# Optimization of a parameterised integral

Nick Hale, 17th October 2011

(Chebfun example opt/OptimInt.m)

This example serves to show how easy it is to solve one of the example problems from the Oxford MSc in Mathematical Modelling and Scientific Computing week 0 MATLAB 'Crash Course' using Chebfun. (And also how easy it is to make a Chebfun Example!).

PROBLEM. For what values of a does I(a) = int_{-1}^{1} sin(x) + sin(a x^2) dx = 1 ?

SOLUTION. Define the integrand as a function of x and a.

F = @(x,a) sin(x) + sin(a*x.^2);


For a given a, we can compute the integral using Chebfun's SUM command.

I = @(a) sum(chebfun(@(x) F(x,a)));


We compute the a chebfun of this result, for a ranging from 0 to 100.

Ia = chebfun(@(a) I(a),'vectorize',[0 100]);


The 'vectorize' flag is used, since I should not be evaluated with a vector.

We use Chebfun's ROOTS command to find where Ia equals 1.

r = roots(Ia-1)

r =
2.0117
3.1995


We plot this, to make sure it looks sensible.

plot(Ia,'linewidth',2), hold on
plot(r,Ia(r),'xr','linewidth',2);


Since we have Ia as a chebfun, we can do other things, like find where I(a) = 0.25

r = roots(Ia-0.25)
plot(r,Ia(r),'xm','linewidth',2); hold off

r =
0.3789
18.2260
19.7612
23.7538
26.9563
29.2915


or the value of a which maximises I

m = max(Ia)

m =
1.0567


or the standard deviation of the gaps between the local minima for a between [0 100].

[y x] = min(Ia,'local');
f = std(diff(x(2:end-1)))

f =
0.0090