Example for interpolation with piecewise cubic functions
Function 
on 
with 7 equidistant nodes
Consider the function 
. The derivative is 
.
. The derivative is .We compute the function values and the derivatives at the nodes.
We will use 1000 equidistant points xe on [-5,5] for plotting and finding the maximal errors.
clearvars; clf % clear variables and figures
a = -5; b = 5;
x = linspace(a,b,7); % 7 equidistant nodes
y = 1./(1+x.^2); % function values at nodes
s = -2*x./(1+x.^2).^2; % derivatives at nodes
xe = linspace(a,b,1e3); % evaluation points
ye = 1./(1+xe.^2); % function values at xe
plot(x,y,'o',xe,ye,'k:')
title('1/(1+x^2)')
Piecwise cubic Hermite interpolation
The piecewise cubic Hermite function uses function values and derivatives at all nodes.
You need to download the m-file hermite.m from the course web page.
yherm = hermite(x,y,s,xe); % find pw cubic Hermite interpolation and evaluate at points xe
subplot(2,1,1);
plot(x,y,'o',xe,ye,'k:',xe,yherm,'b');
title('pw cubic Hermite interpolation')
legend('given points','f(x)','Hermite')
subplot(2,1,2)
plot(x,0*y,'o',xe,yherm-ye,'b'); grid on
title('Errors')
maxerr_herm = max(abs(yherm-ye))
Complete cubic spline
The complete spline uses function values at the nodes, and the derivative at the first and last node.
sa = s(1); sb = s(end); % slopes at left and right endpoint
ycomp = spline(x,[sa,y,sb],xe); % find complete cubic spline and evaluate at points xe
subplot(2,1,1);
plot(x,y,'o',xe,ye,'k:',xe,ycomp,'r');
title('Complete cubic spline')
legend('given points','f(x)','compl. spline')
subplot(2,1,2)
plot(x,0*y,'o',xe,ycomp-ye,'r'); grid on
title('Errors')
maxerr_complete = max(abs(ycomp-ye))
Not-a-knot cubic spline
The not-a-knot spline uses only the function values at the nodes.
ynak = spline(x,y,xe); % find not-a-knot cubic spline and evaluate at points xe
subplot(2,1,1);
plot(x,y,'o',xe,ye,'k:',xe,ynak,'g');
title('Not-a-knot cubic spline')
legend('given points','f(x)','not-a-knot spline')
subplot(2,1,2)
plot(x,0*y,'o',xe,ynak-ye,'g'); grid on
title('Errors')
maxerr_nak = max(abs(ynak-ye))
Comparison
The Hermite function gives the smallest error (it uses the most information about function f).
The not-a-knot spline performs worse than the complete spline near the endpoints (the complete spline uses the derivatives at the endpoints).
subplot(2,1,1)
plot(x,y,'o',xe,ye,'k:',xe,yherm,'b',xe,ycomp,'r',xe,ynak,'g')
legend('given points','f(x)','Hermite','complete spline','not-a-knot spline')
subplot(2,1,2)
plot(x,0*y,'o',xe,yherm-ye,'b',xe,ycomp-ye,'r',xe,ynak-ye,'g'); grid on
title('Errors')
legend('nodes','Hermite','complete spline','not-a-knot spline')
