legendre {mpoly}

Legendre polynomials

Description

Legendre polynomials as computed by orthopolynom.

Usage

legendre(degree, indeterminate = "x", normalized = FALSE)

Arguments

degree

degree of polynomial

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::legendre.polynomials(), http://en.wikipedia.org/wiki/Legendre_polynomials

Examples

legendre(0)
legendre(1)
legendre(2)
legendre(3)
legendre(4)
legendre(5)
legendre(6)

legendre(2)
legendre(2, normalized = TRUE)

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



# visualize the legendre polynomials

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

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

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

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


# legendre polynomials and the QR decomposition
n <- 201
x <- seq(-1, 1, length.out = n)
mat <- cbind(1, x, x^2, x^3, x^4, x^5)
Q <- qr.Q(qr(mat))
df <- as.data.frame(cbind(x, Q))
names(df) <- c("x", 0:5)
mdf <- gather(df, degree, value, -x)
qplot(x, value, data = mdf, geom = "line", color = degree)

Q <- apply(Q, 2, function(x) x / x[n])
df <- as.data.frame(cbind(x, Q))
names(df) <- c("x", paste0("P_", 0:5))
mdf <- gather(df, degree, value, -x)
qplot(x, value, data = mdf, geom = "line", color = degree)


# chebyshev polynomials are orthogonal in two ways:
P2 <- as.function(legendre(2))
P3 <- as.function(legendre(3))
P4 <- as.function(legendre(4))

integrate(function(x) P2(x) * P3(x), lower = -1, upper = 1)
integrate(function(x) P2(x) * P4(x), lower = -1, upper = 1)
integrate(function(x) P3(x) * P4(x), lower = -1, upper = 1)

n <- 10000L
u <- runif(n, -1, 1)
2 * mean(P2(u) * P3(u))
2 * mean(P2(u) * P4(u))
2 * mean(P3(u) * P4(u))

(2/n) * sum(P2(u) * P3(u))
(2/n) * sum(P2(u) * P4(u))
(2/n) * sum(P3(u) * P4(u))