chebpol-package: Methods for creating multivariate interpolations on...
In chebpol: Multivariate Interpolation

Description Details Chebyshev Uniform grids Arbitrary grids Scattered data Support functions See Also Examples

The package contains methods for creating multivariate/multidimensional interpolations for real-valued functions on hypercubes. The methods include classical Chebyshev interpolation, multilinear and Floater-Hormann for arbitrary Cartesian product grids, and simplex linear and polyharmonic spline for scattered multi dimensional data.

The primary method of the package is ipol which dispatches to some other method. All the generated interpolants accept as an argument a matrix of column vectors. The generated functions also accept an argument threads=getOption('chebpol.threads') to utilize more than one CPU if a matrix of column vectors is evaluated. The option chebpol.threads is initialized from the environment variable CHEBPOL_THREADS upon loading of the package. It defaults to 1.

The interpolants are ordinary R-objects and can be saved with save() and loaded later with load() or serialized/unserialized with other tools, just like any R-object. However, they contain calls to functions in the package, and while the author will make efforts to ensure that generated interpolants are compatible with future versions of chebpol, I can issue no such absolute guarantee.

If we are free to evaluate the function to interpolate in arbitrary points, we can use a Chebyshev interpolation. The classical one is available with ipol(...,method='chebyshev').

There are several options if your function must be evaluated in a uniform grid. There is the Floater-Hormann rational interpolation available with ipol(...,method='fh'). There is a transformed Chebyshev variant ipol(..., method='uniform').

For grids which are not uniform, but still Cartesian products of one-dimensional grids, there is the Floater-Hormann interpolation ipol(...,method='fh'), and a transformed Chebyshev variant ipol(...,method='general'), as well as a multilinear ipol(...,method='multilinear'). These methods work on uniform grids as well. There is also a method='stalker' which is a shape preserving spline, and a variant method='hstalker'. Both are described in a vignette. These splines in one dimension attempts to honour monotonicity and local extrema, in particular the 'hstalker' is positivity-preserving in the sense that it will preserve maxima and minima.

For scattered data, not necessarily organised as a Cartesian product grid, there is a simplex linear interpolation available with ipol(...,method='simplexlinear'), and a polyharmonic spline with ipol(...,method='polyharmonic').

There are also functions for producing Chebyshev grids (chebknots) as well as a function for evaluating a function on a grid (evalongrid), and a function for finding the Chebyshev coefficients (chebcoef).

ipol, interpolant

## make some function values on a 50x50x50 grid
dims <- c(x=50,y=50,z=50)
f <- function(x) sqrt(1+x[1])*exp(x[2])*sin(5*x[3])^2
value <- evalongrid(f , dims)
##fit a Chebyshev approximation to it. Note that the value-array contains the
##grid-structure. 
ch <- ipol(value,method='cheb')
## To see the full grid, use the chebknots function and expand.grid
## Not run: 
head(cbind(expand.grid(chebknots(dims)), value=as.numeric(value),
      appx=as.numeric(evalongrid(ch,dims))))

## End(Not run)
## Make a Floater-Hormann approximation on a uniform grid as well
fh <- ipol(f,grid=lapply(dims,function(n) seq(-1,1,length.out=n)),method='fh',k=5)
## evaluate in some random points in R3
m <- matrix(runif(15,-1,1),3)
rbind(true=apply(m,2,f), cheb=ch(m), fh=fh(m))