Convergence of PL interpolation¶
We measure the convergence rate for piecewise linear interpolation of \(e^{\sin 7x}\).
using FundamentalsNumericalComputation
f = x -> exp(sin(7*x))
x = (0:10000)/1e4 # sample the difference at many points
n = @. 2^(3:10)
err = zeros(0)
for n in n
t = (0:n)/n # interpolation nodes
p = FNC.plinterp(t,f.(t))
dif = @.f(x)-p(x)
push!(err,norm(dif,Inf) )
end
pretty_table((n=n,error=err),backend=:html)
| n | error |
|---|---|
| Int64 | Float64 |
| 8 | 0.21602990323384974 |
| 16 | 0.06381730414968168 |
| 32 | 0.016038182942837764 |
| 64 | 0.004058789402869412 |
| 128 | 0.0010155253347479132 |
| 256 | 0.0002540280995928512 |
| 512 | 6.348972346570392e-5 |
| 1024 | 1.5876989992058554e-5 |
Since we expect convergence that is \(O(h^2)=O(n^{-2})\), we use a log-log graph of error and expect a straight line of slope \(-2\).
order2 = @. 0.1*(n/n[1])^(-2)
plot(n,[err order2],m=:o,l=[:solid :dash],label=["error" "2nd order"],
xaxis=(:log10,"n"), yaxis=(:log10,"\$\\| f-p \\|_\\infty\$"),
title="Convergence of PL interpolation")