Fredholm's determinant

I have to calculate some Fredholm's determinant (FD) for that I found one work (arXiv:0904.1581), where the algorithm for calculating the FD was described. I was trying to repeat the following code from the mentioned work (see page 39 from arXiv:0904.1581):

m = 64; 
[w, x] = ClenshawCurtis(0, inf, m); 
w2 = sqrt(w);
[xi, xj] = ndgrid(x, x);
K1 = @(x,y) airy((x + y) / 2) / 2;
F10 = det(eye(m) - (w2' * w2) .* K1(xi, xj))
>> F10 = 0.831908066202953

and

KAi = @AiryKernel;
F20 = det(eye(m) - (w2' * w2) .* KAi(x, x))
>> F20 = 0.969372828355262

The function ClenshawCurtis() has the following code:

function [w,c] = ClenshawCurtis(a, b, m)
        m = m - 1;
        c = cos((0 : m) * pi / m);
        M = [1 : 2 : m-1]'; l = length(M); n = m - l;
        v0 = [2 ./ M ./ (M-2); 1 / M(end); zeros(n, 1)];
        v2 = -v0(1 : end - 1) - v0(end : -1 : 2);
        
        g0 = -ones(m, 1); g0(1 + l) = g0(1 + l) + m; g0(1 + n) = g0(1 + n) + m;
        g = g0 / (m ^ 2 + mod(m, 2));
        w = ifft(v2 + g); w(m + 1) = w(1);
        c = ((1 - c) / 2 * a + (1 + c) / 2 * b)';
        
        w = ((b - a) * w / 2)';
end

And the expression for the @AiryKernel is given in arXiv:0904.1581 on page 2, formula (1.6). In could be defined as

function fun = test(x, y)
eps = 0.00001;
      if(x ~= y) 
         fun = ( airy(x) .* airy(1, y) - airy(y) .* airy(1, x) ) ./ (x-y);
      else
         x = y + eps;
            fun = ( airy(x) .* airy(1, y) - airy(y) .* airy(1, x) ) ./ (x - y);
      end    
end

When I try to run the same code, I get F10 = Nan and F20 = Nan.

What could be the problem?

Answer 1

[Ashutosh Singh Baghel]

Hi Bogdan,

I understand that the problem is related to the inputs you provided to the function 'clenshawCurtis' as 'a' = 0 and 'b' = 'inf'. Please try to provide a scalar number to 'b'.

Please refer to the file on the MATLAB Central File Exchange which performs adaptive Clenshaw-Curtis quadrature. Download the file located at the following URL:

http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=2905&objectType=FILE

Note that MathWorks does not guarantee or warrant the use or content of these submissions. Any questions, issues, or complaints should be directed to the contributing author.

[Bogdan MP]

Yes, the problem was with the limits.

Thank you for the answer!