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update Octave Project Developers copyright for the new year
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author | John W. Eaton <jwe@octave.org> |
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date | Wed, 10 Feb 2021 09:52:15 -0500 |
parents | bd51beb6205e |
children | 7854d5752dd2 |
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//////////////////////////////////////////////////////////////////////// // // Copyright (C) 1996-2021 The Octave Project Developers // // See the file COPYRIGHT.md in the top-level directory of this // distribution or <https://octave.org/copyright/>. // // 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/>. // //////////////////////////////////////////////////////////////////////// #if defined (HAVE_CONFIG_H) # include "config.h" #endif #include "defun.h" #include "error.h" #include "errwarn.h" #include "ovl.h" #include "EIG.h" #include "fEIG.h" #include "oct-string.h" DEFUN (eig, args, nargout, doc: /* -*- texinfo -*- @deftypefn {} {@var{lambda} =} eig (@var{A}) @deftypefnx {} {@var{lambda} =} eig (@var{A}, @var{B}) @deftypefnx {} {[@var{V}, @var{lambda}] =} eig (@var{A}) @deftypefnx {} {[@var{V}, @var{lambda}] =} eig (@var{A}, @var{B}) @deftypefnx {} {[@var{V}, @var{lambda}, @var{W}] =} eig (@var{A}) @deftypefnx {} {[@var{V}, @var{lambda}, @var{W}] =} eig (@var{A}, @var{B}) @deftypefnx {} {[@dots{}] =} eig (@var{A}, @var{balanceOption}) @deftypefnx {} {[@dots{}] =} eig (@var{A}, @var{B}, @var{algorithm}) @deftypefnx {} {[@dots{}] =} eig (@dots{}, @var{eigvalOption}) Compute the eigenvalues (@var{lambda}) and optionally the right eigenvectors (@var{V}) and the left eigenvectors (@var{W}) of a matrix or pair of matrices. The flag @var{balanceOption} can be one of: @table @asis @item @qcode{"balance"} (default) Preliminary balancing is on. @item @qcode{"nobalance"} Disables preliminary balancing. @end table The flag @var{eigvalOption} can be one of: @table @asis @item @qcode{"matrix"} Return the eigenvalues in a diagonal matrix. (default if 2 or 3 outputs are requested) @item @qcode{"vector"} Return the eigenvalues in a column vector. (default if only 1 output is requested, e.g., @var{lambda} = eig (@var{A})) @end table The flag @var{algorithm} can be one of: @table @asis @item @qcode{"chol"} Use the Cholesky factorization of B. (default if @var{A} is symmetric (Hermitian) and @var{B} is symmetric (Hermitian) positive definite) @item @qcode{"qz"} Use the QZ algorithm. (used whenever @var{A} or @var{B} are not symmetric) @end table @multitable @columnfractions .31 .23 .23 .23 @headitem @tab no flag @tab chol @tab qz @item both are symmetric @tab @qcode{"chol"} @tab @qcode{"chol"} @tab @qcode{"qz"} @item at least one is not symmetric @tab @qcode{"qz"} @tab @qcode{"qz"} @tab @qcode{"qz"} @end multitable The eigenvalues returned by @code{eig} are not ordered. @seealso{eigs, svd} @end deftypefn */) { int nargin = args.length (); if (nargin > 4 || nargin == 0) print_usage (); octave_value_list retval; octave_value arg_a, arg_b; arg_a = args(0); if (arg_a.isempty ()) return octave_value_list (2, Matrix ()); if (! arg_a.isfloat ()) err_wrong_type_arg ("eig", arg_a); if (arg_a.rows () != arg_a.columns ()) err_square_matrix_required ("eig", "A"); // determine if it's AEP or GEP bool AEPcase = nargin == 1 || args(1).is_string (); if (! AEPcase) { arg_b = args(1); if (arg_b.isempty ()) return octave_value_list (2, Matrix ()); if (! arg_b.isfloat ()) err_wrong_type_arg ("eig", arg_b); if (arg_b.rows () != arg_b.columns ()) err_square_matrix_required ("eig", "B"); } bool qz_flag = false; bool chol_flag = false; bool balance_flag = false; bool no_balance_flag = false; bool matrix_flag = false; bool vector_flag = false; for (int i = (AEPcase ? 1 : 2); i < args.length (); ++i) { if (! args(i).is_string ()) err_wrong_type_arg ("eig", args(i)); std::string arg_i = args(i).string_value (); if (octave::string::strcmpi (arg_i, "qz")) qz_flag = true; else if (octave::string::strcmpi (arg_i, "chol")) chol_flag = true; else if (octave::string::strcmpi (arg_i, "balance")) balance_flag = true; else if (octave::string::strcmpi (arg_i, "nobalance")) no_balance_flag = true; else if (octave::string::strcmpi (arg_i, "matrix")) matrix_flag = true; else if (octave::string::strcmpi (arg_i, "vector")) vector_flag = true; else error (R"(eig: invalid option "%s")", arg_i.c_str ()); } if (balance_flag && no_balance_flag) error (R"(eig: "balance" and "nobalance" options are mutually exclusive)"); if (vector_flag && matrix_flag) error (R"(eig: "vector" and "matrix" options are mutually exclusive)"); if (qz_flag && chol_flag) error (R"(eig: "qz" and "chol" options are mutually exclusive)"); if (AEPcase) { if (qz_flag) error (R"(eig: invalid "qz" option for algebraic eigenvalue problem)"); if (chol_flag) error (R"(eig: invalid "chol" option for algebraic eigenvalue problem)"); } else { if (balance_flag) error (R"(eig: invalid "balance" option for generalized eigenvalue problem)"); if (no_balance_flag) error (R"(eig: invalid "nobalance" option for generalized eigenvalue problem)"); } // Default is to balance const bool balance = (no_balance_flag ? false : true); const bool force_qz = qz_flag; Matrix tmp_a, tmp_b; ComplexMatrix ctmp_a, ctmp_b; FloatMatrix ftmp_a, ftmp_b; FloatComplexMatrix fctmp_a, fctmp_b; if (arg_a.is_single_type ()) { FloatEIG result; if (AEPcase) { if (arg_a.isreal ()) { ftmp_a = arg_a.float_matrix_value (); result = FloatEIG (ftmp_a, nargout > 1, nargout > 2, balance); } else { fctmp_a = arg_a.float_complex_matrix_value (); result = FloatEIG (fctmp_a, nargout > 1, nargout > 2, balance); } } else { if (arg_a.isreal () && arg_b.isreal ()) { ftmp_a = arg_a.float_matrix_value (); ftmp_b = arg_b.float_matrix_value (); result = FloatEIG (ftmp_a, ftmp_b, nargout > 1, nargout > 2, force_qz); } else { fctmp_a = arg_a.float_complex_matrix_value (); fctmp_b = arg_b.float_complex_matrix_value (); result = FloatEIG (fctmp_a, fctmp_b, nargout > 1, nargout > 2, force_qz); } } if (nargout == 0 || nargout == 1) { if (matrix_flag) retval = ovl (FloatComplexDiagMatrix (result.eigenvalues ())); else retval = ovl (result.eigenvalues ()); } else if (nargout == 2) { if (vector_flag) retval = ovl (result.right_eigenvectors (), result.eigenvalues ()); else retval = ovl (result.right_eigenvectors (), FloatComplexDiagMatrix (result.eigenvalues ())); } else { if (vector_flag) retval = ovl (result.right_eigenvectors (), result.eigenvalues (), result.left_eigenvectors ()); else retval = ovl (result.right_eigenvectors (), FloatComplexDiagMatrix (result.eigenvalues ()), result.left_eigenvectors ()); } } else { EIG result; if (AEPcase) { if (arg_a.isreal ()) { tmp_a = arg_a.matrix_value (); result = EIG (tmp_a, nargout > 1, nargout > 2, balance); } else { ctmp_a = arg_a.complex_matrix_value (); result = EIG (ctmp_a, nargout > 1, nargout > 2, balance); } } else { if (arg_a.isreal () && arg_b.isreal ()) { tmp_a = arg_a.matrix_value (); tmp_b = arg_b.matrix_value (); result = EIG (tmp_a, tmp_b, nargout > 1, nargout > 2, force_qz); } else { ctmp_a = arg_a.complex_matrix_value (); ctmp_b = arg_b.complex_matrix_value (); result = EIG (ctmp_a, ctmp_b, nargout > 1, nargout > 2, force_qz); } } if (nargout == 0 || nargout == 1) { if (matrix_flag) retval = ovl (ComplexDiagMatrix (result.eigenvalues ())); else retval = ovl (result.eigenvalues ()); } else if (nargout == 2) { if (vector_flag) retval = ovl (result.right_eigenvectors (), result.eigenvalues ()); else retval = ovl (result.right_eigenvectors (), ComplexDiagMatrix (result.eigenvalues ())); } else { if (vector_flag) retval = ovl (result.right_eigenvectors (), result.eigenvalues (), result.left_eigenvectors ()); else retval = ovl (result.right_eigenvectors (), ComplexDiagMatrix (result.eigenvalues ()), result.left_eigenvectors ()); } } return retval; } /* %!assert (eig ([1, 2; 2, 1]), [-1; 3], sqrt (eps)) %!test %! [v, d] = eig ([1, 2; 2, 1]); %! x = 1 / sqrt (2); %! assert (d, [-1, 0; 0, 3], sqrt (eps)) %! assert (v, [-x, x; x, x], sqrt (eps)) %!test %! [v, d, w] = eig ([1, 2; 2, 1]); %! x = 1 / sqrt (2); %! assert (w, [-x, x; x, x], sqrt (eps)) %!test %! [v, d] = eig ([1, 2; 2, 1], "balance"); %! x = 1 / sqrt (2); %! assert (d, [-1, 0; 0, 3], sqrt (eps)) %! assert (v, [-x, x; x, x], sqrt (eps)) %!test %! [v, d, w] = eig ([1, 2; 2, 1], "balance"); %! x = 1 / sqrt (2); %! assert (w, [-x, x; x, x], sqrt (eps)); %!assert (eig (single ([1, 2; 2, 1])), single ([-1; 3]), sqrt (eps ("single"))) %!assert (eig (single ([1, 2; 2, 1]), "balance"), %! single ([-1; 3]), sqrt (eps ("single"))) %!test %! [v, d] = eig (single ([1, 2; 2, 1])); %! x = single (1 / sqrt (2)); %! assert (d, single ([-1, 0; 0, 3]), sqrt (eps ("single"))) %! assert (v, [-x, x; x, x], sqrt (eps ("single"))) %!test %! [v, d, w] = eig (single ([1, 2; 2, 1])); %! x = single (1 / sqrt (2)); %! assert (w, [-x, x; x, x], sqrt (eps ("single"))) %!test %! [v, d] = eig (single ([1, 2; 2, 1]), "balance"); %! x = single (1 / sqrt (2)); %! assert (d, single ([-1, 0; 0, 3]), sqrt (eps ("single"))); %! assert (v, [-x, x; x, x], sqrt (eps ("single"))) %!test %! [v, d, w] = eig (single ([1, 2; 2, 1]), "balance"); %! x = single (1 / sqrt (2)); %! assert (w, [-x, x; x, x], sqrt (eps ("single"))) ## If (at least one of) the matrices are non-symmetric, ## regardless the algorithm flag the qz algorithm should be used. ## So the results without algorithm flag, with "qz" and with "chol" ## should be the same. %!function nonsym_chol_2_output (A, B, res = sqrt (eps)) %! [v, d] = eig (A, B); %! [v2, d2] = eig (A, B, "qz"); %! [v3, d3] = eig (A, B, "chol"); %! assert (A * v(:, 1), d(1, 1) * B * v(:, 1), res) %! assert (A * v(:, 2), d(2, 2) * B * v(:, 2), res) %! assert (v, v2) %! assert (v, v3) %! assert (d, d2) %! assert (d, d3) %!endfunction %!test nonsym_chol_2_output ([1, 2; -1, 1], [3, 3; 1, 2]) %!test nonsym_chol_2_output ([1+3i, 2+3i; 3-8i, 8+3i], [8+i, 3+i; 4-9i, 3+i]) %!test nonsym_chol_2_output ([1, 2; 3, 8], [8, 3; 4, 3]) %!test nonsym_chol_2_output (single ([1, 2; -1, 1]), %! single ([3, 3; 1, 2]), sqrt (eps ("single"))) %!test nonsym_chol_2_output (single ([1+3i, 2+3i; 3-8i, 8+3i]), %! single ([8+i, 3+i; 4-9i, 3+i]), %! sqrt (eps ("single"))) %!function nonsym_chol_3_output (A, B, res = sqrt (eps)) %! [v, d, w] = eig (A, B); %! [v2, d2, w2] = eig (A, B, "qz"); %! [v3, d3, w3] = eig (A, B, "chol"); %! wt = w'; %! assert (wt(1, :)* A, d(1, 1) * wt(1, :) * B, res) %! assert (wt(2, :)* A, d(2, 2) * wt(2, :) * B, res) %! assert (v, v2) %! assert (v, v3) %! assert (d, d2) %! assert (d, d3) %! assert (w, w2) %! assert (w, w3) %!endfunction %!test nonsym_chol_3_output ([1, 2; -1, 1], [3, 3; 1, 2]) %!test nonsym_chol_3_output ([1+3i, 2+3i; 3-8i, 8+3i], [8+i, 3+i; 4-9i, 3+i]) %!test nonsym_chol_3_output ([1, 2; 3, 8], [8, 3; 4, 3]) %!test nonsym_chol_3_output (single ([1, 2; -1, 1]), %! single ([3, 3; 1, 2]), sqrt (eps ("single"))) %!test nonsym_chol_3_output (single ([1+3i, 2+3i; 3-8i, 8+3i]), %! single ([8+i, 3+i; 4-9i, 3+i]), %! sqrt (eps ("single"))) ## If the matrices are symmetric, ## then the chol method is default. ## So the results without algorithm flag and with "chol" should be the same. %!function sym_chol_2_input (A, B, res = sqrt (eps)) %! [v, d] = eig (A, B); %! [v2, d2] = eig (A, B, "chol"); %! assert (A * v(:, 1), d(1, 1) * B * v(:, 1), res) %! assert (A * v(:, 2), d(2, 2) * B * v(:, 2), res) %! assert (v, v2) %! assert (d, d2) %!endfunction %!test sym_chol_2_input ([1, 2; 2, 1], [3, -2; -2, 3]) %!test sym_chol_2_input ([1+3i, 2+i; 2-i, 1+3i], [5+9i, 2+i; 2-i, 5+9i]) %!test sym_chol_2_input ([1, 1+i; 1-i, 1], [2, 0; 0, 2]) %!test sym_chol_2_input (single ([1, 2; 2, 1]), single ([3, -2; -2, 3]), %! sqrt (eps ("single"))) %!test sym_chol_2_input (single ([1+3i, 2+i; 2-i, 1+3i]), %! single ([5+9i, 2+i; 2-i, 5+9i]), sqrt (eps ("single"))) %!test sym_chol_2_input (single ([1, 1+i; 1-i, 1]), single ([2, 0; 0, 2]), %! sqrt (eps ("single"))) %!function sym_chol_3_input (A, B, res = sqrt (eps)) %! [v, d, w] = eig (A, B); %! [v2, d2, w2] = eig (A, B, "chol"); %! wt = w'; %! assert (wt(1, :)* A, d(1, 1) * wt(1, :) * B, res) %! assert (wt(2, :)* A, d(2, 2) * wt(2, :) * B, res) %! assert (v, v2) %! assert (d, d2) %! assert (w, w2) %!endfunction %!test sym_chol_3_input ([1, 2; 2, 1], [3, -2; -2, 3]) %!test sym_chol_3_input ([1+3i, 2+i; 2-i, 1+3i], [5+9i, 2+i; 2-i, 5+9i]) %!test sym_chol_3_input ([1, 1+i; 1-i, 1], [2, 0; 0, 2]) %!test sym_chol_3_input (single ([1, 2; 2, 1]), single ([3, -2; -2, 3]), %! sqrt (eps ("single"))) %!test sym_chol_3_input (single ([1+3i, 2+i; 2-i, 1+3i]), %! single ([5+9i, 2+i; 2-i, 5+9i]), sqrt (eps ("single"))) %!test sym_chol_3_input (single ([1, 1+i; 1-i, 1]), single ([2, 0; 0, 2]), %! sqrt (eps ("single"))) ## "balance" is always default ## so the results with and without "balance" should be the same ## while in this case "nobalance" should produce different result %!test %! A = [3 -2 -0.9 0; -2 4 1 -0; -0 0 -1 0; -0.5 -0.5 0.1 1]; %! [V1, D1] = eig (A); %! [V2, D2] = eig (A, "balance"); %! [V3, D3] = eig (A, "nobalance"); %! assert (V1, V2) %! assert (D1, D2) %! assert (isequal (V2, V3), false) ## Testing the flags in all combination. ## If 2 flags are on, than the result should be the same regardless ## of the flags order. ## option1 represents the first order while option2 represents the other order. ## d and d2 should be a diagonal matrix if "matrix" flag is on while ## these should be column vectors if the "vector" flag is on. %!function test_eig_args (args, options1, options2, testd = @() true) %! [v, d, w] = eig (args{:}, options1{:}); %! [v2, d2, w2] = eig (args{:}, options2{:}); %! assert (testd (d)) %! assert (testd (d2)) %! assert (v, v2) %! assert (d, d2) %! assert (w, w2) %!endfunction %!function qz_chol_with_shapes (A, B) %! for shapes = struct ("name", {"vector", "matrix"}, %! "test", {@isvector, @isdiag}) %! test_eig_args ({A, B}, {"qz", shapes.name}, %! {shapes.name, "qz"}, shapes.test); %! test_eig_args ({A, B}, {"chol", shapes.name}, %! {shapes.name, "chol"}, shapes.test); %! endfor %!endfunction %!function balance_nobalance_with_shapes (A) %! for shapes = struct ("name", {"vector", "matrix"}, %! "test", {@isvector, @isdiag}) %! test_eig_args ({A}, {"balance", shapes.name}, %! {shapes.name, "balance"}, shapes.test); %! test_eig_args ({A}, {"nobalance", shapes.name}, %! {shapes.name, "nobalance"}, shapes.test); %! endfor %!endfunction ## Default return format: ## diagonal matrix if 2 or 3 outputs are specified ## column vector if 1 output is specified %!function test_shapes (args) %! d = eig (args{:}); %! assert (isvector(d)) %! d2 = eig (args{:}, "vector"); %! assert (isvector(d2)) %! [v, d3] = eig (args{:}); %! assert (isdiag(d3)) %! d4 = eig (args{:}, "matrix"); %! assert (isdiag(d4)) %! [v, d5, w] = eig (args{:}); %! assert (isdiag(d5)) %! d6 = eig (args{:}, "matrix"); %! assert (isdiag(d6)) %! assert (d, d2) %! assert (d3, d4) %! assert (d5, d6) %! assert (d, diag(d3)) %! assert (d, diag(d5)) %!endfunction %!function shapes_AEP (A) %! test_shapes({A}); %!endfunction %!function shapes_GEP (A, B) %! test_shapes({A, B}); %!endfunction %!test balance_nobalance_with_shapes ([1, 2; 2, 1]); %!test balance_nobalance_with_shapes (single ([1, 2; 2, 1])); %!test shapes_AEP ([1, 2; 2, 1]); %!test shapes_AEP (single ([1, 2; 2, 1])); %!test qz_chol_with_shapes ([1, 1+i; 1-i, 1], [2, 0; 0, 2]); %!test qz_chol_with_shapes ([1, 2; 3, 8], [8, 3; 4, 3]); %!test qz_chol_with_shapes ([1, 2; -1, 1], [3, 3; 1, 2]); %!test qz_chol_with_shapes (single ([1, 1+i; 1-i, 1]), single ([2, 0; 0, 2])); %!test qz_chol_with_shapes (single ([1, 2; 3, 8]), single ([8, 3; 4, 3])); %!test qz_chol_with_shapes (single ([1, 2; -1, 1]), single ([3, 3; 1, 2])); %!test shapes_GEP ([1, 1+i; 1-i, 1], [2, 0; 0, 2]); %!test shapes_GEP ([1, 2; 3, 8], [8, 3; 4, 3]); %!test shapes_GEP ([1, 2; -1, 1], [3, 3; 1, 2]); %!test shapes_GEP (single ([1, 1+i; 1-i, 1]), single ([2, 0; 0, 2])); %!test shapes_GEP (single ([1, 2; 3, 8]), single ([8, 3; 4, 3])); %!test shapes_GEP (single ([1, 2; -1, 1]), single ([3, 3; 1, 2])); ## Check if correct default method is used for symmetric input %!function chol_qz_accuracy (A, B, is_qz_accurate, is_chol_accurate) %! [V1, D1] = eig (A, B, 'qz'); %! [V2, D2] = eig (A, B); #default is chol %! assert (isequal (A*V1, A*V1*D1), is_qz_accurate) %! assert (isequal (A*V2, A*V2*D2), is_chol_accurate) %!endfunction %!test %! minij_100 = gallery ('minij', 100); %! chol_qz_accuracy (minij_100, minij_100, false, true); %! moler_100 = gallery ('moler', 100); %! chol_qz_accuracy (moler_100, moler_100, false, true); %! A = diag([1e-16, 1e-15]); %! chol_qz_accuracy (A, A, true, false); %!error eig () %!error eig (false) %!error eig ([1, 2; 3, 4], [4, 3; 2, 1], 1) %!error <EIG requires same size matrices> %! eig ([1, 2; 3, 4], 2) %!error <must be a square matrix> %! eig ([1, 2; 3, 4; 5, 6]) %!error <wrong type argument> %! eig ("abcd") %!error <invalid option "abcd"> %! eig ([1 2 ; 2 3], "abcd") %!error <invalid "chol" option for algebraic eigenvalue problem> %! eig ([1 2 ; 2 3], "chol") %!error <invalid "qz" option for algebraic eigenvalue problem> %! eig ([1 2 ; 2 3], "qz") %!error <wrong type argument> %! eig (false, [1 2 ; 2 3]) %!error <invalid option "abcd"> %! eig ([1 2 ; 2 3], [1 2 ; 2 3], "abcd") %!error <invalid "qz" option for algebraic eigenvalue problem> %! eig ([1 2 ; 2 3], "balance", "qz") %!error <invalid option "abcd"> %! eig ([1 2 ; 2 3], [1 2 ; 2 3], "vector", "abcd") %!error <invalid option "abcd"> %! eig ([1 2 ; 2 3], "balance", "matrix", "abcd") %!error <"balance" and "nobalance" options are mutually exclusive> %! eig ([1 2 ; 2 3], "balance", "nobalance") %!error <"balance" and "nobalance" options are mutually exclusive> %! eig ([1 2 ; 2 3], "nobalance", "balance") %!error <"vector" and "matrix" options are mutually exclusive> %! eig ([1 2 ; 2 3], "matrix", "vector") %!error <"vector" and "matrix" options are mutually exclusive> %! eig ([1 2 ; 2 3], "vector", "matrix") %!error <"vector" and "matrix" options are mutually exclusive> %! eig ([1 2 ; 2 3], [1 2 ; 2 3], "matrix", "vector") %!error <"vector" and "matrix" options are mutually exclusive> %! eig ([1 2 ; 2 3], [1 2 ; 2 3], "vector", "matrix") %!error <wrong type argument> %! eig ([1 2 ; 2 3], [1 2 ; 2 3], false) %!error <wrong type argument> %! eig ([1 2 ; 2 3], [1 2 ; 2 3], [1 2 ; 2 3]) */