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