Mercurial > octave
annotate scripts/general/del2.m @ 17744:d63878346099
maint: Update copyright notices for release.
| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Wed, 23 Oct 2013 22:09:27 -0400 |
| parents | 1c89599167a6 |
| children | 21e5205679d0 |
| rev | line source |
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1 ## Copyright (C) 2000-2013 Kai Habel |
| 6788 | 2 ## Copyright (C) 2007 David Bateman |
| 3 ## | |
| 4 ## This file is part of Octave. | |
| 5 ## | |
| 6 ## Octave is free software; you can redistribute it and/or modify it | |
| 7 ## under the terms of the GNU General Public License as published by | |
| 7016 | 8 ## the Free Software Foundation; either version 3 of the License, or (at |
| 9 ## your option) any later version. | |
| 6788 | 10 ## |
| 11 ## Octave is distributed in the hope that it will be useful, but | |
| 12 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
| 13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| 14 ## General Public License for more details. | |
| 15 ## | |
| 16 ## You should have received a copy of the GNU General Public License | |
| 7016 | 17 ## along with Octave; see the file COPYING. If not, see |
| 18 ## <http://www.gnu.org/licenses/>. | |
| 6788 | 19 |
| 20 ## -*- texinfo -*- | |
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21 ## @deftypefn {Function File} {@var{d} =} del2 (@var{M}) |
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22 ## @deftypefnx {Function File} {@var{d} =} del2 (@var{M}, @var{h}) |
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23 ## @deftypefnx {Function File} {@var{d} =} del2 (@var{M}, @var{dx}, @var{dy}, @dots{}) |
| 6788 | 24 ## |
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25 ## Calculate the discrete Laplace |
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26 ## @tex |
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27 ## operator $( \nabla^2 )$. |
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28 ## @end tex |
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29 ## @ifnottex |
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30 ## operator. |
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31 ## @end ifnottex |
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32 ## For a 2-dimensional matrix @var{M} this is defined as |
| 6788 | 33 ## @tex |
| 8499 | 34 ## $$d = {1 \over 4} \left( {d^2 \over dx^2} M(x,y) + {d^2 \over dy^2} M(x,y) \right)$$ |
| 6788 | 35 ## @end tex |
| 36 ## @ifnottex | |
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37 ## |
| 6788 | 38 ## @example |
| 39 ## @group | |
| 40 ## 1 / d^2 d^2 \ | |
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41 ## D = --- * | --- M(x,y) + --- M(x,y) | |
| 6788 | 42 ## 4 \ dx^2 dy^2 / |
| 43 ## @end group | |
| 44 ## @end example | |
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45 ## |
| 6788 | 46 ## @end ifnottex |
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47 ## For N-dimensional arrays the sum in parentheses is expanded to include second |
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48 ## derivatives over the additional higher dimensions. |
| 6788 | 49 ## |
| 50 ## The spacing between evaluation points may be defined by @var{h}, which is a | |
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51 ## scalar defining the equidistant spacing in all dimensions. Alternatively, |
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52 ## the spacing in each dimension may be defined separately by @var{dx}, |
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53 ## @var{dy}, etc. A scalar spacing argument defines equidistant spacing, |
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54 ## whereas a vector argument can be used to specify variable spacing. The |
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55 ## length of the spacing vectors must match the respective dimension of |
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56 ## @var{M}. The default spacing value is 1. |
| 6788 | 57 ## |
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58 ## At least 3 data points are needed for each dimension. Boundary points are |
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59 ## calculated from the linear extrapolation of interior points. |
| 6788 | 60 ## |
| 61 ## @seealso{gradient, diff} | |
| 62 ## @end deftypefn | |
| 63 | |
| 64 ## Author: Kai Habel <kai.habel@gmx.de> | |
| 65 | |
| 66 function D = del2 (M, varargin) | |
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67 |
| 6788 | 68 if (nargin < 1) |
| 69 print_usage (); | |
| 70 endif | |
| 71 | |
| 72 nd = ndims (M); | |
| 73 sz = size (M); | |
| 74 dx = cell (1, nd); | |
| 75 if (nargin == 2 || nargin == 1) | |
| 76 if (nargin == 1) | |
| 77 h = 1; | |
| 78 else | |
| 7669 | 79 h = varargin{1}; |
| 6788 | 80 endif |
| 81 for i = 1 : nd | |
| 82 if (isscalar (h)) | |
| 10549 | 83 dx{i} = h * ones (sz (i), 1); |
| 6788 | 84 else |
| 10549 | 85 if (length (h) == sz (i)) |
| 86 dx{i} = diff (h)(:); | |
| 87 else | |
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88 error ("del2: dimensionality mismatch in %d-th spacing vector", i); |
| 10549 | 89 endif |
| 6788 | 90 endif |
| 91 endfor | |
| 92 elseif (nargin - 1 == nd) | |
| 93 ## Reverse dx{1} and dx{2} as the X-dim is the 2nd dim of the ND array | |
| 94 tmp = varargin{1}; | |
| 95 varargin{1} = varargin{2}; | |
| 96 varargin{2} = tmp; | |
| 97 | |
| 98 for i = 1 : nd | |
| 99 if (isscalar (varargin{i})) | |
| 10549 | 100 dx{i} = varargin{i} * ones (sz (i), 1); |
| 6788 | 101 else |
| 10549 | 102 if (length (varargin{i}) == sz (i)) |
| 103 dx{i} = diff (varargin{i})(:); | |
| 104 else | |
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105 error ("del2: dimensionality mismatch in %d-th spacing vector", i); |
| 10549 | 106 endif |
| 6788 | 107 endif |
| 108 endfor | |
| 109 else | |
| 110 print_usage (); | |
| 111 endif | |
| 112 | |
| 113 idx = cell (1, nd); | |
| 114 for i = 1: nd | |
| 115 idx{i} = ":"; | |
| 116 endfor | |
| 117 | |
| 118 D = zeros (sz); | |
| 119 for i = 1: nd | |
| 120 if (sz(i) >= 3) | |
| 121 DD = zeros (sz); | |
| 122 idx1 = idx2 = idx3 = idx; | |
| 123 | |
| 124 ## interior points | |
| 125 idx1{i} = 1 : sz(i) - 2; | |
| 126 idx2{i} = 2 : sz(i) - 1; | |
| 127 idx3{i} = 3 : sz(i); | |
| 128 szi = sz; | |
| 129 szi (i) = 1; | |
| 130 | |
| 131 h1 = repmat (shiftdim (dx{i}(1 : sz(i) - 2), 1 - i), szi); | |
| 132 h2 = repmat (shiftdim (dx{i}(2 : sz(i) - 1), 1 - i), szi); | |
| 133 DD(idx2{:}) = ((M(idx1{:}) - M(idx2{:})) ./ h1 + ... | |
| 10549 | 134 (M(idx3{:}) - M(idx2{:})) ./ h2) ./ (h1 + h2); |
| 6788 | 135 |
| 136 ## left and right boundary | |
| 137 if (sz(i) == 3) | |
| 10549 | 138 DD(idx1{:}) = DD(idx3{:}) = DD(idx2{:}); |
| 6788 | 139 else |
| 10549 | 140 idx1{i} = 1; |
| 141 idx2{i} = 2; | |
| 142 idx3{i} = 3; | |
| 143 DD(idx1{:}) = (dx{i}(1) + dx{i}(2)) / dx{i}(2) * DD (idx2{:}) - ... | |
| 144 dx{i}(1) / dx{i}(2) * DD (idx3{:}); | |
| 6788 | 145 |
| 10549 | 146 idx1{i} = sz(i); |
| 147 idx2{i} = sz(i) - 1; | |
| 148 idx3{i} = sz(i) - 2; | |
| 149 DD(idx1{:}) = (dx{i}(sz(i) - 1) + dx{i}(sz(i) - 2)) / ... | |
| 150 dx{i}(sz(i) - 2) * DD (idx2{:}) - ... | |
| 151 dx{i}(sz(i) - 1) / dx{i}(sz(i) - 2) * DD (idx3{:}); | |
| 6788 | 152 endif |
| 153 | |
| 154 D += DD; | |
| 155 endif | |
| 156 endfor | |
| 157 | |
| 158 D = D ./ nd; | |
| 159 endfunction | |
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