chebCoeff: Chebyshev Polynomials

In pracma: Practical Numerical Math Functions

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.

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).

Value

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

Note

See the “Chebfun Project” <https://www.chebfun.org/> by Nick Trefethen.

References

Weisstein, Eric W. “Chebyshev Polynomial of the First Kind." From MathWorld — A Wolfram Web Resource. https://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

pracma documentation built on March 19, 2024, 3:05 a.m.