Bernstein polynomial approximation

Bernstein polynomial approximation

bernstein_approx(f, n, lower = 0, upper = 1, indeterminate = "x")

bernsteinApprox(...)

Arguments

f	the function to approximate
n	Bernstein polynomial degree
lower	lower bound for approximation
upper	upper bound for approximation
indeterminate	indeterminate
...	...

Value

a mpoly object

Examples

if (FALSE)   # visualize the bernstein polynomials

library(ggplot2); theme_set(theme_bw())
library(reshape2)




f <- function(x) sin(2*pi*x)
p <- bernstein_approx(f, 20)
round(p, 3)
#> 6.18 x  -  5.747 x^2  -  31.108 x^3  +  27.287 x^4  +  32.866 x^5  -  29.411 x^6  -  10.328 x^7  +  10.999 x^8  +  0.755 x^9  -  1.66 x^10  +  0.074 x^11  +  0.105 x^12  -  0.01 x^13  -  0.003 x^14

x <- seq(0, 1, length.out = 101)
df <- data.frame(
  x = rep(x, 2),
  y = c(f(x), as.function(p)(x)),
  which = rep(c("actual", "approx"), each = 101)
)
#> f(.) with . = x
qplot(x, y, data = df, geom = "line", color = which)






p <- bernstein_approx(sin, 20, pi/2, 1.5*pi)
round(p, 4)
#> 1.1704 x  -  0.1818 x^2  -  0.1466 x^3  +  0.021 x^4  +  0.0039 x^5  -  0.0006 x^6  +  0.023

x <- seq(0, 2*pi, length.out = 101)
df <- data.frame(
  x = rep(x, 2),
  y = c(sin(x), as.function(p)(x)),
  which = rep(c("actual", "approx"), each = 101)
)
#> f(.) with . = x
qplot(x, y, data = df, geom = "line", color = which)








p <- bernstein_approx(dnorm, 15, -1.25, 1.25)
round(p, 4)
#> -0.1624 x^2  +  0.0262 x^4  -  0.002 x^6  +  0.0001 x^8  +  0.3796

x <- seq(-3, 3, length.out = 101)
df <- data.frame(
  x = rep(x, 2),
  y = c(dnorm(x), as.function(p)(x)),
  which = rep(c("actual", "approx"), each = 101)
)
#> f(.) with . = x
qplot(x, y, data = df, geom = "line", color = which)