pracma (version 1.9.9)

chebApprox: Chebyshev Approximation

Description

Function approximation through Chebyshev polynomials (of the first kind).

Usage

chebApprox(x, fun, a, b, n)

Arguments

x

Numeric vector of points within interval [a, b].

fun

Function to be approximated.

a, b

Endpoints of the interval.

n

An integer >= 0.

Value

A numeric vector of the same length as x.

Details

Return approximate y-coordinates of points at x by computing the Chebyshev approximation of degree n for fun on the interval [a, b].

References

Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery (1992). Numerical Recipes in C: The Art of Scientific Computing. Second Edition, Cambridge University Press.

See Also

polyApprox

Examples

# Approximate sin(x) on [-pi, pi] with a polynomial of degree 9 !
# This polynomial has to be beaten:
# P(x) = x - 1/6*x^3 + 1/120*x^5 - 1/5040*x^7 + 1/362880*x^9

# Compare these polynomials
p1 <- rev(c(0, 1, 0, -1/6, 0, 1/120, 0, -1/5040, 0, 1/362880))
p2 <- chebCoeff(sin, -pi, pi, 9)

# Estimate the maximal distance
x  <- seq(-pi, pi, length.out = 101)
ys <- sin(x)
yp <- polyval(p1, x)
yc <- chebApprox(x, sin, -pi, pi, 9)
max(abs(ys-yp))                       # 0.006925271
max(abs(ys-yc))                       # 1.151207e-05

## Not run: 
# # Plot the corresponding curves
# plot(x, ys, type = "l", col = "gray", lwd = 5)
# lines(x, yp, col = "navy")
# lines(x, yc, col = "red")
# grid()## End(Not run)