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