Question

I’m trying to use fsolve in matlab to solve a system of nonlinear equations numerically. Here is a test sample of my program, k1 and R are parameters and x0 is the start point.

function y=f(k1, R, x0)

pair=fsolve(@system,x0);

y=pair(1);

    function r=system(v)

        int1=@(x) exp(k1*x);
        int2=@(x) exp(k1*x^2)/(x^4);


        r(1)=exp(v(1))*quadl(int1,0,v(1));     
        r(2)=exp(k1*v(2))*quadl(int2,v(1),20)*k1*R;

    end

end

The strange thing is when I run this program, matlab keeps telling me that I should use .^ instead of ^ in int2=@(x) exp(k1*x^2)/(x^4). I am confused because the x in that function handle is supposed to be a scalar when it is used by quadl. Why should I have to use .^ in this case?

Also I see that a lot of the examples provided in online documentations also use .^ even though they are clearly taking power of a scalar, as in here. Can anybody help explain why?

Thanks in advance.

Answer 1

in the function int2 you have used matrix power (^) where you should use element-wise power (.^). Also, you have used matrix right division (/) where you should use element-wise division (./). This is needed, since quadl (and friends) will evaluate the integrand int2 for a whole array of x‘s at a time for reasons of efficiency.

So, use this:

function y = f(k1, R, x0)

    pair = fsolve(@system,x0);

    y = pair(1);

    function r = system(v)

        int1 = @(x) exp(k1*x);
        int2 = @(x) exp(k1*x.^2)./(x.^4);

        r(1) = exp(   v(1)) * quadl(int1,0,v(1));     
        r(2) = exp(k1*v(2)) * k1*R*quadl(int2,v(1),20);

    end

end

Also, have a look at quadgk or integral (if you’re on newer Matlab).

By the way, I assume your real functions int1 and int2 are different functions? Because these functions are of course trivial to solve analytically…