How to use the Output function of a Cubic Spline Interpolation?

I'm supposed to use cubic spline interpolation to approximate a function such as: exp(-1*(X^2)) then integrate the approximated function using different methods.

What I've been able to do so far is this: x=-10:1:10 y=exp(-1*(x.^2)); plot(x,y)

Then using the Curve fitting toolbox the approximated function shown on the graph is: y=-483e-020*x^3 - 0.00286*x^2+4.54e-018*x+0.18

How can I get the same function as an output in order to use it in the integration? Thanks in Advance!

Answer 1

[Dr. Seis]

Example:

x = 0:10;
y = 5 + 3*x.^2;
pp = spline(x,y);
int_y = quad(@(xx)ppval(pp,xx),x(1),x(end));

[Shady]

That's great!

Thank you very much, that is much appreciated! :)

(Edit: If you could explain the last command line it would be even more helpful as I am not familiar except with the quad function.

[Dr. Seis]

I found (and modified) the example inside "doc ppval"

The part with the "@(xx)" is where a prototype (I am not sure what the technical term for it would be) of the equation is generated. For example, you could define an equation as:

myfun = @(xx)5+3*xx.^2;

Then:

int_y = quad(myfun,0,10); % which will also yield 1050

The equation prototype "myfun" does not store any data. So to see the result of evaluating "myfun" for "x" would give:

myvals = myfun(x); % which would yield [5 8 17 32 53 80 113 152 197 248 305] for x = 1:10

[Shady]

Thank you again :)

Answer 2

[Mike Hosea]

Just wanted to add that the best way of integrating a piecewise-defined function in MATLAB is

quadgk(@(t)ppval(pp,t),x(1),x(end),'Waypoints',x)

where x is, as in the above examples, the vector of locations where the pieces of the function are joined. Supplying these "waypoints" to QUADGK allows it to integrate efficiently over any discontinuities or limited smoothness where the pieces are joined. If there is, in fact, some of that limited smoothness there, you will typically find that using QUADGK like that is significantly faster and more accurate.

Answer 3

[Christopher Crawford]

Piecewise polynomials like cubic splines can be integrated analytically. This is implemented by 'ppint' (8ve), or 'fnint' (Curvefitting Toolbox), see https://www.mathworks.com/matlabcentral/answers/394457-piecewise-polynomial-integration-ppint

diff(ppval( ppint(pp), x([1,end]) ))