chebpol (version 2.1-2)

ucappx: Interpolation on a uniform grid

Description

A poor-man's approximation on uniform grids. If you for some reason can't evaluate your function on a Chebyshev-grid, but instead have a uniform grid, you may use this function to create an interpolation.

Usage

ucappx(...)
ucappxf(...)

Arguments

...

Further arguments to fun.

val

Array. Function values on a grid.

intervals

List of vectors of length two. Specifying the hypercube extent in each dimension

fun

Function to be interpolated.

dims

Integer. Number of grid points in each dimension.

Value

A function(x) defined on the hypercube, approximating the given function.

Details

This does about the same as chebappxg for unform grids, though no grid map function is constructed, as a fixed such function is used.

A Chebyshev-interpolation ch is made for val with chebappx. Upon evaluation the uniform grid in each dimension is mapped differentiably to the Chebyshev-knots so that ch is evaluated in \(sin(\frac{\pi }{sin(0.5*pi*x*(1-n)/n)}\eqn{ x(1-n)}{2n})\) where n is the number of knots in the dimension, possibly after x has been remapped from the hypercube interval to [-1,1].

Thus, the interpolation is not a polynomial.

For ucappx the function values are provided, the number of grid points in each dimension is to be found in dim(val). For ucappxf the function to be interpolated is fun, and the number of grid points is passed in dims.

As the example shows, this approximation is better than the Chebyshev approximation for some functions.

Examples

# NOT RUN {
# }
# NOT RUN {
# Runge function
f <- function(x) 1/(1+25*x^2)
grid <- seq(-1,1,length.out=15)
val <- f(grid)
uc <- Vectorize(ucappx(val))
# and the Chebyshev
ch <- Vectorize(chebappxf(f,15))
# test it at 10 random points
t(replicate(10,{a<-runif(1,-1,1); c(arg=a, uc=uc(a), true=f(a), cheb=ch(a))}))
# }