\name{bernstein}
\alias{bernstein}
\alias{bernsteinb}
\title{
Bernstein Polynomials
}
\description{
Bernstein base polynomials and approximations.
}
\usage{
bernstein(f, n, x)
bernsteinb(k, n, x)
}
\arguments{
\item{f}{function to be approximated by Bernstein polynomials.}
\item{k}{integer between 0 and n, the k-th Bernstein polynomial
of order n.}
\item{n}{order of the Bernstein polynomial(s).}
\item{x}{numeric scalar or vector where the Bernstein polynomials
will be calculated.}
}
\details{
The Bernstein basis polynomials \eqn{B_{k,n}(x)} are defined as
\deqn{ B_{k,n}(x) = {{n}\choose{k}} x^k (1-x)^{n-k} }
and form a basis for the vector space of polynomials of degree
\eqn{n} over the interval \eqn{[0,1]}.
\code{bernstein(f, n, x)} computes the approximation of function
\code{f} through Bernstein polynomials of degree \code{n}, resp.
computes the value of this approximation at \code{x}. The function
is vectorized and applies a brute force calculation.
But if \code{x} is a scalar, the value will be calculated using
De Casteljau's algorithm for higher accuracy. For bigger \code{n}
the binomial coefficients may be in for problems.
}
\value{
Returns a scalar or vector of function values.
}
\references{
See https://en.wikipedia.org/wiki/Bernstein_polynomial
}
\examples{
## Example
f <- function(x) sin(2*pi*x)
xs <- linspace(0, 1)
ys <- f(xs)
\dontrun{
plot(xs, ys, type='l', col="blue",
main="Bernstein Polynomials")
grid()
b10 <- bernstein(f, 10, xs)
b100 <- bernstein(f, 100, xs)
lines(xs, b10, col="magenta")
lines(xs, b100, col="red") }
# Bernstein basis polynomials
\dontrun{
xs <- linspace(0, 1)
plot(c(0,1), c(0,1), type='n',
main="Bernstein Basis Polynomials")
grid()
n = 10
for (i in 0:n) {
bs <- bernsteinb(i, n, xs)
lines(xs, bs, col=i+1)
} }
}
\keyword{ math }