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maint: Periodic merge of stable to default.
author John W. Eaton <jwe@octave.org>
date Tue, 02 Feb 2016 17:06:11 -0500
parents 66a08c3cafe3 2935d56203a4
children dcf8922b724b
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19697
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1 ## Copyright (C) 2014-2015 Nathan Podlich
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2 ##
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3 ## This file is part of Octave.
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4 ##
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5 ## Octave is free software; you can redistribute it and/or modify
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6 ## it under the terms of the GNU General Public License as published by
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7 ## the Free Software Foundation; either version 3 of the License, or
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8 ## (at your option) any later version.
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9 ##
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10 ## Octave is distributed in the hope that it will be useful,
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11 ## but WITHOUT ANY WARRANTY; without even the implied warranty of
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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13 ## GNU General Public License for more details.
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14 ##
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15 ## You should have received a copy of the GNU General Public License
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16 ## along with Octave; If not, see <http://www.gnu.org/licenses/>.
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17
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18 ## -*- texinfo -*-
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19 ## @deftypefn {} {@var{x} =} qmr (@var{A}, @var{b}, @var{rtol}, @var{maxit}, @var{M1}, @var{M2}, @var{x0})
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20 ## @deftypefnx {} {@var{x} =} qmr (@var{A}, @var{b}, @var{rtol}, @var{maxit}, @var{P})
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21 ## @deftypefnx {} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}] =} qmr (@var{A}, @var{b}, @dots{})
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22 ## Solve @code{A x = b} using the Quasi-Minimal Residual iterative method
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23 ## (without look-ahead).
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24 ##
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25 ## @itemize @minus
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26 ## @item @var{rtol} is the relative tolerance, if not given or set to [] the
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27 ## default value 1e-6 is used.
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28 ##
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29 ## @item @var{maxit} the maximum number of outer iterations, if not given or
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30 ## set to [] the default value @code{min (20, numel (b))} is used.
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31 ##
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32 ## @item @var{x0} the initial guess, if not given or set to [] the default
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33 ## value @code{zeros (size (b))} is used.
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34 ## @end itemize
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35 ##
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36 ## @var{A} can be passed as a matrix or as a function handle or inline
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37 ## function @code{f} such that @code{f(x, "notransp") = A*x} and
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38 ## @code{f(x, "transp") = A'*x}.
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39 ##
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40 ## The preconditioner @var{P} is given as @code{P = M1 * M2}. Both @var{M1}
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41 ## and @var{M2} can be passed as a matrix or as a function handle or inline
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42 ## function @code{g} such that @code{g(x, "notransp") = M1 \ x} or
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43 ## @code{g(x, "notransp") = M2 \ x} and @code{g(x, "transp") = M1' \ x} or
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44 ## @code{g(x, "transp") = M2' \ x}.
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45 ##
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46 ## If called with more than one output parameter
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47 ##
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48 ## @itemize @minus
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49 ## @item @var{flag} indicates the exit status:
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50 ##
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51 ## @itemize @minus
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52 ## @item 0: iteration converged to the within the chosen tolerance
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53 ##
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54 ## @item 1: the maximum number of iterations was reached before convergence
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55 ##
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56 ## @item 3: the algorithm reached stagnation
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57 ## @end itemize
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58 ##
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59 ## (the value 2 is unused but skipped for compatibility).
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60 ##
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61 ## @item @var{relres} is the final value of the relative residual.
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62 ##
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63 ## @item @var{iter} is the number of iterations performed.
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64 ##
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65 ## @item @var{resvec} is a vector containing the residual norms at each
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66 ## iteration.
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67 ## @end itemize
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68 ##
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69 ## References:
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70 ##
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71 ## @enumerate
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72 ## @item
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73 ## @nospell{R. Freund and N. Nachtigal}, @cite{QMR: a quasi-minimal residual
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74 ## method for non-Hermitian linear systems}, @nospell{Numerische Mathematik},
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75 ## 1991, 60, pp. 315-339.
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76 ##
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77 ## @item
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78 ## @nospell{ R. Barrett, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra},
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79 ## @nospell{ V. Eijkhour, R. Pozo, C. Romine, and H. van der Vorst},
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80 ## @cite{Templates for the solution of linear systems: Building blocks
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81 ## for iterative methods}, SIAM, 2nd ed., 1994.
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82 ## @end enumerate
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83 ##
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84 ## @seealso{bicg, bicgstab, cgs, gmres, pcg}
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85 ## @end deftypefn
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86
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87 ## Author: Nathan Podlich <nathan.podlich@gmail.com>
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88
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89 function [x, flag, relres, iter, resvec] = qmr (A, b, tol, maxit, M1, M2, x0)
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90
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91 if (nargin >= 2 && isvector (full (b)))
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92
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93 if (ischar (A))
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94 fun = str2func (A);
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95 Ax = @(x) feval (fun, x, "notransp");
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96 Atx = @(x) feval (fun, x, "transp");
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97 elseif (isa (A, "function_handle"))
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98 Ax = @(x) feval (A, x, "notransp");
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99 Atx = @(x) feval (A, x, "transp");
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100 elseif (isnumeric (A) && issquare (A))
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101 Ax = @(x) A * x;
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102 Atx = @(x) A' * x;
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103 else
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104 error ("qmr: A must be a square matrix or function");
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105 endif
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106
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107 if (nargin < 3 || isempty (tol))
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108 tol = 1e-6;
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109 endif
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110
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111 if (nargin < 4 || isempty (maxit))
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112 maxit = min (rows (b), 20);
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113 else
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114 maxit = fix (maxit);
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115 endif
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116
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117 if (nargin < 5 || isempty (M1))
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118 M1m1x = @(x, ignore) x;
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119 M1tm1x = M1m1x;
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120 elseif (ischar (M1))
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121 fun = str2func (M1);
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122 M1m1x = @(x) feval (fun, x, "notransp");
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123 M1tm1x = @(x) feval (fun, x, "transp");
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124 elseif (isa (M1, "function_handle"))
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125 M1m1x = @(x) feval (M1, x, "notransp");
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126 M1tm1x = @(x) feval (M1, x, "transp");
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127 elseif (isnumeric (M1) && ismatrix (M1))
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128 M1m1x = @(x) M1 \ x;
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129 M1tm1x = @(x) M1' \ x;
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130 else
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131 error ("qmr: preconditioner M1 must be a function or matrix");
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132 endif
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133
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134 if (nargin < 6 || isempty (M2))
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135 M2m1x = @(x, ignore) x;
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Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
136 M2tm1x = M2m1x;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
137 elseif (ischar (M2))
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
138 fun = str2func (M2);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
139 M2m1x = @(x) feval (fun, x, "notransp");
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
140 M2tm1x = @(x) feval (fun, x, "transp");
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
141 elseif (isa (M2, "function_handle"))
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
142 M2m1x = @(x) feval (M2, x, "notransp");
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
143 M2tm1x = @(x) feval (M2, x, "transp");
19700
00e31f316a3a Fix Matlab incompatibility of "ismatrix" (bug #42422).
Kai T. Ohlhus <k.ohlhus@gmail.com>
parents: 19697
diff changeset
144 elseif (isnumeric (M2) && ismatrix (M2))
19354
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
145 M2m1x = @(x) M2 \ x;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
146 M2tm1x = @(x) M2' \ x;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
147 else
21171
2935d56203a4 Fix regressions caused by ismatrix definition change (partial fix bug #47036).
Rik <rik@octave.org>
parents: 20164
diff changeset
148 error ("qmr: preconditioner M2 must be a function or matrix");
19354
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
149 endif
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
150
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
151 if (nargin < 7 || isempty (x0))
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
152 x = zeros (size (b));
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
153 else
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
154 x = x0;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
155 endif
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
156
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
157 r = b - Ax (x);
19597
db92e7e28e1f strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 19356
diff changeset
158
19354
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
159 bnorm = norm (b);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
160 res0 = norm (r);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
161 if (nargout > 4)
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
162 resvec(1) = res0;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
163 endif
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
164 vt = r;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
165
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
166 y = M1m1x (vt);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
167
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
168 rho0 = norm (y);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
169 wt = r;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
170
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
171 z = M2tm1x (wt);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
172
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
173 xi1 = norm (z);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
174 gamma0 = 1;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
175 eta0 = -1;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
176 flag = 1;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
177 for iter=1:1:maxit
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
178 ## If rho0 == 0 or xi1 == 0, method fails.
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
179 v = vt / rho0;
20231
83792dd9bcc1 Use in-place operators in m-files where possible.
Rik <rik@octave.org>
parents: 20164
diff changeset
180 y /= rho0;
19354
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
181 w = wt / xi1;
20231
83792dd9bcc1 Use in-place operators in m-files where possible.
Rik <rik@octave.org>
parents: 20164
diff changeset
182 z /= xi1;
19354
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
183
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19725
diff changeset
184 delta1 = z' * y; # If delta1 == 0, method fails.
19354
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
185
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
186 yt = M2m1x (y);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
187 zt = M1tm1x (z);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
188
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
189 if (iter == 1)
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
190 p = yt;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
191 q = zt;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
192 else
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
193 p = yt - (xi1*delta1/eps0) * p;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
194 q = zt - (rho0*delta1/eps0) * q;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
195 endif
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
196 pt = Ax (p);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
197
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19725
diff changeset
198 eps0 = q' * pt; # If eps0 == 0, method fails.
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19725
diff changeset
199 beta1 = eps0 / delta1; # If beta1 == 0, method fails.
19354
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
200 vt = pt - beta1 * v;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
201
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
202 y = M1m1x (vt);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
203 rho1 = norm(y);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
204 wt = Atx (q) - beta1 * w;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
205 z = M2tm1x (wt);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
206
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
207 xi1 = norm(z);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
208 theta1 = rho1 / (gamma0 * abs(beta1));
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19725
diff changeset
209 gamma1 = 1 / sqrt(1 + theta1^2); # If gamma1 == 0, method fails.
19354
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
210 eta1 = -eta0 * rho0 * gamma1^2 / (beta1 * gamma0^2);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
211
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
212 if (iter == 1)
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
213 d = eta1 * p;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
214 s = eta1 * pt;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
215 else
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
216 d = eta1 * p + (theta0*gamma1)^2 * d;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
217 s = eta1 * pt + (theta0 * gamma1)^2 * s;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
218 endif
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
219 x += d;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
220 r -= s;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
221
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
222 res1 = norm (r) / bnorm;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
223 if (nargout > 4)
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
224 resvec(iter + 1, 1) = norm (r);
21031
66a08c3cafe3 maint: Follow Octave coding conventions in m-files.
Rik <rik@octave.org>
parents: 20852
diff changeset
225 endif
19597
db92e7e28e1f strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 19356
diff changeset
226
19354
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
227 if (res1 < tol)
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
228 ## Convergence achieved.
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
229 flag = 0;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
230 break;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
231 elseif (res0 <= res1)
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
232 ## Stagnation encountered.
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
233 flag = 3;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
234 break;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
235 endif
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
236 theta0 = theta1;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
237 eta0 = eta1;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
238 gamma0 = gamma1;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
239 rho0 = rho1;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
240 endfor
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
241
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
242 relres = res1;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
243 if (flag == 1)
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
244 if (nargout < 2)
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
245 printf ("qmr stopped at iteration %i ", iter);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
246 printf ("without converging to the desired tolerance %e\n", tol);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
247 printf ("because the maximum number of iterations was reached. ");
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
248 printf ("The iterate returned (number %i) has ", maxit);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
249 printf ("relative residual %e\n", res1);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
250 endif
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
251 elseif (flag == 3)
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
252 if (nargout < 2)
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
253 printf ("qmr stopped at iteration %i ", iter);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
254 printf (" without converging to the desired tolerance %e\n", tol);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
255 printf ("because the method stagnated.\n");
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
256 printf ("The iterate returned (number %i) ", iter);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
257 printf ("has relative residual %e\n", res1);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
258 endif
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
259 elseif (nargout < 2)
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
260 printf ("qmr converged at iteration %i ", iter);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
261 printf ("to a solution with relative residual %e\n", res1);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
262 endif
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
263 else
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
264 print usage();
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
265 endif
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
266 endfunction
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
267
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
268
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
269 %!demo
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
270 %! % Solve system of A*x=b
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
271 %! A = [5 -1 3;-1 2 -2;3 -2 3];
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
272 %! b = [7;-1;4];
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
273 %! [x, flag, relres, iter, resvec] = qmr (A, b)
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
274
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
275 %!test
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
276 %! n = 100;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
277 %! A = spdiags ([-2*ones(n,1) 4*ones(n,1) -ones(n,1)], -1:1, n, n);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
278 %! b = sum (A, 2);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
279 %! tol = 1e-8;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
280 %! maxit = 15;
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
281 %! M1 = spdiags ([ones(n,1)/(-2) ones(n,1)],-1:0, n, n);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
282 %! M2 = spdiags ([4*ones(n,1) -ones(n,1)], 0:1, n, n);
cc6000ccd627 New function qmr (patch #8569).
Nathan Podlich <nathan.podlich@gmail.com>
parents:
diff changeset
283 %! [x, flag, relres, iter, resvec] = qmr (A, b, tol, maxit, M1, M2);
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284 %! assert (x, ones (size (b)), 1e-7);
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285
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286 %!function y = afun (x, t, a)
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287 %! switch (t)
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288 %! case "notransp"
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289 %! y = a * x;
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290 %! case "transp"
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291 %! y = a' * x;
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292 %! endswitch
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293 %!endfunction
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294 %!
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295 %!test
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296 %! n = 100;
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297 %! A = spdiags ([-2*ones(n,1) 4*ones(n,1) -ones(n,1)], -1:1, n, n);
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298 %! b = sum (A, 2);
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299 %! tol = 1e-8;
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300 %! maxit = 15;
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301 %! M1 = spdiags ([ones(n,1)/(-2) ones(n,1)],-1:0, n, n);
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302 %! M2 = spdiags ([4*ones(n,1) -ones(n,1)], 0:1, n, n);
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303 %!
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304 %! [x, flag, relres, iter, resvec] = qmr (@(x, t) afun (x, t, A),
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305 %! b, tol, maxit, M1, M2);
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306 %! assert (x, ones (size (b)), 1e-7);
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307
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308 %!test
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309 %! n = 100;
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310 %! tol = 1e-8;
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311 %! a = sprand (n, n, .1);
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312 %! A = a' * a + 100 * eye (n);
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313 %! b = sum (A, 2);
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314 %! [x, flag, relres, iter, resvec] = qmr (A, b, tol, [], diag (diag (A)));
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315 %! assert (x, ones (size (b)), 1e-7);
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316
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317 %!test
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318 %! A = [1 + 1i, 1 + 1i; 2 - 1i, 2 + 1i];
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319 %! b = A * [1; 1];
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320 %! [x, flag, relres, iter, resvec] = qmr (A, b);
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321 %! assert (x, [1; 1], 1e-6);