This file is indexed.

/usr/share/octave/packages/specfun-1.1.0/laplacian.m is in octave-specfun 1.1.0-4.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
## 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