pracma (version 1.9.9)

chebCoeff: Chebyshev Polynomials

Description

Chebyshev Coefficients for Chebyshev polynomials of the first kind.

Usage

chebCoeff(fun, a, b, n)

Arguments

fun

function to be approximated.

a, b

endpoints of the interval.

n

an integer >= 0.

Value

Vector of coefficients for the Chebyshev polynomials, from low to high degrees (see the example).

Details

For a function fun on on the interval [a, b] determines the coefficients of the Chebyshev polynomials up to degree n that will approximate the function (in L2 norm).

References

Weisstein, Eric W. ``Chebyshev Polynomial of the First Kind." From MathWorld --- A Wolfram Web Resource. http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html

See Also

chebPoly, chebApprox

Examples

##  Chebyshev coefficients for x^2 + 1
n <- 4
f2 <- function(x) x^2 + 1
cC <- chebCoeff(f2, -1, 1, n)  #  3.0   0  0.5   0   0
cC[1] <- cC[1]/2               # correcting the absolute Chebyshev term
                               # i.e.  1.5*T_0 + 0.5*T_2
cP <- chebPoly(n)              # summing up the polynomial coefficients
p <- cC %*% cP                 #  0 0 1 0 1