function [x_lambda,rho,eta,data,X] = maxent(A,b,lambda,w,x0)%MAXENT Maximum entropy regularization.%% [x_lambda,rho,eta] = maxent(A,b,lambda,w,x0)%% Maximum entropy regularization:%    min { || A x - b ||^2 + lambda^2*x'*log(diag(w)*x) } ,% where -x'*log(diag(w)*x) is the entropy of the solution x.% If no weights w are specified, unit weights are used.%% If lambda is a vector, then x_lambda is a matrix such that%    x_lambda = [x_lambda(1), x_lambda(2), ... ] .%% This routine uses a nonlinear conjugate gradient algorithm with "soft"% line search and a step-length control that insures a positive solution.% If the starting vector x0 is not specified, then the default is%    x0 = norm(b)/norm(A,1)*ones(n,1) .% Per Christian Hansen, IMM and Tommy Elfving, Dept. of Mathematics,% Linkoping University, 06/10/92.% Reference: R. Fletcher, "Practical Methods for Optimization",% Second Edition, Wiley, Chichester, 1987.% Set defaults.flat = 1e-3;     % Measures a flat minimum.flatrange = 10;  % How many iterations before a minimum is considered flat.maxit = 150;     % Maximum number of CG iterations;minstep = 1e-12; % Determines the accuracy of x_lambda.sigma = 0.5;     % Threshold used in descent test.tau0 = 1e-3;     % Initial threshold used in secant root finder.% Initialization.[m,n] = size(A); x_lambda = zeros(n,length(lambda)); F = zeros(maxit,1);if (min(lambda) <= 0)  error('Regularization parameter lambda must be positive')endif (nargin ==3), w  = ones(n,1); endif (nargin < 5), x0 = ones(n,1); end% Treat each lambda separately.for j=1:length(lambda);  % Prepare for nonlinear CG iteration.  l2 = lambda(j)^2;  x  = x0; Ax = A*x;  g  = 2*A'*(Ax - b) + l2*(1 + log(w.*x));  p  = -g;  r  = Ax - b;  % Start the nonlinear CG iteration here.  delta_x = x; dF = 1; it = 0; phi0 = p'*g;  while (norm(delta_x) > minstep*norm(x) & dF > flat & it < maxit & phi0 < 0)    it = it + 1;    % Compute some CG quantities.    Ap = A*p; gamma = Ap'*Ap; v = A'*Ap;    % Determine the steplength alpha by "soft" line search in which    % the minimum of phi(alpha) = p'*g(x + alpha*p) is determined to    % a certain "soft" tolerance.    % First compute initial parameters for the root finder.    alpha_left = 0; phi_left = phi0;    if (min(p) >= 0)      alpha_right = -phi0/(2*gamma);      h = 1 + alpha_right*p./x;    else      % Step-length control to insure a positive x + alpha*p.      I = find(p < 0);      alpha_right = min(-x(I)./p(I));      h = 1 + alpha_right*p./x; delta = eps;      while (min(h) <= 0)        alpha_right = alpha_right*(1 - delta);        h = 1 + alpha_right*p./x;        delta = delta*2;      end    end    z = log(h);    phi_right = phi0 + 2*alpha_right*gamma + l2*p'*z;    alpha = alpha_right; phi = phi_right;    if (phi_right <= 0)      % Special treatment of the case when phi(alpha_right) = 0.      z = log(1 + alpha*p./x);      g_new = g + l2*z + 2*alpha*v; t = g_new'*g_new;      beta = (t - g'*g_new)/(phi - phi0);    else      % The regular case: improve the steplength alpha iteratively      % until the new step is a descent step.      t = 1; u = 1; tau = tau0;      while (u > -sigma*t)        % Use the secant method to improve the root of phi(alpha) = 0        % to within an accuracy determined by tau.        while (abs(phi/phi0) > tau)          alpha = (alpha_left*phi_right - alpha_right*phi_left)/...                  (phi_right - phi_left);          z = log(1 + alpha*p./x);          phi = phi0 + 2*alpha*gamma + l2*p'*z;          if (phi > 0)            alpha_right = alpha; phi_right = phi;          else            alpha_left  = alpha; phi_left  = phi;          end        end        % To check the descent step, compute u = p'*g_new and        % t = norm(g_new)^2, where g_new is the gradient at x + alpha*p.        g_new = g + l2*z + 2*alpha*v; t = g_new'*g_new;        beta = (t - g'*g_new)/(phi - phi0);        u = -t + beta*phi;        tau = tau/10;      end  % End of improvement iteration.    end  % End of regular case.        % Update the iteration vectors.    g = g_new; delta_x = alpha*p;    x = x + delta_x;    p = -g + beta*p;    r = r + alpha*Ap;    phi0 = p'*g;    % Compute some norms and check for flat minimum.    rho(j,1) = norm(r); eta(j,1) = x'*log(w.*x);    F(it) = rho(j,1)^2 + l2*eta(j,1);    if (it <= flatrange)      dF = 1;    else      dF = abs(F(it) - F(it-flatrange))/abs(F(it));    end    data(it,:) = [F(it),norm(delta_x),norm(g)];    X(:,it) = x;  end  % End of iteration for x_lambda(j).  x_lambda(:,j) = x;end