Mercurial > octave-nkf
annotate scripts/sparse/spaugment.m @ 12575:d0b799dafede
Grammarcheck files for 3.4.1 release.
| author | Rik <octave@nomad.inbox5.com> |
|---|---|
| date | Mon, 04 Apr 2011 15:33:46 -0700 |
| parents | b0084095098e |
| children | 72c96de7a403 |
| rev | line source |
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| 11523 | 1 ## Copyright (C) 2008-2011 David Bateman |
| 7681 | 2 ## |
| 3 ## This file is part of Octave. | |
| 4 ## | |
| 5 ## Octave is free software; you can redistribute it and/or modify it | |
| 6 ## under the terms of the GNU General Public License as published by | |
| 7 ## the Free Software Foundation; either version 3 of the License, or (at | |
| 8 ## your option) any later version. | |
| 9 ## | |
| 10 ## Octave is distributed in the hope that it will be useful, but | |
| 11 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
| 12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| 13 ## General Public License for more details. | |
| 14 ## | |
| 15 ## You should have received a copy of the GNU General Public License | |
| 16 ## along with Octave; see the file COPYING. If not, see | |
| 17 ## <http://www.gnu.org/licenses/>. | |
| 18 | |
| 19 ## -*- texinfo -*- | |
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20 ## @deftypefn {Function File} {@var{s} =} spaugment (@var{A}, @var{c}) |
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21 ## Create the augmented matrix of @var{A}. This is given by |
| 7681 | 22 ## |
| 23 ## @example | |
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24 ## @group |
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25 ## [@var{c} * eye(@var{m}, @var{m}),@var{A}; @var{A}', zeros(@var{n}, |
| 7681 | 26 ## @var{n})] |
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27 ## @end group |
| 7681 | 28 ## @end example |
| 29 ## | |
| 30 ## @noindent | |
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31 ## This is related to the least squares solution of |
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32 ## @code{@var{A} \\ @var{b}}, by |
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33 ## |
| 7681 | 34 ## @example |
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35 ## @group |
| 7681 | 36 ## @var{s} * [ @var{r} / @var{c}; x] = [@var{b}, zeros(@var{n}, |
| 37 ## columns(@var{b})] | |
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38 ## @end group |
| 7681 | 39 ## @end example |
| 40 ## | |
| 41 ## @noindent | |
| 42 ## where @var{r} is the residual error | |
| 43 ## | |
| 44 ## @example | |
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45 ## @var{r} = @var{b} - @var{A} * @var{x} |
| 7681 | 46 ## @end example |
| 47 ## | |
| 48 ## As the matrix @var{s} is symmetric indefinite it can be factorized | |
| 49 ## with @code{lu}, and the minimum norm solution can therefore be found | |
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50 ## without the need for a @code{qr} factorization. As the residual |
| 7681 | 51 ## error will be @code{zeros (@var{m}, @var{m})} for under determined |
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52 ## problems, and example can be |
| 7681 | 53 ## |
| 54 ## @example | |
| 55 ## @group | |
| 56 ## m = 11; n = 10; mn = max(m ,n); | |
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57 ## A = spdiags ([ones(mn,1), 10*ones(mn,1), -ones(mn,1)], |
| 8516 | 58 ## [-1, 0, 1], m, n); |
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59 ## x0 = A \ ones (m,1); |
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60 ## s = spaugment (A); |
| 7681 | 61 ## [L, U, P, Q] = lu (s); |
| 62 ## x1 = Q * (U \ (L \ (P * [ones(m,1); zeros(n,1)]))); | |
| 63 ## x1 = x1(end - n + 1 : end); | |
| 64 ## @end group | |
| 65 ## @end example | |
| 66 ## | |
| 67 ## To find the solution of an overdetermined problem needs an estimate | |
| 68 ## of the residual error @var{r} and so it is more complex to formulate | |
| 69 ## a minimum norm solution using the @code{spaugment} function. | |
| 70 ## | |
| 71 ## In general the left division operator is more stable and faster than | |
| 72 ## using the @code{spaugment} function. | |
| 73 ## @end deftypefn | |
| 74 | |
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75 function s = spaugment (A, c) |
| 7681 | 76 if (nargin < 2) |
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77 if (issparse (A)) |
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78 c = max (max (abs (A))) / 1000; |
| 7681 | 79 else |
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80 if (ndims (A) != 2) |
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81 error ("spaugment: expecting 2-dimenisional matrix"); |
| 7681 | 82 else |
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83 c = max (abs (A(:))) / 1000; |
| 7681 | 84 endif |
| 85 endif | |
| 86 elseif (!isscalar (c)) | |
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87 error ("spaugment: C must be a scalar"); |
| 7681 | 88 endif |
| 89 | |
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90 [m, n] = size (A); |
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91 s = [ c * speye(m, m), A; A', sparse(n, n)]; |
| 7681 | 92 endfunction |
| 93 | |
| 8871 | 94 %!testif HAVE_UMFPACK |
| 7681 | 95 %! m = 11; n = 10; mn = max(m ,n); |
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96 %! A = spdiags ([ones(mn,1), 10*ones(mn,1), -ones(mn,1)],[-1,0,1], m, n); |
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97 %! x0 = A \ ones (m,1); |
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98 %! s = spaugment (A); |
| 7681 | 99 %! [L, U, P, Q] = lu (s); |
| 100 %! x1 = Q * (U \ (L \ (P * [ones(m,1); zeros(n,1)]))); | |
| 101 %! x1 = x1(end - n + 1 : end); | |
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102 %! assert (x1, x0, 1e-6) |