function

Description

Simplify a rational function by dropping terms whose coefficients are close to zero, and then reducing it to an irreducible form.

Usage

simplify(x, ...)

Arguments

Value

A new object of class "rationalfun"representing the simplified rational function.

Examples

# (x + 1) / (x^2 + 2 * x + 1) ==> 1 / (x + 1)
r <- rationalfun(c(1, 1), c(1, 2, 1))
simplify(r)