# chebApprox: Chebyshev Approximation

### Description

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

### Usage

 `1` ```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`.

### 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.)

### 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.

`polyApprox`

### Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23``` ```# 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) ```

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