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Stéphane Laurent 2024-07-26
Multivariate polynomials with symbolic parameters.
These notes about the symbolicQspray package assume that the reader is a bit familiar with the qspray package and with the ratioOfQsprays package.
A symbolicQspray object represents a multivariate
polynomial whose coefficients are fractions of polynomials with rational
coefficients. Actually (see our discussion in the next section), a
symbolicQspray object represents a multivariate
polynomial with parameters. The parameters are the variables of the
fractions of polynomials, and so they are symbolically represented.
To construct a symbolicQspray polynomial, use
qlone (from the qspray package) to
introduce the parameters and use Qlone to introduce the
variables of the polynomial:
library(symbolicQspray)
f <- function(a1, a2, X1, X2, X3) {
(a1/(a2^2+1)) * X1^2*X2 + (a2+1) * X3 + a1/a2
}
# parameters, the variables occurring in the coefficients:
a1 <- qlone(1)
a2 <- qlone(2)
# variables:
X1 <- Qlone(1)
X2 <- Qlone(2)
X3 <- Qlone(3)
# the 'symbolicQspray':
( Qspray <- f(a1, a2, X1, X2, X3) )
## { [ a1 ] %//% [ a2^2 + 1 ] } * X^2.Y + { [ a2 + 1 ] } * Z + { [ a1 ] %//% [ a2 ] }The fractions of polynomials such as the first coefficient
a1/(a2^2+1) in the above example are ratioOfQsprays
objects, and the numerator and the denominator of a
ratioOfQsprays are qspray
objects.
Arithmetic on symbolicQspray objects is available:
Qspray^2
## { [ a1^2 ] %//% [ a2^4 + 2*a2^2 + 1 ] } * X^4.Y^2 + { [ 2*a1.a2 + 2*a1 ] %//% [ a2^2 + 1 ] } * X^2.Y.Z + { [ 2*a1^2 ] %//% [ a2^3 + a2 ] } * X^2.Y + { [ a2^2 + 2*a2 + 1 ] } * Z^2 + { [ 2*a1.a2 + 2*a1 ] %//% [ a2 ] } * Z + { [ a1^2 ] %//% [ a2^2 ] }
Qspray - Qspray
## 0
(Qspray - 1)^2
## { [ a1^2 ] %//% [ a2^4 + 2*a2^2 + 1 ] } * X^4.Y^2 + { [ 2*a1.a2 + 2*a1 ] %//% [ a2^2 + 1 ] } * X^2.Y.Z + { [ 2*a1^2 - 2*a1.a2 ] %//% [ a2^3 + a2 ] } * X^2.Y + { [ a2^2 + 2*a2 + 1 ] } * Z^2 + { [ 2*a1.a2 + 2*a1 - 2*a2^2 - 2*a2 ] %//% [ a2 ] } * Z + { [ a1^2 - 2*a1.a2 + a2^2 ] %//% [ a2^2 ] }
Qspray^2 - 2*Qspray + 1
## { [ a1^2 ] %//% [ a2^4 + 2*a2^2 + 1 ] } * X^4.Y^2 + { [ 2*a1.a2 + 2*a1 ] %//% [ a2^2 + 1 ] } * X^2.Y.Z + { [ 2*a1^2 - 2*a1.a2 ] %//% [ a2^3 + a2 ] } * X^2.Y + { [ a2^2 + 2*a2 + 1 ] } * Z^2 + { [ 2*a1.a2 + 2*a1 - 2*a2^2 - 2*a2 ] %//% [ a2 ] } * Z + { [ a1^2 - 2*a1.a2 + a2^2 ] %//% [ a2^2 ] }symbolicQspraySubstituting the “exterior” variables (the variables occurring in the
ratios of polynomials, also called the parameters - see below)
yields a qspray object:
a <- c(2, "3/2")
( qspray <- evalSymbolicQspray(Qspray, a = a) )
## 8/13*X^2.Y + 5/2*Z + 4/3Substituting the “main” variables yields a
ratioOfQsprays object:
X <- c(4, 3, "2/5")
( ratioOfQsprays <- evalSymbolicQspray(Qspray, X = X) )
## [ a1.a2^2 + 48*a1.a2 + a1 + 2/5*a2^4 + 2/5*a2^3 + 2/5*a2^2 + 2/5*a2 ] %//% [ a2^3 + a2 ]There is a discutable point here. A symbolicQspray
object represents a polynomial with ratioOfQsprays
coefficients. So one could consider that the polynomial variables
X, Y and Z represent some
indeterminate ratioOfQsprays fractions, and that it should
be possible to replace them with ratioOfQsprays objects.
However this is not allowed. We will discuss that, just after checking
the consistency:
evalSymbolicQspray(Qspray, a = a, X = X)
## Big Rational ('bigq') :
## [1] 1243/39
evalQspray(qspray, X)
## Big Rational ('bigq') :
## [1] 1243/39
evalRatioOfQsprays(ratioOfQsprays, a)
## Big Rational ('bigq') :
## [1] 1243/39
a <- gmp::as.bigq(a); X <- gmp::as.bigq(X)
f(a[1], a[2], X[1], X[2], X[3])
## Big Rational ('bigq') :
## [1] 1243/39Now let’s turn to our promised discussion. Why is replacing the
values of the polynomial variables with some ratioOfQsprays
objects not allowed?
Actually my motivation to do this package was inspired by the Jack
polynomials. In the context of Jack polynomials, the
variables X, Y and Z represent
indeterminate numbers, and the coefficients are numbers
depending on a parameter (the Jack parameter), and it turns out
that they are fractions of polynomials of this parameter. So I consider
that a symbolicQspray is not a polynomial on the
field of fractions of polynomials: I consider it is a polynomial with
rational coefficients depending on some parameters.
Also note that evaluating the ratioOfQsprays object
evalSymbolicQspray(Qspray, X = X) at a would
make no sense if we took some ratioOfQsprays objects for
the values of X.
symbolicQsprayThe package provides some functions to perform elementary queries on
a symbolicQspray:
numberOfVariables(Qspray)
## [1] 3
numberOfParameters(Qspray)
## [1] 2
numberOfTerms(Qspray)
## [1] 3
getCoefficient(Qspray, c(2, 1)) # coefficient of X^2.Y
## [ a1 ] %//% [ a2^2 + 1 ]
getConstantTerm(Qspray)
## [ a1 ] %//% [ a2 ]
isUnivariate(Qspray)
## [1] FALSE
isConstant(Qspray)
## [1] FALSEsymbolicQsprayYou can differentiate a symbolicQspray polynomial:
derivSymbolicQspray(Qspray, 2) # derivative w.r.t. Y
## { [ a1 ] %//% [ a2^2 + 1 ] } * X^2You can permute its variables:
swapVariables(Qspray, 2, 3) == f(a1, a2, X1, X3, X2)
## [1] TRUEYou can perform polynomial transformations of its variables:
changeVariables(Qspray, list(X1+1, X2^2, X1+X2+X3)) ==
f(a1, a2, X1+1, X2^2, X1+X2+X3)
## [1] TRUEYou can also perform polynomial transformations of its parameters:
changeParameters(Qspray, list(a1^2, a2^2)) == f(a1^2, a2^2, X1, X2, X3)
## [1] TRUEsymbolicQsprayYou can change the way a symbolicQspray is printed by
using showSymbolicQsprayOption:
showSymbolicQsprayOption(Qspray, "a") <- "x"
showSymbolicQsprayOption(Qspray, "showMonomial") <-
showMonomialXYZ(c("A", "B", "C"))
showSymbolicQsprayOption(Qspray, "quotientBar") <- " / "
Qspray
## { [ x1 ] / [ x2^2 + 1 ] } * A^2.B + { [ x2 + 1 ] } * C + { [ x1 ] / [ x2 ] }When this is possible, the result of an arithmetic operation between
two symbolicQspray objects inherits the show options of the
first operand:
set.seed(421)
( Q <- rSymbolicQspray() ) # a random symbolicQspray
## { [ -a1^2.a3^3 - 5/2*a1^2 - 5/2*a3 ] %//% [ a1^4.a3^3 - 3/2*a2^4 ] } * X^3.Y + { [ 5/3*a1^4.a3^4 - 1/3*a1^2.a2.a3^3 ] %//% [ a1^2 - a2^2.a3^3 ] } * Y^3
Qspray + Q
## { [ -x1^2.x3^3 - 5/2*x1^2 - 5/2*x3 ] / [ x1^4.x3^3 - 3/2*x2^4 ] } * A^3.B + { [ x1 ] / [ x2^2 + 1 ] } * A^2.B + { [ 5/3*x1^4.x3^4 - 1/3*x1^2.x2.x3^3 ] / [ x1^2 - x2^2.x3^3 ] } * B^3 + { [ x2 + 1 ] } * C + { [ x1 ] / [ x2 ] }This behavior is the same as the ones implemented in qspray and in ratioOfQsprays. You should be familiar with these two packages in order to use symbolicQspray.
The Jacobi polynomials are
univariate polynomials depending on two parameters that we will denote
by alpha and beta. They are implemented in
this package:
JP <- JacobiPolynomial(2)
isUnivariate(JP)
## [1] TRUE
numberOfParameters(JP)
## [1] 2
showSymbolicQsprayOption(JP, "showRatioOfQsprays") <-
showRatioOfQspraysXYZ(c("alpha", "beta"))
JP
## { [ 1/8*alpha^2 + 1/4*alpha.beta + 7/8*alpha + 1/8*beta^2 + 7/8*beta + 3/2 ] } * X^2 + { [ 1/4*alpha^2 + 3/4*alpha - 1/4*beta^2 - 3/4*beta ] } * X + { [ 1/8*alpha^2 - 1/4*alpha.beta - 1/8*alpha + 1/8*beta^2 - 1/8*beta - 1/2 ] }The implementation constructs these polynomials by using the recurrence relation. This is a child game, one just has to copy the first two terms and this recurrence relation:
JacobiPolynomial <- function(n) {
stopifnot(isPositiveInteger(n))
if(n == 0) {
Qone()
} else if(n == 1) {
alpha <- qlone(1)
beta <- qlone(2)
X <- Qlone(1)
(alpha + 1) + (alpha + beta + 2) * (X - 1)/2
} else {
alpha <- qlone(1)
beta <- qlone(2)
X <- Qlone(1)
a <- n + alpha
b <- n + beta
c <- a + b
K <- 2 * n * (c - n) * (c - 2)
lambda1 <- ((c - 1) * (c * (c - 2) * X + (a - b) * (c - 2*n))) / K
lambda2 <- (2 * (a - 1) * (b - 1) * c) / K
(lambda1 * JacobiPolynomial(n - 1) - lambda2 * JacobiPolynomial(n - 2))
}
}It is clearly visible from the recurrence relation that the
coefficients of the Jacobi polynomials are indeed fractions of
polynomials in alpha and beta. But they
actually are polynomials in alpha and
beta. Actually I don’t know, this is a conjecture I made
because I observed this fact for some small values of n. We
can check it with the function
hasPolynomialCoefficientsOnly:
JP <- JacobiPolynomial(7)
hasPolynomialCoefficientsOnly(JP)
## [1] TRUEUp to a factor, the Gegenbauer polynomials
with parameter alpha coincide with the Jacobi polynomials
with parameters alpha - 1/2 and alpha - 1/2.
Let’s derive them from the Jacobi polynomials, as an exercise. The
factor can be implemented as follows (see Wikipedia for its
formula):
risingFactorial <- function(theta, n) {
toMultiply <- c(theta, lapply(seq_len(n-1), function(i) theta + i))
Reduce(`*`, toMultiply)
}
theFactor <- function(alpha, n) {
risingFactorial(2*alpha, n) / risingFactorial((2*alpha + 1)/2, n)
}Now let’s apply the formula given in the Wikipedia article:
GegenbauerPolynomial <- function(n) {
alpha <- qlone(1)
P <- changeParameters(
JacobiPolynomial(n), list(alpha - "1/2", alpha - "1/2")
)
theFactor(alpha, n) * P
}Let’s check that the recurrence relation given in the Wikipedia article is fulfilled:
n <- 5
alpha <- qlone(1)
X <- Qlone(1)
(n + 1) * GegenbauerPolynomial(n+1) ==
2*(n + alpha) * X * GegenbauerPolynomial(n) -
(n + 2*alpha - 1) * GegenbauerPolynomial(n-1)
## [1] TRUEThe symbolicQspray package is used in the jack
package to compute the Jack polynomials with a symbolic Jack
parameter. The Jack polynomials exactly fit to the polynomials
represented by the symbolicQspray objects: their
coefficients are fractions of polynomials by definition, of one
variable: the Jack parameter.
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.