Numerical Integration Problems with Product Rule due to differnet resolution

Question

I am facing some problem during calculation of Numerical Integration with two data set. For integration i am using simpsons 1/3 rule.

    function I = Simpsons(f,a,b,n)
    if numel(f)>1 % If the input provided is a vector
        n=numel(f)-1; h=(b-a)/n;
        I= h/3*(f(1)+2*sum(f(3:2:end-2))+4*sum(f(2:2:end))+f(end));
    else 
        h=(b-a)/n; xi=a:h:b;
        I= h/3*(f(xi(1))+2*sum(f(xi(3:2:end-2)))+4*sum(f(xi(2:2:end)))+f(xi(end)));
    end

This code correctly calculates the integration.

Now the problem occurs during the calculation of multiplied values.

for example I have two functions f and g both are depending on same variable. the variables is in the same ranges. SO lower Limit and Upper Limit is same.

$\int_a^b \! f(x) *g(x) \, \mathrm{d} x.$

here the resolution of x is different. Means for f(x) we have 1000 data where as for g(x) we have 1700 data points. so element by element multiplication cant be done.

How to Solve this integration ..

Answer 1

you'll need to interpolate one of your functions, f or g, to the other function points, for 1D functions that is achievable using interp1.

For example:

% x1 an x2 have the same limits but different # of elements

 x1 = linspace(-10,10,100); 
 x2 = sort(rand(1,170)*20-10); # non-unifrom points from -10 to 10

 f1 = sin(x1);
 f2 = cos(x2);

now say we want to multiply f1*f2, we need them to have the # of elements, so

 f2i= interp1(x2,f2,x1,'spline');

will make f2 to have the same # of elements as f1, or instead

 f1i= interp1(x1,f1,x2,'spline');

will make f1 to have the same # of elements as f2.