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
:
<- JackPol(2, lambda = c(3, 1), alpha = gmp::as.bigq("2/5"))
jpol
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:
::evalQspray(jpol, c("2", "3/2"))
qspray## Big Rational ('bigq') :
## [1] 1239/10
By default, ZonalPol
, ZonalQPol
and
SchurPol
return exact symbolic polynomials.
<- ZonalPol(2, lambda = c(3, 1))
zpol
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:
<- as.function(zpol) zyacas
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)"
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.
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.
<- c("1/2", "2/3", "1", "2/3", "1", "5/4")
x <- c(5, 3, 2, 2, 1)
lambda <- "3"
alpha 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