Mercurial > octave
view libinterp/corefcn/lu.cc @ 31192:7401a2501be4
maint: merge stable to default
author | Rik <rik@octave.org> |
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date | Mon, 15 Aug 2022 21:06:21 -0700 |
parents | 83f9f8bda883 |
children | e88a07dec498 |
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//////////////////////////////////////////////////////////////////////// // // Copyright (C) 1996-2022 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 "lu.h" #include "sparse-lu.h" #include "defun.h" #include "error.h" #include "errwarn.h" #include "ovl.h" #include "ov-re-sparse.h" #include "ov-cx-sparse.h" OCTAVE_NAMESPACE_BEGIN template <typename MT> static octave_value get_lu_l (const math::lu<MT>& fact) { MT L = fact.L (); if (L.issquare ()) return octave_value (L, MatrixType (MatrixType::Lower)); else return L; } template <typename MT> static octave_value get_lu_u (const math::lu<MT>& fact) { MT U = fact.U (); if (U.issquare () && fact.regular ()) return octave_value (U, MatrixType (MatrixType::Upper)); else return U; } DEFUN (lu, args, nargout, doc: /* -*- texinfo -*- @deftypefn {} {[@var{L}, @var{U}] =} lu (@var{A}) @deftypefnx {} {[@var{L}, @var{U}, @var{P}] =} lu (@var{A}) @deftypefnx {} {[@var{L}, @var{U}, @var{P}, @var{Q}] =} lu (@var{S}) @deftypefnx {} {[@var{L}, @var{U}, @var{P}, @var{Q}, @var{R}] =} lu (@var{S}) @deftypefnx {} {[@dots{}] =} lu (@var{S}, @var{thresh}) @deftypefnx {} {@var{y} =} lu (@dots{}) @deftypefnx {} {[@dots{}] =} lu (@dots{}, "vector") @cindex LU decomposition Compute the LU@tie{}decomposition of @var{A}. If @var{A} is full then subroutines from @sc{lapack} are used, and if @var{A} is sparse then @sc{umfpack} is used. The result is returned in a permuted form, according to the optional return value @var{P}. For example, given the matrix @code{@var{A} = [1, 2; 3, 4]}, @example [@var{L}, @var{U}, @var{P}] = lu (@var{A}) @end example @noindent returns @example @group L = 1.00000 0.00000 0.33333 1.00000 U = 3.00000 4.00000 0.00000 0.66667 P = 0 1 1 0 @end group @end example The matrix is not required to be square. When called with two or three output arguments and a sparse input matrix, @code{lu} does not attempt to perform sparsity preserving column permutations. Called with a fourth output argument, the sparsity preserving column transformation @var{Q} is returned, such that @code{@var{P} * @var{A} * @var{Q} = @var{L} * @var{U}}. This is the @strong{preferred} way to call @code{lu} with sparse input matrices. Called with a fifth output argument and a sparse input matrix, @code{lu} attempts to use a scaling factor @var{R} on the input matrix such that @code{@var{P} * (@var{R} \ @var{A}) * @var{Q} = @var{L} * @var{U}}. This typically leads to a sparser and more stable factorization. An additional input argument @var{thresh} that defines the pivoting threshold can be given. @var{thresh} can be a scalar, in which case it defines the @sc{umfpack} pivoting tolerance for both symmetric and unsymmetric cases. If @var{thresh} is a 2-element vector, then the first element defines the pivoting tolerance for the unsymmetric @sc{umfpack} pivoting strategy and the second for the symmetric strategy. By default, the values defined by @code{spparms} are used ([0.1, 0.001]). Given the string argument @qcode{"vector"}, @code{lu} returns the values of @var{P} and @var{Q} as vector values, such that for full matrix, @code{@var{A}(@var{P},:) = @var{L} * @var{U}}, and @code{@var{R}(@var{P},:) * @var{A}(:,@var{Q}) = @var{L} * @var{U}}. With two output arguments, returns the permuted forms of the upper and lower triangular matrices, such that @code{@var{A} = @var{L} * @var{U}}. With one output argument @var{y}, then the matrix returned by the @sc{lapack} routines is returned. If the input matrix is sparse then the matrix @var{L} is embedded into @var{U} to give a return value similar to the full case. For both full and sparse matrices, @code{lu} loses the permutation information. @seealso{luupdate, ilu, chol, hess, qr, qz, schur, svd} @end deftypefn */) { int nargin = args.length (); bool issparse = (nargin > 0 && args(0).issparse ()); if (nargin < 1 || (issparse && nargin > 3) || (! issparse && nargin > 2)) print_usage (); bool vecout = false; Matrix thresh; int n = 1; while (n < nargin) { if (args(n).is_string ()) { std::string tmp = args(n++).string_value (); if (tmp == "vector") vecout = true; else error ("lu: unrecognized string argument"); } else { if (! issparse) error ("lu: can not define pivoting threshold THRESH for full matrices"); Matrix tmp = args(n++).matrix_value (); if (tmp.numel () == 1) { thresh.resize (1, 2); thresh(0) = tmp(0); thresh(1) = tmp(0); } else if (tmp.numel () == 2) thresh = tmp; else error ("lu: THRESH must be a 1- or 2-element vector"); } } octave_value_list retval; bool scale = (nargout == 5); octave_value arg = args(0); octave_idx_type nr = arg.rows (); octave_idx_type nc = arg.columns (); if (issparse) { if (arg.isempty ()) return octave_value_list (5, SparseMatrix ()); if (arg.isreal ()) { SparseMatrix m = arg.sparse_matrix_value (); if (nargout < 4) { warning_with_id ("Octave:lu:sparse_input", "lu: function may fail when called with less than 4 output arguments and a sparse input"); ColumnVector Qinit (nc); for (octave_idx_type i = 0; i < nc; i++) Qinit(i) = i; math::sparse_lu<SparseMatrix> fact (m, Qinit, thresh, false, true); if (nargout < 2) retval(0) = fact.Y (); else { retval.resize (nargout == 3 ? 3 : 2); retval(1) = octave_value (fact.U () * fact.Pc_mat ().transpose (), MatrixType (MatrixType::Permuted_Upper, nc, fact.col_perm ())); PermMatrix P = fact.Pr_mat (); SparseMatrix L = fact.L (); if (nargout == 2) retval(0) = octave_value (P.transpose () * L, MatrixType (MatrixType::Permuted_Lower, nr, fact.row_perm ())); else { retval(0) = L; if (vecout) retval(2) = fact.Pr_vec(); else retval(2) = P; } } } else { retval.resize (scale ? 5 : 4); math::sparse_lu<SparseMatrix> fact (m, thresh, scale); retval(0) = octave_value (fact.L (), MatrixType (MatrixType::Lower)); retval(1) = octave_value (fact.U (), MatrixType (MatrixType::Upper)); if (vecout) { retval(2) = fact.Pr_vec (); retval(3) = fact.Pc_vec (); } else { retval(2) = fact.Pr_mat (); retval(3) = fact.Pc_mat (); } if (scale) retval(4) = fact.R (); } } else if (arg.iscomplex ()) { SparseComplexMatrix m = arg.sparse_complex_matrix_value (); if (nargout < 4) { warning_with_id ("Octave:lu:sparse_input", "lu: function may fail when called with less than 4 output arguments and a sparse input"); ColumnVector Qinit (nc); for (octave_idx_type i = 0; i < nc; i++) Qinit(i) = i; math::sparse_lu<SparseComplexMatrix> fact (m, Qinit, thresh, false, true); if (nargout < 2) retval(0) = fact.Y (); else { retval.resize (nargout == 3 ? 3 : 2); retval(1) = octave_value (fact.U () * fact.Pc_mat ().transpose (), MatrixType (MatrixType::Permuted_Upper, nc, fact.col_perm ())); PermMatrix P = fact.Pr_mat (); SparseComplexMatrix L = fact.L (); if (nargout == 2) retval(0) = octave_value (P.transpose () * L, MatrixType (MatrixType::Permuted_Lower, nr, fact.row_perm ())); else { retval(0) = L; if (vecout) retval(2) = fact.Pr_vec(); else retval(2) = P; } } } else { retval.resize (scale ? 5 : 4); math::sparse_lu<SparseComplexMatrix> fact (m, thresh, scale); retval(0) = octave_value (fact.L (), MatrixType (MatrixType::Lower)); retval(1) = octave_value (fact.U (), MatrixType (MatrixType::Upper)); if (vecout) { retval(2) = fact.Pr_vec (); retval(3) = fact.Pc_vec (); } else { retval(2) = fact.Pr_mat (); retval(3) = fact.Pc_mat (); } if (scale) retval(4) = fact.R (); } } else err_wrong_type_arg ("lu", arg); } else { if (arg.isempty ()) return octave_value_list (3, Matrix ()); if (arg.isreal ()) { if (arg.is_single_type ()) { FloatMatrix m = arg.float_matrix_value (); math::lu<FloatMatrix> fact (m); switch (nargout) { case 0: case 1: retval = ovl (fact.Y ()); break; case 2: { PermMatrix P = fact.P (); FloatMatrix L = P.transpose () * fact.L (); retval = ovl (L, get_lu_u (fact)); } break; case 3: default: { if (vecout) retval = ovl (get_lu_l (fact), get_lu_u (fact), fact.P_vec ()); else retval = ovl (get_lu_l (fact), get_lu_u (fact), fact.P ()); } break; } } else { Matrix m = arg.matrix_value (); math::lu<Matrix> fact (m); switch (nargout) { case 0: case 1: retval = ovl (fact.Y ()); break; case 2: { PermMatrix P = fact.P (); Matrix L = P.transpose () * fact.L (); retval = ovl (L, get_lu_u (fact)); } break; case 3: default: { if (vecout) retval = ovl (get_lu_l (fact), get_lu_u (fact), fact.P_vec ()); else retval = ovl (get_lu_l (fact), get_lu_u (fact), fact.P ()); } break; } } } else if (arg.iscomplex ()) { if (arg.is_single_type ()) { FloatComplexMatrix m = arg.float_complex_matrix_value (); math::lu<FloatComplexMatrix> fact (m); switch (nargout) { case 0: case 1: retval = ovl (fact.Y ()); break; case 2: { PermMatrix P = fact.P (); FloatComplexMatrix L = P.transpose () * fact.L (); retval = ovl (L, get_lu_u (fact)); } break; case 3: default: { if (vecout) retval = ovl (get_lu_l (fact), get_lu_u (fact), fact.P_vec ()); else retval = ovl (get_lu_l (fact), get_lu_u (fact), fact.P ()); } break; } } else { ComplexMatrix m = arg.complex_matrix_value (); math::lu<ComplexMatrix> fact (m); switch (nargout) { case 0: case 1: retval = ovl (fact.Y ()); break; case 2: { PermMatrix P = fact.P (); ComplexMatrix L = P.transpose () * fact.L (); retval = ovl (L, get_lu_u (fact)); } break; case 3: default: { if (vecout) retval = ovl (get_lu_l (fact), get_lu_u (fact), fact.P_vec ()); else retval = ovl (get_lu_l (fact), get_lu_u (fact), fact.P ()); } break; } } } else err_wrong_type_arg ("lu", arg); } return retval; } /* %!assert(lu ([1, 2; 3, 4]), [3, 4; 1/3, 2/3], eps) %!test %! [l, u] = lu ([1, 2; 3, 4]); %! assert (l, [1/3, 1; 1, 0], sqrt (eps)); %! assert (u, [3, 4; 0, 2/3], sqrt (eps)); %!test %! [l, u, p] = lu ([1, 2; 3, 4]); %! assert (l, [1, 0; 1/3, 1], sqrt (eps)); %! assert (u, [3, 4; 0, 2/3], sqrt (eps)); %! assert (p(:,:), [0, 1; 1, 0], sqrt (eps)); %!test %! [l, u, p] = lu ([1, 2; 3, 4], "vector"); %! assert (l, [1, 0; 1/3, 1], sqrt (eps)); %! assert (u, [3, 4; 0, 2/3], sqrt (eps)); %! assert (p, [2;1], sqrt (eps)); %!test %! [l, u, p] = lu ([1, 2; 3, 4; 5, 6]); %! assert (l, [1, 0; 1/5, 1; 3/5, 1/2], sqrt (eps)); %! assert (u, [5, 6; 0, 4/5], sqrt (eps)); %! assert (p(:,:), [0, 0, 1; 1, 0, 0; 0 1 0], sqrt (eps)); %!assert (lu (single ([1, 2; 3, 4])), single ([3, 4; 1/3, 2/3]), eps ("single")) %!test %! [l, u] = lu (single ([1, 2; 3, 4])); %! assert (l, single ([1/3, 1; 1, 0]), sqrt (eps ("single"))); %! assert (u, single ([3, 4; 0, 2/3]), sqrt (eps ("single"))); %!test %! [l, u, p] = lu (single ([1, 2; 3, 4])); %! assert (l, single ([1, 0; 1/3, 1]), sqrt (eps ("single"))); %! assert (u, single ([3, 4; 0, 2/3]), sqrt (eps ("single"))); %! assert (p(:,:), single ([0, 1; 1, 0]), sqrt (eps ("single"))); %!test %! [l, u, p] = lu (single ([1, 2; 3, 4]), "vector"); %! assert (l, single ([1, 0; 1/3, 1]), sqrt (eps ("single"))); %! assert (u, single ([3, 4; 0, 2/3]), sqrt (eps ("single"))); %! assert (p, single ([2;1]), sqrt (eps ("single"))); %!test %! [l u p] = lu (single ([1, 2; 3, 4; 5, 6])); %! assert (l, single ([1, 0; 1/5, 1; 3/5, 1/2]), sqrt (eps ("single"))); %! assert (u, single ([5, 6; 0, 4/5]), sqrt (eps ("single"))); %! assert (p(:,:), single ([0, 0, 1; 1, 0, 0; 0 1 0]), sqrt (eps ("single"))); %!testif HAVE_UMFPACK %! Bi = [1 2 3 4 5 2 3 6 7 8 4 5 7 8 9]; %! Bj = [1 3 4 5 6 7 8 9 11 12 13 14 15 16 17]; %! Bv = [1 1 1 1 1 1 -1 1 1 1 1 -1 1 -1 1]; %! B = sparse (Bi, Bj, Bv); %! warning ("off", "Octave:lu:sparse_input", "local"); %! [L, U] = lu (B); %! assert (L*U, B); %! [L, U, P] = lu(B); %! assert (P'*L*U, B); %! [L, U, P, Q] = lu (B); %! assert (P'*L*U*Q', B); %!error lu () %!testif HAVE_UMFPACK %! fail ("[l,u] = lu (sparse (magic (3)))", "warning", "function may fail"); %!error <can not define pivoting threshold> lu ([1, 2; 3, 4], 2) */ static bool check_lu_dims (const octave_value& l, const octave_value& u, const octave_value& p) { octave_idx_type m = l.rows (); octave_idx_type k = u.rows (); octave_idx_type n = u.columns (); return ((l.ndims () == 2 && u.ndims () == 2 && k == l.columns ()) && k == std::min (m, n) && (p.is_undefined () || p.rows () == m)); } DEFUN (luupdate, args, , doc: /* -*- texinfo -*- @deftypefn {} {[@var{L}, @var{U}] =} luupdate (@var{L}, @var{U}, @var{x}, @var{y}) @deftypefnx {} {[@var{L}, @var{U}, @var{P}] =} luupdate (@var{L}, @var{U}, @var{P}, @var{x}, @var{y}) Given an LU@tie{}factorization of a real or complex matrix @w{@var{A} = @var{L}*@var{U}}, @var{L}@tie{}lower unit trapezoidal and @var{U}@tie{}upper trapezoidal, return the LU@tie{}factorization of @w{@var{A} + @var{x}*@var{y}.'}, where @var{x} and @var{y} are column vectors (rank-1 update) or matrices with equal number of columns (rank-k update). Optionally, row-pivoted updating can be used by supplying a row permutation (pivoting) matrix @var{P}; in that case, an updated permutation matrix is returned. Note that if @var{L}, @var{U}, @var{P} is a pivoted LU@tie{}factorization as obtained by @code{lu}: @example [@var{L}, @var{U}, @var{P}] = lu (@var{A}); @end example @noindent then a factorization of @tcode{@var{A}+@var{x}*@var{y}.'} can be obtained either as @example [@var{L1}, @var{U1}] = lu (@var{L}, @var{U}, @var{P}*@var{x}, @var{y}) @end example @noindent or @example [@var{L1}, @var{U1}, @var{P1}] = lu (@var{L}, @var{U}, @var{P}, @var{x}, @var{y}) @end example The first form uses the unpivoted algorithm, which is faster, but less stable. The second form uses a slower pivoted algorithm, which is more stable. The matrix case is done as a sequence of rank-1 updates; thus, for large enough k, it will be both faster and more accurate to recompute the factorization from scratch. @seealso{lu, cholupdate, qrupdate} @end deftypefn */) { int nargin = args.length (); if (nargin != 4 && nargin != 5) print_usage (); bool pivoted = (nargin == 5); octave_value argl = args(0); octave_value argu = args(1); octave_value argp = (pivoted ? args(2) : octave_value ()); octave_value argx = args(2 + pivoted); octave_value argy = args(3 + pivoted); if (! (argl.isnumeric () && argu.isnumeric () && argx.isnumeric () && argy.isnumeric () && (! pivoted || argp.is_perm_matrix ()))) error ("luupdate: L, U, X, and Y must be numeric"); if (! check_lu_dims (argl, argu, argp)) error ("luupdate: dimension mismatch"); PermMatrix P = (pivoted ? argp.perm_matrix_value () : PermMatrix::eye (argl.rows ())); if (argl.isreal () && argu.isreal () && argx.isreal () && argy.isreal ()) { // all real case if (argl.is_single_type () || argu.is_single_type () || argx.is_single_type () || argy.is_single_type ()) { FloatMatrix L = argl.float_matrix_value (); FloatMatrix U = argu.float_matrix_value (); FloatMatrix x = argx.float_matrix_value (); FloatMatrix y = argy.float_matrix_value (); math::lu<FloatMatrix> fact (L, U, P); if (pivoted) fact.update_piv (x, y); else fact.update (x, y); if (pivoted) return ovl (get_lu_l (fact), get_lu_u (fact), fact.P ()); else return ovl (get_lu_l (fact), get_lu_u (fact)); } else { Matrix L = argl.matrix_value (); Matrix U = argu.matrix_value (); Matrix x = argx.matrix_value (); Matrix y = argy.matrix_value (); math::lu<Matrix> fact (L, U, P); if (pivoted) fact.update_piv (x, y); else fact.update (x, y); if (pivoted) return ovl (get_lu_l (fact), get_lu_u (fact), fact.P ()); else return ovl (get_lu_l (fact), get_lu_u (fact)); } } else { // complex case if (argl.is_single_type () || argu.is_single_type () || argx.is_single_type () || argy.is_single_type ()) { FloatComplexMatrix L = argl.float_complex_matrix_value (); FloatComplexMatrix U = argu.float_complex_matrix_value (); FloatComplexMatrix x = argx.float_complex_matrix_value (); FloatComplexMatrix y = argy.float_complex_matrix_value (); math::lu<FloatComplexMatrix> fact (L, U, P); if (pivoted) fact.update_piv (x, y); else fact.update (x, y); if (pivoted) return ovl (get_lu_l (fact), get_lu_u (fact), fact.P ()); else return ovl (get_lu_l (fact), get_lu_u (fact)); } else { ComplexMatrix L = argl.complex_matrix_value (); ComplexMatrix U = argu.complex_matrix_value (); ComplexMatrix x = argx.complex_matrix_value (); ComplexMatrix y = argy.complex_matrix_value (); math::lu<ComplexMatrix> fact (L, U, P); if (pivoted) fact.update_piv (x, y); else fact.update (x, y); if (pivoted) return ovl (get_lu_l (fact), get_lu_u (fact), fact.P ()); else return ovl (get_lu_l (fact), get_lu_u (fact)); } } } /* %!shared A, u, v, Ac, uc, vc %! A = [0.091364 0.613038 0.999083; %! 0.594638 0.425302 0.603537; %! 0.383594 0.291238 0.085574; %! 0.265712 0.268003 0.238409; %! 0.669966 0.743851 0.445057 ]; %! %! u = [0.85082; %! 0.76426; %! 0.42883; %! 0.53010; %! 0.80683 ]; %! %! v = [0.98810; %! 0.24295; %! 0.43167 ]; %! %! Ac = [0.620405 + 0.956953i 0.480013 + 0.048806i 0.402627 + 0.338171i; %! 0.589077 + 0.658457i 0.013205 + 0.279323i 0.229284 + 0.721929i; %! 0.092758 + 0.345687i 0.928679 + 0.241052i 0.764536 + 0.832406i; %! 0.912098 + 0.721024i 0.049018 + 0.269452i 0.730029 + 0.796517i; %! 0.112849 + 0.603871i 0.486352 + 0.142337i 0.355646 + 0.151496i ]; %! %! uc = [0.20351 + 0.05401i; %! 0.13141 + 0.43708i; %! 0.29808 + 0.08789i; %! 0.69821 + 0.38844i; %! 0.74871 + 0.25821i ]; %! %! vc = [0.85839 + 0.29468i; %! 0.20820 + 0.93090i; %! 0.86184 + 0.34689i ]; %! %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (A); %! [L,U] = luupdate (L,U,P*u,v); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - A - u*v.'), Inf) < norm (A)*1e1*eps); %! %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (Ac); %! [L,U] = luupdate (L,U,P*uc,vc); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - Ac - uc*vc.'), Inf) < norm (Ac)*1e1*eps); %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (single (A)); %! [L,U] = luupdate (L,U,P*single (u), single (v)); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - single (A) - single (u)*single (v).'), Inf) %! < norm (single (A))*1e1*eps ("single")); %! %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (single (Ac)); %! [L,U] = luupdate (L,U,P*single (uc),single (vc)); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - single (Ac) - single (uc)*single (vc).'), Inf) %! < norm (single (Ac))*1e1*eps ("single")); %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (A); %! [L,U,P] = luupdate (L,U,P,u,v); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - A - u*v.'), Inf) < norm (A)*1e1*eps); %! %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (A); %! [~,ordcols] = max (P,[],1); %! [~,ordrows] = max (P,[],2); %! P1 = eye (size (P))(:,ordcols); %! P2 = eye (size (P))(ordrows,:); %! assert (P1 == P); %! assert (P2 == P); %! [L,U,P] = luupdate (L,U,P,u,v); %! [L,U,P1] = luupdate (L,U,P1,u,v); %! [L,U,P2] = luupdate (L,U,P2,u,v); %! assert (P1 == P); %! assert (P2 == P); %! %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (Ac); %! [L,U,P] = luupdate (L,U,P,uc,vc); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - Ac - uc*vc.'), Inf) < norm (Ac)*1e1*eps); %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (single (A)); %! [L,U,P] = luupdate (L,U,P,single (u),single (v)); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - single (A) - single (u)*single (v).'), Inf) %! < norm (single (A))*1e1*eps ("single")); %! %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (single (Ac)); %! [L,U,P] = luupdate (L,U,P,single (uc),single (vc)); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - single (Ac) - single (uc)*single (vc).'), Inf) %! < norm (single (Ac))*1e1*eps ("single")); */ OCTAVE_NAMESPACE_END