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view libinterp/corefcn/sqrtm.cc @ 33170:82ae00b40c2c
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author | John W. Eaton <jwe@octave.org> |
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date | Thu, 07 Mar 2024 14:16:22 -0500 |
parents | f1cb625c8eb5 |
children | 786d203458bd |
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//////////////////////////////////////////////////////////////////////// // // Copyright (C) 2001-2024 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 "schur.h" #include "lo-ieee.h" #include "lo-mappers.h" #include "oct-norm.h" #include "defun.h" #include "error.h" #include "errwarn.h" #include "utils.h" #include "xnorm.h" OCTAVE_BEGIN_NAMESPACE(octave) template <typename T> static void sqrtm_utri_inplace (T& m) { typedef typename T::element_type element_type; typedef typename T::real_matrix_type real_matrix_type; typedef typename T::real_elt_type real_elt_type; const element_type zero = element_type (); bool singular = false; bool diagonal = true; // The Schur matrix of Hermitian matrices is diagonal. // check for off-diagonal elements above tolerance const octave_idx_type n = m.rows (); real_matrix_type abs_m = m.abs (); real_elt_type max_abs_diag = 0; for (octave_idx_type i = 0; i < n; i++) max_abs_diag = std::max (max_abs_diag, abs_m(i,i)); const real_elt_type tol = n * max_abs_diag * std::numeric_limits<real_elt_type>::epsilon (); for (octave_idx_type j = 0; j < n; j++) { for (octave_idx_type i = j-1; i >= 0; i--) { if (abs_m(i,j) > tol) { diagonal = false; break; } } if (! diagonal) break; } element_type *mp = m.rwdata (); if (diagonal) { // shortcut for diagonal Schur matrices for (octave_idx_type i = 0; i < n; i++) { octave_idx_type idx_diag = i*(n+1); if (mp[idx_diag] != zero) mp[idx_diag] = sqrt (mp[idx_diag]); else singular = true; } } else { // The following code is equivalent to this triple loop: // // n = rows (m); // for j = 1:n // m(j,j) = sqrt (m(j,j)); // for i = j-1:-1:1 // if m(i,j) != 0 // m(i,j) /= (m(i,i) + m(j,j)); // endif // k = 1:i-1; // m(k,j) -= m(k,i) * m(i,j); // endfor // endfor // // this is an in-place, cache-aligned variant of the code // given in Higham's paper. for (octave_idx_type j = 0; j < n; j++) { element_type *colj = mp + n*j; if (colj[j] != zero) colj[j] = sqrt (colj[j]); else singular = true; for (octave_idx_type i = j-1; i >= 0; i--) { const element_type *coli = mp + n*i; if (colj[i] != zero) colj[i] /= (coli[i] + colj[j]); const element_type colji = colj[i]; for (octave_idx_type k = 0; k < i; k++) colj[k] -= coli[k] * colji; } } } if (singular) warning_with_id ("Octave:sqrtm:SingularMatrix", "sqrtm: matrix is singular, may not have a square root"); } template <typename Matrix, typename ComplexMatrix, typename ComplexSCHUR> static octave_value do_sqrtm (const octave_value& arg) { octave_value retval; MatrixType mt = arg.matrix_type (); bool iscomplex = arg.iscomplex (); typedef typename Matrix::element_type real_type; real_type cutoff = 0; real_type one = 1; real_type eps = std::numeric_limits<real_type>::epsilon (); if (! iscomplex) { Matrix x = octave_value_extract<Matrix> (arg); if (mt.is_unknown ()) // if type is not known, compute it now. arg.matrix_type (mt = MatrixType (x)); switch (mt.type ()) { case MatrixType::Upper: case MatrixType::Diagonal: if (! x.diag ().any_element_is_negative ()) { // Do it in real arithmetic. sqrtm_utri_inplace (x); retval = x; retval.matrix_type (mt); } else iscomplex = true; break; case MatrixType::Lower: if (! x.diag ().any_element_is_negative ()) { x = x.transpose (); sqrtm_utri_inplace (x); retval = x.transpose (); retval.matrix_type (mt); } else iscomplex = true; break; default: iscomplex = true; break; } if (iscomplex) cutoff = 10 * x.rows () * eps * xnorm (x, one); } if (iscomplex) { ComplexMatrix x = octave_value_extract<ComplexMatrix> (arg); if (mt.is_unknown ()) // if type is not known, compute it now. arg.matrix_type (mt = MatrixType (x)); switch (mt.type ()) { case MatrixType::Upper: case MatrixType::Diagonal: sqrtm_utri_inplace (x); retval = x; retval.matrix_type (mt); break; case MatrixType::Lower: x = x.transpose (); sqrtm_utri_inplace (x); retval = x.transpose (); retval.matrix_type (mt); break; default: { ComplexMatrix u; do { ComplexSCHUR schur_fact (x, "", true); x = schur_fact.schur_matrix (); u = schur_fact.unitary_schur_matrix (); } while (0); // schur_fact no longer needed. sqrtm_utri_inplace (x); x = u * x; // original x no longer needed. ComplexMatrix res = xgemm (x, u, blas_no_trans, blas_conj_trans); if (cutoff > 0 && xnorm (imag (res), one) <= cutoff) retval = real (res); else retval = res; } break; } } return retval; } DEFUN (sqrtm, args, nargout, doc: /* -*- texinfo -*- @deftypefn {} {@var{s} =} sqrtm (@var{A}) @deftypefnx {} {[@var{s}, @var{error_estimate}] =} sqrtm (@var{A}) Compute the matrix square root of the square matrix @var{A}. Ref: @nospell{N.J. Higham}. @cite{A New sqrtm for @sc{matlab}}. Numerical Analysis Report No.@: 336, Manchester @nospell{Centre} for Computational Mathematics, Manchester, England, January 1999. @seealso{expm, logm} @end deftypefn */) { if (args.length () != 1) print_usage (); octave_value arg = args(0); octave_idx_type n = arg.rows (); octave_idx_type nc = arg.columns (); if (n != nc || arg.ndims () > 2) err_square_matrix_required ("sqrtm", "A"); octave_value_list retval (nargout > 1 ? 3 : 1); if (nargout > 1) { // FIXME: Octave does not calculate a condition number with respect to // sqrtm. Should this return NaN instead of -1? retval(2) = -1.0; } if (arg.is_diag_matrix ()) // sqrtm of a diagonal matrix is just sqrt. retval(0) = arg.sqrt (); else if (arg.is_single_type ()) retval(0) = do_sqrtm<FloatMatrix, FloatComplexMatrix, math::schur<FloatComplexMatrix>> (arg); else if (arg.isnumeric ()) retval(0) = do_sqrtm<Matrix, ComplexMatrix, math::schur<ComplexMatrix>> (arg); if (nargout > 1) { // This corresponds to generic code // // norm (s*s - x, "fro") / norm (x, "fro"); octave_value s = retval(0); retval(1) = xfrobnorm (s*s - arg) / xfrobnorm (arg); } return retval; } /* %!assert (sqrtm (2* ones (2)), ones (2), 3*eps) %!assert <*60797> (sqrtm (ones (4))^2, ones (4), 5*eps) ## The following two tests are from the reference in the docstring above. %!test %! warning ("off", "Octave:sqrtm:SingularMatrix", "local"); %! x = [0 1; 0 0]; %! assert (any (isnan (sqrtm (x))(:))); %!test %! x = eye (4); x(2,2) = x(3,3) = 2^-26; x(1,4) = 1; %! z = eye (4); z(2,2) = z(3,3) = 2^-13; z(1,4) = 0.5; %! [y, err] = sqrtm (x); %! assert (y, z); %! assert (err, 0); # Yes, this one has to hold exactly */ OCTAVE_END_NAMESPACE(octave)