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author | John W. Eaton <jwe@octave.org> |
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date | Mon, 06 Jan 2020 22:29:51 -0500 |
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/* Copyright (C) 1997-2020 The Octave Project Developers See the file COPYRIGHT.md in the top-level directory of this distribution or <https://octave.org/COPYRIGHT.html/>. 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 <https://www.gnu.org/licenses/>. */ #ifdef HAVE_CONFIG_H # include <config.h> #endif #include "dMatrix.h" #include "CMatrix.h" #include "dDiagMatrix.h" #include "gsvd.h" #include "defun.h" #include "defun-int.h" #include "error.h" #include "errwarn.h" #include "utils.h" #include "ovl.h" #include "ov.h" template <typename T> static typename octave::math::gsvd<T>::Type gsvd_type (int nargout) { return ((nargout == 0 || nargout == 1) ? octave::math::gsvd<T>::Type::sigma_only : (nargout > 5) ? octave::math::gsvd<T>::Type::std : octave::math::gsvd<T>::Type::economy); } // Named like this to avoid conflicts with the gsvd class. template <typename T> static octave_value_list do_gsvd (const T& A, const T& B, const octave_idx_type nargout, bool is_single = false) { octave::math::gsvd<T> result (A, B, gsvd_type<T> (nargout)); octave_value_list retval (nargout); if (nargout < 2) { if (is_single) { FloatDiagMatrix sigA = result.singular_values_A (); FloatDiagMatrix sigB = result.singular_values_B (); for (int i = sigA.rows () - 1; i >= 0; i--) sigA.dgxelem(i) /= sigB.dgxelem(i); retval(0) = sigA.diag (); } else { DiagMatrix sigA = result.singular_values_A (); DiagMatrix sigB = result.singular_values_B (); for (int i = sigA.rows () - 1; i >= 0; i--) sigA.dgxelem(i) /= sigB.dgxelem(i); retval(0) = sigA.diag (); } } else { retval(0) = result.left_singular_matrix_A (); retval(1) = result.left_singular_matrix_B (); if (nargout > 2) retval(2) = result.right_singular_matrix (); if (nargout > 3) retval(3) = result.singular_values_A (); if (nargout > 4) retval(4) = result.singular_values_B (); if (nargout > 5) retval(5) = result.R_matrix (); } return retval; } DEFUN (gsvd, args, nargout, doc: /* -*- texinfo -*- @deftypefn {} {@var{S} =} gsvd (@var{A}, @var{B}) @deftypefnx {} {[@var{U}, @var{V}, @var{X}, @var{C}, @var{S}] =} gsvd (@var{A}, @var{B}) @deftypefnx {} {[@var{U}, @var{V}, @var{X}, @var{C}, @var{S}] =} gsvd (@var{A}, @var{B}, 0) Compute the generalized singular value decomposition of (@var{A}, @var{B}). The generalized singular value decomposition is defined by the following relations: @tex $$ A = U C X^\dagger $$ $$ B = V S X^\dagger $$ $$ C^\dagger C + S^\dagger S = eye (columns (A)) $$ @end tex @ifnottex @example @group A = U*C*X' B = V*S*X' C'*C + S'*S = eye (columns (A)) @end group @end example @end ifnottex The function @code{gsvd} normally returns just the vector of generalized singular values @tex $$ \sqrt{{{diag (C^\dagger C)} \over {diag (S^\dagger S)}}} $$ @end tex @ifnottex @code{sqrt (diag (C'*C) ./ diag (S'*S))}. @end ifnottex If asked for five return values, it also computes @tex $U$, $V$, $X$, and $C$. @end tex @ifnottex U, V, X, and C. @end ifnottex If the optional third input is present, @code{gsvd} constructs the "economy-sized" decomposition where the number of columns of @var{U}, @var{V} and the number of rows of @var{C}, @var{S} is less than or equal to the number of columns of @var{A}. This option is not yet implemented. Programming Note: the code is a wrapper to the corresponding @sc{lapack} dggsvd and zggsvd routines. @seealso{svd} @end deftypefn */) { int nargin = args.length (); if (nargin < 2 || nargin > 3) print_usage (); else if (nargin == 3) warning ("gsvd: economy-sized decomposition is not yet implemented, returning full decomposition"); octave_value_list retval; octave_value argA = args(0); octave_value argB = args(1); octave_idx_type nr = argA.rows (); octave_idx_type nc = argA.columns (); octave_idx_type np = argB.columns (); // FIXME: This "special" case should be handled in the gsvd class, not here if (nr == 0 || nc == 0) { retval = octave_value_list (nargout); if (nargout < 2) // S = gsvd (A, B) { if (argA.is_single_type () || argB.is_single_type ()) retval(0) = FloatMatrix (0, 1); else retval(0) = Matrix (0, 1); } else // [U, V, X, C, S, R] = gsvd (A, B) { if (argA.is_single_type () || argB.is_single_type ()) { retval(0) = octave::float_identity_matrix (nc, nc); retval(1) = octave::float_identity_matrix (nc, nc); if (nargout > 2) retval(2) = octave::float_identity_matrix (nr, nr); if (nargout > 3) retval(3) = FloatMatrix (nr, nc); if (nargout > 4) retval(4) = octave::float_identity_matrix (nr, nr); if (nargout > 5) retval(5) = octave::float_identity_matrix (nr, nr); } else { retval(0) = octave::identity_matrix (nc, nc); retval(1) = octave::identity_matrix (nc, nc); if (nargout > 2) retval(2) = octave::identity_matrix (nr, nr); if (nargout > 3) retval(3) = Matrix (nr, nc); if (nargout > 4) retval(4) = octave::identity_matrix (nr, nr); if (nargout > 5) retval(5) = octave::identity_matrix (nr, nr); } } } else { if (nc != np) print_usage (); if (argA.is_single_type () || argB.is_single_type ()) { if (argA.isreal () && argB.isreal ()) { FloatMatrix tmpA = argA.xfloat_matrix_value ("gsvd: A must be a real or complex matrix"); FloatMatrix tmpB = argB.xfloat_matrix_value ("gsvd: B must be a real or complex matrix"); if (tmpA.any_element_is_inf_or_nan ()) error ("gsvd: A cannot have Inf or NaN values"); if (tmpB.any_element_is_inf_or_nan ()) error ("gsvd: B cannot have Inf or NaN values"); retval = do_gsvd (tmpA, tmpB, nargout, true); } else if (argA.iscomplex () || argB.iscomplex ()) { FloatComplexMatrix ctmpA = argA.xfloat_complex_matrix_value ("gsvd: A must be a real or complex matrix"); FloatComplexMatrix ctmpB = argB.xfloat_complex_matrix_value ("gsvd: B must be a real or complex matrix"); if (ctmpA.any_element_is_inf_or_nan ()) error ("gsvd: A cannot have Inf or NaN values"); if (ctmpB.any_element_is_inf_or_nan ()) error ("gsvd: B cannot have Inf or NaN values"); retval = do_gsvd (ctmpA, ctmpB, nargout, true); } else error ("gsvd: A and B must be real or complex matrices"); } else { if (argA.isreal () && argB.isreal ()) { Matrix tmpA = argA.xmatrix_value ("gsvd: A must be a real or complex matrix"); Matrix tmpB = argB.xmatrix_value ("gsvd: B must be a real or complex matrix"); if (tmpA.any_element_is_inf_or_nan ()) error ("gsvd: A cannot have Inf or NaN values"); if (tmpB.any_element_is_inf_or_nan ()) error ("gsvd: B cannot have Inf or NaN values"); retval = do_gsvd (tmpA, tmpB, nargout); } else if (argA.iscomplex () || argB.iscomplex ()) { ComplexMatrix ctmpA = argA.xcomplex_matrix_value ("gsvd: A must be a real or complex matrix"); ComplexMatrix ctmpB = argB.xcomplex_matrix_value ("gsvd: B must be a real or complex matrix"); if (ctmpA.any_element_is_inf_or_nan ()) error ("gsvd: A cannot have Inf or NaN values"); if (ctmpB.any_element_is_inf_or_nan ()) error ("gsvd: B cannot have Inf or NaN values"); retval = do_gsvd (ctmpA, ctmpB, nargout); } else error ("gsvd: A and B must be real or complex matrices"); } } return retval; } /* ## Basic test of decomposition %!test <48807> %! A = reshape (1:15,5,3); %! B = magic (3); %! [U,V,X,C,S] = gsvd (A,B); %! assert (U*C*X', A, 50*eps); %! assert (V*S*X', B, 50*eps); %! S0 = gsvd (A, B); %! S1 = svd (A / B); %! assert (S0, S1, 10*eps); ## a few tests for gsvd.m %!shared A, A0, B, B0, U, V, C, S, X, R, D1, D2 %! A0 = randn (5, 3); %! B0 = diag ([1 2 4]); %! A = A0; %! B = B0; ## A (5x3) and B (3x3) are full rank %!test <48807> %! [U, V, X, C, S, R] = gsvd (A, B); %! D1 = zeros (5, 3); D1(1:3, 1:3) = C; %! D2 = S; %! assert (norm (diag (C).^2 + diag (S).^2 - ones (3, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*R) <= 1e-6); %! assert (norm ((V'*B*X) - D2*R) <= 1e-6); ## A: 5x3 full rank, B: 3x3 rank deficient %!test <48807> %! B(2, 2) = 0; %! [U, V, X, C, S, R] = gsvd (A, B); %! D1 = zeros (5, 3); D1(1, 1) = 1; D1(2:3, 2:3) = C; %! D2 = [zeros(2, 1) S; zeros(1, 3)]; %! assert (norm (diag (C).^2 + diag (S).^2 - ones (2, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*R) <= 1e-6); %! assert (norm ((V'*B*X) - D2*R) <= 1e-6); ## A: 5x3 rank deficient, B: 3x3 full rank %!test <48807> %! B = B0; %! A(:, 3) = 2*A(:, 1) - A(:, 2); %! [U, V, X, C, S, R] = gsvd (A, B); %! D1 = zeros(5, 3); D1(1:3, 1:3) = C; %! D2 = S; %! assert (norm (diag (C).^2 + diag (S).^2 - ones (3, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*R) <= 1e-6); %! assert (norm ((V'*B*X) - D2*R) <= 1e-6); ## A and B are both rank deficient %!test <48807> %! B(:, 3) = 2*B(:, 1) - B(:, 2); %! [U, V, X, C, S, R] = gsvd (A, B); %! D1 = zeros(5, 2); D1(1:2, 1:2) = C; %! D2 = [S; zeros(1, 2)]; %! assert (norm (diag (C).^2 + diag (S).^2 - ones (2, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*[zeros(2, 1) R]) <= 1e-6); %! assert (norm ((V'*B*X) - D2*[zeros(2, 1) R]) <= 1e-6); ## A (now 3x5) and B (now 5x5) are full rank %!test <48807> %! A = A0.'; %! B0 = diag ([1 2 4 8 16]); %! B = B0; %! [U, V, X, C, S, R] = gsvd (A, B); %! D1 = [C zeros(3,2)]; %! D2 = [S zeros(3,2); zeros(2, 3) eye(2)]; %! assert (norm (diag (C).^2 + diag (S).^2 - ones (3, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*R) <= 1e-6); %! assert (norm ((V'*B*X) - D2*R) <= 1e-6); ## A: 3x5 full rank, B: 5x5 rank deficient %!test <48807> %! B(2, 2) = 0; %! [U, V, X, C, S, R] = gsvd (A, B); %! D1 = zeros(3, 5); D1(1, 1) = 1; D1(2:3, 2:3) = C; %! D2 = zeros(5, 5); D2(1:2, 2:3) = S; D2(3:4, 4:5) = eye (2); %! assert (norm (diag (C).^2 + diag (S).^2 - ones (2, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*R) <= 1e-6); %! assert (norm ((V'*B*X) - D2*R) <= 1e-6); ## A: 3x5 rank deficient, B: 5x5 full rank %!test <48807> %! B = B0; %! A(3, :) = 2*A(1, :) - A(2, :); %! [U, V, X, C, S, R] = gsvd (A, B); %! D1 = zeros (3, 5); D1(1:3, 1:3) = C; %! D2 = zeros (5, 5); D2(1:3, 1:3) = S; D2(4:5, 4:5) = eye (2); %! assert (norm (diag (C).^2 + diag (S).^2 - ones (3, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*R) <= 1e-6); %! assert (norm ((V'*B*X) - D2*R) <= 1e-6); ## A and B are both rank deficient %!test <48807> %! A = A0.'; B = B0.'; %! A(:, 3) = 2*A(:, 1) - A(:, 2); %! B(:, 3) = 2*B(:, 1) - B(:, 2); %! [U, V, X, C, S, R]=gsvd (A, B); %! D1 = zeros(3, 4); D1(1:3, 1:3) = C; %! D2 = eye (4); D2(1:3, 1:3) = S; D2(5,:) = 0; %! assert (norm (diag (C).^2 + diag (S).^2 - ones (3, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*[zeros(4, 1) R]) <= 1e-6); %! assert (norm ((V'*B*X) - D2*[zeros(4, 1) R]) <= 1e-6); ## A: 5x3 complex full rank, B: 3x3 complex full rank %!test <48807> %! A0 = A0 + j*randn (5, 3); %! B0 = diag ([1 2 4]) + j*diag ([4 -2 -1]); %! A = A0; %! B = B0; %! [U, V, X, C, S, R] = gsvd (A, B); %! D1 = zeros(5, 3); D1(1:3, 1:3) = C; %! D2 = S; %! assert (norm (diag (C).^2 + diag (S).^2 - ones (3, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*R) <= 1e-6); %! assert (norm ((V'*B*X) - D2*R) <= 1e-6); ## A: 5x3 complex full rank, B: 3x3 complex rank deficient %!test <48807> %! B(2, 2) = 0; %! [U, V, X, C, S, R] = gsvd (A, B); %! D1 = zeros(5, 3); D1(1, 1) = 1; D1(2:3, 2:3) = C; %! D2 = [zeros(2, 1) S; zeros(1, 3)]; %! assert (norm (diag (C).^2 + diag (S).^2 - ones (2, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*R) <= 1e-6); %! assert (norm ((V'*B*X) - D2*R) <= 1e-6); ## A: 5x3 complex rank deficient, B: 3x3 complex full rank %!test <48807> %! B = B0; %! A(:, 3) = 2*A(:, 1) - A(:, 2); %! [U, V, X, C, S, R] = gsvd (A, B); %! D1 = zeros(5, 3); D1(1:3, 1:3) = C; %! D2 = S; %! assert (norm (diag (C).^2 + diag (S).^2 - ones (3, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*R) <= 1e-6); %! assert (norm ((V'*B*X) - D2*R) <= 1e-6); ## A (5x3) and B (3x3) are both complex rank deficient %!test <48807> %! B(:, 3) = 2*B(:, 1) - B(:, 2); %! [U, V, X, C, S, R] = gsvd (A, B); %! D1 = zeros(5, 2); D1(1:2, 1:2) = C; %! D2 = [S; zeros(1, 2)]; %! assert (norm (diag (C).^2 + diag (S).^2 - ones (2, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*[zeros(2, 1) R]) <= 1e-6); %! assert (norm ((V'*B*X) - D2*[zeros(2, 1) R]) <= 1e-6); ## A (now 3x5) complex and B (now 5x5) complex are full rank ## now, A is 3x5 %!test <48807> %! A = A0.'; %! B0 = diag ([1 2 4 8 16]) + j*diag ([-5 4 -3 2 -1]); %! B = B0; %! [U, V, X, C, S, R] = gsvd (A, B); %! D1 = [C zeros(3,2)]; %! D2 = [S zeros(3,2); zeros(2, 3) eye(2)]; %! assert (norm (diag (C).^2 + diag (S).^2 - ones (3, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*R) <= 1e-6); %! assert (norm ((V'*B*X) - D2*R) <= 1e-6); ## A: 3x5 complex full rank, B: 5x5 complex rank deficient %!test <48807> %! B(2, 2) = 0; %! [U, V, X, C, S, R] = gsvd (A, B); %! D1 = zeros(3, 5); D1(1, 1) = 1; D1(2:3, 2:3) = C; %! D2 = zeros(5,5); D2(1:2, 2:3) = S; D2(3:4, 4:5) = eye (2); %! assert (norm (diag (C).^2 + diag (S).^2 - ones (2, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*R) <= 1e-6); %! assert (norm ((V'*B*X) - D2*R) <= 1e-6); ## A: 3x5 complex rank deficient, B: 5x5 complex full rank %!test <48807> %! B = B0; %! A(3, :) = 2*A(1, :) - A(2, :); %! [U, V, X, C, S, R] = gsvd (A, B); %! D1 = zeros(3, 5); D1(1:3, 1:3) = C; %! D2 = zeros(5,5); D2(1:3, 1:3) = S; D2(4:5, 4:5) = eye (2); %! assert (norm (diag (C).^2 + diag (S).^2 - ones (3, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*R) <= 1e-6); %! assert (norm ((V'*B*X) - D2*R) <= 1e-6); ## A and B are both complex rank deficient %!test <48807> %! A = A0.'; %! B = B0.'; %! A(:, 3) = 2*A(:, 1) - A(:, 2); %! B(:, 3) = 2*B(:, 1) - B(:, 2); %! [U, V, X, C, S, R] = gsvd (A, B); %! D1 = zeros(3, 4); D1(1:3, 1:3) = C; %! D2 = eye (4); D2(1:3, 1:3) = S; D2(5,:) = 0; %! assert (norm (diag (C).^2 + diag (S).^2 - ones (3, 1)) <= 1e-6); %! assert (norm ((U'*A*X) - D1*[zeros(4, 1) R]) <= 1e-6); %! assert (norm ((V'*B*X) - D2*[zeros(4, 1) R]) <= 1e-6); ## Test that single inputs produce single outputs %!test %! s = gsvd (single (ones (0,1)), B); %! assert (class (s), "single"); %! s = gsvd (single (ones (1,0)), B); %! assert (class (s), "single"); %! s = gsvd (single (ones (1,0)), B); %! [U,V,X,C,S,R] = gsvd (single ([]), B); %! assert (class (U), "single"); %! assert (class (V), "single"); %! assert (class (X), "single"); %! assert (class (C), "single"); %! assert (class (S), "single"); %! assert (class (R), "single"); %! %! s = gsvd (single (A), B); %! assert (class (s), "single"); %! [U,V,X,C,S,R] = gsvd (single (A), B); %! assert (class (U), "single"); %! assert (class (V), "single"); %! assert (class (X), "single"); %! assert (class (C), "single"); %! assert (class (S), "single"); %! assert (class (R), "single"); */