Chebyshev Approximation

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

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.

Details

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

Value

A numeric vector of the same length as x.

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.

Note

TODO: Evaluate the Chebyshev approximative polynomial by using the Clenshaw recurrence formula. (Not yet vectorized, that's why we still use the Horner scheme.)

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
#> [1] 0.006925271
max(abs(ys-yc))                       # 1.151207e-05
#> [1] 1.151207e-05

if (FALSE) { # \dontrun{
# Plot the corresponding curves
plot(x, ys, type = "l", col = "gray", lwd = 5)
lines(x, yp, col = "navy")
lines(x, yc, col = "red")
grid()} # }