How do I find the derivative of a spline curve in MATLAB 7.9 (R2009b)?

I am trying to find a derivative of a spline using a POLYDER function. The result looks wrong.

x=[0 600 1000 1381 1887 3087 4265 5367 6267 6987 ; 40 100 100 200 200 800 700 349 400 40];

pp=spline(x(1,:),x(2,:));

xx=linspace(x(1,1),x(1,10),2000);

erg=ppval(pp,xx);

plot(xx,erg,'k');

hold on

[breaks,coefs,l,k,d] = unmkpp(pp);

pp2 = mkpp(breaks,repmat(k-1:-1:1,d*l,1).*coefs(:,1:k-1),d);

y_der=polyder(ppval(pp2,linspace(x(1,1),x(1,10),2000)));

plot(linspace(x(1,1),x(1,10),2000-1),y_der,'r');

Answer 1

[MathWorks Support Team]

The POLYDER function is meant to be used to differentiate polynomials, and not piece-wise polynomials, such as splines.

If you have Spline Toolbox, you can use FNDER function to find the derivative of a spline:

pp=spline(x,y);

p_der=fnder(pp,1);

y_prime=ppval(p_der,x);

For more information on the FNDER function, please refer to the documentation by executing the following in the MATLAB command prompt:

doc fnder

If you do not have access to the Spline Toolbox, you can use UNMKPP function to break down your polynomial and then use MKPP function to assemble a new polynomial that will be a derivative of the first polynomial as in the following example:

% create points we will use to construct the spline

x = 0:10;

y = (x-5).^3+3*3+x+5;

% points the spline will be plotted at

xx = linspace(0,10,20);

% create the spline

pp = spline(x,y);

% plot the spline and the initial points used to create it

figure

plot(x,y,'o',xx,ppval(pp,xx))

hold on
% extract details from piece-wise polynomial by breaking it apart

[breaks,coefs,l,k,d] = unmkpp(pp);

% make the polynomial that describes the derivative

pp2 = mkpp(breaks,repmat(k-1:-1:1,d*l,1).*coefs(:,1:k-1),d);

% plot the derivative of the polynomial

plot(xx,ppval(pp2,xx),'-r')
% to calculate 2nd derivative differentiate the 1st derivative

[breaks,coefs,l,k,d] = unmkpp(pp2);

pp3 = mkpp(breaks,repmat(k-1:-1:1,d*l,1).*coefs(:,1:k-1),d);

plot(xx,ppval(pp3,xx),'-g')

For more information on the UNMKPP and MKPP functions please refer to the documentation by executing the following in the MATLAB command prompt:

doc unmkpp

doc mkpp

[Walter Roberson]

See also https://www.mathworks.com/matlabcentral/fileexchange/71225-splinefit for ppdiff.m