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view scripts/linear-algebra/qzhess.m @ 33577:2506c2d30b32 bytecode-interpreter tip
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author | Arun Giridhar <arungiridhar@gmail.com> |
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date | Sat, 11 May 2024 18:49:01 -0400 |
parents | 2e484f9f1f18 |
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######################################################################## ## ## Copyright (C) 1993-2024 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/>. ## ######################################################################## ## -*- texinfo -*- ## @deftypefn {} {[@var{aa}, @var{bb}, @var{q}, @var{z}] =} qzhess (@var{A}, @var{B}) ## Compute the Hessenberg-triangular decomposition of the matrix pencil ## @code{(@var{A}, @var{B})}, returning ## @code{@var{aa} = @var{q} * @var{A} * @var{z}}, ## @code{@var{bb} = @var{q} * @var{B} * @var{z}}, with @var{q} and @var{z} ## orthogonal. ## ## For example: ## ## @example ## @group ## [aa, bb, q, z] = qzhess ([1, 2; 3, 4], [5, 6; 7, 8]) ## @result{} aa = ## -3.02244 -4.41741 ## 0.92998 0.69749 ## @result{} bb = ## -8.60233 -9.99730 ## 0.00000 -0.23250 ## @result{} q = ## -0.58124 -0.81373 ## -0.81373 0.58124 ## @result{} z = ## Diagonal Matrix ## 1 0 ## 0 1 ## @end group ## @end example ## ## The Hessenberg-triangular decomposition is the first step in ## @nospell{Moler and Stewart's} QZ@tie{}decomposition algorithm. ## ## Algorithm taken from @nospell{Golub and Van Loan}, ## @cite{Matrix Computations, 2nd edition}. ## ## @seealso{lu, chol, hess, qr, qz, schur, svd} ## @end deftypefn function [aa, bb, q, z] = qzhess (A, B) if (nargin != 2) print_usage (); endif [na, ma] = size (A); [nb, mb] = size (B); if (na != ma || na != nb || nb != mb) error ("qzhess: incompatible dimensions"); endif ## Reduce to hessenberg-triangular form. [q, bb] = qr (B); aa = q' * A; q = q'; z = eye (na); for j = 1:(na-2) for i = na:-1:(j+2) ## disp (["zero out aa(", num2str(i), ",", num2str(j), ")"]) rot = givens (aa (i-1, j), aa (i, j)); aa((i-1):i, :) = rot *aa((i-1):i, :); bb((i-1):i, :) = rot *bb((i-1):i, :); q((i-1):i, :) = rot * q((i-1):i, :); ## disp (["now zero out bb(", num2str(i), ",", num2str(i-1), ")"]) rot = givens (bb (i, i), bb (i, i-1))'; bb(:, (i-1):i) = bb(:, (i-1):i) * rot'; aa(:, (i-1):i) = aa(:, (i-1):i) * rot'; z(:, (i-1):i) = z(:, (i-1):i) * rot'; endfor endfor bb(2, 1) = 0.0; for i = 3:na bb (i, 1:(i-1)) = zeros (1, i-1); aa (i, 1:(i-2)) = zeros (1, i-2); endfor endfunction %!test %! a = [1 2 1 3; %! 2 5 3 2; %! 5 5 1 0; %! 4 0 3 2]; %! b = [0 4 2 1; %! 2 3 1 1; %! 1 0 2 1; %! 2 5 3 2]; %! mask = [0 0 0 0; %! 0 0 0 0; %! 1 0 0 0; %! 1 1 0 0]; %! [aa, bb, q, z] = qzhess (a, b); %! assert (inv (q) - q', zeros (4), 2e-8); %! assert (inv (z) - z', zeros (4), 2e-8); %! assert (q * a * z, aa, 2e-8); %! assert (aa .* mask, zeros (4), 2e-8); %! assert (q * b * z, bb, 2e-8); %! assert (bb .* mask, zeros (4), 2e-8); %!test %! a = [1 2 3 4 5; %! 3 2 3 1 0; %! 4 3 2 1 1; %! 0 1 0 1 0; %! 3 2 1 0 5]; %! b = [5 0 4 0 1; %! 1 1 1 2 5; %! 0 3 2 1 0; %! 4 3 0 3 5; %! 2 1 2 1 3]; %! mask = [0 0 0 0 0; %! 0 0 0 0 0; %! 1 0 0 0 0; %! 1 1 0 0 0; %! 1 1 1 0 0]; %! [aa, bb, q, z] = qzhess (a, b); %! assert (inv (q) - q', zeros (5), 2e-8); %! assert (inv (z) - z', zeros (5), 2e-8); %! assert (q * a * z, aa, 2e-8); %! assert (aa .* mask, zeros (5), 2e-8); %! assert (q * b * z, bb, 2e-8); %! assert (bb .* mask, zeros (5), 2e-8); ## Test input validation %!error <Invalid call> qzhess () %!error <Invalid call> qzhess (1)