Actual source code: laplacian.m
petsc-3.10.5 2019-03-28
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      1: function [lambda, V, A] = laplacian(varargin)

      3: % LAPLACIAN   Sparse Negative Laplacian in 1D, 2D, or 3D
      4: %
      5: %    [~,~,A]=LAPLACIAN(N) generates a sparse negative 3D Laplacian matrix
      6: %    with Dirichlet boundary conditions, from a rectangular cuboid regular
      7: %    grid with j x k x l interior grid points if N = [j k l], using the
      8: %    standard 7-point finite-difference scheme,  The grid size is always
      9: %    one in all directions.
     10: %
     11: %    [~,~,A]=LAPLACIAN(N,B) specifies boundary conditions with a cell array
     12: %    B. For example, B = {'DD' 'DN' 'P'} will Dirichlet boundary conditions
     13: %    ('DD') in the x-direction, Dirichlet-Neumann conditions ('DN') in the
     14: %    y-direction and period conditions ('P') in the z-direction. Possible
     15: %    values for the elements of B are 'DD', 'DN', 'ND', 'NN' and 'P'.
     16: %
     17: %    LAMBDA = LAPLACIAN(N,B,M) or LAPLACIAN(N,M) outputs the m smallest
     18: %    eigenvalues of the matrix, computed by an exact known formula, see
     19: %    http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors_of_the_second_derivative
     20: %    It will produce a warning if the mth eigenvalue is equal to the
     21: %    (m+1)th eigenvalue. If m is absebt or zero, lambda will be empty.
     22: %
     23: %    [LAMBDA,V] = LAPLACIAN(N,B,M) also outputs orthonormal eigenvectors
     24: %    associated with the corresponding m smallest eigenvalues.
     25: %
     26: %    [LAMBDA,V,A] = LAPLACIAN(N,B,M) produces a 2D or 1D negative
     27: %    Laplacian matrix if the length of N and B are 2 or 1 respectively.
     28: %    It uses the standard 5-point scheme for 2D, and 3-point scheme for 1D.
     29: %
     30: %    % Examples:
     31: %    [lambda,V,A] = laplacian([100,45,55],{'DD' 'NN' 'P'}, 20);
     32: %    % Everything for 3D negative Laplacian with mixed boundary conditions.
     33: %    laplacian([100,45,55],{'DD' 'NN' 'P'}, 20);
     34: %    % or
     35: %    lambda = laplacian([100,45,55],{'DD' 'NN' 'P'}, 20);
     36: %    % computes the eigenvalues only
     37: %
     38: %    [~,V,~] = laplacian([200 200],{'DD' 'DN'},30);
     39: %    % Eigenvectors of 2D negative Laplacian with mixed boundary conditions.
     40: %
     41: %    [~,~,A] = laplacian(200,{'DN'},30);
     42: %    % 1D negative Laplacian matrix A with mixed boundary conditions.
     43: %
     44: %    % Example to test if outputs correct eigenvalues and vectors:
     45: %    [lambda,V,A] = laplacian([13,10,6],{'DD' 'DN' 'P'},30);
     46: %    [Veig D] = eig(full(A)); lambdaeig = diag(D(1:30,1:30));
     47: %    max(abs(lambda-lambdaeig))  %checking eigenvalues
     48: %    subspace(V,Veig(:,1:30))    %checking the invariant subspace
     49: %    subspace(V(:,1),Veig(:,1))  %checking selected eigenvectors
     50: %    subspace(V(:,29:30),Veig(:,29:30)) %a multiple eigenvalue
     51: %
     52: %    % Example showing equivalence between laplacian.m and built-in MATLAB
     53: %    % DELSQ for the 2D case. The output of the last command shall be 0.
     54: %    A1 = delsq(numgrid('S',32)); % input 'S' specifies square grid.
     55: %    [~,~,A2] = laplacian([30,30]);
     56: %    norm(A1-A2,inf)
     57: %
     58: %    Class support for inputs:
     59: %    N - row vector float double
     60: %    B - cell array
     61: %    M - scalar float double
     62: %
     63: %    Class support for outputs:
     64: %    lambda and V  - full float double, A - sparse float double.
     65: %
     66: %    Note: the actual numerical entries of A fit int8 format, but only
     67: %    double data class is currently (2010) supported for sparse matrices.
     68: %
     69: %    This program is designed to efficiently compute eigenvalues,
     70: %    eigenvectors, and the sparse matrix of the (1-3)D negative Laplacian
     71: %    on a rectangular grid for Dirichlet, Neumann, and Periodic boundary
     72: %    conditions using tensor sums of 1D Laplacians. For more information on
     73: %    tensor products, see
     74: %    http://en.wikipedia.org/wiki/Kronecker_sum_of_discrete_Laplacians
     75: %    For 2D case in MATLAB, see
     76: %    http://www.mathworks.com/access/helpdesk/help/techdoc/ref/kron.html.
     77: %
     78: %    This code is a part of the BLOPEX package:
     79: %    http://en.wikipedia.org/wiki/BLOPEX or directly
     80: %    http://code.google.com/p/blopex/

     82: %    Revision 1.1 changes: rearranged the output variables, always compute
     83: %    the eigenvalues, compute eigenvectors and/or the matrix on demand only.

     85: %    License: BSD
     86: %    Copyright 2010-2011 Bryan C. Smith, Andrew V. Knyazev
     87: %    $Revision: 1.1 $ $Date: 1-Aug-2011
     88: %    Tested in MATLAB 7.11.0 (R2010b) and Octave 3.4.0.

     90: tic

     92: % Input/Output handling.
     93: if nargin > 3
     94:     error('BLOPEX:laplacian:TooManyInputs',...
     95:         '%s','Too many input arguments.');
     96: elseif nargin == 0
     97:     error('BLOPEX:laplacian:NoInputArguments',...
     98:         '%s','Must have at least one input argument.');
     99: end

    101: if nargout > 3
    102:     error('BLOPEX:laplacian:TooManyOutputs',...
    103:         '%s','Maximum number of outputs is 3.');
    104: end

    106: u = varargin{1};
    107: dim2 = size(u);
    108: if dim2(1) ~= 1
    109:     error('BLOPEX:laplacian:WrongVectorOfGridPoints',...
    110:         '%s','Number of grid points must be in a row vector.')
    111: end
    112: if dim2(2) > 3
    113:     error('BLOPEX:laplacian:WrongNumberOfGridPoints',...
    114:         '%s','Number of grid points must be a 1, 2, or 3')
    115: end
    116: dim=dim2(2); clear dim2;

    118: uint = round(u);
    119: if max(uint~=u)
    120:     warning('BLOPEX:laplacian:NonIntegerGridSize',...
    121:         '%s','Grid sizes must be integers. Rounding...');
    122:     u = uint; clear uint
    123: end
    124: if max(u<=0 )
    125:     error('BLOPEX:laplacian:NonIntegerGridSize',...
    126:         '%s','Grid sizes must be positive.');
    127: end

    129: if nargin == 3
    130:     m = varargin{3};
    131:     B = varargin{2};
    132: elseif nargin == 2
    133:     f = varargin{2};
    134:     a = whos('regep','f');
    135:     if sum(a.class(1:4)=='cell') == 4
    136:         B = f;
    137:         m = 0;
    138:     elseif sum(a.class(1:4)=='doub') == 4
    139:         if dim == 1
    140:             B = {'DD'};
    141:         elseif dim == 2
    142:             B = {'DD' 'DD'};
    143:         else
    144:             B = {'DD' 'DD' 'DD'};
    145:         end
    146:         m = f;
    147:     else
    148:         error('BLOPEX:laplacian:InvalidClass',...
    149:             '%s','Second input must be either class double or a cell array.');
    150:     end
    151: else
    152:     if dim == 1
    153:         B = {'DD'};
    154:     elseif dim == 2
    155:         B = {'DD' 'DD'};
    156:     else
    157:         B = {'DD' 'DD' 'DD'};
    158:     end
    159:     m = 0;
    160: end

    162: if max(size(m) - [1 1]) ~= 0
    163:     error('BLOPEX:laplacian:WrongNumberOfEigenvalues',...
    164:         '%s','The requested number of eigenvalues must be a scalar.');
    165: end

    167: maxeigs = prod(u);
    168: mint = round(m);
    169: if mint ~= m || mint > maxeigs
    170:     error('BLOPEX:laplacian:InvalidNumberOfEigs',...
    171:         '%s','Number of eigenvalues output must be a nonnegative ',...
    172:         'integer no bigger than number of grid points.');
    173: end
    174: m = mint;

    176: bdryerr = 0;
    177: a = whos('regep','B');
    178: if sum(a.class(1:4)=='cell') ~= 4 || sum(a.size == [1 dim]) ~= 2
    179:     bdryerr = 1;
    180: else
    181:     BB = zeros(1, 2*dim);
    182:     for i = 1:dim
    183:         if (length(B{i}) == 1)
    184:             if B{i} == 'P'
    185:                 BB(i) = 3;
    186:                 BB(i + dim) = 3;
    187:             else
    188:                 bdryerr = 1;
    189:             end
    190:         elseif (length(B{i}) == 2)
    191:             if B{i}(1) == 'D'
    192:                 BB(i) = 1;
    193:             elseif B{i}(1) == 'N'
    194:                 BB(i) = 2;
    195:             else
    196:                 bdryerr = 1;
    197:             end
    198:             if B{i}(2) == 'D'
    199:                 BB(i + dim) = 1;
    200:             elseif B{i}(2) == 'N'
    201:                 BB(i + dim) = 2;
    202:             else
    203:                 bdryerr = 1;
    204:             end
    205:         else
    206:             bdryerr = 1;
    207:         end
    208:     end
    209: end

    211: if bdryerr == 1
    212:     error('BLOPEX:laplacian:InvalidBdryConds',...
    213:         '%s','Boundary conditions must be a cell of length 3 for 3D, 2',...
    214:         ' for 2D, 1 for 1D, with values ''DD'', ''DN'', ''ND'', ''NN''',...
    215:         ', or ''P''.');
    216: end

    218: % Set the component matrices. SPDIAGS converts int8 into double anyway.
    219: e1 = ones(u(1),1); %e1 = ones(u(1),1,'int8');
    220: if dim > 1
    221:     e2 = ones(u(2),1);
    222: end
    223: if dim > 2
    224:     e3 = ones(u(3),1);
    225: end

    227: % Calculate m smallest exact eigenvalues.
    228: if m > 0
    229:     if (BB(1) == 1) && (BB(1+dim) == 1)
    230:         a1 = pi/2/(u(1)+1);
    231:         N = (1:u(1))';
    232:     elseif (BB(1) == 2) && (BB(1+dim) == 2)
    233:         a1 = pi/2/u(1);
    234:         N = (0:(u(1)-1))';
    235:     elseif ((BB(1) == 1) && (BB(1+dim) == 2)) || ((BB(1) == 2)...
    236:             && (BB(1+dim) == 1))
    237:         a1 = pi/4/(u(1)+0.5);
    238:         N = 2*(1:u(1))' - 1;
    239:     else
    240:         a1 = pi/u(1);
    241:         N = floor((1:u(1))/2)';
    242:     end
    243:
    244:     lambda1 = 4*sin(a1*N).^2;
    245:
    246:     if dim > 1
    247:         if (BB(2) == 1) && (BB(2+dim) == 1)
    248:             a2 = pi/2/(u(2)+1);
    249:             N = (1:u(2))';
    250:         elseif (BB(2) == 2) && (BB(2+dim) == 2)
    251:             a2 = pi/2/u(2);
    252:             N = (0:(u(2)-1))';
    253:         elseif ((BB(2) == 1) && (BB(2+dim) == 2)) || ((BB(2) == 2)...
    254:                 && (BB(2+dim) == 1))
    255:             a2 = pi/4/(u(2)+0.5);
    256:             N = 2*(1:u(2))' - 1;
    257:         else
    258:             a2 = pi/u(2);
    259:             N = floor((1:u(2))/2)';
    260:         end
    261:         lambda2 = 4*sin(a2*N).^2;
    262:     else
    263:         lambda2 = 0;
    264:     end
    265:
    266:     if dim > 2
    267:         if (BB(3) == 1) && (BB(6) == 1)
    268:             a3 = pi/2/(u(3)+1);
    269:             N = (1:u(3))';
    270:         elseif (BB(3) == 2) && (BB(6) == 2)
    271:             a3 = pi/2/u(3);
    272:             N = (0:(u(3)-1))';
    273:         elseif ((BB(3) == 1) && (BB(6) == 2)) || ((BB(3) == 2)...
    274:                 && (BB(6) == 1))
    275:             a3 = pi/4/(u(3)+0.5);
    276:             N = 2*(1:u(3))' - 1;
    277:         else
    278:             a3 = pi/u(3);
    279:             N = floor((1:u(3))/2)';
    280:         end
    281:         lambda3 = 4*sin(a3*N).^2;
    282:     else
    283:         lambda3 = 0;
    284:     end
    285:
    286:     if dim == 1
    287:         lambda = lambda1;
    288:     elseif dim == 2
    289:         lambda = kron(e2,lambda1) + kron(lambda2, e1);
    290:     else
    291:         lambda = kron(e3,kron(e2,lambda1)) + kron(e3,kron(lambda2,e1))...
    292:             + kron(lambda3,kron(e2,e1));
    293:     end
    294:     [lambda, p] = sort(lambda);
    295:     if m < maxeigs - 0.1
    296:         w = lambda(m+1);
    297:     else
    298:         w = inf;
    299:     end
    300:     lambda = lambda(1:m);
    301:     p = p(1:m)';
    302: else
    303:     lambda = [];
    304: end

    306: V = [];
    307: if nargout > 1 && m > 0 % Calculate eigenvectors if specified in output.
    308:
    309:     p1 = mod(p-1,u(1))+1;
    310:
    311:     if (BB(1) == 1) && (BB(1+dim) == 1)
    312:         V1 = sin(kron((1:u(1))'*(pi/(u(1)+1)),p1))*(2/(u(1)+1))^0.5;
    313:     elseif (BB(1) == 2) && (BB(1+dim) == 2)
    314:         V1 = cos(kron((0.5:1:u(1)-0.5)'*(pi/u(1)),p1-1))*(2/u(1))^0.5;
    315:         V1(:,p1==1) = 1/u(1)^0.5;
    316:     elseif ((BB(1) == 1) && (BB(1+dim) == 2))
    317:         V1 = sin(kron((1:u(1))'*(pi/2/(u(1)+0.5)),2*p1 - 1))...
    318:             *(2/(u(1)+0.5))^0.5;
    319:     elseif ((BB(1) == 2) && (BB(1+dim) == 1))
    320:         V1 = cos(kron((0.5:1:u(1)-0.5)'*(pi/2/(u(1)+0.5)),2*p1 - 1))...
    321:             *(2/(u(1)+0.5))^0.5;
    322:     else
    323:         V1 = zeros(u(1),m);
    324:         a = (0.5:1:u(1)-0.5)';
    325:         V1(:,mod(p1,2)==1) = cos(a*(pi/u(1)*(p1(mod(p1,2)==1)-1)))...
    326:             *(2/u(1))^0.5;
    327:         pp = p1(mod(p1,2)==0);
    328:         if ~isempty(pp)
    329:             V1(:,mod(p1,2)==0) = sin(a*(pi/u(1)*p1(mod(p1,2)==0)))...
    330:                 *(2/u(1))^0.5;
    331:         end
    332:         V1(:,p1==1) = 1/u(1)^0.5;
    333:         if mod(u(1),2) == 0
    334:             V1(:,p1==u(1)) = V1(:,p1==u(1))/2^0.5;
    335:         end
    336:     end
    337:
    338:
    339:     if dim > 1
    340:         p2 = mod(p-p1,u(1)*u(2));
    341:         p3 = (p - p2 - p1)/(u(1)*u(2)) + 1;
    342:         p2 = p2/u(1) + 1;
    343:         if (BB(2) == 1) && (BB(2+dim) == 1)
    344:             V2 = sin(kron((1:u(2))'*(pi/(u(2)+1)),p2))*(2/(u(2)+1))^0.5;
    345:         elseif (BB(2) == 2) && (BB(2+dim) == 2)
    346:             V2 = cos(kron((0.5:1:u(2)-0.5)'*(pi/u(2)),p2-1))*(2/u(2))^0.5;
    347:             V2(:,p2==1) = 1/u(2)^0.5;
    348:         elseif ((BB(2) == 1) && (BB(2+dim) == 2))
    349:             V2 = sin(kron((1:u(2))'*(pi/2/(u(2)+0.5)),2*p2 - 1))...
    350:                 *(2/(u(2)+0.5))^0.5;
    351:         elseif ((BB(2) == 2) && (BB(2+dim) == 1))
    352:             V2 = cos(kron((0.5:1:u(2)-0.5)'*(pi/2/(u(2)+0.5)),2*p2 - 1))...
    353:                 *(2/(u(2)+0.5))^0.5;
    354:         else
    355:             V2 = zeros(u(2),m);
    356:             a = (0.5:1:u(2)-0.5)';
    357:             V2(:,mod(p2,2)==1) = cos(a*(pi/u(2)*(p2(mod(p2,2)==1)-1)))...
    358:                 *(2/u(2))^0.5;
    359:             pp = p2(mod(p2,2)==0);
    360:             if ~isempty(pp)
    361:                 V2(:,mod(p2,2)==0) = sin(a*(pi/u(2)*p2(mod(p2,2)==0)))...
    362:                     *(2/u(2))^0.5;
    363:             end
    364:             V2(:,p2==1) = 1/u(2)^0.5;
    365:             if mod(u(2),2) == 0
    366:                 V2(:,p2==u(2)) = V2(:,p2==u(2))/2^0.5;
    367:             end
    368:         end
    369:     else
    370:         V2 = ones(1,m);
    371:     end
    372:
    373:     if dim > 2
    374:         if (BB(3) == 1) && (BB(3+dim) == 1)
    375:             V3 = sin(kron((1:u(3))'*(pi/(u(3)+1)),p3))*(2/(u(3)+1))^0.5;
    376:         elseif (BB(3) == 2) && (BB(3+dim) == 2)
    377:             V3 = cos(kron((0.5:1:u(3)-0.5)'*(pi/u(3)),p3-1))*(2/u(3))^0.5;
    378:             V3(:,p3==1) = 1/u(3)^0.5;
    379:         elseif ((BB(3) == 1) && (BB(3+dim) == 2))
    380:             V3 = sin(kron((1:u(3))'*(pi/2/(u(3)+0.5)),2*p3 - 1))...
    381:                 *(2/(u(3)+0.5))^0.5;
    382:         elseif ((BB(3) == 2) && (BB(3+dim) == 1))
    383:             V3 = cos(kron((0.5:1:u(3)-0.5)'*(pi/2/(u(3)+0.5)),2*p3 - 1))...
    384:                 *(2/(u(3)+0.5))^0.5;
    385:         else
    386:             V3 = zeros(u(3),m);
    387:             a = (0.5:1:u(3)-0.5)';
    388:             V3(:,mod(p3,2)==1) = cos(a*(pi/u(3)*(p3(mod(p3,2)==1)-1)))...
    389:                 *(2/u(3))^0.5;
    390:             pp = p1(mod(p3,2)==0);
    391:             if ~isempty(pp)
    392:                 V3(:,mod(p3,2)==0) = sin(a*(pi/u(3)*p3(mod(p3,2)==0)))...
    393:                     *(2/u(3))^0.5;
    394:             end
    395:             V3(:,p3==1) = 1/u(3)^0.5;
    396:             if mod(u(3),2) == 0
    397:                 V3(:,p3==u(3)) = V3(:,p3==u(3))/2^0.5;
    398:             end
    399:
    400:         end
    401:     else
    402:         V3 = ones(1,m);
    403:     end
    404:
    405:     if dim == 1
    406:         V = V1;
    407:     elseif dim == 2
    408:         V = kron(e2,V1).*kron(V2,e1);
    409:     else
    410:         V = kron(e3, kron(e2, V1)).*kron(e3, kron(V2, e1))...
    411:             .*kron(kron(V3,e2),e1);
    412:     end
    413:
    414:     if m ~= 0
    415:         if abs(lambda(m) - w) < maxeigs*eps('double')
    416:             sprintf('\n%s','Warning: (m+1)th eigenvalue is  nearly equal',...
    417:                 ' to mth.')
    418:
    419:         end
    420:     end
    421:
    422: end

    424: A = [];
    425: if nargout > 2 %also calulate the matrix if specified in the output
    426:
    427:     % Set the component matrices. SPDIAGS converts int8 into double anyway.
    428:     %    e1 = ones(u(1),1); %e1 = ones(u(1),1,'int8');
    429:     D1x = spdiags([-e1 2*e1 -e1], [-1 0 1], u(1),u(1));
    430:     if dim > 1
    431:         %        e2 = ones(u(2),1);
    432:         D1y = spdiags([-e2 2*e2 -e2], [-1 0 1], u(2),u(2));
    433:     end
    434:     if dim > 2
    435:         %        e3 = ones(u(3),1);
    436:         D1z = spdiags([-e3 2*e3 -e3], [-1 0 1], u(3),u(3));
    437:     end
    438:
    439:
    440:     % Set boundary conditions if other than Dirichlet.
    441:     for i = 1:dim
    442:         if BB(i) == 2
    443:             eval(['D1' char(119 + i) '(1,1) = 1;'])
    444:         elseif BB(i) == 3
    445:             eval(['D1' char(119 + i) '(1,' num2str(u(i)) ') = D1'...
    446:                 char(119 + i) '(1,' num2str(u(i)) ') -1;']);
    447:             eval(['D1' char(119 + i) '(' num2str(u(i)) ',1) = D1'...
    448:                 char(119 + i) '(' num2str(u(i)) ',1) -1;']);
    449:         end
    450:
    451:         if BB(i+dim) == 2
    452:             eval(['D1' char(119 + i)...
    453:                 '(',num2str(u(i)),',',num2str(u(i)),') = 1;'])
    454:         end
    455:     end
    456:
    457:     % Form A using tensor products of lower dimensional Laplacians
    458:     Ix = speye(u(1));
    459:     if dim == 1
    460:         A = D1x;
    461:     elseif dim == 2
    462:         Iy = speye(u(2));
    463:         A = kron(Iy,D1x) + kron(D1y,Ix);
    464:     elseif dim == 3
    465:         Iy = speye(u(2));
    466:         Iz = speye(u(3));
    467:         A = kron(Iz, kron(Iy, D1x)) + kron(Iz, kron(D1y, Ix))...
    468:             + kron(kron(D1z,Iy),Ix);
    469:     end
    470: end

    472: disp('  ')
    473: toc
    474: if ~isempty(V)
    475:     a = whos('regep','V');
    476:     disp(['The eigenvectors take ' num2str(a.bytes) ' bytes'])
    477: end
    478: if  ~isempty(A)
    479:     a = whos('regexp','A');
    480:     disp(['The Laplacian matrix takes ' num2str(a.bytes) ' bytes'])
    481: end
    482: disp('  ')