function [A,b] = ursell(n) %URSELL Test problem: integral equation wiht no square integrable solution. % % [A,b] = ursell(n) % % Discretization of a first kind Fredholm integral equation with % kernel K and right-hand side g given by %    K(s,t) = 1/(s+t+1) ,  g(s) = 1 , % where both integration itervals are [0,1]. % % Note: this integral equation has NO square integrable solution.  % Reference: F. Ursell, "Introduction to the theory of linear % integral equations", Chapter 1 in L. M. Delves & J. Walsh (Eds.), % "Numerical Solution of Integral Equations", Clarendon Press, 1974.  % Discretized by Galerkin method with orthonormal box functions.  % Per Christian Hansen, IMM, 09/16/92.  % Compute the matrix A. for k = 1:n   d1 = 1 + (1+k)/n; d2 = 1 + k/n; d3 = 1 + (k-1)/n;   c(k) = n*(d1*log(d1) + d3*log(d3) - 2*d2*log(d2));   e1 = 1 + (n+k)/n; e2 = 1 + (n+k-1)/n; e3 = 1 + (n+k-2)/n;   r(k) = n*(e1*log(e1) + e3*log(e3) - 2*e2*log(e2)); end A = hankel(c,r);  % Compute the right-hand side b. b = ones(n,1)/sqrt(n);