The ‘jack’ package: Jack polynomials

library(jack)
library(microbenchmark)

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
microbenchmark(
      R = Jack(x, lambda, alpha),
  Julia = julia$Jack(x, lambda, alpha),
  times = 5
)
## Unit: milliseconds
##   expr          min           lq       mean     median         uq       max
##      R 15767.690900 16131.731601 16546.8662 16381.9088 17123.5399 17329.460
##  Julia     8.395101     9.733001   458.8062    11.0415    90.6334  2174.228
##  neval
##      5
##      5

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
## mvp object algebraically equal to
## 3.92 x1 x2^3  +  5.6 x1^2 x2^2  +  3.92 x1^3 x2
## 
## Exact expression:
## 98/25 * x1 * x2^3  +  28/5 * x1^2 * x2^2  +  98/25 * x1^3 * x2

To evaluate a polynomial, use as.function:

jp <- as.function(JP)

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

jp(2, "3/2")
## [1] "1239/10"

The evaluation is performed by the Ryacas package. 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:

jp("2 + 2*I", "2/3")
## [1] "Complex((-26656)/675,43232/675)"

Two functions are provided to print the polynomials with an exact expression:

prettyForm(JP)
## 
##   3               2          2               3
## x2  * 98 * x1   x2  * 28 * x1    x2 * 98 * x1 
## ------------- + -------------- + -------------
##      25               5               25

toLaTeX(JP)
## $\frac{x_{2}^{3} 98 x_{1}}{25}  + \frac{x_{2}^{2} 28 x_{1}^{2}}{5}  + \frac{x_{2} 98 x_{1}^{3}}{25} $