chebappxg: Interpolation on a non-Chebyshev grid

In chebpol: Multivariate Interpolation

Description

A poor-man's approximation on non-Chebyshev grids. If you for some reason can't evaluate your function on a Chebyshev-grid, but instead have some other grid which still is a Cartesian product of one-dimensional grids, you may use this function to create an interpolation.

Usage

chebappxg(...)

chebappxgf(...)

Arguments

...	Further arguments to fun.
val	Array. Function values on a grid.
grid	A list. Each element is a sorted vector of grid-points for a dimension. These need not be Chebyshev-knots, nor evenly spaced.
fun	The function to be approximated.
mapdim	Deprecated.

Details

A call fun <- chebappxg(val,grid) does the following. A Chebyshev interpolation ch for val is created, on the [-1,1] hypercube. For each dimension a grid-map function gm is created which maps the grid-points monotonically into Chebyshev knots. For this, the function splinefun with method='hyman' is used. When fun(x) is called, it translates to ch(gm(x)). For uniform grids, the function ucappx will produce a faster interpolation in that a closed form gm is used.

chebappxgf is used if the function, rather than its values, is available. The function will be evaluated on the grid.

Even though this approach works in simple cases it is not a panacea. The grid in each dimension should probably not be too irregularly spaced. I.e. short and long gaps interspersed is likely to cause problems.

Value

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

Examples

## Not run: 
## evenly spaced grid-points
su <- seq(0,1,length.out=10)
## irregularly spaced grid-points
s <- su^3
## create approximation on the irregularly spaced grid
ch <- Vectorize(chebappxg(exp(s),list(s)))
## test it:
ch(su) - exp(su)
# try one with three variables
f <- function(x) exp(-sum(x^2))
grid <- list(s,su,su^2)
ch2 <- chebappxg(evalongrid(f,grid=grid),grid)
# test it at 10 random points
replicate(10,{a<-runif(3); ch2(a)-f(a)})

# Try Runge's function on a uniformly spaced grid.
# Ordinary polynomial fitting of high degree of Runge's function on a uniform grid
# creates large oscillations near the end of the interval. Not so with chebappxgf
f <- function(x) 1/(1+25*x^2)
chg <- Vectorize(chebappxgf(f,seq(-1,1,length.out=15)))
# also compare with Chebyshev interpolation
ch <- Vectorize(chebappxf(f,15))
\dontrun{
 # plot it
 s <- seq(-1,1,length.out=200)
 plot(s, f(s), type='l', col='black')
 lines(s, chg(s), col='blue')
 lines(s, ch(s), col='red')
 legend('topright',
        legend=c('Runge function','chebappxg on uniform grid','Chebyshev'),
        col=c('black','blue','red'), lty=1)
}

## End(Not run)

chebpol documentation built on Dec. 9, 2019, 5:08 p.m.