1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
|
// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
// Copyright (C) XXXX-2008 - INRIA
// Copyright (C) 2005 - IRISA - Sage Group
//
// This file must be used under the terms of the CeCILL.
// This source file is licensed as described in the file COPYING, which
// you should have received as part of this distribution. The terms
// are also available at
// http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt
// [x, flag, err, iter, res] = qmr( A, Ap, b, x, M1, M1p, M2, M2p, max_it, tol )
//
// QMR solves the linear system Ax=b using the
// Quasi Minimal Residual method with preconditioning.
//
// input A REAL matrix or function
// x REAL initial guess vector
// b REAL right hand side vector
// M1 REAL left preconditioner matrix
// M2 REAL right preconditioner matrix
// max_it INTEGER maximum number of iterations
// tol REAL error tolerance
//
// output x REAL solution vector
// flag INTEGER: 0: solution found to tolerance
// 1: no convergence given max_it
// breakdown:
// -1: rho
// -2: Beta
// -3: gam
// -4: delta
// -5: ep
// -6: xi
// err REAL final residual norm
// iter INTEGER number of iterations performed
// res REAL residual vector
// 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).
//
// "Iterative Methods for Sparse Linear Systems, Second Edition"
// Saad, SIAM Publications, 2003
// (ftp ftp.cs.umn.edu; cd dept/users/saad/PS; get all_ps.zip).
// Sage Group (IRISA, 2005)
function [x, flag, err, iter, res] = qmr( A, varargin)
// -----------------------
// Parsing input arguments
// -----------------------
[lhs,rhs]=argn(0);
if ( rhs < 2 ),
error(msprintf(gettext("%s: Wrong number of input arguments: At least %d expected.\n"),"qmr",2));
end
// Parsing the matrix A
select type(A)
case 1 then
cpt=1;
case 5 then
cpt=1;
case 13 then
cpt=0;
else
error(msprintf(gettext("%s: Wrong type for input argument #%d : A real or complex matrix or a sparse matrix or a function expected.\n"),"qmr",1));
end
// If A is a matrix (dense or sparse)
if (cpt==1),
if (size(A,1) ~= size(A,2)),
error(msprintf(gettext("%s: Wrong size for input argument #%d: Square matrix expected.\n"),"qmr",1));
end
fct=0;
cptmat = 1;
end
// If A is a function
if (cpt==0),
fct = 0;
if rhs >= 2,
funcorvec=varargin(1);
if and(type(funcorvec) <> [1 5 13]) then
error(msprintf(gettext("%s: Wrong value for input argument #%d: Transpose of the function %s expected.\n"),"qmr",2,"A"));
// if the following input argument is a sparse or dense matrix
elseif or(type(funcorvec) == [1 5]) then
if size(getfield(1,macrovar(A)),"*") == 1 then
error(msprintf(gettext("%s: Wrong type for input argument #%d: A transpose of the function %s expected.\n"), "qmr", 2, "A"));
end
matvec = A;
cptmat = 1;
// if the following input argument is a function
else
matvec = A;
matvecp = funcorvec;
cptmat = 2;
fct = 1;
warning(msprintf(gettext("%s : Calling qmr(A,Ap) is deprecated. Please see qmr documentation for more details.\n"),"qmr"));
end
end
end
// Parsing right hand side b
if ( rhs >= fct+2 ),
b=varargin(fct+1);
// if b is not constant or sparse
if and(type(b) <> [1 5]) then
error(msprintf(gettext("%s: Wrong type for input argument #%d: A real or complex, full or sparse column vector expected.\n"), "qmr", fct+2));
end
if ( size(b,2) ~= 1),
error(msprintf(gettext("%s: Wrong size for input argument #%d: Column vector expected.\n"),"qmr",fct+2));
end
end
// Parsing initial vector x
if ( rhs >= fct+3),
x=varargin(fct+2);
// if x is not constant or sparse
if and(type(x) <> [1 5]) then
error(msprintf(gettext("%s: Wrong type for input argument #%d: A real or complex, full or sparse column vector expected.\n"), "qmr", fct + 3));
end
if (size(x,2) ~= 1),
error(msprintf(gettext("%s: Wrong size for input argument #%d: Column vector expected.\n"),"qmr",fct+3));
end
if ( size(x,1) ~= size(b,1)),
error(msprintf(gettext("%s: Wrong size for input argument #%d: Same size as input argument #%d expected.\n"),"qmr",fct+3,fct+2));
end
// By default
else
x=zeros(size(b,1),1);
end
//--------------------------------------------------------
// Parsing of the preconditioner matrix M1
//--------------------------------------------------------
if (rhs >=fct+4),
Prec_g=varargin(fct+3);
select type(Prec_g)
case 1 then
cpt=1;
case 5 then
cpt=1;
case 13 then
cpt=0;
else
error(msprintf(gettext("%s: Wrong type for input argument #%d: A real or complex, full or sparse, square matrix or a function expected.\n"), "qmr", fct + 4));
end
// if M1 is a matrix
if ( cpt==1 ),
if (size(Prec_g,1) ~= size(Prec_g,2)),
error(msprintf(gettext("%s: Wrong size for input argument #%d: Square matrix expected.\n"),"qmr",fct+4));
end
if (size(Prec_g,1)~=size(b,1)),
error(msprintf(gettext("%s: Wrong size for input argument #%d: Same size as input argument #%d expected.\n"),"qmr",fct+4,fct+2));
end
cptmatM1 = 1;
end
// if M1 is a function
if ( cpt==0 ),
precond_g = Prec_g;
cptmatM1 = 1;
if ( rhs >= fct+5 & size(getfield(1,macrovar(precond_g)),"*") == 1),
Precp_g = varargin(fct+4);
if (type(Precp_g) == 13 & size(getfield(1,macrovar(Precp_g)),"*")==1) then
//precond_g = Prec_g;
precondp_g = Precp_g;
cptmatM1 = 2;
fct = fct+1;
warning(msprintf(gettext("%s : Calling qmr(...,M1,M1p) is deprecated. Please see qmr documentation for more details.\n"),"qmr"));
else
error(msprintf(gettext("%s: Wrong type for input argument #%d: A transpose of the function expected.\n"), "qmr", fct + 5));
end
elseif rhs < fct+5 & size(getfield(1, macrovar(precond_g)), "*") == 1 then
error(msprintf(gettext("%s: Wrong prototype of input argument #%d: If M1 is function, use the header M1(x,t) instead M1(x).\n"), "qmr", fct+4));
end
end
// By default
else
deff("y=precond_g(x)","y=x");
deff("y=precondp_g(x)","y=x");
cptmatM1 = 2;
end
//--------------------------------------------------------
// Parsing of the preconditioner matrix M2
//--------------------------------------------------------
if (rhs >=fct+5),
Prec_d=varargin(fct+4);
select type(Prec_d)
case 1 then
cpt=1;
case 5 then
cpt=1;
case 13 then
cpt=0;
else
error(msprintf(gettext("%s: Wrong type for input argument #%d: A real or complex, full or sparse, square matrix or a function expected.\n"), "qmr", fct + 5));
end
// M2 matrix
if ( cpt==1 ),
if (size(Prec_d,1) ~= size(Prec_d,2)),
error(msprintf(gettext("%s: Wrong size for input argument #%d: Square matrix expected.\n"),"qmr",fct+5));
end
if (size(Prec_d,1)~=size(b,1)),
error(msprintf(gettext("%s: Wrong size for input argument #%d: Same size as input argument #%d expected.\n"),"qmr",fct+5,fct+2));
end
cptmatM2 = 1;
end
// M2 function
if ( cpt==0 )
precond_d = Prec_d;
cptmatM2 = 1;
if ( rhs >= fct+6 & size(getfield(1,macrovar(precond_d)),"*") == 1),
Precp_d=varargin(fct+5);
if (type(Precp_d) == 13 & size(getfield(1,macrovar(Precp_d)),"*") == 1) then
precond_d = Prec_d;
precondp_d = Precp_d;
cptmatM2 = 2;
fct = fct+1;
warning(msprintf(gettext("%s : Calling qmr(...,M2,M2p) is deprecated. Please see qmr documentation for more details.\n"),"qmr"));
else
error(msprintf(gettext("%s: Wrong type for input argument #%d: A transpose of the function expected.\n"), "qmr", fct + 6));
end
elseif rhs < fct+6 & size(getfield(1, macrovar(precond_d)), "*") == 1 then
error(msprintf(gettext("%s: Wrong prototype of input argument #%d: If M2 is function, use the header M2(x,t) instead M2(x).\n"), "qmr", fct+5));
end
end
// By default
else
deff("y=precond_d(x)","y=x");
deff("y=precondp_d(x)","y=x");
cptmatM2 = 2;
end
//--------------------------------------------------------
// Parsing of the maximum number of iterations max_it
//--------------------------------------------------------
if (rhs >= fct+6),
max_it=varargin(fct+5);
// if max_it is not constant
if type(max_it) <> 1 then
error(msprintf(gettext("%s: Wrong type for input argument #%d: Scalar expected.\n"),"qmr",fct + 6));
end
if ~isreal(max_it) then
error(msprintf(gettext("%s: Wrong type for input argument #%d: A real scalar expected.\n"),"qmr",fct + 6));
end
if (size(max_it,1) ~= 1 | size(max_it,2) ~=1),
error(msprintf(gettext("%s: Wrong size for input argument #%d: Scalar expected.\n"),"qmr",fct+6));
end
// By default
else
max_it=size(b,1);
end
//--------------------------------------------------------
// Parsing of the error tolerance tol
//--------------------------------------------------------
if (rhs == fct+7),
tol=varargin(fct+6);
// if tol is not constant
if type(tol) <> 1 then
error(msprintf(gettext("%s: Wrong type for input argument #%d: Scalar expected.\n"),"qmr",fct + 7));
end
if ~isreal(tol) then
error(msprintf(gettext("%s: Wrong type for input argument #%d: A real scalar expected.\n"),"qmr",fct + 7));
end
if (size(tol,1) ~= 1 | size(tol,2) ~=1),
error(msprintf(gettext("%s: Wrong size for input argument #%d: Scalar expected.\n"),"qmr",fct+7));
end
// By default
else
tol=1000*%eps;
end
//--------------------------------------------------------
// test about input arguments number
//--------------------------------------------------------
if (rhs > fct+7),
error(msprintf(gettext("%s: Wrong number of input arguments: %d to %d expected.\n"),"qmr",2,fct+7));
end
// ------------
// Computations
// ------------
// initialization
i = 0;
flag = 0;
iter = 0;
bnrm2 = norm( b );
if (bnrm2 == 0.0),
bnrm2 = 1.0;
end
// r = b - A*x;
if (cptmat == 1) then
r = b - matvec(x,"notransp");
elseif (cptmat==2) then // If A is a function
r = b - matvec(x);
end
err = norm( r ) / bnrm2;
res = err;
if ( err < tol ), return; end
// [M1,M2] = lu( M );
v_tld = r;
// y = M1 \ v_tld;
if (cptmatM1 == 1) then
y = precond_g(v_tld,"notransp");
elseif (cptmatM1==2) then
y = precond_g(v_tld);
end
rho = norm( y );
w_tld = r;
// z = M2' \ w_tld;
if (cptmatM2 == 1) then
z = precond_d(w_tld,"transp");
elseif (cptmatM2 == 2) then
z = precondp_d(w_tld);
end
xi = norm( z );
gam = 1.0;
eta = -1.0;
theta = 0.0;
for i = 1:max_it, // begin iteration
if ( rho == 0.0 | xi == 0.0 ), iter=i; break; end
v = v_tld / rho;
y = y / rho;
w = w_tld / xi;
z = z / xi;
delta = z'*y;
if ( delta == 0.0 ), iter=i; break; end
// y_tld = M2 \ y;
if (cptmatM2 == 1) then
y_tld = precond_d(y,"notransp");
elseif (cptmatM2 == 2) then
y_tld = precond_d(y);
end
// z_tld = M1'\ z;
if (cptmatM1 == 1) then
z_tld = precond_g(z,"transp");
elseif (cptmatM1 == 2) then
z_tld = precondp_g(z);
end
if ( i > 1 ), // direction vector
p = y_tld - ( xi*delta / ep )*p;
q = z_tld - ( rho*delta / ep )*q;
else
p = y_tld;
q = z_tld;
end
// p_tld = A*p;
if (cptmat == 1) then
p_tld = matvec(p,"notransp");
elseif (cptmat == 2) then
p_tld = matvec(p);
end
ep = q'*p_tld;
if ( ep == 0.0 ), iter=i; break; end
Beta = ep / delta;
if ( Beta == 0.0 ), iter=i; break; end
v_tld = p_tld - Beta*v;
// y = M1 \ v_tld;
if (cptmatM1 == 1) then
y = precond_g(v_tld,"notransp");
elseif (cptmatM1==2) then
y = precond_g(v_tld);
end
rho_1 = rho;
rho = norm( y );
// w_tld = ( A'*q ) - ( Beta*w );
if (cptmat == 1) then
w_tld = ( matvec(q,"transp") ) - ( Beta*w );
elseif (cptmat == 2) then
w_tld = ( matvecp(q) ) - ( Beta*w );
end
// z = M2' \ w_tld;
if (cptmatM2 == 1) then
z = precond_d(w_tld,"transp");
elseif (cptmatM2 == 2) then
z = precondp_d(w_tld);
end
xi = norm( z );
gamma_1 = gam;
theta_1 = theta;
theta = rho / ( gamma_1*Beta );
gam = 1.0 / sqrt( 1.0 + (theta^2) );
if ( gam == 0.0 ), iter=i; break; end
eta = -eta*rho_1*(gam^2) / ( Beta*(gamma_1^2) );
if ( i > 1 ), // compute adjustment
d = eta*p + (( theta_1*gam )^2)*d;
s = eta*p_tld + (( theta_1*gam )^2)*s;
else
d = eta*p;
s = eta*p_tld;
end
x = x + d; // update approximation
r = r - s; // update residual
err = norm( r ) / bnrm2; // check convergence
res = [res;err];
if ( err <= tol ), iter=i; break; end
if ( i == max_it ), iter=i; end
end
if ( err <= tol ), // converged
flag = 0;
elseif ( rho == 0.0 ), // breakdown
flag = -1;
elseif ( Beta == 0.0 ),
flag = -2;
elseif ( gam == 0.0 ),
flag = -3;
elseif ( delta == 0.0 ),
flag = -4;
elseif ( ep == 0.0 ),
flag = -5;
elseif ( xi == 0.0 ),
flag = -6;
else // no convergence
flag = 1;
end
endfunction
function y = matvec(x,t)
if (t=="notransp") then
y = A*x;
elseif (t=="transp") then
y = A'*x;
end
endfunction
function y = precond_g(x,t)
if (t=="notransp") then
y = Prec_g*x;
elseif (t=="transp") then
y = Prec_g'*x;
end
endfunction
function y = precond_d(x,t)
if (t=="notransp") then
y = Prec_d*x;
elseif (t=="transp") then
y = Prec_d'*x;
end
endfunction
|
