function [A,b,x] = shaw(n) %SHAW Test problem: one-dimensional image restoration model. % % [A,b,x] = shaw(n) % % Discretization of a first kind Fredholm integral equation with % [-pi/2,pi/2] as both integration intervals. The kernel K and % the solution f, which are given by % K(s,t) = (cos(s) + cos(t))*(sin(u)/u)^2 % u = pi*(sin(s) + sin(t)) % f(t) = a1*exp(-c1*(t - t1)^2) + a2*exp(-c2*(t - t2)^2) , % are discretized by simple quadrature to produce A and x. % Then the discrete right-hand b side is produced as b = A*x. % % The order n must be even. % Reference: C. B. Shaw, Jr., "Improvements of the resolution of % an instrument by numerical solution of an integral equation", % J. Math. Anal. Appl. 37 (1972), 83-112. % Per Christian Hansen, IMM, 08/20/91. % Check input. if (rem(n,2)~=0), error('The order n must be even'), end % Initialization. h = pi/n; A = zeros(n,n); % Compute the matrix A. co = cos(-pi/2 + [.5:n-.5]*h); psi = pi*sin(-pi/2 + [.5:n-.5]*h); for i=1:n/2 for j=i:n-i ss = psi(i) + psi(j); A(i,j) = ((co(i) + co(j))*sin(ss)/ss)^2; A(n-j+1,n-i+1) = A(i,j); end A(i,n-i+1) = (2*co(i))^2; end A = A + triu(A,1)'; A = A*h; % Compute the vectors x and b. a1 = 2; c1 = 6; t1 = .8; a2 = 1; c2 = 2; t2 = -.5; if (nargout>1) x = a1*exp(-c1*(-pi/2 + [.5:n-.5]'*h - t1).^2) ... + a2*exp(-c2*(-pi/2 + [.5:n-.5]'*h - t2).^2); b = A*x; end