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author | Rik <rik@octave.org> |
---|---|
date | Wed, 28 Aug 2013 08:33:02 -0700 |
parents | 72c96de7a403 |
children | d63878346099 |
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## Copyright (C) 2000-2012 Kai Habel ## Copyright (C) 2007 David Bateman ## ## This file is part of Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## <http://www.gnu.org/licenses/>. ## -*- texinfo -*- ## @deftypefn {Function File} {@var{d} =} del2 (@var{M}) ## @deftypefnx {Function File} {@var{d} =} del2 (@var{M}, @var{h}) ## @deftypefnx {Function File} {@var{d} =} del2 (@var{M}, @var{dx}, @var{dy}, @dots{}) ## ## Calculate the discrete Laplace ## @tex ## operator $( \nabla^2 )$. ## @end tex ## @ifnottex ## operator. ## @end ifnottex ## For a 2-dimensional matrix @var{M} this is defined as ## @tex ## $$d = {1 \over 4} \left( {d^2 \over dx^2} M(x,y) + {d^2 \over dy^2} M(x,y) \right)$$ ## @end tex ## @ifnottex ## ## @example ## @group ## 1 / d^2 d^2 \ ## D = --- * | --- M(x,y) + --- M(x,y) | ## 4 \ dx^2 dy^2 / ## @end group ## @end example ## ## @end ifnottex ## For N-dimensional arrays the sum in parentheses is expanded to include second ## derivatives over the additional higher dimensions. ## ## The spacing between evaluation points may be defined by @var{h}, which is a ## scalar defining the equidistant spacing in all dimensions. Alternatively, ## the spacing in each dimension may be defined separately by @var{dx}, ## @var{dy}, etc. A scalar spacing argument defines equidistant spacing, ## whereas a vector argument can be used to specify variable spacing. The ## length of the spacing vectors must match the respective dimension of ## @var{M}. The default spacing value is 1. ## ## At least 3 data points are needed for each dimension. Boundary points are ## calculated from the linear extrapolation of interior points. ## ## @seealso{gradient, diff} ## @end deftypefn ## Author: Kai Habel <kai.habel@gmx.de> function D = del2 (M, varargin) if (nargin < 1) print_usage (); endif nd = ndims (M); sz = size (M); dx = cell (1, nd); if (nargin == 2 || nargin == 1) if (nargin == 1) h = 1; else h = varargin{1}; endif for i = 1 : nd if (isscalar (h)) dx{i} = h * ones (sz (i), 1); else if (length (h) == sz (i)) dx{i} = diff (h)(:); else error ("del2: dimensionality mismatch in %d-th spacing vector", i); endif endif endfor elseif (nargin - 1 == nd) ## Reverse dx{1} and dx{2} as the X-dim is the 2nd dim of the ND array tmp = varargin{1}; varargin{1} = varargin{2}; varargin{2} = tmp; for i = 1 : nd if (isscalar (varargin{i})) dx{i} = varargin{i} * ones (sz (i), 1); else if (length (varargin{i}) == sz (i)) dx{i} = diff (varargin{i})(:); else error ("del2: dimensionality mismatch in %d-th spacing vector", i); endif endif endfor else print_usage (); endif idx = cell (1, nd); for i = 1: nd idx{i} = ":"; endfor D = zeros (sz); for i = 1: nd if (sz(i) >= 3) DD = zeros (sz); idx1 = idx2 = idx3 = idx; ## interior points idx1{i} = 1 : sz(i) - 2; idx2{i} = 2 : sz(i) - 1; idx3{i} = 3 : sz(i); szi = sz; szi (i) = 1; h1 = repmat (shiftdim (dx{i}(1 : sz(i) - 2), 1 - i), szi); h2 = repmat (shiftdim (dx{i}(2 : sz(i) - 1), 1 - i), szi); DD(idx2{:}) = ((M(idx1{:}) - M(idx2{:})) ./ h1 + ... (M(idx3{:}) - M(idx2{:})) ./ h2) ./ (h1 + h2); ## left and right boundary if (sz(i) == 3) DD(idx1{:}) = DD(idx3{:}) = DD(idx2{:}); else idx1{i} = 1; idx2{i} = 2; idx3{i} = 3; DD(idx1{:}) = (dx{i}(1) + dx{i}(2)) / dx{i}(2) * DD (idx2{:}) - ... dx{i}(1) / dx{i}(2) * DD (idx3{:}); idx1{i} = sz(i); idx2{i} = sz(i) - 1; idx3{i} = sz(i) - 2; DD(idx1{:}) = (dx{i}(sz(i) - 1) + dx{i}(sz(i) - 2)) / ... dx{i}(sz(i) - 2) * DD (idx2{:}) - ... dx{i}(sz(i) - 1) / dx{i}(sz(i) - 2) * DD (idx3{:}); endif D += DD; endif endfor D = D ./ nd; endfunction