function
   
petsc-3.7.6 2017-04-24
[lambda, V, A] = laplacian(varargin) % 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 a 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. % License: BSD % Copyright 2010-2011 Bryan C. Smith, Andrew V. Knyazev % $Revision: 1.1 $ $Date: 1-Aug-2011 % Tested in MATLAB 7.11.0 (R2010b) and Octave 3.4.0. tic % Input/Output handling. if nargin > 3 error('BLOPEX:laplacian:TooManyInputs',... '%s','Too many input arguments.'); elseif nargin == 0 error('BLOPEX:laplacian:NoInputArguments',... '%s','Must have at least one input argument.'); end if nargout > 3 error('BLOPEX:laplacian:TooManyOutputs',... '%s','Maximum number of outputs is 3.'); end 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(' ') toc 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(' ')