--- /dev/null
+## Copyright (c) 2010-2011 Andrew V. Knyazev <andrew.knyazev@ucdenver.edu>
+## Copyright (c) 2010-2011 Bryan C. Smith <bryan.c.smith@ucdenver.edu>
+## All rights reserved.
+##
+## Redistribution and use in source and binary forms, with or without
+## modification, are permitted provided that the following conditions are met:
+## * Redistributions of source code must retain the above copyright
+## notice, this list of conditions and the following disclaimer.
+## * Redistributions in binary form must reproduce the above copyright
+## notice, this list of conditions and the following disclaimer in the
+## documentation and/or other materials provided with the distribution.
+## * Neither the name of the <organization> nor the
+## names of its contributors may be used to endorse or promote products
+## derived from this software without specific prior written permission.
+##
+## THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
+## ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
+## WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
+## DISCLAIMED. IN NO EVENT SHALL <COPYRIGHT HOLDER> BE LIABLE FOR ANY
+## DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
+## (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
+## LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
+## ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+## (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+## SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+% LAPLACIAN Sparse Negative Laplacian in 1D, 2D, or 3D
+%
+% [~,~,A]=LAPLACIAN(N) generates a sparse negative 3D Laplacian matrix
+% with Dirichlet boundary conditions, from a rectangular cuboid regular
+% grid with j x k x l interior grid points if N = [j k l], using the
+% standard 7-point finite-difference scheme, The grid size is always
+% one in all directions.
+%
+% [~,~,A]=LAPLACIAN(N,B) specifies boundary conditions with a cell array
+% B. For example, B = {'DD' 'DN' 'P'} will Dirichlet boundary conditions
+% ('DD') in the x-direction, Dirichlet-Neumann conditions ('DN') in the
+% y-direction and period conditions ('P') in the z-direction. Possible
+% values for the elements of B are 'DD', 'DN', 'ND', 'NN' and 'P'.
+%
+% LAMBDA = LAPLACIAN(N,B,M) or LAPLACIAN(N,M) outputs the m smallest
+% eigenvalues of the matrix, computed by an exact known formula, see
+% http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors_of_the_second_derivative
+% It will produce a warning if the mth eigenvalue is equal to the
+% (m+1)th eigenvalue. If m is absebt or zero, lambda will be empty.
+%
+% [LAMBDA,V] = LAPLACIAN(N,B,M) also outputs orthonormal eigenvectors
+% associated with the corresponding m smallest eigenvalues.
+%
+% [LAMBDA,V,A] = LAPLACIAN(N,B,M) produces a 2D or 1D negative
+% Laplacian matrix if the length of N and B are 2 or 1 respectively.
+% It uses the standard 5-point scheme for 2D, and 3-point scheme for 1D.
+%
+% % Examples:
+% [lambda,V,A] = laplacian([100,45,55],{'DD' 'NN' 'P'}, 20);
+% % Everything for 3D negative Laplacian with mixed boundary conditions.
+% laplacian([100,45,55],{'DD' 'NN' 'P'}, 20);
+% % or
+% lambda = laplacian([100,45,55],{'DD' 'NN' 'P'}, 20);
+% % computes the eigenvalues only
+%
+% [~,V,~] = laplacian([200 200],{'DD' 'DN'},30);
+% % Eigenvectors of 2D negative Laplacian with mixed boundary conditions.
+%
+% [~,~,A] = laplacian(200,{'DN'},30);
+% % 1D negative Laplacian matrix A with mixed boundary conditions.
+%
+% % Example to test if outputs correct eigenvalues and vectors:
+% [lambda,V,A] = laplacian([13,10,6],{'DD' 'DN' 'P'},30);
+% [Veig D] = eig(full(A)); lambdaeig = diag(D(1:30,1:30));
+% max(abs(lambda-lambdaeig)) %checking eigenvalues
+% subspace(V,Veig(:,1:30)) %checking the invariant subspace
+% subspace(V(:,1),Veig(:,1)) %checking selected eigenvectors
+% subspace(V(:,29:30),Veig(:,29:30)) %a multiple eigenvalue
+%
+% % Example showing equivalence between laplacian.m and built-in MATLAB
+% % DELSQ for the 2D case. The output of the last command shall be 0.
+% A1 = delsq(numgrid('S',32)); % input 'S' specifies square grid.
+% [~,~,A2] = laplacian([30,30]);
+% norm(A1-A2,inf)
+%
+% Class support for inputs:
+% N - row vector float double
+% B - cell array
+% M - scalar float double
+%
+% Class support for outputs:
+% lambda and V - full float double, A - sparse float double.
+%
+% Note: the actual numerical entries of A fit int8 format, but only
+% double data class is currently (2010) supported for sparse matrices.
+%
+% This program is designed to efficiently compute eigenvalues,
+% eigenvectors, and the sparse matrix of the (1-3)D negative Laplacian
+% on a rectangular grid for Dirichlet, Neumann, and Periodic boundary
+% conditions using tensor sums of 1D Laplacians. For more information on
+% tensor products, see
+% http://en.wikipedia.org/wiki/Kronecker_sum_of_discrete_Laplacians
+% For 2D case in MATLAB, see
+% http://www.mathworks.com/access/helpdesk/help/techdoc/ref/kron.html.
+%
+% This code is also part of the BLOPEX package:
+% http://en.wikipedia.org/wiki/BLOPEX or directly
+% http://code.google.com/p/blopex/
+
+% Revision 1.1 changes: rearranged the output variables, always compute
+% the eigenvalues, compute eigenvectors and/or the matrix on demand only.
+
+% $Revision: 1.1 $ $Date: 1-Aug-2011
+% Tested in MATLAB 7.11.0 (R2010b) and Octave 3.4.0.
+
+function [lambda, V, A] = laplacian(varargin)
+
+ % Input/Output handling.
+ if (nargin < 1 || nargin > 3)
+ print_usage;
+ endif
+
+ u = varargin{1};
+ dim2 = size(u);
+ if dim2(1) ~= 1
+ error('BLOPEX:laplacian:WrongVectorOfGridPoints',...
+ '%s','Number of grid points must be in a row vector.')
+ end
+ if dim2(2) > 3
+ error('BLOPEX:laplacian:WrongNumberOfGridPoints',...
+ '%s','Number of grid points must be a 1, 2, or 3')
+ end
+ dim=dim2(2); clear dim2;
+
+ uint = round(u);
+ if max(uint~=u)
+ warning('BLOPEX:laplacian:NonIntegerGridSize',...
+ '%s','Grid sizes must be integers. Rounding...');
+ u = uint; clear uint
+ end
+ if max(u<=0 )
+ error('BLOPEX:laplacian:NonIntegerGridSize',...
+ '%s','Grid sizes must be positive.');
+ end
+
+ if nargin == 3
+ m = varargin{3};
+ B = varargin{2};
+ elseif nargin == 2
+ f = varargin{2};
+ a = whos('regep','f');
+ if sum(a.class(1:4)=='cell') == 4
+ B = f;
+ m = 0;
+ elseif sum(a.class(1:4)=='doub') == 4
+ if dim == 1
+ B = {'DD'};
+ elseif dim == 2
+ B = {'DD' 'DD'};
+ else
+ B = {'DD' 'DD' 'DD'};
+ end
+ m = f;
+ else
+ error('BLOPEX:laplacian:InvalidClass',...
+ '%s','Second input must be either class double or a cell array.');
+ end
+ else
+ if dim == 1
+ B = {'DD'};
+ elseif dim == 2
+ B = {'DD' 'DD'};
+ else
+ B = {'DD' 'DD' 'DD'};
+ end
+ m = 0;
+ end
+
+ if max(size(m) - [1 1]) ~= 0
+ error('BLOPEX:laplacian:WrongNumberOfEigenvalues',...
+ '%s','The requested number of eigenvalues must be a scalar.');
+ end
+
+ maxeigs = prod(u);
+ mint = round(m);
+ if mint ~= m || mint > maxeigs
+ error('BLOPEX:laplacian:InvalidNumberOfEigs',...
+ '%s','Number of eigenvalues output must be a nonnegative ',...
+ 'integer no bigger than number of grid points.');
+ end
+ m = mint;
+
+ bdryerr = 0;
+ a = whos('regep','B');
+ if sum(a.class(1:4)=='cell') ~= 4 || sum(a.size == [1 dim]) ~= 2
+ bdryerr = 1;
+ else
+ BB = zeros(1, 2*dim);
+ for i = 1:dim
+ if (length(B{i}) == 1)
+ if B{i} == 'P'
+ BB(i) = 3;
+ BB(i + dim) = 3;
+ else
+ bdryerr = 1;
+ end
+ elseif (length(B{i}) == 2)
+ if B{i}(1) == 'D'
+ BB(i) = 1;
+ elseif B{i}(1) == 'N'
+ BB(i) = 2;
+ else
+ bdryerr = 1;
+ end
+ if B{i}(2) == 'D'
+ BB(i + dim) = 1;
+ elseif B{i}(2) == 'N'
+ BB(i + dim) = 2;
+ else
+ bdryerr = 1;
+ end
+ else
+ bdryerr = 1;
+ end
+ end
+ end
+
+ if bdryerr == 1
+ error('BLOPEX:laplacian:InvalidBdryConds',...
+ '%s','Boundary conditions must be a cell of length 3 for 3D, 2',...
+ ' for 2D, 1 for 1D, with values ''DD'', ''DN'', ''ND'', ''NN''',...
+ ', or ''P''.');
+ end
+
+ % Set the component matrices. SPDIAGS converts int8 into double anyway.
+ e1 = ones(u(1),1); %e1 = ones(u(1),1,'int8');
+ if dim > 1
+ e2 = ones(u(2),1);
+ end
+ if dim > 2
+ e3 = ones(u(3),1);
+ end
+
+ % Calculate m smallest exact eigenvalues.
+ if m > 0
+ if (BB(1) == 1) && (BB(1+dim) == 1)
+ a1 = pi/2/(u(1)+1);
+ N = (1:u(1))';
+ elseif (BB(1) == 2) && (BB(1+dim) == 2)
+ a1 = pi/2/u(1);
+ N = (0:(u(1)-1))';
+ elseif ((BB(1) == 1) && (BB(1+dim) == 2)) || ((BB(1) == 2)...
+ && (BB(1+dim) == 1))
+ a1 = pi/4/(u(1)+0.5);
+ N = 2*(1:u(1))' - 1;
+ else
+ a1 = pi/u(1);
+ N = floor((1:u(1))/2)';
+ end
+
+ lambda1 = 4*sin(a1*N).^2;
+
+ if dim > 1
+ if (BB(2) == 1) && (BB(2+dim) == 1)
+ a2 = pi/2/(u(2)+1);
+ N = (1:u(2))';
+ elseif (BB(2) == 2) && (BB(2+dim) == 2)
+ a2 = pi/2/u(2);
+ N = (0:(u(2)-1))';
+ elseif ((BB(2) == 1) && (BB(2+dim) == 2)) || ((BB(2) == 2)...
+ && (BB(2+dim) == 1))
+ a2 = pi/4/(u(2)+0.5);
+ N = 2*(1:u(2))' - 1;
+ else
+ a2 = pi/u(2);
+ N = floor((1:u(2))/2)';
+ end
+ lambda2 = 4*sin(a2*N).^2;
+ else
+ lambda2 = 0;
+ end
+
+ if dim > 2
+ if (BB(3) == 1) && (BB(6) == 1)
+ a3 = pi/2/(u(3)+1);
+ N = (1:u(3))';
+ elseif (BB(3) == 2) && (BB(6) == 2)
+ a3 = pi/2/u(3);
+ N = (0:(u(3)-1))';
+ elseif ((BB(3) == 1) && (BB(6) == 2)) || ((BB(3) == 2)...
+ && (BB(6) == 1))
+ a3 = pi/4/(u(3)+0.5);
+ N = 2*(1:u(3))' - 1;
+ else
+ a3 = pi/u(3);
+ N = floor((1:u(3))/2)';
+ end
+ lambda3 = 4*sin(a3*N).^2;
+ else
+ lambda3 = 0;
+ end
+
+ if dim == 1
+ lambda = lambda1;
+ elseif dim == 2
+ lambda = kron(e2,lambda1) + kron(lambda2, e1);
+ else
+ lambda = kron(e3,kron(e2,lambda1)) + kron(e3,kron(lambda2,e1))...
+ + kron(lambda3,kron(e2,e1));
+ end
+ [lambda, p] = sort(lambda);
+ if m < maxeigs - 0.1
+ w = lambda(m+1);
+ else
+ w = inf;
+ end
+ lambda = lambda(1:m);
+ p = p(1:m)';
+ else
+ lambda = [];
+ end
+
+ V = [];
+ if nargout > 1 && m > 0 % Calculate eigenvectors if specified in output.
+
+ p1 = mod(p-1,u(1))+1;
+
+ if (BB(1) == 1) && (BB(1+dim) == 1)
+ V1 = sin(kron((1:u(1))'*(pi/(u(1)+1)),p1))*(2/(u(1)+1))^0.5;
+ elseif (BB(1) == 2) && (BB(1+dim) == 2)
+ V1 = cos(kron((0.5:1:u(1)-0.5)'*(pi/u(1)),p1-1))*(2/u(1))^0.5;
+ V1(:,p1==1) = 1/u(1)^0.5;
+ elseif ((BB(1) == 1) && (BB(1+dim) == 2))
+ V1 = sin(kron((1:u(1))'*(pi/2/(u(1)+0.5)),2*p1 - 1))...
+ *(2/(u(1)+0.5))^0.5;
+ elseif ((BB(1) == 2) && (BB(1+dim) == 1))
+ V1 = cos(kron((0.5:1:u(1)-0.5)'*(pi/2/(u(1)+0.5)),2*p1 - 1))...
+ *(2/(u(1)+0.5))^0.5;
+ else
+ V1 = zeros(u(1),m);
+ a = (0.5:1:u(1)-0.5)';
+ V1(:,mod(p1,2)==1) = cos(a*(pi/u(1)*(p1(mod(p1,2)==1)-1)))...
+ *(2/u(1))^0.5;
+ pp = p1(mod(p1,2)==0);
+ if ~isempty(pp)
+ V1(:,mod(p1,2)==0) = sin(a*(pi/u(1)*p1(mod(p1,2)==0)))...
+ *(2/u(1))^0.5;
+ end
+ V1(:,p1==1) = 1/u(1)^0.5;
+ if mod(u(1),2) == 0
+ V1(:,p1==u(1)) = V1(:,p1==u(1))/2^0.5;
+ end
+ end
+
+
+ if dim > 1
+ p2 = mod(p-p1,u(1)*u(2));
+ p3 = (p - p2 - p1)/(u(1)*u(2)) + 1;
+ p2 = p2/u(1) + 1;
+ if (BB(2) == 1) && (BB(2+dim) == 1)
+ V2 = sin(kron((1:u(2))'*(pi/(u(2)+1)),p2))*(2/(u(2)+1))^0.5;
+ elseif (BB(2) == 2) && (BB(2+dim) == 2)
+ V2 = cos(kron((0.5:1:u(2)-0.5)'*(pi/u(2)),p2-1))*(2/u(2))^0.5;
+ V2(:,p2==1) = 1/u(2)^0.5;
+ elseif ((BB(2) == 1) && (BB(2+dim) == 2))
+ V2 = sin(kron((1:u(2))'*(pi/2/(u(2)+0.5)),2*p2 - 1))...
+ *(2/(u(2)+0.5))^0.5;
+ elseif ((BB(2) == 2) && (BB(2+dim) == 1))
+ V2 = cos(kron((0.5:1:u(2)-0.5)'*(pi/2/(u(2)+0.5)),2*p2 - 1))...
+ *(2/(u(2)+0.5))^0.5;
+ else
+ V2 = zeros(u(2),m);
+ a = (0.5:1:u(2)-0.5)';
+ V2(:,mod(p2,2)==1) = cos(a*(pi/u(2)*(p2(mod(p2,2)==1)-1)))...
+ *(2/u(2))^0.5;
+ pp = p2(mod(p2,2)==0);
+ if ~isempty(pp)
+ V2(:,mod(p2,2)==0) = sin(a*(pi/u(2)*p2(mod(p2,2)==0)))...
+ *(2/u(2))^0.5;
+ end
+ V2(:,p2==1) = 1/u(2)^0.5;
+ if mod(u(2),2) == 0
+ V2(:,p2==u(2)) = V2(:,p2==u(2))/2^0.5;
+ end
+ end
+ else
+ V2 = ones(1,m);
+ end
+
+ if dim > 2
+ if (BB(3) == 1) && (BB(3+dim) == 1)
+ V3 = sin(kron((1:u(3))'*(pi/(u(3)+1)),p3))*(2/(u(3)+1))^0.5;
+ elseif (BB(3) == 2) && (BB(3+dim) == 2)
+ V3 = cos(kron((0.5:1:u(3)-0.5)'*(pi/u(3)),p3-1))*(2/u(3))^0.5;
+ V3(:,p3==1) = 1/u(3)^0.5;
+ elseif ((BB(3) == 1) && (BB(3+dim) == 2))
+ V3 = sin(kron((1:u(3))'*(pi/2/(u(3)+0.5)),2*p3 - 1))...
+ *(2/(u(3)+0.5))^0.5;
+ elseif ((BB(3) == 2) && (BB(3+dim) == 1))
+ V3 = cos(kron((0.5:1:u(3)-0.5)'*(pi/2/(u(3)+0.5)),2*p3 - 1))...
+ *(2/(u(3)+0.5))^0.5;
+ else
+ V3 = zeros(u(3),m);
+ a = (0.5:1:u(3)-0.5)';
+ V3(:,mod(p3,2)==1) = cos(a*(pi/u(3)*(p3(mod(p3,2)==1)-1)))...
+ *(2/u(3))^0.5;
+ pp = p1(mod(p3,2)==0);
+ if ~isempty(pp)
+ V3(:,mod(p3,2)==0) = sin(a*(pi/u(3)*p3(mod(p3,2)==0)))...
+ *(2/u(3))^0.5;
+ end
+ V3(:,p3==1) = 1/u(3)^0.5;
+ if mod(u(3),2) == 0
+ V3(:,p3==u(3)) = V3(:,p3==u(3))/2^0.5;
+ end
+
+ end
+ else
+ V3 = ones(1,m);
+ end
+
+ if dim == 1
+ V = V1;
+ elseif dim == 2
+ V = kron(e2,V1).*kron(V2,e1);
+ else
+ V = kron(e3, kron(e2, V1)).*kron(e3, kron(V2, e1))...
+ .*kron(kron(V3,e2),e1);
+ end
+
+ if m ~= 0
+ if abs(lambda(m) - w) < maxeigs*eps('double')
+ sprintf('\n%s','Warning: (m+1)th eigenvalue is nearly equal',...
+ ' to mth.')
+
+ end
+ end
+
+ end
+
+ A = [];
+ if nargout > 2 %also calulate the matrix if specified in the output
+
+ % Set the component matrices. SPDIAGS converts int8 into double anyway.
+ % e1 = ones(u(1),1); %e1 = ones(u(1),1,'int8');
+ D1x = spdiags([-e1 2*e1 -e1], [-1 0 1], u(1),u(1));
+ if dim > 1
+ % e2 = ones(u(2),1);
+ D1y = spdiags([-e2 2*e2 -e2], [-1 0 1], u(2),u(2));
+ end
+ if dim > 2
+ % e3 = ones(u(3),1);
+ D1z = spdiags([-e3 2*e3 -e3], [-1 0 1], u(3),u(3));
+ end
+
+
+ % Set boundary conditions if other than Dirichlet.
+ for i = 1:dim
+ if BB(i) == 2
+ eval(['D1' char(119 + i) '(1,1) = 1;'])
+ elseif BB(i) == 3
+ eval(['D1' char(119 + i) '(1,' num2str(u(i)) ') = D1'...
+ char(119 + i) '(1,' num2str(u(i)) ') -1;']);
+ eval(['D1' char(119 + i) '(' num2str(u(i)) ',1) = D1'...
+ char(119 + i) '(' num2str(u(i)) ',1) -1;']);
+ end
+
+ if BB(i+dim) == 2
+ eval(['D1' char(119 + i)...
+ '(',num2str(u(i)),',',num2str(u(i)),') = 1;'])
+ end
+ end
+
+ % Form A using tensor products of lower dimensional Laplacians
+ Ix = speye(u(1));
+ if dim == 1
+ A = D1x;
+ elseif dim == 2
+ Iy = speye(u(2));
+ A = kron(Iy,D1x) + kron(D1y,Ix);
+ elseif dim == 3
+ Iy = speye(u(2));
+ Iz = speye(u(3));
+ A = kron(Iz, kron(Iy, D1x)) + kron(Iz, kron(D1y, Ix))...
+ + kron(kron(D1z,Iy),Ix);
+ end
+ end
+
+ disp(' ')
+ if ~isempty(V)
+ a = whos('regep','V');
+ disp(['The eigenvectors take ' num2str(a.bytes) ' bytes'])
+ end
+ if ~isempty(A)
+ a = whos('regexp','A');
+ disp(['The Laplacian matrix takes ' num2str(a.bytes) ' bytes'])
+ end
+ disp(' ')
+endfunction
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff