mpoly (version 1.1.1)

laguerre: Generalized Laguerre polynomials

Description

Generalized Laguerre polynomials as computed by orthopolynom.

Usage

laguerre(degree, alpha = 0, indeterminate = "x", normalized = FALSE)

Arguments

degree

degree of polynomial

alpha

generalization constant

indeterminate

indeterminate

normalized

provide normalized coefficients

Value

a mpoly object or mpolyList object

See Also

orthopolynom::glaguerre.polynomials(), http://en.wikipedia.org/wiki/Laguerre_polynomials

Examples

# NOT RUN {
laguerre(0)
laguerre(1)
laguerre(2)
laguerre(3)
laguerre(4)
laguerre(5)
laguerre(6)

laguerre(2)
laguerre(2, normalized = TRUE)

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



# visualize the laguerre polynomials

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

s <- seq(-5, 20, length.out = 201)
N <- 5 # number of laguerre polynomials to plot
(lagPolys <- laguerre(0:N))

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

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

qplot(x, value, data = mdf, geom = "line", color = degree) +
  coord_cartesian(ylim = c(-25, 25))


# laguerre polynomials are orthogonal with respect to the exponential kernel:
L2 <- as.function(laguerre(2))
L3 <- as.function(laguerre(3))
L4 <- as.function(laguerre(4))

w <- dexp
integrate(function(x) L2(x) * L3(x) * w(x), lower = 0, upper = Inf)
integrate(function(x) L2(x) * L4(x) * w(x), lower = 0, upper = Inf)
integrate(function(x) L3(x) * L4(x) * w(x), lower = 0, upper = Inf)


# }