The ‘jack’ package: Jack polynomials

library(jack)
library(microbenchmark)

Schur polynomials have applications in combinatorics and zonal polynomials have applications in multivariate statistics. They are particular cases of Jack polynomials. This package allows to evaluate these polynomials. It can also compute their symbolic form.

The functions JackPol, ZonalPol, ZonalQPol and SchurPol respectively return the Jack polynomial, the zonal polynomial, the quaternionic zonal polynomial, and the Schur polynomial.

Each of these polynomials corresponds is given by a positive integer, the number of variables, and an integer partition, the lambda argument; the Jack polynomial has one more parameter, the alpha argument, a positive number.

To get an exact symbolic polynomial with JackPol, you have to supply a bigq rational number for the parameter alpha:

jpol <- JackPol(2, lambda = c(3, 1), alpha = gmp::as.bigq("2/5"))
jpol
## 98/25*x^(3, 1) + 98/25*x^(1, 3) + 28/5*x^(2, 2)

This is a qspray object, from the qspray package. Here is how you can evaluate this polynomial:

qspray::evalQspray(jpol, c("2", "3/2"))
## Big Rational ('bigq') :
## [1] 1239/10

By default, ZonalPol, ZonalQPol and SchurPol return exact symbolic polynomials.

zpol <- ZonalPol(2, lambda = c(3, 1))
zpol
## 24/7*x^(3, 1) + 24/7*x^(1, 3) + 16/7*x^(2, 2)

It is also possible to convert a qspray polynomial to a function whose evaluation is performed by the Ryacas package:

zyacas <- as.function(zpol)

You can provide the values of the variables of this function as numbers or character strings:

zyacas(2, "3/2")
## [1] "594/7"

You can even pass a variable name to this function:

zyacas("x", "x")
## [1] "(64*x^4)/7"

If you want to substitute a variable with a complex number, use a character string which represents this number, with I denoting the imaginary unit:

zyacas("2 + 2*I", "2/3")
## [1] "Complex((-2176)/63,2944/63)"

Jack polynomials with Julia

As of version 2.0.0, it was possible to calculate the Jack polynomials with Julia. This feature has been removed in version 5.3.0. Use the Julia package JackPolynomials.jl instead.

‘Rcpp’ implementation of the polynomials

As of version 5.0.0, a ‘Rcpp’ implementation of the polynomials is provided by the package.

As of version 5.1.0, there’s also a ‘Rcpp’ implementation of the evaluation of the polynomials.

x <- c("1/2", "2/3", "1", "2/3", "1", "5/4")
lambda <- c(5, 3, 2, 2, 1)
alpha <- "3"
print(
  microbenchmark(
        R = Jack(gmp::as.bigq(x), lambda, gmp::as.bigq(alpha)),
     Rcpp = JackCPP(x, lambda, alpha),
    times = 6L,
    unit  = "seconds"
  ),
  signif = 2L
)
## Unit: seconds
##  expr    min    lq  mean median    uq   max neval
##     R 110.00 130.0 140.0  130.0 160.0 160.0     6
##  Rcpp   0.98   1.2   1.3    1.2   1.5   1.6     6