R: Chebyshev Polynomials

chebPoly {pracma} | R Documentation

## Chebyshev Polynomials

### Description

Chebyshev polynomials and their values.

### Usage

    chebPoly(n, x = NULL)

### Arguments

`n` | an integer `>= 0`.
`x` | a numeric vector, possibly empty; default `NULL`.

### Details

Determines an (n+1)-ny-(n+1)-Matrix of Chebyshev polynomials up to degree n.

The coefficients of the first `n` Chebyshev polynomials are computed using the recursion formula. For computing any values at points the well known Horner schema is applied.

### Value

If `x` is `NULL`, returns an `(n+1)`-by-`(n+1)` matrix with the coefficients of the first Chebyshev polynomials from `0` to `n`, one polynomial per row with coefficients from highest to lowest order.

If `x` is a numeric vector, returns the values of the `n`-th Chebyshev polynomial at the points of `x`.

### Note

See the “Chebfun Project” <http://www.maths.ox.ac.uk/chebfun/> by Nick Trefethen.

### References

Carothers, N. L. (1998). A Short Course on Approximation Theory. Bowling Green State University, URL: http://personal.bgsu.edu/~carother/Approx.html.

### See Also

`chebCoeff`, `chebApprox`

### Examples

    chebPoly(6)

    ## Not run:
    ##  Plot 6 Chebyshev Polynomials
    plot(0, 0, type="n", xlim=c(-1, 1), ylim=c(-1.2, 1.2),
        main="Chebyshev Polynomials for n=1..6", xlab="x", ylab="y")
    grid()
    x <- seq(-1, 1, length.out = 101)
    for (i in 1:6) {
        y <- chebPoly(i, x)
        lines(x, y, col=i)
    }
    legend(x = 0.55, y = 1.2, c("n=1", "n=2", "n=3", "n=4", "n=5", "n=6"),
        col = 1:6, lty = 1, bg="whitesmoke", cex = 0.75)

    ## End(Not run)

[Package pracma version 2.0.4 Index]