chebyshevPoly: Chebyshev Polynomial Evaluation

In DPQ: Density, Probability, Quantile ('DPQ') Computations

Chebyshev Polynomial Evaluation

Description

Provides (evaluation of) Chebyshev polynomials, given their coefficients vector coef (using 2 c_0, i.e., 2*coef[1] as the base R mathlib chebyshev*() functions. Specifically, the following sum is evaluated:

\sum_{j=0}^n c_j T_j(x)

where c_0 :=coef[1] and c_j :=coef[j+1] for j \ge 1. n := chebyshev_nc(coef, .) is the maximal degree and hence one less than the number of terms, and T_j() is the Chebyshev polynomial (of the first kind) of degree j.

Usage

chebyshevPoly(coef, nc = chebyshev_nc(coef, eta), eta = .Machine$double.eps/20)

chebyshev_nc(coef, eta = .Machine$double.eps/20)
chebyshevEval(x, coef,
              nc = chebyshev_nc(coef, eta), eta = .Machine$double.eps/20)

Arguments

coef	a numeric vector of coefficients for the Chebyshev polynomial.
nc	the maximal degree, i.e., one less than the number of polynomial terms to use; typically use the default.
eta	a positive number; typically keep the default.
x	for chebyshevEval(): numeric vector of abscissa values at which the polynomial should be evaluated. Typically x values are inside the interval [-1, 1].

Value

chebyshevPoly() returns function(x) which computes the values of the underlying Chebyshev polynomial at x.

chebyshev_nc() returns an integer, and chebyshevEval(x, coef) returns a numeric “like” x with the values of the polynomial at x.

Author(s)

R Core team, notably Ross Ihaka; Martin Maechler provided the R interface.

References

https://en.wikipedia.org/wiki/Chebyshev_polynomials

See Also

polyn.eval() from CRAN package sfsmisc; as one example of many more.

Examples

## The first 5 (base) Chebyshev polynomials:
T0 <- chebyshevPoly(2)  # !! 2, not 1
T1 <- chebyshevPoly(0:1)
T2 <- chebyshevPoly(c(0,0,1))
T3 <- chebyshevPoly(c(0,0,0,1))
T4 <- chebyshevPoly(c(0,0,0,0,1))
curve(T0(x), -1,1, col=1, lwd=2, ylim=c(-1,1))
abline(h=0, lty=2)
curve(T1(x), col=2, lwd=2, add=TRUE)
curve(T2(x), col=3, lwd=2, add=TRUE)
curve(T3(x), col=4, lwd=2, add=TRUE)
curve(T4(x), col=5, lwd=2, add=TRUE)

(Tv <- vapply(c(T0=T0, T1=T1, T2=T2, T3=T3, T4=T4),
              function(Tp) Tp(-1:1), numeric(3)))
x <- seq(-1,1, by = 1/64)
stopifnot(exprs = {
   all.equal(chebyshevPoly(1:5)(x),
             0.5*T0(x) + 2*T1(x) + 3*T2(x) + 4*T3(x) + 5*T4(x))
   all.equal(unname(Tv), rbind(c(1,-1), c(1:-1,0:1), rep(1,5)))# warning on rbind()
})

DPQ documentation built on Nov. 3, 2024, 3 a.m.