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# baart

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

### Function Synopsis

    [A,b,x] = baart(n)

### Help Text

    BAART Test problem: Fredholm integral equation of the first kind.

     [A,b,x] = baart(n)

     Discretization of a first kind Fredholm integral equation with
     kernel K and right-hand side g given by
        K(s,t) = exp(s*cos(t)) ,  g(s) = 2*sinh(s)/s ,
     and with integration intervals  s in [0,pi/2] ,  t in [0,pi] .
     The solution is given by
        f(t) = sin(t) .

     The order n must be even.

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

    function [A,b,x] = baart(n)
    %BAART Test problem: Fredholm integral equation of the first kind.
    %
    % [A,b,x] = baart(n)
    %
    % Discretization of a first kind Fredholm integral equation with
    % kernel K and right-hand side g given by
    %    K(s,t) = exp(s*cos(t)) ,  g(s) = 2*sinh(s)/s ,
    % and with integration intervals  s in [0,pi/2] ,  t in [0,pi] .
    % The solution is given by
    %    f(t) = sin(t) .
    %
    % The order n must be even.

    % Reference: M. L. Baart, "The use of auto-correlation for pseudo-
    % rank determination in noisy ill-conditioned linear least-squares
    % problems", IMA J. Numer. Anal. 2 (1982), 241-247.

    % Discretized by Galerkin method with orthonormal box functions;
    % one integration is exact, the other is done by Simpson's rule.

    % Per Christian Hansen, UNI-C, 09/16/92.

    % Check input.
    if (rem(n,2)~=0), error('The order n must be even'), end

    % Generate the matrix.
    hs = pi/(2*n); ht = pi/n; c = 1/(3*sqrt(2));
    A = zeros(n,n); ihs = [0:n]'*hs; n1 = n+1; nh = n/2;
    f3 = exp(ihs(2:n1)) - exp(ihs(1:n));
    for j=1:n
      f1 = f3; co2 = cos((j-.5)*ht); co3 = cos(j*ht);
      f2 = (exp(ihs(2:n1)*co2) - exp(ihs(1:n)*co2))/co2;
      if (j==nh)
        f3 = hs*ones(n,1);
      else
        f3 = (exp(ihs(2:n1)*co3) - exp(ihs(1:n)*co3))/co3;
      end
      A(:,j) = c*(f1 + 4*f2 + f3);
    end

    % Generate the right-hand side.
    if (nargout>1)
      si(1:2*n) = [.5:.5:n]'*hs; si = sinh(si)./si;
      b = zeros(n,1);
      b(1) = 1 + 4*si(1) + si(2);
      b(2:n) = si(2:2:2*n-2) + 4*si(3:2:2*n-1) + si(4:2:2*n);
      b = b*sqrt(hs)/3;
    end

    % Generate the solution.
    if (nargout==3)
      x = -diff(cos([0:n]'*ht))/sqrt(ht);
    end

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