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+## Copyright (C) 2014 Nathan Podlich
+##
+## This file is part of Octave.
+##
+## Octave is free software; you can redistribute it and/or modify
+## it under the terms of the GNU General Public License as published by
+## the Free Software Foundation; either version 3 of the License, or
+## (at your option) any later version.
+##
+## Octave is distributed in the hope that it will be useful,
+## but WITHOUT ANY WARRANTY; without even the implied warranty of
+## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+## GNU General Public License for more details.
+##
+## You should have received a copy of the GNU General Public License
+## along with Octave; If not, see <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn  {Function File} {@var{x} =} qmr (@var{A}, @var{b}, @var{rtol}, @var{maxit}, @var{M1}, @var{M2}, @var{x0})
+## @deftypefnx {Function File} {@var{x} =} qmr (@var{A}, @var{b}, @var{rtol}, @var{maxit}, @var{P})
+## @deftypefnx {Function File} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}] =} qmr (@var{A}, @var{b}, @dots{})
+## Solve @code{A x = b} using the Quasi-Minimal Residual iterative
+## method (without look-ahead).
+##
+## @itemize @minus
+## @item @var{rtol} is the relative tolerance, if not given
+## or set to [] the default value 1e-6 is used.
+##
+## @item @var{maxit} the maximum number of outer iterations,
+## if not given or set to [] the default value
+## @code{min (20, numel (b))} is used.
+##
+## @item @var{x0} the initial guess, if not given or set to []
+## the default value @code{zeros (size (b))} is used.
+## @end itemize
+##
+## @var{A} can be passed as a matrix or as a function handle or
+## inline function @code{f} such that @code{f(x, "notransp") = A*x}
+## and @code{f(x, "transp") = A'*x}.
+##
+## The preconditioner @var{P} is given as @code{P = M1 * M2}.
+## Both @var{M1} and @var{M2} can be passed as a matrix or as
+## a function handle or inline function @code{g} such that
+## @code{g(x, "notransp") = M1 \ x} or @code{g(x, "notransp") = M2 \ x} and
+## @code{g(x, "transp") = M1' \ x} or @code{g(x, "transp") = M2' \ x}.
+##
+## If called with more than one output parameter
+##
+## @itemize @minus
+## @item @var{flag} indicates the exit status:
+##
+## @itemize @minus
+## @item 0: iteration converged to the within the chosen tolerance
+##
+## @item 1: the maximum number of iterations was reached before convergence
+##
+## @item 3: the algorithm reached stagnation
+## @end itemize
+##
+## (the value 2 is unused but skipped for compatibility).
+##
+## @item @var{relres} is the final value of the relative residual.
+##
+## @item @var{iter} is the number of iterations performed.
+##
+## @item @var{resvec} is a vector containing the residual norms at each
+##       iteration.
+## @end itemize
+##
+## References:
+##
+## @enumerate
+## @item
+## R. Freund and N. Nachtigal, @cite{QMR: a quasi-minimal residual
+## method for non-Hermitian linear systems}, Numerische Mathematik,
+## 1991, 60, pp. 315-339.
+##
+## @item
+## R. Barrett, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra,
+## V. Eijkhour, R. Pozo, C. Romine, and H. van der Vorst,
+## @cite{Templates for the solution of linear systems: Building blocks
+## for iterative methods}, SIAM, 2nd ed., 1994.
+##
+## @seealso{bicg, bicgstab, cgs, gmres}
+##
+## @end deftypefn
+
+## Author: Nathan Podlich <nathan.podlich@gmail.com>
+
+function [x, flag, relres, iter, resvec] = qmr (A, b, tol, maxit, M1, M2, x0)
+
+  if (nargin >= 2 && isvector (full (b)))
+
+    if (ischar (A))
+      fun = str2func (A);
+      Ax  = @(x) feval (fun, x, "notransp");
+      Atx = @(x) feval (fun, x, "transp");
+    elseif (isa (A, "function_handle"))
+      Ax  = @(x) feval (A, x, "notransp");
+      Atx = @(x) feval (A, x, "transp");
+    elseif (ismatrix (A))
+      Ax  = @(x) A  * x;
+      Atx = @(x) A' * x;
+    else
+      error (["qmr: first argument is expected to " ...
+                "be a function or a square matrix"]);
+    endif
+
+    if (nargin < 3 || isempty (tol))
+      tol = 1e-6;
+    endif
+
+    if (nargin < 4 || isempty (maxit))
+      maxit = min (rows (b), 20);
+    endif
+
+    if (nargin < 5 || isempty (M1))
+      M1m1x = @(x, ignore) x;
+      M1tm1x = M1m1x;
+    elseif (ischar (M1))
+      fun = str2func (M1);
+      M1m1x  = @(x) feval (fun, x, "notransp");
+      M1tm1x = @(x) feval (fun, x, "transp");
+    elseif (isa (M1, "function_handle"))
+      M1m1x  = @(x) feval (M1, x, "notransp");
+      M1tm1x = @(x) feval (M1, x, "transp");
+    elseif (ismatrix (M1))
+      M1m1x  = @(x) M1  \ x;
+      M1tm1x = @(x) M1' \ x;
+    else
+      error (["qmr: preconditioner is expected to " ...
+                "be a function or matrix"]);
+    endif
+
+    if (nargin < 6 || isempty (M2))
+      M2m1x = @(x, ignore) x;
+      M2tm1x = M2m1x;
+    elseif (ischar (M2))
+      fun = str2func (M2);
+      M2m1x  = @(x) feval (fun, x, "notransp");
+      M2tm1x = @(x) feval (fun, x, "transp");
+    elseif (isa (M2, "function_handle"))
+      M2m1x  = @(x) feval (M2, x, "notransp");
+      M2tm1x = @(x) feval (M2, x, "transp");
+    elseif (ismatrix (M2))
+      M2m1x  = @(x) M2  \ x;
+      M2tm1x = @(x) M2' \ x;
+    else
+      error (["qmr: preconditioner is expected to " ...
+                "be a function or matrix"]);
+    endif
+
+
+    if (nargin < 7 || isempty (x0))
+      x = zeros (size (b));
+    else
+      x = x0;
+    endif
+
+    r = b - Ax (x);
+    
+    bnorm = norm (b);
+    res0 = norm (r);
+    if (nargout > 4)
+      resvec(1) = res0;
+    endif
+    vt = r;
+
+    y = M1m1x (vt);
+
+    rho0 = norm (y);
+    wt = r;
+
+    z = M2tm1x (wt);
+
+    xi1 = norm (z);
+    gamma0 = 1;
+    eta0 = -1;
+    flag = 1;
+    for iter=1:1:maxit
+      ## If rho0 == 0 or xi1 == 0, method fails.
+      v = vt / rho0;
+      y = y / rho0;
+      w = wt / xi1;
+      z = z / xi1;
+
+      delta1 = z' * y; ## If delta1 == 0, method fails.
+
+      yt = M2m1x (y);
+      zt = M1tm1x (z);
+
+      if (iter == 1)
+        p = yt;
+        q = zt;
+      else
+        p = yt - (xi1*delta1/eps0) * p;
+        q = zt - (rho0*delta1/eps0) * q;
+      endif
+      pt = Ax (p);
+
+      eps0 = q' * pt; ## If eps0 == 0, method fails.
+      beta1 = eps0 / delta1; ## If beta1 == 0, method fails.
+      vt = pt - beta1 * v;
+
+      y = M1m1x (vt);
+      rho1 = norm(y);
+      wt = Atx (q) - beta1 * w;
+      z = M2tm1x (wt);
+
+      xi1 = norm(z);
+      theta1 = rho1 / (gamma0 * abs(beta1));
+      gamma1 = 1 / sqrt(1 + theta1^2); ## If gamma1 == 0, method fails.
+      eta1 = -eta0 * rho0 * gamma1^2 / (beta1 * gamma0^2);
+
+      if (iter == 1)
+        d = eta1 * p;
+        s = eta1 * pt;
+      else
+        d = eta1 * p + (theta0*gamma1)^2 * d;
+        s = eta1 * pt + (theta0 * gamma1)^2 * s;
+      endif
+      x += d;
+      r -= s;
+
+      res1 = norm (r) / bnorm;
+      if (nargout > 4)
+        resvec(iter + 1, 1) = norm (r);
+      end
+      
+      if (res1 < tol)
+        ## Convergence achieved.
+        flag = 0;
+        break;
+      elseif (res0 <= res1)
+        ## Stagnation encountered.
+        flag = 3;
+        break;
+      endif
+      theta0 = theta1;
+      eta0 = eta1;
+      gamma0 = gamma1;
+      rho0 = rho1;
+    endfor
+
+    relres = res1;
+    if (flag == 1)
+      if (nargout < 2)
+        printf ("qmr stopped at iteration %i ", iter);
+        printf ("without converging to the desired tolerance %e\n", tol);
+        printf ("because the maximum number of iterations was reached. ");
+        printf ("The iterate returned (number %i) has ", maxit);
+        printf ("relative residual %e\n", res1);
+      endif
+    elseif (flag == 3)
+      if (nargout < 2)
+        printf ("qmr stopped at iteration %i ", iter);
+        printf (" without converging to the desired tolerance %e\n", tol);
+        printf ("because the method stagnated.\n");
+        printf ("The iterate returned (number %i) ", iter);
+        printf ("has relative residual %e\n", res1);
+      endif
+    elseif (nargout < 2)
+      printf ("qmr converged at iteration %i ", iter);
+      printf ("to a solution with relative residual %e\n", res1);
+    endif
+  else
+    print usage();
+  endif
+endfunction
+
+
+%!demo
+%! % Solve system of A*x=b
+%! A = [5 -1 3;-1 2 -2;3 -2 3];
+%! b = [7;-1;4];
+%! [x, flag, relres, iter, resvec] = qmr (A, b)
+
+%!test
+%! n = 100;
+%! A = spdiags ([-2*ones(n,1) 4*ones(n,1) -ones(n,1)], -1:1, n, n);
+%! b = sum (A, 2);
+%! tol = 1e-8;
+%! maxit = 15;
+%! M1 = spdiags ([ones(n,1)/(-2) ones(n,1)],-1:0, n, n);
+%! M2 = spdiags ([4*ones(n,1) -ones(n,1)], 0:1, n, n);
+%! [x, flag, relres, iter, resvec] = qmr (A, b, tol, maxit, M1, M2);
+%! assert (x, ones (size (b)), 1e-7);
+
+%!function y = afun (x, t, a)
+%!  switch (t)
+%!    case "notransp"
+%!      y = a * x;
+%!    case "transp"
+%!      y = a' * x;
+%!  endswitch
+%!endfunction
+%!
+%!test
+%! n = 100;
+%! A = spdiags ([-2*ones(n,1) 4*ones(n,1) -ones(n,1)], -1:1, n, n);
+%! b = sum (A, 2);
+%! tol = 1e-8;
+%! maxit = 15;
+%! M1 = spdiags ([ones(n,1)/(-2) ones(n,1)],-1:0, n, n);
+%! M2 = spdiags ([4*ones(n,1) -ones(n,1)], 0:1, n, n);
+%!
+%! [x, flag, relres, iter, resvec] = qmr (@(x, t) afun (x, t, A),
+%!                                         b, tol, maxit, M1, M2);
+%! assert (x, ones (size (b)), 1e-7);
+
+%!test
+%! n = 100;
+%! tol = 1e-8;
+%! a = sprand (n, n, .1);
+%! A = a' * a + 100 * eye (n);
+%! b = sum (A, 2);
+%! [x, flag, relres, iter, resvec] = qmr (A, b, tol, [], diag (diag (A)));
+%! assert (x, ones (size (b)), 1e-7);
+
+%!test
+%! A = [1 + 1i, 1 + 1i; 2 - 1i, 2 + 1i];
+%! b = A * [1; 1];
+%! [x, flag, relres, iter, resvec] = qmr (A, b);
+%! assert (x, [1; 1], 1e-6);