Package 'qspray' reference manual

Title:	Multivariate Polynomials with Rational Coefficients
Description:	Symbolic calculation and evaluation of multivariate polynomials with rational coefficients. This package is strongly inspired by the 'spray' package. It provides a function to compute Gröbner bases (reference <doi:10.1007/978-3-319-16721-3>). It also includes some features for symmetric polynomials, such as the Hall inner product. The header file of the C++ code can be used by other packages. It provides the templated class 'Qspray' that can be used to represent and to deal with multivariate polynomials with another type of coefficients.
Authors:	Stéphane Laurent [aut, cre], Robin Hankin [ctb, cph] (author of the 'spray' package, which strongly inspired this package)
Maintainer:	Stéphane Laurent <[email protected]>
License:	GPL-3
Version:	3.1.0
Built:	2024-11-25 06:55:15 UTC
Source:	CRAN

Multivariate polynomial as function

Description

Coerces a qspray polynomial into a function.

Usage

## S3 method for class 'qspray'
as.function(x, N = FALSE, ...)
## S3 method for class 'qspray'
as.function(x, N = FALSE, ...)

Arguments

`x`	object of class `qspray`
`N`	Boolean, whether the function must numerically approximate the result
`...`	ignored

Value

A function having the same variables as the polynomial. If N=FALSE, it returns a string. If N=TRUE, it returns a number if the result does not contain any variable, otherwise it returns a R expression.

Examples

library(qspray)
P <- (qlone(1) + "1/2"*qlone(2))^2 + 5
f <- as.function(P)
g <- as.function(P, N = TRUE)
f(2, "3/7")
g(2, "3/7")
f("x", "y")
g("x", "y")
# the evaluation is performed by (R)yacas and complex numbers are
# allowed; the imaginary unit is denoted by `I`
f("2 + 2*I", "Sqrt(2)")
g("2 + 2*I", "Sqrt(2)")
library(qspray)
P <- (qlone(1) + "1/2"*qlone(2))^2 + 5
f <- as.function(P)
g <- as.function(P, N = TRUE)
f(2, "3/7")
g(2, "3/7")
f("x", "y")
g("x", "y")
# the evaluation is performed by (R)yacas and complex numbers are
# allowed; the imaginary unit is denoted by `I`
f("2 + 2*I", "Sqrt(2)")
g("2 + 2*I", "Sqrt(2)")

Coercion to a 'qspray' object

Description

Coercion to a 'qspray' object

Usage

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

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

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

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

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

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

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

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

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

Arguments

`x`	a `qspray` object or an object yielding a quoted integer or a quoted fraction after an application of `as.character`, e.g. a `bigq` number

Value

A qspray object.

Examples

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

Change of variables in a 'qspray' polynomial

Description

Replaces the variables of a qspray polynomial with some qspray polynomials. E.g. you have a polynomial $P(x, y)$ and you want the polynomial $P(x^2, x+y+1)$ . This is an alias of composeQspray.

Usage

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

Arguments

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

Value

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

Examples

library(qspray)
f <- function(x, y) x*y/2 + 4*y
x <- qlone(1)
y <- qlone(2)
P <- f(x, y)
X <- x^2
Y <- x + y + 1
changeVariables(P, list(X, Y)) == f(X, Y) # should be TRUE
library(qspray)
f <- function(x, y) x*y/2 + 4*y
x <- qlone(1)
y <- qlone(2)
P <- f(x, y)
X <- x^2
Y <- x + y + 1
changeVariables(P, list(X, Y)) == f(X, Y) # should be TRUE

Characteristic polynomial

Description

Characteristic polynomial of a matrix.

Usage

characteristicPolynomial(A)
characteristicPolynomial(A)

Arguments

`A`	a square matrix with numeric, character, or `bigq` entries

Value

A univariate qspray polynomial.

Examples

set.seed(666)
A <- matrix(rpois(9L, 10), nrow = 3, ncol = 3)
( P <- characteristicPolynomial(A) )
# check the roots are the eigen values:
f <- as.function(P, N = TRUE)
sapply(eigen(A)$values, f) # approx c(0, 0, 0)
set.seed(666)
A <- matrix(rpois(9L, 10), nrow = 3, ncol = 3)
( P <- characteristicPolynomial(A) )
# check the roots are the eigen values:
f <- as.function(P, N = TRUE)
sapply(eigen(A)$values, f) # approx c(0, 0, 0)

Whether two 'qspray' polynomials are collinear

Description

Checks whether the polynomials represented by two qspray objects are collinear, that is, whether they are equal up to a scalar factor.

Usage

collinearQsprays(qspray1, qspray2)
collinearQsprays(qspray1, qspray2)

Arguments

qspray1, qspray2

two qspray objects

Value

A Boolean value.

Examples

library(qspray)
qspray1 <- qsprayMaker(string = "1/2 x^(1, 1) + 4 x^(0, 2) + 5")
qspray2 <- "4/7" * qspray1
collinearQsprays(qspray1, qspray2)
library(qspray)
qspray1 <- qsprayMaker(string = "1/2 x^(1, 1) + 4 x^(0, 2) + 5")
qspray2 <- "4/7" * qspray1
collinearQsprays(qspray1, qspray2)

Compact symmetric qspray

Description

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

Usage

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

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

## S4 method for signature 'qspray,ANY'
compactSymmetricQspray(qspray)

Arguments

`qspray`	a `qspray` object, which should correspond to a symmetric polynomial
`check`	Boolean, whether to check the symmetry (default `TRUE`)

Value

A character string.

Examples

library(qspray)
( qspray <- PSFpoly(4, c(3, 1)) - ESFpoly(4, c(2, 2)) + 4L )
compactSymmetricQspray(qspray, check = TRUE)
library(qspray)
( qspray <- PSFpoly(4, c(3, 1)) - ESFpoly(4, c(2, 2)) + 4L )
compactSymmetricQspray(qspray, check = TRUE)

Compose 'qspray' polynomials

Description

Substitutes the variables of a qspray polynomial with some qspray polynomials. E.g. you have a polynomial $P(x, y)$ and you want the polynomial $P(x^2, x+y+1)$ (see example).

Usage

composeQspray(qspray, listOfQsprays)
composeQspray(qspray, listOfQsprays)

Arguments

`qspray`	a `qspray` polynomial
`listOfQsprays`	a list containing at least `n` `qspray` polynomials where `n` is the number of variables of the polynomial given in the `qspray` argument

Value

The qspray polynomial obtained by composing the polynomial given in the qspray argument with the polynomials given in the listOfQsprays argument.

Examples

library(qspray)
x <- qlone(1)
y <- qlone(2)
P <- x*y/2 + 4*y
X <- x^2
Y <- x + y + 1
composeQspray(P, list(X, Y)) # this is P(x^2, x+y+1)
library(qspray)
x <- qlone(1)
y <- qlone(2)
P <- x*y/2 + 4*y
X <- x^2
Y <- x + y + 1
composeQspray(P, list(X, Y)) # this is P(x^2, x+y+1)

Complete homogeneous symmetric function

Description

Returns a complete homogeneous symmetric function as a qspray polynomial.

Usage

CSHFpoly(m, lambda)
CSHFpoly(m, lambda)

Arguments

`m`	integer, the number of variables
`lambda`	an integer partition, given as a vector of decreasing positive integers

Value

A qspray object.

Examples

library(qspray)
CSHFpoly(3, c(3, 1))
library(qspray)
CSHFpoly(3, c(3, 1))

Partial derivative

Description

Partial derivative of a qspray polynomial.

Usage

derivQspray(qspray, i, derivative = 1)
derivQspray(qspray, i, derivative = 1)

Arguments

`qspray`	object of class `qspray`
`i`	integer, the dimension to differentiate with respect to, e.g. `1` to differentiate with respect to $x$
`derivative`	integer, how many times to differentiate

Value

A qspray object.

Examples

library(qspray)
x <- qlone(1)
y <- qlone(2)
qspray <- 2*x  + 3*x*y
derivQspray(qspray, 2) # derivative w.r.t. y
library(qspray)
x <- qlone(1)
y <- qlone(2)
qspray <- 2*x  + 3*x*y
derivQspray(qspray, 2) # derivative w.r.t. y

Partial differentiation

Description

Partial differentiation of a qspray polynomial.

Usage

dQspray(qspray, orders)
dQspray(qspray, orders)

Arguments

`qspray`	object of class `qspray`
`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 qspray object.

Examples

library(qspray)
x <- qlone(1)
y <- qlone(2)
qspray <- x + 2*y  + 3*x*y
dQspray(qspray, c(1, 1))
derivQspray(derivQspray(qspray, 1), 2)
library(qspray)
x <- qlone(1)
y <- qlone(2)
qspray <- x + 2*y  + 3*x*y
dQspray(qspray, c(1, 1))
derivQspray(derivQspray(qspray, 1), 2)

Elementary symmetric polynomial

Description

Returns an elementary symmetric function as a polynomial.

Usage

ESFpoly(m, lambda)
ESFpoly(m, lambda)

Arguments

`m`	integer, the number of variables
`lambda`	an integer partition, given as a vector of decreasing positive integers

Value

A qspray object.

Examples

library(qspray)
ESFpoly(3, c(3, 1))
library(qspray)
ESFpoly(3, c(3, 1))

Evaluate a 'qspray' object

Description

Evaluation of the multivariate polynomial represented by a qspray object.

Usage

evalQspray(qspray, values_re, values_im = NULL)
evalQspray(qspray, values_re, values_im = NULL)

Arguments

`qspray`	a `qspray` object
`values_re`	vector of the real parts of the values; each element of `as.character(values_re)` must be a quoted integer or a quoted fraction
`values_im`	vector of the imaginary parts of the values; each element of `as.character(values_im)` must be a quoted integer or a quoted fraction

Value

A bigq number if values_im=NULL, a pair of bigq numbers otherwise: the real part and the imaginary part of the result.

Examples

x <- qlone(1); y <- qlone(2)
P <- 2*x + "1/2"*y
evalQspray(P, c("2", "5/2", "99999")) # "99999" will be ignored
x <- qlone(1); y <- qlone(2)
P <- 2*x + "1/2"*y
evalQspray(P, c("2", "5/2", "99999")) # "99999" will be ignored

Get a coefficient in a 'qspray' polynomial

Description

Get the coefficient of the term with the given monomial.

Usage

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

Arguments

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

Value

The coefficient as a bigq number.

Examples

library(qspray)
x <- qlone(1)
y <- qlone(2)
p <- 4*x^2 + 3*y - 5
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, integer(0)) # the constant term 
getCoefficient(p, 3)          # there's no x^3
library(qspray)
x <- qlone(1)
y <- qlone(2)
p <- 4*x^2 + 3*y - 5
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, integer(0)) # the constant term 
getCoefficient(p, 3)          # there's no x^3

Get the constant term of a 'qspray' polynomial

Description

Get the constant term of a qspray polynomial.

Usage

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

Arguments

qspray

a qspray object

Value

A bigq number.

Gröbner basis

Description

Returns a Gröbner basis following Buchberger's algorithm using the lexicographical order.

Usage

groebner(G, minimal = TRUE, reduced = TRUE)
groebner(G, minimal = TRUE, reduced = TRUE)

Arguments

`G`	a list of `qspray` polynomials, the generators of the ideal
`minimal`	Boolean, whether to return a minimal basis
`reduced`	Boolean, whether to return the reduced basis

Value

A Gröbner basis of the ideal generated by G, given as a list of qspray polynomials.

References

Cox, Little & O'Shea. Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. Fourth edition, Springer 2015.

Examples

library(qspray)
f <- qsprayMaker(string = "x^(3) - 2 x^(1,1)")
g <- qsprayMaker(string = "x^(2,1) - 2 x^(0,2) + x^(1)")
groebner(list(f, g), FALSE, FALSE)
# other example
x <- qlone(1); y <- qlone(2); z <- qlone(3)
f1 <- x^2 + y + z^2 - 1
f2 <- x^2 + y + z - 1
f3 <- x + y^2 + z - 1
groebner(list(f1, f2, f3))
library(qspray)
f <- qsprayMaker(string = "x^(3) - 2 x^(1,1)")
g <- qsprayMaker(string = "x^(2,1) - 2 x^(0,2) + x^(1)")
groebner(list(f, g), FALSE, FALSE)
# other example
x <- qlone(1); y <- qlone(2); z <- qlone(3)
f1 <- x^2 + y + z^2 - 1
f2 <- x^2 + y + z - 1
f3 <- x + y^2 + z - 1
groebner(list(f1, f2, f3))

Hall inner product

Description

Hall inner product of two symmetric polynomials. It has a parameter alpha and the standard Hall inner product is the case when alpha=1. It is possible to get the Hall inner product with a symbolic alpha parameter.

Usage

HallInnerProduct(qspray1, qspray2, alpha = 1)
HallInnerProduct(qspray1, qspray2, alpha = 1)

Arguments

`qspray1`, `qspray2`	two symmetric `qspray` polynomials
`alpha`	parameter equal to `1` for the usual Hall inner product, otherwise this is the "Jack parameter"; it must be either a single value coercible to a `bigq` number, e.g. `"2/5"`, or `NULL` to get the Hall product with a symbolic `alpha`

Value

A bigq number if alpha is not NULL, otherwise a univariate qspray polynomial.

Implicitization with Gröbner bases

Description

Implicitization of a system of parametric equations (see example).

Usage

implicitization(nvariables, parameters, equations, relations)
implicitization(nvariables, parameters, equations, relations)

Arguments

`nvariables`	number of variables
`parameters`	character vector of the names of the parameters, or `NULL` if there's no parameter
`equations`	named list of `qspray` polynomials representing the parametric equations
`relations`	list of `qspray` polynomials representing the relations between the variables and between the parameters, or `NULL` if there is none

Value

A list of qspray polynomials.

Examples

library(qspray)
# ellipse example ####
# variables 
cost <- qlone(1)
sint <- qlone(2)
nvariables <- 2
# parameters
a <- qlone(3)
b <- qlone(4)
parameters <- c("a", "b")
#
equations <- list(
  "x" = a * cost,
  "y" = b * sint
)
relations <- list(
  cost^2 + sint^2 - 1
)
# 
implicitization(nvariables, parameters, equations, relations)
library(qspray)
# ellipse example ####
# variables 
cost <- qlone(1)
sint <- qlone(2)
nvariables <- 2
# parameters
a <- qlone(3)
b <- qlone(4)
parameters <- c("a", "b")
#
equations <- list(
  "x" = a * cost,
  "y" = b * sint
)
relations <- list(
  cost^2 + sint^2 - 1
)
# 
implicitization(nvariables, parameters, equations, relations)

Integral of a multivariate polynomial over a simplex

Description

Returns the exact value of the integral of a multivariate polynomial with rational coefficients over a simplex whose vertices have rational coordinates.

Usage

integratePolynomialOnSimplex(P, S)
integratePolynomialOnSimplex(P, S)

Arguments

`P`	a `qspray` object
`S`	the simplex, a `(n+1)xn` matrix such that each entry of the matrix `as.character(S)` is a quoted integer or a quoted fraction

Value

A bigq number, the exact value of the integral.

Examples

library(qspray)
x <- qlone(1); y <- qlone(2)
P <- x/2 + x*y
S <- rbind(c("0", "0"), c("1", "0"), c("1", "1")) # a triangle
integratePolynomialOnSimplex(P, S)
library(qspray)
x <- qlone(1); y <- qlone(2)
P <- x/2 + x*y
S <- rbind(c("0", "0"), c("1", "0"), c("1", "1")) # a triangle
integratePolynomialOnSimplex(P, S)

Variables involved in a 'qspray' polynomial

Description

Variables involved in a qspray object.

Usage

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

Arguments

`x`	a `qspray` object

Value

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

Examples

x <- qlone(1); z <- qlone(3)
involvedVariables(x^2 + x*z + 1) # should be c(1L, 3L)
x <- qlone(1); z <- qlone(3)
involvedVariables(x^2 + x*z + 1) # should be c(1L, 3L)

Whether a 'qspray' polynomial is constant

Description

Checks whether a qspray object defines a constant polynomial.

Usage

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

Arguments

`x`	a `qspray` object

Value

A Boolean value.

Whether a 'qspray' polynomial is homogeneous

Description

Checks whether the polynomial defined by a qspray object is homogeneous, and also returns the degree if this is true.

Usage

isHomogeneousQspray(qspray)
isHomogeneousQspray(qspray)

Arguments

qspray

a qspray object

Value

A Boolean value indicating whether the polynomial defined by qspray is homogeneous. Moreover, if it is homogeneous, the degree is given in the attribute "degree" of the output.

Examples

lambda <- c(3, 2, 1)
p <- PSFpoly(4, lambda)
( homogeneous <- isHomogeneousQspray(p) ) # should be TRUE
attr(homogeneous, "degree") == sum(lambda) # should be TRUE
lambda <- c(3, 2, 1)
p <- PSFpoly(4, lambda)
( homogeneous <- isHomogeneousQspray(p) ) # should be TRUE
attr(homogeneous, "degree") == sum(lambda) # should be TRUE

Whether a 'qspray' is a polynomial of some given 'qsprays'

Description

Checks whether a qspray polynomial can be written as a polynomial of some given qspray polynomials. If TRUE, this polynomial is returned.

Usage

isPolynomialOf(qspray, qsprays)
isPolynomialOf(qspray, qsprays)

Arguments

`qspray`	a `qspray` object
`qsprays`	a list of `qspray` objects

Value

A Boolean value indicating whether the polynomial defined by qspray can be written as a polynomial of the polynomials defined by the qspray objects given in the qsprays list. If this is TRUE, this polynomial is returned as an attribute named "polynomial".

Examples

library(qspray)
x <- qlone(1); y <- qlone(2); z <- qlone(3)
q1 <- x + y
q2 <- x*z^2 + 4
qspray <- q1^2*q2 + 2*q1 + 3
( check <- isPolynomialOf(qspray, list(q1, q2)) )
POLYNOMIAL <- attr(check, "polynomial")
changeVariables(POLYNOMIAL, list(q1, q2)) == qspray # should be TRUE
library(qspray)
x <- qlone(1); y <- qlone(2); z <- qlone(3)
q1 <- x + y
q2 <- x*z^2 + 4
qspray <- q1^2*q2 + 2*q1 + 3
( check <- isPolynomialOf(qspray, list(q1, q2)) )
POLYNOMIAL <- attr(check, "polynomial")
changeVariables(POLYNOMIAL, list(q1, q2)) == qspray # should be TRUE

Whether a 'qspray' polynomial is the unit polynomial

Description

Checks whether a qspray object defines the unit polynomial.

Usage

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

Arguments

qspray

a qspray object

Value

A Boolean value.

Whether a 'qspray' polynomial is null

Description

Checks whether a qspray object defines the zero polynomial.

Usage

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

Arguments

qspray

a qspray object

Value

A Boolean value.

Check symmetry of a polynomial

Description

Check whether a qspray polynomial is symmetric.

Usage

isSymmetricQspray(qspray)
isSymmetricQspray(qspray)

Arguments

qspray

a qspray polynomial

Value

A Boolean value indicating whether the polynomial defined by qspray is symmetric.

Examples

e1 <- ESFpoly(3, 1)
e2 <- ESFpoly(3, 2)
e3 <- ESFpoly(3, 3)
q <- e1 + 2*e2 + 3*e3 + 4*e1*e3
isSymmetricQspray(q)
e1 <- ESFpoly(3, 1)
e2 <- ESFpoly(3, 2)
e3 <- ESFpoly(3, 3)
q <- e1 + 2*e2 + 3*e3 + 4*e1*e3
isSymmetricQspray(q)

Whether a 'qspray' is univariate

Description

Checks whether a qspray object defines a univariate polynomial.

Usage

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

Arguments

`x`	a `qspray` object

Value

A Boolean value.

Note

It is considered that a constant qspray is univariate, and that the qspray object y^2+1 where y=qlone(2) is not univariate, although only one variable is present (see the note in the documentation of numberOfVariables).

Leading coefficient of a 'qspray' polynomial

Description

Returns the leading coefficient of a qspray polynomial.

Usage

leadingCoefficient(qspray)
leadingCoefficient(qspray)

Arguments

qspray

a qspray object

Value

The coefficient of the leading term of qspray, a bigq rational number.

Leading term of a 'qspray' polynomial

Description

Returns the leading term of a qspray polynomial.

Usage

leadingTerm(qspray)
leadingTerm(qspray)

Arguments

qspray

a qspray object

Value

A list providing the exponents of the leading term in the field powers, an integer vector, and the coefficient of the leading term in the field coeff, a bigq rational number.

Monomial symmetric function

Description

Returns a monomial symmetric function as a polynomial.

Usage

MSFpoly(m, lambda)
MSFpoly(m, lambda)

Arguments

`m`	integer, the number of variables
`lambda`	an integer partition, given as a vector of decreasing positive integers

Value

A qspray object.

Examples

library(qspray)
MSFpoly(3, c(3, 1))
library(qspray)
MSFpoly(3, c(3, 1))

Symmetric polynomial in terms of the monomial symmetric polynomials

Description

Expression of a symmetric polynomial as a linear combination of the monomial symmetric polynomials.

Usage

MSPcombination(qspray, check = TRUE)
MSPcombination(qspray, check = TRUE)

Arguments

`qspray`	a `qspray` object defining a symmetric polynomial
`check`	Boolean, whether to check the symmetry

Value

A list defining the combination. Each element of this list is a list with two elements: coeff, a bigq number, and lambda, an integer partition; then this list corresponds to the term coeff * MSFpoly(n, lambda), where n is the number of variables in the symmetric polynomial.

Examples

qspray <- PSFpoly(4, c(3, 1)) + ESFpoly(4, c(2, 2)) + 4L
MSPcombination(qspray)
qspray <- PSFpoly(4, c(3, 1)) + ESFpoly(4, c(2, 2)) + 4L
MSPcombination(qspray)

Number of terms in a 'qspray' polynomial

Description

Number of terms of the polynomial defined by a qspray object.

Usage

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

Arguments

qspray

a qspray object

Value

An integer.

Number of variables in a 'qspray' polynomial

Description

Number of variables involved in a qspray object (see the note for the precise meaning).

Usage

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

Arguments

`x`	a `qspray` object

Value

An integer.

Note

The number of variables in the qspray object y^2+1 where y=qlone(2) is 2, not 1, although only one variable is present. Strictly speaking, the function returns the maximal integer d such that the variable qlone(d) occurs in the polynomial.

Ordered 'qspray'

Description

Reorders the terms of a qspray object according to the lexicographic order of the powers. This function is rather used internally only but it is exported for internal usage in other packages.

Usage

orderedQspray(qspray)
orderedQspray(qspray)

Arguments

qspray

a qspray object

Value

A qspray object. It defines the same polynomial as the input qspray object but it is ordered.

Examples

qspray <- rQspray()
qspray == orderedQspray(qspray) # should be TRUE
qspray <- rQspray()
qspray == orderedQspray(qspray) # should be TRUE

Permute variables

Description

Permute the variables of a qspray polynomial.

Usage

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

Arguments

`x`	a `qspray` object
`permutation`	a permutation

Value

A qspray object.

Examples

library(qspray)
f <- function(x, y, z) {
  x^2 + 5*y + z - 1
}
x <- qlone(1)
y <- qlone(2)
z <- qlone(3)
P <- f(x, y, z)
permutation <- c(3, 1, 2)
Q <- permuteVariables(P, permutation)
Q == f(z, x, y) # should be TRUE
library(qspray)
f <- function(x, y, z) {
  x^2 + 5*y + z - 1
}
x <- qlone(1)
y <- qlone(2)
z <- qlone(3)
P <- f(x, y, z)
permutation <- c(3, 1, 2)
Q <- permuteVariables(P, permutation)
Q == f(z, x, y) # should be TRUE

Pretty polynomial

Description

Pretty form of a qspray polynomial.

Usage

prettyQspray(qspray, vars = NULL)
prettyQspray(qspray, vars = NULL)

Arguments

`qspray`	a `qspray` object
`vars`	variable names; `NULL` for `"x1"`, `"x2"`, ...

Value

A character string.

Examples

library(qspray)
P <- (qlone(1) + "1/2"*qlone(2))^2 + 5
prettyP <- prettyQspray(P, vars = c("x", "y"))
prettyP
cat(Ryacas::yac_str(sprintf("PrettyForm(%s)", prettyP)))
Ryacas::yac_str(sprintf("TeXForm(%s)", prettyP))
library(qspray)
P <- (qlone(1) + "1/2"*qlone(2))^2 + 5
prettyP <- prettyQspray(P, vars = c("x", "y"))
prettyP
cat(Ryacas::yac_str(sprintf("PrettyForm(%s)", prettyP)))
Ryacas::yac_str(sprintf("TeXForm(%s)", prettyP))

Power sum polynomial

Description

Returns a power sum function as a polynomial.

Usage

PSFpoly(m, lambda)
PSFpoly(m, lambda)

Arguments

`m`	integer, the number of variables
`lambda`	an integer partition, given as a vector of decreasing positive integers

Value

A qspray object.

Examples

library(qspray)
PSFpoly(3, c(3, 1))
library(qspray)
PSFpoly(3, c(3, 1))

Symmetric polynomial as a linear combination of some power sum polynomials

Description

Expression of a symmetric qspray polynomial as a linear combination of some power sum polynomials.

Usage

PSPcombination(qspray)
PSPcombination(qspray)

Arguments

qspray

a symmetric qspray polynomial; symmetry is not checked

Value

A list of pairs. Each pair is made of a bigq number, the coefficient of the term of the linear combination, and an integer partition, corresponding to a power sum polynomial.

Examples

# take a symmetric polynomial
( qspray <- ESFpoly(4, c(2, 1)) + ESFpoly(4, c(2, 2)) )
# compute the power sum combination
( pspCombo <- PSPcombination(qspray) )
# then the polynomial can be reconstructed as follows:
Reduce(`+`, lapply(pspCombo, function(term) {
  term[["coeff"]] * PSFpoly(4, term[["lambda"]])
}))
# take a symmetric polynomial
( qspray <- ESFpoly(4, c(2, 1)) + ESFpoly(4, c(2, 2)) )
# compute the power sum combination
( pspCombo <- PSPcombination(qspray) )
# then the polynomial can be reconstructed as follows:
Reduce(`+`, lapply(pspCombo, function(term) {
  term[["coeff"]] * PSFpoly(4, term[["lambda"]])
}))

Symmetric polynomial in terms of the power sum polynomials

Description

Expression of a symmetric qspray polynomial as a polynomial in the power sum polynomials.

Usage

PSPexpression(qspray)
PSPexpression(qspray)

Arguments

qspray

a symmetric qspray polynomial; symmetry is not checked

Value

A qspray polynomial, say $P$ , such that $P(p_1, ..., p_n)$ equals the input symmetric polynomial, where $p_i$ is the i-th power sum polynomial (PSFpoly(n, i)).

Examples

# take a symmetric polynomial
( qspray <- ESFpoly(4, c(2, 1)) + ESFpoly(4, c(2, 2)) )
# compute the power sum expression
( pspExpr <- PSPexpression(qspray) )
# take the involved power sum polynomials
psPolys <- lapply(1:numberOfVariables(pspExpr), function(i) PSFpoly(4, i))
# then this should be TRUE:
qspray == changeVariables(pspExpr, psPolys)
# take a symmetric polynomial
( qspray <- ESFpoly(4, c(2, 1)) + ESFpoly(4, c(2, 2)) )
# compute the power sum expression
( pspExpr <- PSPexpression(qspray) )
# take the involved power sum polynomials
psPolys <- lapply(1:numberOfVariables(pspExpr), function(i) PSFpoly(4, i))
# then this should be TRUE:
qspray == changeVariables(pspExpr, psPolys)

Division of a qspray polynomial

Description

Division of a qspray polynomial by a list of qspray polynomials. See the reference for the definition.

Usage

qdivision(qspray, divisors)
qdivision(qspray, divisors)

Arguments

`qspray`	the dividend, a `qspray` object
`divisors`	the divisors, a list of `qspray` objects

Value

The remainder of the division, a qspray object.

References

Michael Weiss, 2010. Computing Gröbner Bases in Python with Buchberger’s Algorithm.

Examples

# a univariate example
library(qspray)
x <- qlone(1)
f <- x^4 - 4*x^3 + 4*x^2 - x # 0 and 1 are trivial roots
g <- x * (x - 1)
qdivision(f, list(g)) # should be zero
# a univariate example
library(qspray)
x <- qlone(1)
f <- x^4 - 4*x^3 + 4*x^2 - x # 0 and 1 are trivial roots
g <- x * (x - 1)
qdivision(f, list(g)) # should be zero

Polynomial variable

Description

Creates a polynomial variable. Using this function is the main way to build qspray objects.

Usage

qlone(n)
qlone(n)

Arguments

`n`	positive integer, the index of the variable

Value

A qspray object.

Examples

x <- qlone(1)
y <- qlone(2)
(x + y) * (x - y)
x <- qlone(1)
y <- qlone(2)
(x + y) * (x - y)

The unit 'qspray' polynomial

Description

Returns the qspray polynomial identically equal to 1.

Usage

qone()
qone()

Value

A qspray object.

(internal) Make a 'qspray' 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

qspray_from_list(qspray_as_list)
qspray_from_list(qspray_as_list)

Arguments

qspray_as_list

list returned by the Rcpp function returnQspray

Value

A qspray object.

Unary operators for qspray objects

Description

Unary operators for qspray objects.

Usage

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

## S4 method for signature 'qspray,missing'
e1 - e2
## S4 method for signature 'qspray,missing'
e1 + e2

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

Arguments

`e1`	object of class `qspray`
`e2`	nothing

Value

A qspray object.

Division of two polynomials

Description

Division of two polynomials

Usage

qsprayDivision(qsprayA, qsprayB)
qsprayDivision(qsprayA, qsprayB)

Arguments

`qsprayA`	a `qspray` object, the dividend
`qsprayB`	a `qspray` object, the divisor

Value

A list with two qspray objects, the quotient and the remainder.

Examples

library(qspray)
x <- qlone(1)
y <- qlone(2)
z <- qlone(3)
B <- x*y^2 + z*x^2 + 1
A <- B * (x^2*y^2*z^2 - 3) + x*y
divis <- qsprayDivision(A, B)
B * divis[["Q"]] + divis[["R"]] == A # should be TRUE
library(qspray)
x <- qlone(1)
y <- qlone(2)
z <- qlone(3)
B <- x*y^2 + z*x^2 + 1
A <- B * (x^2*y^2*z^2 - 3) + x*y
divis <- qsprayDivision(A, B)
B * divis[["Q"]] + divis[["R"]] == A # should be TRUE

Make a 'qspray' object

Description

Make a qspray object from a list of exponents and a vector of coefficients.

Usage

qsprayMaker(powers, coeffs, string = NULL)
qsprayMaker(powers, coeffs, string = NULL)

Arguments

`powers`	list of positive integer vectors
`coeffs`	a vector such that each element of `as.character(coeffs)` is a quoted integer or a quoted fraction; it must have the same length as the `powers` list
`string`	if not `NULL`, this argument takes precedence over `powers` and `coeffs`; it must be a string representing a multivariate polynomial; see the example

Value

A qspray object.

Examples

powers <- list(c(1, 1), c(0, 2))
coeffs <- c("1/2", "4")
qsprayMaker(powers, coeffs)
qsprayMaker(string = "1/2 x^(1, 1) + 4 x^(0, 2)")
powers <- list(c(1, 1), c(0, 2))
coeffs <- c("1/2", "4")
qsprayMaker(powers, coeffs)
qsprayMaker(string = "1/2 x^(1, 1) + 4 x^(0, 2)")

The null 'qspray' polynomial

Description

Returns the qspray polynomial identically equal to 0.

Usage

qzero()
qzero()

Value

A qspray object.

Random 'qspray'

Description

Generates a random qspray object.

Usage

rQspray()
rQspray()

Value

A qspray object with at most 4 terms and at most 3 variables.

Print a monomial

Description

Prints a monomial like "x^(1, 0, 2)". This way of showing a monomial was used by default in previous versions of this package.

Usage

showMonomialOld(x = "x")
showMonomialOld(x = "x")

Arguments

`x`	a string, usually a letter such as `"x"` or `"X"`, to denote the variable

Value

A function which takes as argument a sequence of exponents and which prints the corresponding monomial.

Examples

showMonomialOld("X")(c(1, 0, 2))
showMonomialOld("X")(NULL)
showMonomialOld("X")(c(1, 0, 2))
showMonomialOld("X")(NULL)

Print a monomial

Description

Prints a monomial in the style of "x1.x3^2".

Usage

showMonomialX1X2X3(x = "x", collapse = ".")
showMonomialX1X2X3(x = "x", collapse = ".")

Arguments

`x`	a string, usually a letter such as `"x"` or `"X"`, to denote the non-indexed variables
`collapse`	a string to denote the symbol representing the multiplication, e.g. `"*"` or `""`

Value

A function which takes as argument a sequence of exponents and which prints the corresponding monomial.

Note

The function returned by this function can be used as the option "showMonomial" in the showQsprayOption<- function. But if you are happy with the default collapse argument, then you can equivalently set the "x" option instead, thereby typing less code. See the example.

Examples

showMonomialX1X2X3("X")(c(1, 0, 2))
showMonomialX1X2X3("X", collapse = "*")(c(1, 0, 2))
showMonomialX1X2X3("X")(c(1, 0, 2)) == 
  showMonomialXYZ(c("X1", "X2", "X3"))(c(1, 0, 2))
showMonomialX1X2X3()(NULL)
# setting a show option:
set.seed(3141)
( qspray <- rQspray() )
showQsprayOption(qspray, "showMonomial") <- showMonomialX1X2X3("X")
qspray
# this is equivalent to:
showQsprayOption(qspray, "showQspray") <- showQsprayX1X2X3("X")
# and also equivalent to:
showQsprayOption(qspray, "x") <- "X" 
showMonomialX1X2X3("X")(c(1, 0, 2))
showMonomialX1X2X3("X", collapse = "*")(c(1, 0, 2))
showMonomialX1X2X3("X")(c(1, 0, 2)) == 
  showMonomialXYZ(c("X1", "X2", "X3"))(c(1, 0, 2))
showMonomialX1X2X3()(NULL)
# setting a show option:
set.seed(3141)
( qspray <- rQspray() )
showQsprayOption(qspray, "showMonomial") <- showMonomialX1X2X3("X")
qspray
# this is equivalent to:
showQsprayOption(qspray, "showQspray") <- showQsprayX1X2X3("X")
# and also equivalent to:
showQsprayOption(qspray, "x") <- "X"

Print a monomial

Description

Prints a monomial like "x.z^2" if possible (see details).

Usage

showMonomialXYZ(letters = c("x", "y", "z"), collapse = ".")
showMonomialXYZ(letters = c("x", "y", "z"), collapse = ".")

Arguments

`letters`	a vector of strings, usually some letters such as `"x"` and `"y"`, to denote the variables
`collapse`	a string to denote the symbol representing the multiplication, e.g. `"*"` or `"."`

Details

If the function returned by this function is applied to a vector of exponents whose length is higher than the length of the letters vector, then showMonomialX1X2X3(x=letters[1]) is applied (see the last example).

Value

A function which takes as argument a sequence of exponents and which prints the corresponding monomial.

Note

The function returned by this function can be used as the option "showMonomial" in the showQsprayOption<- function.

Examples

showMonomialXYZ()(c(1, 0, 2))
showMonomialXYZ(collapse = "*")(c(1, 0, 2))
showMonomialXYZ()(NULL)
# what happens if there are more exponents than letters:
showMonomialXYZ(c("a", "b"), collapse = "*")(c(1, 2, 3))
# same as:
showMonomialX1X2X3("a", collapse = "*")(c(1, 2, 3))
# setting a show option:
set.seed(3141)
( qspray <- rQspray() )
showQsprayOption(qspray, "showMonomial") <- showMonomialXYZ(c("A", "B", "C"))
qspray
# this is equivalent to:
showQsprayOption(qspray, "showQspray") <- showQsprayXYZ(c("A", "B", "C"))
showMonomialXYZ()(c(1, 0, 2))
showMonomialXYZ(collapse = "*")(c(1, 0, 2))
showMonomialXYZ()(NULL)
# what happens if there are more exponents than letters:
showMonomialXYZ(c("a", "b"), collapse = "*")(c(1, 2, 3))
# same as:
showMonomialX1X2X3("a", collapse = "*")(c(1, 2, 3))
# setting a show option:
set.seed(3141)
( qspray <- rQspray() )
showQsprayOption(qspray, "showMonomial") <- showMonomialXYZ(c("A", "B", "C"))
qspray
# this is equivalent to:
showQsprayOption(qspray, "showQspray") <- showQsprayXYZ(c("A", "B", "C"))

Print a 'qspray' object

Description

Prints a qspray object given a function which prints the monomials.

Usage

showQspray(showMonomial, compact = FALSE, multiplication = "*")
showQspray(showMonomial, compact = FALSE, multiplication = "*")

Arguments

`showMonomial`	a function which takes as argument a sequence of exponents and which returns a string representing the corresponding monomial
`compact`	a Boolean value; if `TRUE`, then the `+` sign and the `-` sign will not be surrounded by spaces
`multiplication`	used to separate the coefficient and the monomial within a term

Value

A function which prints a qspray object.

Note

The function returned by this function can be used as the option "showQspray" in the showQsprayOption<- function. But one generally prefers to use showQsprayX1X2X3 or showQsprayXYZ instead, which are both built with showQspray.

Examples

set.seed(3141)
( qspray <- rQspray() )
f <- showQspray(showMonomialX1X2X3("X"), compact = TRUE)
f(qspray)
# this is equivalent to:
f <- showQsprayX1X2X3("X", compact = TRUE)
f(qspray)
# if you want to adopt this way to show a qspray, use 
# the setter function \code{\link{showQsprayOption<-}}: 
showQsprayOption(qspray, "showQspray") <- 
  showQsprayX1X2X3("X", compact = TRUE)
qspray
# then this show option will be preserved by some operations on the qspray:
qspray^2
set.seed(3141)
( qspray <- rQspray() )
f <- showQspray(showMonomialX1X2X3("X"), compact = TRUE)
f(qspray)
# this is equivalent to:
f <- showQsprayX1X2X3("X", compact = TRUE)
f(qspray)
# if you want to adopt this way to show a qspray, use 
# the setter function \code{\link{showQsprayOption<-}}: 
showQsprayOption(qspray, "showQspray") <- 
  showQsprayX1X2X3("X", compact = TRUE)
qspray
# then this show option will be preserved by some operations on the qspray:
qspray^2

Set a show option to a 'qspray' object

Description

Set a show option to a qspray object

Usage

showQsprayOption(x, which) <- value
showQsprayOption(x, which) <- value

Arguments

`x`	a `qspray` object
`which`	which option to set; this can be `"x"`, `"showMonomial"`, or `"showQspray"`
`value`	the value of the option

Value

This returns the updated qspray.

Note

The interest of setting some show options to a 'qspray' is that these options are preserved by some operations. See the examples and the README.

Examples

set.seed(3141)
( qspray <- rQspray() )
showQsprayOption(qspray, "x") <- "a"
qspray
# this is identical to:
showQsprayOption(qspray, "showMonomial") <- showMonomialX1X2X3("a")
# and also identical to:
showQsprayOption(qspray, "showQspray") <- showQsprayX1X2X3("a")
# old show method:
showQsprayOption(qspray, "showMonomial") <- showMonomialOld()
qspray
# show options are preserved by some operations:
qspray^2
3*qspray
derivQspray(qspray, 1)
swapVariables(qspray, 1, 2)
substituteQspray(qspray, c(NA, NA, "3/2"))
# for the binary arithmetic operations, the show options of the first 
# operand are transferred to the result when possible:
( qspray2 <- rQspray() )
qspray + qspray2 
set.seed(3141)
( qspray <- rQspray() )
showQsprayOption(qspray, "x") <- "a"
qspray
# this is identical to:
showQsprayOption(qspray, "showMonomial") <- showMonomialX1X2X3("a")
# and also identical to:
showQsprayOption(qspray, "showQspray") <- showQsprayX1X2X3("a")
# old show method:
showQsprayOption(qspray, "showMonomial") <- showMonomialOld()
qspray
# show options are preserved by some operations:
qspray^2
3*qspray
derivQspray(qspray, 1)
swapVariables(qspray, 1, 2)
substituteQspray(qspray, c(NA, NA, "3/2"))
# for the binary arithmetic operations, the show options of the first 
# operand are transferred to the result when possible:
( qspray2 <- rQspray() )
qspray + qspray2

Print a 'qspray' object

Description

Prints a qspray object given a string for the variable.

Usage

showQsprayX1X2X3(x = "x", collapse = ".", ...)
showQsprayX1X2X3(x = "x", collapse = ".", ...)

Arguments

`x`, `collapse`	see `showMonomialX1X2X3`
`...`	arguments passed to `showQspray`, such as `compact=TRUE`

Value

A function which prints a qspray object.

Note

The way qspray objects are displayed can be controlled with the help of the function showQsprayOption<-, and showQsprayX1X2X3() is a possible option to pass in showQsprayOption<-.

Examples

set.seed(3141)
( qspray <- rQspray() )
showQsprayX1X2X3("X")(qspray)
# setting a show option:
showQsprayOption(qspray, "showQspray") <- showQsprayX1X2X3("A")
qspray
# this is equivalent to:
showQsprayOption(qspray, "showMonomial") <- showMonomialX1X2X3("A")
# and also equivalent to:
showQsprayOption(qspray, "x") <- "A"
set.seed(3141)
( qspray <- rQspray() )
showQsprayX1X2X3("X")(qspray)
# setting a show option:
showQsprayOption(qspray, "showQspray") <- showQsprayX1X2X3("A")
qspray
# this is equivalent to:
showQsprayOption(qspray, "showMonomial") <- showMonomialX1X2X3("A")
# and also equivalent to:
showQsprayOption(qspray, "x") <- "A"

Print a polynomial

Description

Prints a polynomial by printing monomials like "x^2.yz".

Usage

showQsprayXYZ(letters = c("x", "y", "z"), collapse = ".", ...)
showQsprayXYZ(letters = c("x", "y", "z"), collapse = ".", ...)

Arguments

`letters`, `collapse`	see `showMonomialXYZ`
`...`	arguments passed to `showQspray`, such as `compact=TRUE`

Value

A function which prints a qspray object. It is constructed with showQspray and showMonomialXYZ.

Note

The function returned by this function can be used as the option "showQspray" in the showQsprayOption<- function.

Examples

set.seed(3141)
( qspray <- rQspray() )
showQsprayXYZ(c("X", "Y", "Z"))(qspray)
showQsprayXYZ(c("X", "Y", "Z"))(qlone(1) + qlone(2) + qlone(3) + qlone(4))
# setting a show option:
showQsprayOption(qspray, "showQspray") <- showQsprayXYZ(c("A", "B", "C"))
qspray
# this is equivalent to:
showQsprayOption(qspray, "showMonomial") <- showMonomialXYZ(c("A", "B", "C"))
set.seed(3141)
( qspray <- rQspray() )
showQsprayXYZ(c("X", "Y", "Z"))(qspray)
showQsprayXYZ(c("X", "Y", "Z"))(qlone(1) + qlone(2) + qlone(3) + qlone(4))
# setting a show option:
showQsprayOption(qspray, "showQspray") <- showQsprayXYZ(c("A", "B", "C"))
qspray
# this is equivalent to:
showQsprayOption(qspray, "showMonomial") <- showMonomialXYZ(c("A", "B", "C"))

Substitutions in a 'qspray' polynomial

Description

Substitute some variables in a qspray polynomial.

Usage

substituteQspray(qspray, values)
substituteQspray(qspray, values)

Arguments

`qspray`	a `qspray` object
`values`	the values to be substituted; this must be a vector whose length equals the number of variables of `qspray`, and whose each entry is either `NA` (for non-substitution) or a 'scalar' `x` such that `as.character(x)` is a quoted integer or a quoted fraction

Value

A qspray object.

Examples

library(qspray)
x <- qlone(1)
y <- qlone(2)
z <- qlone(3)
p <- x^2 + y^2 + x*y*z - 1
substituteQspray(p, c("2", NA, "3/2"))
library(qspray)
x <- qlone(1)
y <- qlone(2)
z <- qlone(3)
p <- x^2 + y^2 + x*y*z - 1
substituteQspray(p, c("2", NA, "3/2"))

Swap variables

Description

Swap two variables in a qspray polynomial.

Usage

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

Arguments

`x`	a `qspray` object
`i`, `j`	indices of the variables to be swapped

Value

A qspray object.

Examples

library(qspray)
f <- function(x, y, z) {
  x^2 + 5*y + z - 1
}
x <- qlone(1)
y <- qlone(2)
z <- qlone(3)
P <- f(x, y, z)
Q <- swapVariables(P, 2, 3)
Q == f(x, z, y) # should be TRUE
library(qspray)
f <- function(x, y, z) {
  x^2 + 5*y + z - 1
}
x <- qlone(1)
y <- qlone(2)
z <- qlone(3)
P <- f(x, y, z)
Q <- swapVariables(P, 2, 3)
Q == f(x, z, y) # should be TRUE

Package 'qspray'

Help Index

Multivariate polynomial as function

Description

Usage

Arguments

Value

Examples

Coercion to a 'qspray' object

Description

Usage

Arguments

Value

Examples

Change of variables in a 'qspray' polynomial

Description

Usage

Arguments

Value

Examples

Characteristic polynomial

Description

Usage

Arguments

Value

Examples

Whether two 'qspray' polynomials are collinear

Description

Usage

Arguments

Value

Examples

Compact symmetric qspray

Description

Usage

Arguments

Value

See Also

Examples

Compose 'qspray' polynomials

Description

Usage

Arguments

Value

Examples

Complete homogeneous symmetric function

Description

Usage

Arguments

Value

Examples

Partial derivative

Description

Usage

Arguments

Value

Examples

Partial differentiation

Description

Usage

Arguments

Value

Examples

Elementary symmetric polynomial

Description

Usage

Arguments

Value

Examples

Evaluate a 'qspray' object

Description

Usage

Arguments

Value

Examples

Get a coefficient in a 'qspray' polynomial

Description

Usage

Arguments

Value