Title: Multivariate Polynomials with Symbolic Parameters in their Coefficients
Version: 1.1.0
Description: Introduces the 'symbolicQspray' objects. Such an object represents a multivariate polynomial whose coefficients are fractions of multivariate polynomials with rational coefficients. The package allows arithmetic on such polynomials. It is based on the 'qspray' and 'ratioOfQsprays' packages. Some functions for 'qspray' polynomials have their counterpart for 'symbolicQspray' polynomials. A 'symbolicQspray' polynomial should not be seen as a polynomial on the field of fractions of rational polynomials, but should rather be seen as a polynomial with rational coefficients depending on some parameters, symbolically represented, with a dependence given by fractions of rational polynomials.
License: GPL-3
URL: https://github.com/stla/symbolicQspray
BugReports: https://github.com/stla/symbolicQspray/issues
Depends: qspray (≥ 3.1.0), ratioOfQsprays (≥ 1.1.0)
Imports: gmp, methods, Rcpp, utils
Suggests: testthat (≥ 3.0.0)
LinkingTo: BH, qspray, ratioOfQsprays, Rcpp, RcppCGAL
Config/testthat/edition: 3
Encoding: UTF-8
RoxygenNote: 7.3.1
SystemRequirements: C++17, gmp, mpfr
Collate: 'JacobiPolynomial.R' 'RcppExports.R' 'creation.R' 'evaluation.R' 'internal.R' 'symbolicQspray.R' 'queries.R' 'show.R' 'symmetricPolynomials.R' 'transformation.R'
NeedsCompilation: yes
Packaged: 2024-07-27 17:07:10 UTC; User
Author: Stéphane Laurent [aut, cre]
Maintainer: Stéphane Laurent <laurent_step@outlook.fr>
Repository: CRAN
Date/Publication: 2024-07-28 16:50:02 UTC

Jacobi polynomial

Description

Computes the n-th Jacobi polynomial as a symbolicQspray.

Usage

JacobiPolynomial(n)

Arguments

n

index (corresponding to the degree), a positive integer

Details

The Jacobi polynomials are univariate polynomials whose coefficients depend on two parameters.

Value

A symbolicQspray object representing the n-th Jacobi polynomial.

Examples

JP1 <- JacobiPolynomial(1)
showSymbolicQsprayOption(JP1, "showRatioOfQsprays") <-
  showRatioOfQspraysXYZ(c("alpha", "beta"))
JP1

Polynomial variable

Description

Creates a polynomial variable for a symbolicQspray.

Usage

Qlone(n)

Arguments

n

positive integer, the index of the variable

Value

A symbolicQspray object.

Examples

X <- Qlone(1)
Y <- Qlone(2)
(X + Y)^2
Qlone(0) == 1

The unit 'symbolicQspray' polynomial

Description

Returns the symbolicQspray polynomial identically equal to 1.

Usage

Qone()

Value

A symbolicQspray object.

The null 'symbolicQspray' polynomial

Description

Returns the symbolicQspray polynomial identically equal to 0.

Usage

Qzero()

Value

A symbolicQspray object.

Coercion to a 'symbolicQspray' object

Description

Coercion to a 'symbolicQspray' object

Usage

## S4 method for signature 'character'
as.symbolicQspray(x)

## S4 method for signature 'qspray'
as.symbolicQspray(x)

## S4 method for signature 'ratioOfQsprays'
as.symbolicQspray(x)

## S4 method for signature 'symbolicQspray'
as.symbolicQspray(x)

## S4 method for signature 'numeric'
as.symbolicQspray(x)

## S4 method for signature 'bigz'
as.symbolicQspray(x)

## S4 method for signature 'bigq'
as.symbolicQspray(x)

Arguments

x

a symbolicQspray object or an object for which as.ratioOfQsprays is applicable

Value

A symbolicQspray object.

Examples

as.symbolicQspray(2)
as.symbolicQspray("1/3")

Change of parameters in a 'symbolicQspray' polynomial

Description

Replaces the parameters of a symbolicQspray polynomial (which are qspray objects) with some qspray polynomials. E.g. you have a polynomial with two parameters P_{a,b}(x) and you want the polynomial P_{a+1,b+1}(x) (see example).

Usage

changeParameters(Qspray, newParameters)

Arguments

Qspray

a symbolicQspray polynomial

newParameters

a list containing at least n qspray objects, or objects coercible to qspray objects, where n is the number of parameters in the symbolic polynomial given in the Qspray argument; if this list is named, then its names will be used in the show options of the result

Value

The symbolicQspray polynomial obtained by replacing the parameters of the symbolic polynomial given in the Qspray argument with the polynomials given in the newParameters argument.

See Also

If you want to change the variables of a symbolic qspray, use changeVariables. If you want to assign some values to its parameters, use substituteParameters.

Examples

library(symbolicQspray)
( JP <- JacobiPolynomial(2) ) # a univariate polynomial with two parameters
a1 <- qlone(1)
a2 <- qlone(2)
changeParameters(JP, list(a1, a2)) == JP # should be TRUE
changeParameters(JP, list(a1+1, a2+1))

Change of variables in a 'symbolicQspray' polynomial

Description

Replaces the variables of a symbolicQspray polynomial with some symbolicQspray polynomials. E.g. you have a polynomial P_a(x, y) and you want the polynomial P_a(x+a, y+a) (see example).

Usage

## S4 method for signature 'symbolicQspray,list'
changeVariables(x, listOfQsprays)

Arguments

x

a symbolicQspray polynomial

listOfQsprays

a list containing at least n symbolicQspray objects, or objects coercible to symbolicQspray objects, where n is the number of variables in the polynomial given in the x argument; if this list is named, their its names will be used in the show options of the result

Value

The symbolicQspray polynomial obtained by replacing the variables of the polynomial given in the x argument with the polynomials given in the listOfQsprays argument.

See Also

If you want to change the parameters of a symbolic qspray, use changeParameters. If you want to assign some values to its variables, see substituteVariables.

Examples

library(symbolicQspray)
f <- function(a, X, Y) {
  a^2 / (a + 1) * X^2*Y  +  (3*a - 2) / a * Y^2
}
a <- qlone(1)
X <- Qlone(1)
Y <- Qlone(2)
Qspray <- f(a, X, Y)
U <- X + a
V <- Y + a
changeVariables(Qspray, list(U, V)) == f(a, U, V) # should be TRUE

Compact symmetric qspray

Description

Prints a symmetric symbolicQspray polynomial as a linear combination of the monomial symmetric polynomials.

Usage

## S4 method for signature 'symbolicQspray,logical'
compactSymmetricQspray(qspray, check)

## S4 method for signature 'symbolicQspray,missing'
compactSymmetricQspray(qspray, check)

Arguments

qspray

a symbolicQspray object which should correspond to a symmetric polynomial

check

Boolean, whether to check the symmetry

Value

A character string.

See Also

MSPcombination

Partial differentiation

Description

Partial differentiation of a symbolicQspray polynomial.

Usage

dSymbolicQspray(Qspray, orders)

Arguments

Qspray

object of class symbolicQspray

orders

integer vector, the orders of the differentiation; e.g. c(2, 0, 1) means that you differentiate two times with respect to x, you do not differentiate with respect to y, and you differentiate one time with respect to z

Value

A symbolicQspray object.

Partial derivative

Description

Partial derivative of a symbolicQspray polynomial.

Usage

derivSymbolicQspray(Qspray, i, derivative = 1)

Arguments

Qspray

object of class symbolicQspray

i

integer, the dimension to differentiate with respect to, e.g. 2 to differentiate w.r.t. y

derivative

positive integer, how many times to differentiate

Value

A symbolicQspray object.

Evaluation of a 'symbolicQspray' polynomial

Description

Evaluates a symbolicQspray polynomial by substituting some values to the parameters (same as substituteParameters) or to the variables (same as substituteVariables) or both.

Usage

evalSymbolicQspray(Qspray, a = NULL, X = NULL)

Arguments

Qspray

a symbolicQspray object

a

vector of values to be substituted to the parameters; these values must be coercible to bigq numbers

X

vector of values to be substituted to the variables; these values must be coercible to bigq numbers

Value

If both a and X are NULL, this returns the input symbolicQspray object; otherwise, if a is not NULL, this returns a qspray object, and if X is not NULL, this returns a ratioOfQsprays object.

Examples

library(symbolicQspray)
a1 <- qlone(1); a2 <- qlone(2)
X1 <- Qlone(1); X2 <- Qlone(2); X3 <- Qlone(3)
( Qspray <- (a1 + 2)*X1^2*X2 + (a2/(a1^2+a2))*X1*X2*X3 )
a <- c(2, 3)
X <- c(4, 3, 2)
( qspray <- evalSymbolicQspray(Qspray, a = a) )
( rOQ <- evalSymbolicQspray(Qspray, X = X) )
evalSymbolicQspray(Qspray, a = a, X = X)
evalQspray(qspray, X)
evalRatioOfQsprays(rOQ, a)

Get a coefficient in a 'symbolicQspray' polynomial

Description

Get the coefficient of the term with the given monomial.

Usage

## S4 method for signature 'symbolicQspray,numeric'
getCoefficient(qspray, exponents)

Arguments

qspray

a symbolicQspray object

exponents

a vector of exponents, thereby defining a monomial; trailing zeros are ignored

Value

The coefficient, ratioOfQsprays object.

Examples

a1 <- qlone(1); a2 <- qlone(2)
X <- Qlone(1); Y <- Qlone(2)
p <- 2*(a1/a2)*X^2 + (a1/(a1+a2))*Y + a2^2/a1
getCoefficient(p, 2)       # coefficient of X^2
getCoefficient(p, c(2, 0)) # same as getCoefficient(p, 2)
getCoefficient(p, c(0, 1)) # coefficient of Y (because Y=X^0.Y^1)
getCoefficient(p, 0)       # the constant term
getCoefficient(p, 3)       # coefficient of X^3

Get the constant term of a 'symbolicQspray' polynomial

Description

Get the constant term of a symbolicQspray polynomial.

Usage

## S4 method for signature 'symbolicQspray'
getConstantTerm(qspray)

Arguments

qspray

a symbolicQspray object

Value

A ratioOfQsprays object.

Whether the coefficients of a 'symbolicQspray' polynomially depend on its parameters

Description

Checks whether the dependence of the coefficients of a symbolicQspray polynomial on their parameters is polynomial.

Usage

hasPolynomialCoefficientsOnly(Qspray)

Arguments

Qspray

a symbolicQspray object

Value

A Boolean value. The coefficients of a symbolicQspray polynomial always are fractions of polynomials. This function checks whether they are polynomials.

Examples

JP <- JacobiPolynomial(4)
hasPolynomialCoefficientsOnly(JP)

Variables involved in a 'symbolicQspray' polynomial

Description

Variables involved in a symbolicQspray object.

Usage

## S4 method for signature 'symbolicQspray'
involvedVariables(x)

Arguments

x

a symbolicQspray object

Value

A vector of integers. Each integer represents the index of a variable involved in x.

See Also

numberOfVariables.

Examples

a1 <- qlone(1); a2 <- qlone(2)
X <- Qlone(1); Z <- Qlone(3)
Qspray <- (a1/a2)*X^2 + (a1/(a1+a2))*X*Z + a2^2/a1
involvedVariables(Qspray) # should be c(1L, 3L)

Whether a 'symbolicQspray' polynomial is constant

Description

Checks whether a symbolicQspray object defines a constant polynomial.

Usage

## S4 method for signature 'symbolicQspray'
isConstant(x)

Arguments

x

a symbolicQspray object

Value

A Boolean value.

Whether a 'symbolicQspray' polynomial is the unit polynomial

Description

Checks whether a symbolicQspray object defines the unit polynomial.

Usage

## S4 method for signature 'symbolicQspray'
isQone(qspray)

Arguments

qspray

a symbolicQspray object

Value

A Boolean value.

Whether a 'symbolicQspray' polynomial is null

Description

Checks whether a symbolicQspray object defines the zero polynomial.

Usage

## S4 method for signature 'symbolicQspray'
isQzero(qspray)

Arguments

qspray

a symbolicQspray object

Value

A Boolean value.

Whether a 'symbolicQspray' polynomial is univariate

Description

Checks whether a symbolicQspray object defines a univariate polynomial.

Usage

## S4 method for signature 'symbolicQspray'
isUnivariate(x)

Arguments

x

a symbolicQspray object

Value

A Boolean value.

Note

It is considered that a constant symbolicQspray is univariate.

Number of parameters

Description

Number of parameters of a symbolicQspray polynomial, i.e. the number of variables occurring in its coefficients.

Usage

numberOfParameters(Qspray)

Arguments

Qspray

a symbolicQspray object

Value

An integer, the number of parameters involved in (the coefficients of) Qspray.

Examples

JP <- JacobiPolynomial(4) # Jacobi polynomials have two parameters
numberOfParameters(JP)

Number of terms in a 'symbolicQspray' polynomial

Description

Number of terms in the polynomial defined by a symbolicQspray object.

Usage

## S4 method for signature 'symbolicQspray'
numberOfTerms(qspray)

Arguments

qspray

a symbolicQspray object

Value

An integer.

Number of variables of a 'symbolicQspray' polynomial

Description

Number of variables involved in a symbolicQspray object.

Usage

## S4 method for signature 'symbolicQspray'
numberOfVariables(x)

Arguments

x

a symbolicQspray object

Value

An integer.

Note

The number of variables in the symbolicQspray object Qlone(d) is d, not 1.

See Also

involvedVariables.

Permute variables

Description

Permute the variables of a symbolicQspray polynomial.

Usage

## S4 method for signature 'symbolicQspray,numeric'
permuteVariables(x, permutation)

Arguments

x

a symbolicQspray object

permutation

a permutation

Value

A symbolicQspray object.

Examples

f <- function(a1, a2, X, Y, Z) {
  (a1^2 + 5*a2) / (a1 + 1) * X^2*Y  +  (3*a1 - a2) / a2 * Y^3
}
a1 <- qlone(1)
a2 <- qlone(2)
X <- Qlone(1)
Y <- Qlone(2)
Z <- Qlone(3)
Qspray <- f(a1, a2, X, Y, Z)
perm <- c(3, 1, 2)
permuteVariables(Qspray, perm) == f(a1, a2, Z, X, Y) # should be TRUE

Random 'symbolicQspray'

Description

Generates a random symbolicQspray object.

Usage

rSymbolicQspray()

Value

A symbolicQspray object.

Print a 'symbolicQspray' object

Description

Prints a symbolicQspray object given a function to print a ratioOfQsprays object.

Usage

showSymbolicQspray(
  showRatioOfQsprays,
  showMonomial,
  lbrace = "{ ",
  rbrace = " }",
  addition = "  +  ",
  multiplication = " * "
)

Arguments

showRatioOfQsprays

a function which prints a ratioOfQsprays object

showMonomial

a function which prints a monomial, such as showMonomialXYZ() (and not showMonomialXYZ!)

lbrace, rbrace

used to enclose the coefficients

addition

used to separate the terms

multiplication

used to separate the coefficient and the monomial within a term

Value

A function which prints a symbolicQspray object.

Note

The function returned by this function is appropriate for usage in showSymbolicQsprayOption<- as the option "showSymbolicQspray" but in general we would rather use showSymbolicQsprayX1X2X3 or showSymbolicQsprayXYZ, or rather set the options "a", "X" and "quotientBar".

See Also

showSymbolicQsprayX1X2X3, showSymbolicQsprayXYZ.

Examples

set.seed(421)
( Qspray <- rSymbolicQspray() )
showRatioOfQsprays <-
  showRatioOfQspraysXYZ(c("a", "b", "c"), quotientBar = " / ")
showMonomial <- showMonomialX1X2X3("X")
f <- showSymbolicQspray(showRatioOfQsprays, showMonomial, "{{{", "}}}")
f(Qspray)
# setting a show option:
showSymbolicQsprayOption(Qspray, "showSymbolicQspray") <- f
Qspray
# the show options are preserved by certain operations, e.g.:
2*Qspray

Print a 'symbolicQspray' object

Description

Prints a symbolicQspray object.

Usage

showSymbolicQsprayABCXYZ(
  params,
  vars = c("X", "Y", "Z"),
  quotientBar = " %//% ",
  ...
)

Arguments

params

vector of strings, usually some letters, to denote the parameters of the polynomial

vars

a vector of strings, usually some letters, to denote the variables of the polynomial

quotientBar

a string for the quotient bar between the numerator and the denominator of a ratioOfQsprays object, including surrounding spaces, e.g. " / "

...

arguments other than showRatioOfQsprays and showMonomial passed to showSymbolicQspray

Value

A function which prints symbolicQspray objects.

Note

This function is built by applying showSymbolicQspray to showRatioOfQspraysXYZ(params) and showMonomialXYZ(vars).

Examples

set.seed(421)
( Qspray <- rSymbolicQspray() )
showSymbolicQsprayABCXYZ(c("a", "b", "c"), c("U", "V"))(Qspray)

Set a show option to a 'symbolicQspray' object

Description

Set show option to a symbolicQspray object

Usage

showSymbolicQsprayOption(x, which) <- value

Arguments

x

a symbolicQspray object

which

which option to set; this can be "a", "X", "quotientBar", "showMonomial", "showRatioOfQsprays" or "showSymbolicQspray"

value

the value for the option

Value

This returns the updated symbolicQspray.

Examples

set.seed(421)
Qspray <- rSymbolicQspray()
showSymbolicQsprayOption(Qspray, "a") <- "x"
showSymbolicQsprayOption(Qspray, "X") <- "A"
showSymbolicQsprayOption(Qspray, "quotientBar") <- " / "
Qspray
showSymbolicQsprayOption(Qspray, "showRatioOfQsprays") <-
  showRatioOfQspraysXYZ()
Qspray

Print a 'symbolicQspray' object

Description

Prints a symbolicQspray object.

Usage

showSymbolicQsprayX1X2X3(a = "a", X = "X", quotientBar = " %//% ", ...)

Arguments

a

a string, usually a letter, to denote the non-indexed variables of the ratioOfQsprays coefficients

X

a string, usually a letter, to denote the non-indexed variables

quotientBar

a string for the quotient bar between the numerator and the denominator of a ratioOfQsprays object, including surrounding spaces, e.g. "/"

...

arguments other than showRatioOfQsprays and showMonomial passed to showSymbolicQspray

Value

A function which prints symbolicQspray objects.

Note

This function is built by applying showSymbolicQspray to showRatioOfQspraysX1X2X3(a) and showMonomialX1X2X3(X).

Examples

set.seed(421)
Qspray <- rSymbolicQspray()
showSymbolicQsprayX1X2X3(quotientBar = " / ")(Qspray)

Print a 'symbolicQspray' object

Description

Prints a symbolicQspray object.

Usage

showSymbolicQsprayXYZ(
  a = "a",
  letters = c("X", "Y", "Z"),
  quotientBar = " %//% ",
  ...
)

Arguments

a

a string, usually a letter, to denote the non-indexed variables of the ratioOfQsprays coefficients

letters

a vector of strings, usually some letters, to denote the variables of the polynomial

quotientBar

a string for the quotient bar between the numerator and the denominator of a ratioOfQsprays object, including surrounding spaces, e.g. " / "

...

arguments other than showRatioOfQsprays and showMonomial passed to showSymbolicQspray

Value

A function which prints symbolicQspray objects.

Note

This function is built by applying showSymbolicQspray to showRatioOfQspraysX1X2X3(a) and showMonomialXYZ(letters).

Examples

set.seed(421)
Qspray <- rSymbolicQspray()
showSymbolicQsprayX1X2X3(quotientBar = " / ")(Qspray)

Assign values to the parameters of a 'symbolicQspray'

Description

Substitutes some values to the parameters of a symbolicQspray polynomial.

Usage

substituteParameters(Qspray, values)

Arguments

Qspray

a symbolicQspray object

values

vector of values to be substituted to the parameters; these values must be coercible to bigq numbers

Value

A qspray object.

See Also

Use changeParameters to apply a transformation of the parameters. Use substituteVariables to substitute some values to the variables.

Examples

library(symbolicQspray)
f <- function(a1, a2, X, Y) {
  (a1 + 2)*X^2*Y + (a2/(a1^2+a2))*X*Y
}
Qspray <- f(qlone(1), qlone(2), Qlone(1), Qlone(2))
a <- c(2, "2/3")
( qspray <- substituteParameters(Qspray, values = a) )
a <- gmp::as.bigq(a)
qspray == f(a[1], a[2], qlone(1), qlone(2)) ## should be TRUE

Assign values to the variables of a 'symbolicQspray'

Description

Substitutes some values to the variables of a symbolicQspray polynomial.

Usage

substituteVariables(Qspray, values)

Arguments

Qspray

a symbolicQspray object

values

vector of values to be substituted to the variables; these values must be coercible to bigq numbers

Value

A ratioOfQsprays object.

See Also

Use changeVariables to apply a transformation of the variables. Use substituteParameters to substitute some values to the parameters.

Examples

library(symbolicQspray)
f <- function(a1, a2, X, Y) {
  (a1 + 2)*X^2*Y + (a2/(a1^2+a2))*X*Y
}
a1 <- qlone(1); a2 <- qlone(2)
Qspray <- f(a1, a2, Qlone(1), Qlone(2))
values <- c(3, "2/3")
( rOQ <- substituteVariables(Qspray, values) )
values <- gmp::as.bigq(values)
rOQ == f(a1, a2, values[1], values[2]) ## should be TRUE

Swap variables

Description

Swap two variables of a symbolicQspray.

Usage

## S4 method for signature 'symbolicQspray,numeric,numeric'
swapVariables(x, i, j)

Arguments

x

a symbolicQspray object

i, j

indices of the variables to be swapped

Value

A symbolicQspray object.

Examples

library(symbolicQspray)
f <- function(a1, a2, X, Y, Z) {
  (a1^2 + 5*a2) / (a1 + 1) * X^2*Y  +  (3*a1 - a2) / a2 * Y^3
}
a1 <- qlone(1)
a2 <- qlone(2)
X <- Qlone(1)
Y <- Qlone(2)
Z <- Qlone(3)
Qspray <- f(a1, a2, X, Y, Z)
swapVariables(Qspray, 2, 3) == f(a1, a2, X, Z, Y) # should be TRUE

Unary operators for 'symbolicQspray objects

Description

Unary operators for symbolicQspray objects.

Usage

## S4 method for signature 'symbolicQspray,missing'
e1 + e2

## S4 method for signature 'symbolicQspray,missing'
e1 - e2

Arguments

e1

object of class symbolicQspray

e2

nothing

Value

A symbolicQspray object.

(internal) Make a 'symbolicQspray' object from a list

Description

This function is for internal usage. It is exported because it is also used for internal usage in others packages.

Usage

symbolicQspray_from_list(x)

Arguments

x

list returned by the Rcpp function returnSymbolicQspray

Value

A symbolicQspray object.