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Complete homogeneous symmetric function

Description

Returns a complete homogeneous symmetric function as aqspray polynomial.

Usage

CSHFpoly(m, lambda)

Arguments

Value

Aqspray object.

Examples

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

Elementary symmetric polynomial

Description

Returns an elementary symmetric function as a polynomial.

Usage

ESFpoly(m, lambda)

Arguments

Value

Aqspray object.

Examples

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

Hall inner product

Description

Hall inner product of two symmetric polynomials. It has a parameteralpha and the standard Hall inner product is the case whenalpha=1. It is possible to get the Hall inner product with a symbolicalpha parameter.

Usage

HallInnerProduct(qspray1, qspray2, alpha = 1)

Arguments

Value

Abigq number ifalpha is notNULL, otherwise a univariateqspray polynomial.

Monomial symmetric function

Description

Returns a monomial symmetric function as a polynomial.

Usage

MSFpoly(m, lambda)

Arguments

Value

Aqspray object.

Examples

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)

Arguments

Value

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

Examples

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

Power sum polynomial

Description

Returns a power sum function as a polynomial.

Usage

PSFpoly(m, lambda)

Arguments

Value

Aqspray object.

Examples

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

Symmetric polynomial as a linear combination of some power sum polynomials

Description

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

Usage

PSPcombination(qspray)

Arguments

Value

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

See Also

PSPexpression.

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"]])}))

Symmetric polynomial in terms of the power sum polynomials

Description

Expression of a symmetricqspray polynomial as a polynomial in the power sum polynomials.

Usage

PSPexpression(qspray)

Arguments

Value

Aqspray polynomial, sayP, such thatP(p_1, ..., p_n) equals the input symmetric polynomial, wherep_i is the i-th power sum polynomial (PSFpoly(n, i)).

See Also

PSPcombination

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 polynomialspsPolys <- lapply(1:numberOfVariables(pspExpr), function(i) PSFpoly(4, i))# then this should be TRUE:qspray == changeVariables(pspExpr, psPolys)

Multivariate polynomial as function

Description

Coerces aqspray polynomial into a function.

Usage

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

Arguments

Value

A function having the same variables as the polynomial. IfN=FALSE, it returns a string. IfN=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 + 5f <- 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)

Arguments

Value

Aqspray object.

Examples

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

Change of variables in a 'qspray' polynomial

Description

Replaces the variables of aqspray polynomial with someqspray polynomials. E.g. you have a polynomialP(x, y) and you want the polynomialP(x^2, x+y+1). This is an alias ofcomposeQspray.

Usage

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

Arguments

Value

Theqspray polynomial obtained by replacing the variables of the polynomial given in thex argument with the polynomials given in thelistOfQsprays argument.

Examples

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

Characteristic polynomial

Description

Characteristic polynomial of a matrix.

Usage

characteristicPolynomial(A)

Arguments

Value

A univariateqspray 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)

Whether two 'qspray' polynomials are collinear

Description

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

Usage

collinearQsprays(qspray1, qspray2)

Arguments

Value

A Boolean value.

Examples

library(qspray)qspray1 <- qsprayMaker(string = "1/2 x^(1, 1) + 4 x^(0, 2) + 5")qspray2 <- "4/7" * qspray1collinearQsprays(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)

Arguments

Value

A character string.

See Also

MSPcombination

Examples

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 aqspray polynomial with someqspray polynomials. E.g. you have a polynomialP(x, y) and you want the polynomialP(x^2, x+y+1) (see example).

Usage

composeQspray(qspray, listOfQsprays)

Arguments

Value

Theqspray polynomial obtained by composing the polynomial given in theqspray argument with the polynomials given in thelistOfQsprays argument.

Examples

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

Partial differentiation

Description

Partial differentiation of aqspray polynomial.

Usage

dQspray(qspray, orders)

Arguments

Value

Aqspray object.

Examples

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

Partial derivative

Description

Partial derivative of aqspray polynomial.

Usage

derivQspray(qspray, i, derivative = 1)

Arguments

Value

Aqspray object.

Examples

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

Evaluate a 'qspray' object

Description

Evaluation of the multivariate polynomial represented by aqspray object.

Usage

evalQspray(qspray, values_re, values_im = NULL)

Arguments

Value

Abigq number ifvalues_im=NULL, a pair ofbigq numbers otherwise: the real part and the imaginary part of the result.

Examples

x <- qlone(1); y <- qlone(2)P <- 2*x + "1/2"*yevalQspray(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)

Arguments

Value

The coefficient as abigq number.

Examples

library(qspray)x <- qlone(1)y <- qlone(2)p <- 4*x^2 + 3*y - 5getCoefficient(p, 2)          # coefficient of x^2getCoefficient(p, c(2, 0))    # same as getCoefficient(p, 2)getCoefficient(p, c(0, 1))    # coefficient of y because y=x^0*y^1getCoefficient(p, 0)          # the constant termgetCoefficient(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 aqspray polynomial.

Usage

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

Arguments

Value

Abigq number.

Gröbner basis

Description

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

Usage

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

Arguments

Value

A Gröbner basis of the ideal generated byG, given as a list ofqspray 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 examplex <- qlone(1); y <- qlone(2); z <- qlone(3)f1 <- x^2 + y + z^2 - 1f2 <- x^2 + y + z - 1f3 <- x + y^2 + z - 1groebner(list(f1, f2, f3))

Implicitization with Gröbner bases

Description

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

Usage

implicitization(nvariables, parameters, equations, relations)

Arguments

Value

A list ofqspray polynomials.

Examples

library(qspray)# ellipse example ##### variables cost <- qlone(1)sint <- qlone(2)nvariables <- 2# parametersa <- 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)

Arguments

Value

Abigq number, the exact value of the integral.

Examples

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

Variables involved in a 'qspray' polynomial

Description

Variables involved in aqspray object.

Usage

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

Arguments

Value

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

Examples

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 aqspray object defines a constant polynomial.

Usage

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

Arguments

Value

A Boolean value.

Whether a 'qspray' polynomial is homogeneous

Description

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

Usage

isHomogeneousQspray(qspray)

Arguments

Value

A Boolean value indicating whether the polynomial defined byqspray 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 TRUEattr(homogeneous, "degree") == sum(lambda) # should be TRUE

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

Description

Checks whether aqspray polynomial can be written as a polynomial of some givenqspray polynomials. IfTRUE, this polynomial is returned.

Usage

isPolynomialOf(qspray, qsprays)

Arguments

Value

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

Examples

library(qspray)x <- qlone(1); y <- qlone(2); z <- qlone(3)q1 <- x + yq2 <- x*z^2 + 4qspray <- 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 aqspray object defines the unit polynomial.

Usage

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

Arguments

Value

A Boolean value.

Whether a 'qspray' polynomial is null

Description

Checks whether aqspray object defines the zero polynomial.

Usage

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

Arguments

Value

A Boolean value.

Check symmetry of a polynomial

Description

Check whether aqspray polynomial is symmetric.

Usage

isSymmetricQspray(qspray)

Arguments

Value

A Boolean value indicating whether the polynomial defined byqspray is symmetric.

See Also

MSPcombination,compactSymmetricQspray

Examples

e1 <- ESFpoly(3, 1)e2 <- ESFpoly(3, 2)e3 <- ESFpoly(3, 3)q <- e1 + 2*e2 + 3*e3 + 4*e1*e3isSymmetricQspray(q)

Whether a 'qspray' is univariate

Description

Checks whether aqspray object defines aunivariate polynomial.

Usage

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

Arguments

Value

A Boolean value.

Note

It is considered that a constantqspray is univariate, and that theqspray objecty^2+1 wherey=qlone(2) is not univariate, although only one variable is present (see the note in the documentation ofnumberOfVariables).

Leading coefficient of a 'qspray' polynomial

Description

Returns the leading coefficient of aqspray polynomial.

Usage

leadingCoefficient(qspray)

Arguments

Value

The coefficient of the leading term ofqspray,abigq rational number.

Leading term of a 'qspray' polynomial

Description

Returns the leading term of aqspray polynomial.

Usage

leadingTerm(qspray)

Arguments

Value

A list providing the exponents of the leading term in the fieldpowers, an integer vector, and the coefficient of the leading term in the fieldcoeff, abigq rational number.

Number of terms in a 'qspray' polynomial

Description

Number of terms of the polynomial defined by aqspray object.

Usage

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

Arguments

Value

An integer.

Number of variables in a 'qspray' polynomial

Description

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

Usage

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

Arguments

Value

An integer.

Note

The number of variables in theqspray objecty^2+1 wherey=qlone(2) is2, not1, although only one variable is present. Strictly speaking, the function returns the maximal integerd such that the variableqlone(d) occurs in the polynomial.

See Also

involvedVariables.

Ordered 'qspray'

Description

Reorders the terms of aqspray 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)

Arguments

Value

Aqspray object. It defines the same polynomial as the inputqspray object but it is ordered.

Examples

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

Permute variables

Description

Permute the variables of aqspray polynomial.

Usage

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

Arguments

Value

Aqspray 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

Pretty polynomial

Description

Pretty form of aqspray polynomial.

Usage

prettyQspray(qspray, vars = NULL)

Arguments

Value

A character string.

Examples

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

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)

Arguments

Value

The remainder of the division, aqspray object.

References

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

Examples

# a univariate examplelibrary(qspray)x <- qlone(1)f <- x^4 - 4*x^3 + 4*x^2 - x # 0 and 1 are trivial rootsg <- 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 buildqspray objects.

Usage

qlone(n)

Arguments

Value

Aqspray object.

Examples

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

The unit 'qspray' polynomial

Description

Returns theqspray polynomial identically equal to 1.

Usage

qone()

Value

Aqspray 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

Arguments

Value

Aqspray object.

Division of two polynomials

Description

Division of two polynomials

Usage

qsprayDivision(qsprayA, qsprayB)

Arguments

Value

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

Make a 'qspray' object

Description

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

Usage

qsprayMaker(powers, coeffs, string = NULL)

Arguments

Value

Aqspray 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)")

(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)

Arguments

Value

Aqspray object.

The null 'qspray' polynomial

Description

Returns theqspray polynomial identically equal to 0.

Usage

qzero()

Value

Aqspray object.

Random 'qspray'

Description

Generates a randomqspray object.

Usage

rQspray()

Value

Aqspray 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")

Arguments

Value

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

See Also

showMonomialX1X2X3,showMonomialXYZ,showQspray,showQsprayOption<-.

Examples

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 = ".")

Arguments

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 theshowQsprayOption<- function.But if you are happy with the defaultcollapse argument, then you can equivalently set the"x" option instead, thereby typing less code. See the example.

See Also

showQsprayX1X2X3,showMonomialXYZ,showQsprayOption<-.

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"

Print a monomial

Description

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

Usage

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

Arguments

Details

If the function returned by this function is applied to a vector of exponents whose length is higher than the length of theletters vector, thenshowMonomialX1X2X3(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 theshowQsprayOption<- function.

See Also

showQsprayXYZ,showMonomialX1X2X3,showQsprayOption<-.

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"))

Print a 'qspray' object

Description

Prints aqspray object given a function which prints the monomials.

Usage

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

Arguments

Value

A function which prints aqspray object.

Note

The function returned by this function can be used as the option"showQspray" in theshowQsprayOption<- function.But one generally prefers to useshowQsprayX1X2X3 orshowQsprayXYZ instead, which are both built withshowQspray.

See Also

showQsprayX1X2X3,showQsprayXYZ,showQsprayOption<-.

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 a show option to a 'qspray' object

Description

Set a show option to aqspray object

Usage

showQsprayOption(x, which) <- value

Arguments

Value

This returns the updatedqspray.

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^23*qsprayderivQspray(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 aqspray object given a string for the variable.

Usage

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

Arguments

Value

A function which prints aqspray object.

Note

The wayqspray objects are displayed can be controlled with the help of the functionshowQsprayOption<-, andshowQsprayX1X2X3() is a possible option to pass inshowQsprayOption<-.

See Also

showMonomialX1X2X3,showQspray,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"

Print a polynomial

Description

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

Usage

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

Arguments

Value

A function which prints aqspray object. It is constructed withshowQspray andshowMonomialXYZ.

Note

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

See Also

showMonomialXYZ,showQspray,showQsprayOption<-.

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"))

Substitutions in a 'qspray' polynomial

Description

Substitute some variables in aqspray polynomial.

Usage

substituteQspray(qspray, values)

Arguments

Value

Aqspray object.

Examples

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

Swap variables

Description

Swap two variables in aqspray polynomial.

Usage

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

Arguments

Value

Aqspray 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