Jacobi polynomials

Jacobi polynomials as computed by orthopolynom.

jacobi(
  degree,
  alpha = 1,
  beta = 1,
  kind = "p",
  indeterminate = "x",
  normalized = FALSE
)

Arguments

degree	degree of polynomial
alpha	the first parameter, also called p
beta	the second parameter, also called q
kind	"g" or "p"
indeterminate	indeterminate
normalized	provide normalized coefficients

Value

a mpoly object or mpolyList object

See also

orthopolynom::jacobi.g.polynomials(), orthopolynom::jacobi.p.polynomials(), http://en.wikipedia.org/wiki/Jacobi_polynomials

Examples

jacobi(0)
#> 1
jacobi(1)
#> 3 x
jacobi(2)
#> -1.5  +  7.5 x^2
jacobi(3)
#> -7.5 x  +  17.5 x^3
jacobi(4)
#> 1.875  -  26.25 x^2  +  39.375 x^4
jacobi(5)
#> 13.125 x  -  78.75 x^3  +  86.625 x^5
jacobi(6)
#> -2.1875  +  59.0625 x^2  -  216.5625 x^4  +  187.6875 x^6
jacobi(10, 2, 2, normalized = TRUE)
#> -0.802729  +  60.20468 x^2  -  682.3197 x^4  +  2592.815 x^6  -  3889.222 x^8  +  1987.825 x^10

jacobi(0:5)
#> 1
#> 3 x
#> 7.5 x^2  -  1.5
#> 17.5 x^3  -  7.5 x
#> 39.375 x^4  -  26.25 x^2  +  1.875
#> 86.625 x^5  -  78.75 x^3  +  13.125 x
jacobi(0:5, normalized = TRUE)
#> 0.866025403784439
#> 1.936492 x
#> 4.050463 x^2  -  0.8100926
#> 8.300979 x^3  -  3.557562 x
#> 16.85937 x^4  -  11.23958 x^2  +  0.802827
#> 34.07809 x^5  -  30.98008 x^3  +  5.163347 x
jacobi(0:5, kind = "g")
#> 1
#> x  -  0.5
#> x^2  -  x  +  0.1666667
#> x^3  -  1.5 x^2  +  0.6 x  -  0.05
#> x^4  -  2 x^3  +  1.285714 x^2  -  0.2857143 x  +  0.01428571
#> x^5  -  2.5 x^4  +  2.222222 x^3  -  0.8333333 x^2  +  0.1190476 x  -  0.003968254
jacobi(0:5, indeterminate = "t")
#> 1
#> 3 t
#> 7.5 t^2  -  1.5
#> 17.5 t^3  -  7.5 t
#> 39.375 t^4  -  26.25 t^2  +  1.875
#> 86.625 t^5  -  78.75 t^3  +  13.125 t



# visualize the jacobi polynomials

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

s <- seq(-1, 1, length.out = 201)
N <- 5 # number of jacobi polynomials to plot
(jacPolys <- jacobi(0:N, 2, 2))
#> 1
#> 5 x
#> 17.5 x^2  -  2.5
#> 52.5 x^3  -  17.5 x
#> 144.375 x^4  -  78.75 x^2  +  4.375
#> 375.375 x^5  -  288.75 x^3  +  39.375 x

df <- data.frame(s, as.function(jacPolys)(s))
names(df) <- c("x", paste0("P_", 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(-30, 30))