hermite {mpoly}

Hermite polynomials

Description

Hermite polynomials as computed by orthopolynom.

Usage

hermite(degree, kind = "he", indeterminate = "x", normalized = FALSE)

Arguments

degree

degree of polynomial

kind

"he" (default, probabilists', see Wikipedia article) or "h" (physicists)

indeterminate

indeterminate

normalized

provide normalized coefficients

Value

a mpoly object or mpolyList object

Author(s)

David Kahle calling code from the orthopolynom package

See Also

orthopolynom::hermite.h.polynomials(), orthopolynom::hermite.he.polynomials(), http://en.wikipedia.org/wiki/Hermite_polynomials

Examples

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

hermite(0:5)
hermite(0:5, normalized = TRUE)
hermite(0:5, indeterminate = "t")



# visualize the hermite polynomials

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

s <- seq(-3, 3, length.out = 201)
N <- 5 # number of hermite polynomials to plot
(hermPolys <- hermite(0:N))

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

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


# hermite polynomials are orthogonal with respect to the gaussian kernel:
He2 <- as.function(hermite(2))
He3 <- as.function(hermite(3))
He4 <- as.function(hermite(4))

w <- dnorm
integrate(function(x) He2(x) * He3(x) * w(x), lower = -Inf, upper = Inf)
integrate(function(x) He2(x) * He4(x) * w(x), lower = -Inf, upper = Inf)
integrate(function(x) He3(x) * He4(x) * w(x), lower = -Inf, upper = Inf)