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Fix warnings
author LYH <lyh.kernel@gmail.com>
date Fri, 27 Sep 2013 17:43:27 +0800
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1 ## Copyright (C) 2008-2012 David Bateman
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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 it
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6 ## 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 (at
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8 ## 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, but
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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13 ## 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; see the file COPYING. If not, see
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17 ## <http://www.gnu.org/licenses/>.
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18
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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
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22 ##
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23 ## @example
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24 ## @group
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25 ## [@var{c} * eye(@var{m}, @var{m}), @var{A};
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26 ## @var{A}', zeros(@var{n}, @var{n})]
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27 ## @end group
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28 ## @end example
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29 ##
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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 ##
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34 ## @example
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35 ## @group
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36 ## @var{s} * [ @var{r} / @var{c}; x] = [ @var{b}, zeros(@var{n}, columns(@var{b})) ]
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37 ## @end group
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38 ## @end example
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39 ##
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40 ## @noindent
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41 ## where @var{r} is the residual error
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42 ##
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43 ## @example
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44 ## @var{r} = @var{b} - @var{A} * @var{x}
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45 ## @end example
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46 ##
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47 ## As the matrix @var{s} is symmetric indefinite it can be factorized
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48 ## with @code{lu}, and the minimum norm solution can therefore be found
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49 ## without the need for a @code{qr} factorization. As the residual
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50 ## error will be @code{zeros (@var{m}, @var{m})} for under determined
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51 ## problems, and example can be
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52 ##
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53 ## @example
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54 ## @group
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55 ## m = 11; n = 10; mn = max (m, n);
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56 ## A = spdiags ([ones(mn,1), 10*ones(mn,1), -ones(mn,1)],
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57 ## [-1, 0, 1], m, n);
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58 ## x0 = A \ ones (m,1);
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59 ## s = spaugment (A);
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60 ## [L, U, P, Q] = lu (s);
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61 ## x1 = Q * (U \ (L \ (P * [ones(m,1); zeros(n,1)])));
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62 ## x1 = x1(end - n + 1 : end);
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63 ## @end group
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64 ## @end example
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65 ##
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66 ## To find the solution of an overdetermined problem needs an estimate
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67 ## of the residual error @var{r} and so it is more complex to formulate
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68 ## a minimum norm solution using the @code{spaugment} function.
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69 ##
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70 ## In general the left division operator is more stable and faster than
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71 ## using the @code{spaugment} function.
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72 ## @end deftypefn
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73
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74 function s = spaugment (A, c)
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75 if (nargin < 2)
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76 if (issparse (A))
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77 c = max (max (abs (A))) / 1000;
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78 else
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79 if (ndims (A) != 2)
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80 error ("spaugment: expecting 2-dimenisional matrix");
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81 else
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82 c = max (abs (A(:))) / 1000;
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83 endif
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84 endif
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85 elseif (!isscalar (c))
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86 error ("spaugment: C must be a scalar");
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87 endif
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88
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89 [m, n] = size (A);
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90 s = [ c * speye(m, m), A; A', sparse(n, n)];
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91 endfunction
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92
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93
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94 %!testif HAVE_UMFPACK
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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);
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99 %! [L, U, P, Q] = lu (s);
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100 %! x1 = Q * (U \ (L \ (P * [ones(m,1); zeros(n,1)])));
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101 %! x1 = x1(end - n + 1 : end);
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102 %! assert (x1, x0, 1e-6);
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103