mpoly (version 1.1.2)

chebyshev: Chebyshev polynomials

Description

Chebyshev polynomials as computed by orthopolynom.

Usage

chebyshev(degree, kind = "t", indeterminate = "x", normalized = FALSE)
chebyshev_roots(k, n)

Value

a mpoly object or mpolyList object

Arguments

degree

degree of polynomial

kind

"t" or "u" (Chebyshev polynomials of the first and second kinds), or "c" or "s"

indeterminate

indeterminate

normalized

provide normalized coefficients

k, n

the k'th root of the n'th chebyshev polynomial

Author

David Kahle calling code from the orthopolynom package

See Also

orthopolynom::chebyshev.t.polynomials(), orthopolynom::chebyshev.u.polynomials(), orthopolynom::chebyshev.c.polynomials(), orthopolynom::chebyshev.s.polynomials(), https://en.wikipedia.org/wiki/Chebyshev_polynomials

Examples

chebyshev(0)
chebyshev(1)
chebyshev(2)
chebyshev(3)
chebyshev(4)
chebyshev(5)
chebyshev(6)
chebyshev(10)

chebyshev(0:5)
chebyshev(0:5, normalized = TRUE)
chebyshev(0:5, kind = "u")
chebyshev(0:5, kind = "c")
chebyshev(0:5, kind = "s")
chebyshev(0:5, indeterminate = "t")



# visualize the chebyshev polynomials

library(ggplot2); theme_set(theme_classic())
library(tidyr)

s <- seq(-1, 1, length.out = 201)
N <- 5 # number of chebyshev polynomials to plot
(cheb_polys <- chebyshev(0:N))

# see ?bernstein for a better understanding of
# how the code below works

df <- data.frame(s, as.function(cheb_polys)(s))
names(df) <- c("x", paste0("T_", 0:N))
mdf <- gather(df, degree, value, -x)
qplot(x, value, data = mdf, geom = "line", color = degree)



# roots of chebyshev polynomials
N <- 5
cheb_roots <- chebyshev_roots(1:N, N)
cheb_fun <- as.function(chebyshev(N))
cheb_fun(cheb_roots)



# chebyshev polynomials are orthogonal in two ways:
T2 <- as.function(chebyshev(2))
T3 <- as.function(chebyshev(3))
T4 <- as.function(chebyshev(4))

w <- function(x) 1 / sqrt(1 - x^2)
integrate(function(x) T2(x) * T3(x) * w(x), lower = -1, upper = 1)
integrate(function(x) T2(x) * T4(x) * w(x), lower = -1, upper = 1)
integrate(function(x) T3(x) * T4(x) * w(x), lower = -1, upper = 1)

(cheb_roots <- chebyshev_roots(1:4, 4))
sum(T2(cheb_roots) * T3(cheb_roots) * w(cheb_roots))
sum(T2(cheb_roots) * T4(cheb_roots) * w(cheb_roots))
sum(T3(cheb_roots) * T4(cheb_roots) * w(cheb_roots))

sum(T2(cheb_roots) * T3(cheb_roots))
sum(T2(cheb_roots) * T4(cheb_roots))
sum(T3(cheb_roots) * T4(cheb_roots))