Mercurial > octave-antonio
annotate scripts/sparse/spaugment.m @ 8920:eb63fbe60fab
update copyright notices
| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Sat, 07 Mar 2009 10:41:27 -0500 |
| parents | fb1c929dbbb7 |
| children | 1bf0ce0930be |
| rev | line source |
|---|---|
| 8920 | 1 ## Copyright (C) 2008, 2009 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 -*- | |
| 20 ## @deftypefn {Function File} {@var{s} =} spaugment (@var{a}, @var{c}) | |
| 21 ## Creates the augmented matrix of @var{a}. This is given by | |
| 22 ## | |
| 23 ## @example | |
| 24 ## [@var{c} * eye(@var{m}, @var{m}),@var{a}; @var{a}', zeros(@var{n}, | |
| 25 ## @var{n})] | |
| 26 ## @end example | |
| 27 ## | |
| 28 ## @noindent | |
| 29 ## This is related to the leasted squared solution of | |
| 30 ## @code{@var{a} \\ @var{b}}, by | |
| 31 ## | |
| 32 ## @example | |
| 33 ## @var{s} * [ @var{r} / @var{c}; x] = [@var{b}, zeros(@var{n}, | |
| 34 ## columns(@var{b})] | |
| 35 ## @end example | |
| 36 ## | |
| 37 ## @noindent | |
| 38 ## where @var{r} is the residual error | |
| 39 ## | |
| 40 ## @example | |
| 41 ## @var{r} = @var{b} - @var{a} * @var{x} | |
| 42 ## @end example | |
| 43 ## | |
| 44 ## As the matrix @var{s} is symmetric indefinite it can be factorized | |
| 45 ## with @code{lu}, and the minimum norm solution can therefore be found | |
| 46 ## without the need for a @code{qr} factorization. As the residual | |
| 47 ## error will be @code{zeros (@var{m}, @var{m})} for under determined | |
| 48 ## problems, and example can be | |
| 49 ## | |
| 50 ## @example | |
| 51 ## @group | |
| 52 ## m = 11; n = 10; mn = max(m ,n); | |
| 8516 | 53 ## a = spdiags ([ones(mn,1), 10*ones(mn,1), -ones(mn,1)], |
| 54 ## [-1, 0, 1], m, n); | |
| 7681 | 55 ## x0 = a \ ones (m,1); |
| 56 ## s = spaugment (a); | |
| 57 ## [L, U, P, Q] = lu (s); | |
| 58 ## x1 = Q * (U \ (L \ (P * [ones(m,1); zeros(n,1)]))); | |
| 59 ## x1 = x1(end - n + 1 : end); | |
| 60 ## @end group | |
| 61 ## @end example | |
| 62 ## | |
| 63 ## To find the solution of an overdetermined problem needs an estimate | |
| 64 ## of the residual error @var{r} and so it is more complex to formulate | |
| 65 ## a minimum norm solution using the @code{spaugment} function. | |
| 66 ## | |
| 67 ## In general the left division operator is more stable and faster than | |
| 68 ## using the @code{spaugment} function. | |
| 69 ## @end deftypefn | |
| 70 | |
| 71 function s = spaugment (a, c) | |
| 72 if (nargin < 2) | |
| 73 if (issparse (a)) | |
| 74 c = max (max (abs (a))) / 1000; | |
| 75 else | |
| 76 if (ndims (a) != 2) | |
| 77 error ("spaugment: expecting 2-dimenisional matrix") | |
| 78 else | |
| 79 c = max (abs (a(:))) / 1000; | |
| 80 endif | |
| 81 endif | |
| 82 elseif (!isscalar (c)) | |
| 83 error ("spaugment: c must be a scalar"); | |
| 84 endif | |
| 85 | |
| 86 [m, n] = size (a); | |
| 87 s = [ c * speye(m, m), a; a', sparse(n, n)]; | |
| 88 endfunction | |
| 89 | |
| 8871 | 90 %!testif HAVE_UMFPACK |
| 7681 | 91 %! m = 11; n = 10; mn = max(m ,n); |
| 92 %! a = spdiags ([ones(mn,1), 10*ones(mn,1), -ones(mn,1)],[-1,0,1], m, n); | |
| 93 %! x0 = a \ ones (m,1); | |
| 94 %! s = spaugment (a); | |
| 95 %! [L, U, P, Q] = lu (s); | |
| 96 %! x1 = Q * (U \ (L \ (P * [ones(m,1); zeros(n,1)]))); | |
| 97 %! x1 = x1(end - n + 1 : end); | |
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7687
795be0215bf7
spaugment: reduce test script tolerance
Ben Abbott <bpabbott@mac.com>
parents:
7681
diff
changeset
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98 %! assert (x1, x0, 1e-6) |