function [x, error, iter, flag] = bicg(A, x, b, M, max_it, tol)%  -- Iterative template routine --%     Univ. of Tennessee and Oak Ridge National Laboratory%     October 1, 1993%     Details of this algorithm are described in "Templates for the%     Solution of Linear Systems: Building Blocks for Iterative%     Methods", Barrett, Berry, Chan, Demmel, Donato, Dongarra,%     Eijkhout, Pozo, Romine, and van der Vorst, SIAM Publications,%     1993. (ftp netlib2.cs.utk.edu; cd linalg; get templates.ps).%%  [x, error, iter, flag] = bicg(A, x, b, M, max_it, tol)%% bicg.m solves the linear system Ax=b using the % BiConjugate Gradient Method with preconditioning.%% input   A        REAL matrix%         M        REAL preconditioner matrix%         x        REAL initial guess vector%         b        REAL right hand side vector%         max_it   INTEGER maximum number of iterations%         tol      REAL error tolerance%% output  x        REAL solution vector%         error    REAL error norm%         iter     INTEGER number of iterations performed%         flag     INTEGER: 0 = solution found to tolerance%                           1 = no convergence given max_it%                          -1 = breakdown  iter = 0;                              % initialization  flag = 0;  bnrm2 = norm( b );  if  ( bnrm2 == 0.0 ), bnrm2 = 1.0; end  r = b - A*x;  error = norm( r ) / bnrm2;  if ( error < tol ) return, end  r_tld  = r;  for iter = 1:max_it                    % begin iteration     z = M \ r;     z_tld = M' \ r_tld;     rho   = ( z'*r_tld );     if ( rho == 0.0 ),        break     end     if ( iter > 1 ),                    % direction vectors        beta = rho / rho_1;        p   = z  + beta*p;        p_tld = z_tld + beta*p_tld;     else        p = z;        p_tld = z_tld;     end     q = A*p;                            % compute residual pair     q_tld = A'*p_tld;     alpha = rho / (p_tld'*q );     x = x + alpha*p;                    % update approximation     r = r - alpha*q;     r_tld = r_tld - alpha*q_tld;     error = norm( r ) / bnrm2;          % check convergence     if ( error <= tol ), break, end     rho_1 = rho;  end  if ( error <= tol ),                   % converged     flag =  0;  elseif ( rho == 0.0 ),                 % breakdown     flag = -1;  else     flag = 1;                           % no convergence  end% END bicg.m