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1 ## Copyright (C) 2008 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 ## Creates 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 ## [@var{c} * eye(@var{m}, @var{m}),@var{a}; @var{a}', zeros(@var{n}, |
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25 ## @var{n})] |
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26 ## @end example |
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27 ## |
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28 ## @noindent |
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29 ## This is related to the leasted squared solution of |
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30 ## @code{@var{a} \\ @var{b}}, by |
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31 ## |
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32 ## @example |
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33 ## @var{s} * [ @var{r} / @var{c}; x] = [@var{b}, zeros(@var{n}, |
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34 ## columns(@var{b})] |
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35 ## @end example |
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36 ## |
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37 ## @noindent |
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38 ## where @var{r} is the residual error |
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39 ## |
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40 ## @example |
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41 ## @var{r} = @var{b} - @var{a} * @var{x} |
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42 ## @end example |
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43 ## |
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44 ## As the matrix @var{s} is symmetric indefinite it can be factorized |
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45 ## with @code{lu}, and the minimum norm solution can therefore be found |
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46 ## without the need for a @code{qr} factorization. As the residual |
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47 ## error will be @code{zeros (@var{m}, @var{m})} for under determined |
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48 ## problems, and example can be |
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49 ## |
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50 ## @example |
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51 ## @group |
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52 ## m = 11; n = 10; mn = max(m ,n); |
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53 ## a = spdiags ([ones(mn,1), 10*ones(mn,1), -ones(mn,1)],[-1,0,1], m, n); |
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54 ## x0 = a \ ones (m,1); |
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55 ## s = spaugment (a); |
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56 ## [L, U, P, Q] = lu (s); |
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57 ## x1 = Q * (U \ (L \ (P * [ones(m,1); zeros(n,1)]))); |
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58 ## x1 = x1(end - n + 1 : end); |
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59 ## @end group |
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60 ## @end example |
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61 ## |
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62 ## To find the solution of an overdetermined problem needs an estimate |
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63 ## of the residual error @var{r} and so it is more complex to formulate |
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64 ## a minimum norm solution using the @code{spaugment} function. |
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65 ## |
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66 ## In general the left division operator is more stable and faster than |
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67 ## using the @code{spaugment} function. |
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68 ## @end deftypefn |
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69 |
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70 function s = spaugment (a, c) |
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71 if (nargin < 2) |
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72 if (issparse (a)) |
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73 c = max (max (abs (a))) / 1000; |
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74 else |
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75 if (ndims (a) != 2) |
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76 error ("spaugment: expecting 2-dimenisional matrix") |
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77 else |
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78 c = max (abs (a(:))) / 1000; |
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79 endif |
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80 endif |
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81 elseif (!isscalar (c)) |
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82 error ("spaugment: c must be a scalar"); |
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83 endif |
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84 |
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85 [m, n] = size (a); |
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86 s = [ c * speye(m, m), a; a', sparse(n, n)]; |
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87 endfunction |
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88 |
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89 %!test |
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90 %! m = 11; n = 10; mn = max(m ,n); |
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91 %! a = spdiags ([ones(mn,1), 10*ones(mn,1), -ones(mn,1)],[-1,0,1], m, n); |
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92 %! x0 = a \ ones (m,1); |
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93 %! s = spaugment (a); |
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94 %! [L, U, P, Q] = lu (s); |
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95 %! x1 = Q * (U \ (L \ (P * [ones(m,1); zeros(n,1)]))); |
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96 %! x1 = x1(end - n + 1 : end); |
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97 %! assert (x1, x0, 1e-10) |