Why do I receive a "Singularity likely" warning when I use the QUAD function in MATLAB

For example when I execute the following command:

for i=1:10

    for j=1:10

        K(i,j)=quad('func',0,1,[],[],i,j);

    end

end

function k=func(x,i,j)

B=[4.73 7.905 11 14.14 17.28 20.42 23.56 26.72 29.86 32.98];

k=(B(i)*(-sin(B(i)*x)+sinh(B(i)*x)-((cos(B(i))-cosh(B(i)))/((sin(B(i))-sinh(B(i)))))*(cos(B(i)*x)+cosh(B(i)*x)))).*(B(j)*(-sin(B(j)*x)+sinh(B(j)*x)-((cos(B(j))-cosh(B(j)))/((sin(B(j))-sinh(B(j)))))*(cos(B(j)*x)+cosh(B(j)*x))));

I receive the following warning message:

Warning: Maximum function count exceeded; singularity likely

what is the problem?

Answer 1

[John D'Errico]

One problem is you have provided us with code for func, but you have quad calling sunc. I imagine that would leave MATLAB "singularly" confused. :)

OK, so assuming you simply cannot type well, look at your function. I'll modify it to be a function handle. I also changed some of the operators from * to .* and / to ./, just to be careful.

B=[4.73 7.905 11 14.14 17.28 20.42 23.56 26.72 29.86 32.98];

k = @(x,i,j) (B(i)*(-sin(B(i)*x)+sinh(B(i)*x)-((cos(B(i))-cosh(B(i)))./((sin(B(i))-sinh(B(i))))).*(cos(B(i)*x)+cosh(B(i)*x)))).*(B(j)*(-sin(B(j)*x)+sinh(B(j)*x)-((cos(B(j))-cosh(B(j)))./((sin(B(j))-sinh(B(j))))).*(cos(B(j)*x)+cosh(B(j)*x))));

So see what happens when you try to do the integration. Here, I plot each function as well as integrate it.

for i = 1:10
  for j = 1:10
    disp([i,j,integral(@(x) k(x,i,j),0,1,'RelTol',1e-10)])
    ezplot(@(x) k(x,i,j),[0 1])
    pause
  end
end

These are sufficiently oscillatory functions, with what appears to be a fair amount of subtractive cancellation involved for larger i and j, that numerical integration tools end up with problems.