# runge2.s                         J F Monahan, June 2001
 # second example of runge phenomenon 
 #  with unequally spaced points
 npts <- 100             # number of points to evaluate
 n <- 5                  # number of interpolation points
 x <- 4*cos( ( 2*seq(n)-1 )*pi/(2*n) )
 "interpolation points"
 x
 b <- 1/(1 + x*x)
 "values of f(x)"
 b
 # make matrix of polynomial interpolation
 A <- matrix(0,n,n)
 A[,1] <- 1
 for(j in 2:n) A[,j] <- x*A[,j-1]
 # solve to get polynomial coefficients in c
 c <- solve(A,b)
 xi <- 4.1*( 2*seq(npts)-npts-1)/(npts-1)
 fxi <- 1/(1 + xi*xi)
 # Horner's rule for evaluating polynomial
 horner <- function(u) {       # actually degree n-1 polynomial
   v <- 0
   for( k in n:1) v <- u*v + c[k]
   v
                         }
 y <- c( sapply(xi,horner) )
 xy <- paste( 'xi', format(xi),'fxi',format(fxi),'y',format(y) )
 cat(xy,sep="\n")
 rm(list=ls())