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# ursell

## (PublicToolbox/regutools/ursell.m in BrainStorm 2.0 (Alpha))

### Function Synopsis

    [A,b] = ursell(n)

### Help Text

    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.

### Cross-Reference Information

This function is called by

    regudemo                           C:\BrainStorm_2001\PublicToolbox\regutools\regudemo.m

### Listing of function C:\BrainStorm_2001\PublicToolbox\regutools\ursell.m

    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, UNI-C, 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);

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