function [A,b,x] = spikes(n,t_max) %SPIKES Test problem with a "spiky" solution. % % [A,b,x] = spikes(n,t_max) % % Artificially generated discrete ill-posed problem. % % The solution x consists of a unit step at t = .5, and a pulse train % of spikes of decrasing magnitude at t = .5, 1.5, 2.5, ... % % The parameter t_max is optional; its default value is 5. % It controls the length of the pulse train.  % Per Christian Hansen, IMM, 04/21/97.  % Initialization. if (nargin == 1), t_max = 5; end t = t_max*[1:n]/n; del = t_max/n;  % Compute the matrix A. [t,sigma] = meshgrid(del:del:t_max,del:del:t_max); A = sigma./(2*sqrt(pi*t.^3)).*exp(-(sigma.^2)./(4*t));  % Compute the right-hand side b and the solution x. if (nargout > 1)   heights = 2*ones(t_max,1); heights(1) = 25;   heights(2) = 9; heights(3) = 5; heights(4) = 4; heights(5) = 3;   x = zeros(n,1); n_h = 1;   peak = 0.5/t_max; peak_dist = 1/t_max;   if (peak < 1)     n_peak = round(peak*n); x(n_peak) = heights(n_h);     x(n_peak+1:n) = ones(n-n_peak,1);     peak = peak + peak_dist; n_h = n_h + 1;   end   while (peak < 1)     x(round(peak*n)) = heights(n_h);     peak = peak + peak_dist; n_h = n_h + 1;   end   b = A*x; end