For illustration, here is a spline interpolant using just a few nodes.

f = @(x) exp(sin(7*x));

fplot(f,[0,1])

t = [0, 0.075, 0.25, 0.55, 0.7, 1]'; % nodes

y = f(t); % values at nodes

hold on, plot(t,y,'.')

S = spinterp(t,y);

fplot(S,[0,1])

xlabel('x'), ylabel('f(x)') % ignore this line

title('Function and cubic spline interpolant') % ignore this line

Now we look at the convergence rate as the number of nodes increases.

x = linspace(0,1,10001)'; % sample the difference at many points

n_ = 2.^(3:0.5:8)';

err_ = 0*n_;

for i = 1:length(n_)

n = n_(i);

t = linspace(0,1,n+1)'; % interpolation nodes

S = spinterp(t,f(t));

err = norm( f(x) - S(x), inf );

err_(i) = err;

end

Since we expect convergence that is , we use a log-log graph of error and expect a straight line of slope .

clf, loglog( n_, err_, '.-' )

hold on, loglog( n_, (n_/n_(1)).^(-4), '--' )

xlabel('n'), ylabel('||f-S||_\infty'), axis tight % ignore this line

title('Convergence of spline interpolation') % ignore this line

legend('error','4th order') % ignore this line