chebpol (version 1.3-952)

chebpol-package: Methods for creating multivariate Chebyshev interpolations on hypercubes

Description

The package contains methods for creating multivariate Chebyshev interpolations for real-valued functions on hypercubes. Some methods for non-Chebyshev grids are also provided.

concept

• Multivariate Chebyshev approximation
• Discrete Cosine Transform

Details

Given a real-valued function on a hypercube, or hyper-rectangle, it may be approximated by a (multivariate) Chebyshev polynomial. In the one-dimensional case, the Chebyshev approximation is constructed by evaluating the function in certain points, a Chebyshev grid, and fitting a polynomial to these values. Thus, one needs the function values on a set of prespecified points. The multivariate case is similar, the grid is the Cartesian product of one-dimensional grids. I.e. all combinations of all grid-points.

The Chebyshev coefficients for the interpolating polynomial in the one-dimensional case is a simple linear transform of the function values. The Chebyshev-transform, or Discrete Cosine Transform, being a variant of the Fourier transform, factors over tensor products, thus the multivariate transform is just a tensor product of several one-dimensional transforms. If http://fftw.org{FFTW} was available at compile time, chebpol uses it to generate the Chebyshev coefficients, otherwise a slower and more memory-demanding matrix method is used, and a warning message is issued when the package is attached. The Chebyshev-approximation is defined on the interval [-1,1], but it is straightforward to map any interval into [-1,1], thus making Chebyshev approximation on an interval of choice. Or, a hypercube of choice.

The primary method of the package is chebappx which takes as input the function values on the grid, possibly together with a hypercube specification in the form of a list of intervals. It produces a function which interpolates on the hypercube. There is also a wrapper called chebappxf which may be used if one has the function to be approximated rather than only its values in the grid-points. There is even an interpolation for uniform grids in ucappx, with a wrapper in ucappxf with interesting examples. And a more general for arbitrary Cartesian-product grids in chebappxg with a wrapper in chebappxgf. These are based on transforms of Chebyshev-polynomials. A multilinear interpolation is available in mlappx. And a polyharmonic spline interpolation in polyh. There are also functions for producing Chebyshev grids (chebknots) as well as a support function for evaluating a function on a grid (evalongrid), and a function for finding the Chebyshev coefficients (chebcoef).

Examples

## make some function values on a 3x3x4 grid
dims <- c(x=3,y=3,z=4)
value <- array(runif(36),dims)
##fit a Chebyshev approximation to it. Note that the value-array contains the
##grid-structure.  (However, we don't really approximate the runif-function :)
ch <- chebappx(value)
## To see the full grid, use the chebknots function and expand.grid
cbind(expand.grid(chebknots(dims)), value=as.numeric(value),
      appx=as.numeric(evalongrid(ch,dims)))
## evaluate between the grid points
ch(c(-0.3,0.2,0.16))