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
|
########################################################################
##
## Copyright (C) 1993-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## 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; see the file COPYING. If not, see
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {[@var{u}, @var{h}, @var{nu}] =} krylov (@var{A}, @var{V}, @var{k}, @var{eps1}, @var{pflg})
## Construct an orthogonal basis @var{u} of a block Krylov subspace.
##
## The block Krylov subspace has the following form:
##
## @example
## [v a*v a^2*v @dots{} a^(k+1)*v]
## @end example
##
## @noindent
## The construction is made with Householder reflections to guard against loss
## of orthogonality.
##
## If @var{V} is a vector, then @var{h} contains the Hessenberg matrix
## such that @nospell{@tcode{a*u == u*h+rk*ek'}}, in which
## @code{rk = a*u(:,k)-u*h(:,k)}, and @nospell{@tcode{ek'}} is the vector
## @code{[0, 0, @dots{}, 1]} of length @var{k}. Otherwise, @var{h} is
## meaningless.
##
## If @var{V} is a vector and @var{k} is greater than @code{length (A) - 1},
## then @var{h} contains the Hessenberg matrix such that @code{a*u == u*h}.
##
## The value of @var{nu} is the dimension of the span of the Krylov subspace
## (based on @var{eps1}).
##
## If @var{b} is a vector and @var{k} is greater than @var{m-1}, then @var{h}
## contains the Hessenberg decomposition of @var{A}.
##
## The optional parameter @var{eps1} is the threshold for zero. The default
## value is 1e-12.
##
## If the optional parameter @var{pflg} is nonzero, row pivoting is used to
## improve numerical behavior. The default value is 0.
##
## Reference: @nospell{A. Hodel, P. Misra}, @cite{Partial Pivoting in the
## Computation of Krylov Subspaces of Large Sparse Systems}, Proceedings of
## the 42nd IEEE Conference on Decision and Control, December 2003.
## @end deftypefn
function [Uret, H, nu] = krylov (A, V, k, eps1, pflg)
if (isa (A, "single") || isa (V, "single"))
defeps = 1e-6;
else
defeps = 1e-12;
endif
if (nargin < 3)
print_usage ();
elseif (nargin < 5)
## Default permutation flag.
pflg = 0;
endif
if (nargin < 4)
## Default tolerance parameter.
eps1 = defeps;
endif
if (isempty (eps1))
eps1 = defeps;
endif
if (! issquare (A) || isempty (A))
error ("krylov: A(%d x %d) must be a non-empty square matrix", rows (A), columns (A));
endif
na = rows (A);
[m, kb] = size (V);
if (m != na)
error ("krylov: A(%d x %d), V(%d x %d): argument dimensions do not match",
na, na, m, kb);
endif
if (! isscalar (k))
error ("krylov: K must be a scalar integer");
endif
Vnrm = norm (V, Inf);
## check for trivial solution.
if (Vnrm == 0)
Uret = [];
H = [];
nu = 0;
return;
endif
## Identify trivial null space.
abm = max (abs ([A, V]'));
zidx = find (abm == 0);
## Set up vector of pivot points.
pivot_vec = 1:na;
iter = 0;
alpha = [];
nh = 0;
while (length (alpha) < na) && (columns (V) > 0) && (iter < k)
iter += 1;
## Get orthogonal basis of V.
jj = 1;
while (jj <= columns (V) && length (alpha) < na)
## Index of next Householder reflection.
nu = length (alpha)+1;
short_pv = pivot_vec(nu:na);
q = V(:,jj);
short_q = q(short_pv);
if (norm (short_q) < eps1)
## Insignificant column; delete.
nv = columns (V);
if (jj != nv)
[V(:,jj), V(:,nv)] = swap (V(:,jj), V(:,nv));
## FIXME: H columns should be swapped too.
## Not done since Block Hessenberg structure is lost anyway.
endif
V = V(:,1:(nv-1));
## One less reflection.
nu -= 1;
else
## New householder reflection.
if (pflg)
## Locate max magnitude element in short_q.
asq = abs (short_q);
maxv = max (asq);
maxidx = find (asq == maxv, 1);
pivot_idx = short_pv(maxidx);
## See if need to change the pivot list.
if (pivot_idx != pivot_vec(nu))
swapidx = maxidx + (nu-1);
[pivot_vec(nu), pivot_vec(swapidx)] = ...
swap (pivot_vec(nu), pivot_vec(swapidx));
endif
endif
## Isolate portion of vector for reflection.
idx = pivot_vec(nu:na);
jdx = pivot_vec(1:nu);
[hv, av, z] = housh (q(idx), 1, 0);
alpha(nu) = av;
U(idx,nu) = hv;
## Reduce V per the reflection.
V(idx,:) = V(idx,:) - av*hv*(hv' * V(idx,:));
if (iter > 1)
## FIXME: not done correctly for block case.
H(nu,nu-1) = V(pivot_vec(nu),jj);
endif
## Advance to next column of V.
jj += 1;
endif
endwhile
## Check for oversize V (due to full rank).
if ((columns (V) > na) && (length (alpha) == na))
## Trim to size.
V = V(:,1:na);
elseif (columns (V) > na)
krylov_V = V;
krylov_na = na;
krylov_length_alpha = length (alpha);
error ("krylov: this case should never happen; submit a bug report");
endif
if (columns (V) > 0)
## Construct next Q and multiply.
Q = zeros (size (V));
for kk = 1:columns (Q)
Q(pivot_vec(nu-columns (Q)+kk),kk) = 1;
endfor
## Apply Householder reflections.
for ii = nu:-1:1
idx = pivot_vec(ii:na);
hv = U(idx,ii);
av = alpha(ii);
Q(idx,:) = Q(idx,:) - av*hv*(hv'*Q(idx,:));
endfor
endif
## Multiply to get new vector.
V = A*Q;
## Project off of previous vectors.
nu = length (alpha);
for i = 1:nu
hv = U(:,i);
av = alpha(i);
V -= av*hv*(hv'*V);
H(i,nu-columns (V)+(1:columns (V))) = V(pivot_vec(i),:);
endfor
endwhile
## Back out complete U matrix.
## back out U matrix.
j1 = columns (U);
for i = j1:-1:1
idx = pivot_vec(i:na);
hv = U(idx,i);
av = alpha(i);
U(:,i) = zeros (na, 1);
U(idx(1),i) = 1;
U(idx,i:j1) = U(idx,i:j1)-av*hv*(hv'*U(idx,i:j1));
endfor
nu = length (alpha);
Uret = U;
if (max (max (abs (Uret(zidx,:)))) > 0)
warning ("krylov: trivial null space corrupted; set pflg = 1 or eps1 > %e",
eps1);
endif
endfunction
function [a1, b1] = swap (a, b)
a1 = b;
b1 = a;
endfunction
|
