ursell.m

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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);