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, the Jack polynomials can be calculated with Julia. The speed is amazing:

julia <- Jack_julia()
## Starting Julia ...
x <- c(1/2, 2/3, 1, 2/3, -1, -2, 1)
lambda <- c(5, 3, 2, 2, 1)
alpha <- 3
print(
  microbenchmark(
        R = Jack(x, lambda, alpha),
    Julia = julia$Jack(x, lambda, alpha),
    times = 6L,
    unit  = "seconds"
  ),
  signif = 2L
)
## Unit: seconds
##   expr   min   lq mean median    uq  max neval
##      R 6.800 6.90 6.90  6.900 6.900 7.00     6
##  Julia 0.046 0.05 0.21  0.076 0.096 0.93     6

Jack_julia() returns a list of functions. ZonalPol, ZonalQPol and SchurPol always return an exact expression of the polynomial, i.e. with rational coefficients (integers for SchurPol). If you want an exact expression with JackPol, you have to give a rational number for the argument alpha, as a character string:

JP <- julia$JackPol(m = 2, lambda = c(3, 1), alpha = "2/5")
JP
## 98/25*x^(1, 3) + 28/5*x^(2, 2) + 98/25*x^(3, 1)

Again, Julia is faster:

n <- 5
lambda <- c(4, 3, 3)
alpha <- "2/3"
alphaq <- gmp::as.bigq(alpha)
print(
  microbenchmark(
        R = JackPol(n, lambda, alphaq),
    Julia = julia$JackPol(n, lambda, alpha),
    times = 6L
  ),
signif = 2L)
## Unit: milliseconds
##   expr min  lq mean median   uq  max neval
##      R 980 980 1000    990 1000 1100     6
##  Julia 470 500  540    510  510  720     6

‘Rcpp’ implementation of the polynomials

As of version 5.0.0, a ‘Rcpp’ implementation of the polynomials is provided by the package. It is faster than Julia (though I didn’t compare in pure Julia - the Julia execution time is slowed down by the ‘JuliaConnectoR’ package):

n <- 5
lambda <- c(4, 3, 3, 2)
print(
  microbenchmark(
     Rcpp = SchurPolCPP(n, lambda),
    Julia = julia$SchurPol(n, lambda),
    times = 6L
  ), 
signif = 2L)
## Unit: milliseconds
##   expr   min    lq  mean median    uq    max neval
##   Rcpp   6.2   6.4   6.6    6.5   6.6    7.3     6
##  Julia 530.0 530.0 640.0  540.0 540.0 1200.0     6

n <- 5
lambda <- c(4, 3, 3, 2)
alpha <- "2/3"
print(
  microbenchmark(
     Rcpp = JackPolCPP(n, lambda, alpha),
    Julia = julia$JackPol(n, lambda, alpha),
    times = 6L
  ), 
signif = 2L)
## Unit: milliseconds
##   expr min  lq mean median  uq max neval
##   Rcpp  24  24   25     25  25  25     6
##  Julia 370 410  430    420 470 500     6

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),
    Julia = julia$Jack(x, lambda, alpha),
    times = 6L,
    unit  = "seconds"
  ),
  signif = 2L
)
## Unit: seconds
##   expr   min    lq  mean median    uq   max neval
##      R 16.00 16.00 16.00  16.00 16.00 17.00     6
##   Rcpp  0.15  0.16  0.16   0.16  0.17  0.17     6
##  Julia  0.31  0.33  0.42   0.39  0.42  0.66     6

JuliaConnectoR::stopJulia()