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1 ## Copyright (C) 1993, 1995, 1996, 1997, 1999, 2000, 2005, 2006, 2007 |
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2 ## John W. Eaton |
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3 ## |
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4 ## This file is part of Octave. |
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5 ## |
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6 ## Octave is free software; you can redistribute it and/or modify it |
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7 ## under the terms of the GNU General Public License as published by |
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8 ## the Free Software Foundation; either version 3 of the License, or (at |
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9 ## your option) any later version. |
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10 ## |
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11 ## Octave is distributed in the hope that it will be useful, but |
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12 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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14 ## General Public License for more details. |
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15 ## |
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16 ## You should have received a copy of the GNU General Public License |
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17 ## along with Octave; see the file COPYING. If not, see |
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18 ## <http://www.gnu.org/licenses/>. |
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19 |
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20 ## -*- texinfo -*- |
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21 ## @deftypefn {Function File} {[@var{aa}, @var{bb}, @var{q}, @var{z}] =} qzhess (@var{a}, @var{b}) |
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22 ## Compute the Hessenberg-triangular decomposition of the matrix pencil |
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23 ## @code{(@var{a}, @var{b})}, returning |
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24 ## @code{@var{aa} = @var{q} * @var{a} * @var{z}}, |
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25 ## @code{@var{bb} = @var{q} * @var{b} * @var{z}}, with @var{q} and @var{z} |
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26 ## orthogonal. For example, |
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27 ## |
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28 ## @example |
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29 ## @group |
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30 ## [aa, bb, q, z] = qzhess ([1, 2; 3, 4], [5, 6; 7, 8]) |
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31 ## @result{} aa = [ -3.02244, -4.41741; 0.92998, 0.69749 ] |
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32 ## @result{} bb = [ -8.60233, -9.99730; 0.00000, -0.23250 ] |
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33 ## @result{} q = [ -0.58124, -0.81373; -0.81373, 0.58124 ] |
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34 ## @result{} z = [ 1, 0; 0, 1 ] |
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35 ## @end group |
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36 ## @end example |
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37 ## |
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38 ## The Hessenberg-triangular decomposition is the first step in |
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39 ## Moler and Stewart's QZ decomposition algorithm. |
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40 ## |
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41 ## Algorithm taken from Golub and Van Loan, @cite{Matrix Computations, 2nd |
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42 ## edition}. |
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43 ## @end deftypefn |
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44 |
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45 ## Author: A. S. Hodel <scotte@eng.auburn.edu> |
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46 ## Created: August 1993 |
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47 ## Adapted-By: jwe |
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48 |
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49 function [aa, bb, q, z] = qzhess (a, b) |
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50 |
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51 if (nargin != 2) |
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52 print_usage (); |
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53 endif |
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54 |
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55 [na, ma] = size (a); |
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56 [nb, mb] = size (b); |
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57 if (na != ma || na != nb || nb != mb) |
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58 error ("qzhess: incompatible dimensions"); |
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59 endif |
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60 |
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61 ## Reduce to hessenberg-triangular form. |
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62 |
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63 [q, bb] = qr (b); |
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64 aa = q' * a; |
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65 q = q'; |
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66 z = eye (na); |
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67 for j = 1:(na-2) |
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68 for i = na:-1:(j+2) |
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69 |
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70 ## disp (["zero out aa(", num2str(i), ",", num2str(j), ")"]) |
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71 |
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72 rot = givens (aa (i-1, j), aa (i, j)); |
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73 aa ((i-1):i, :) = rot *aa ((i-1):i, :); |
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74 bb ((i-1):i, :) = rot *bb ((i-1):i, :); |
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75 q ((i-1):i, :) = rot *q ((i-1):i, :); |
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76 |
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77 ## disp (["now zero out bb(", num2str(i), ",", num2str(i-1), ")"]) |
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78 |
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79 rot = givens (bb (i, i), bb (i, i-1))'; |
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80 bb (:, (i-1):i) = bb (:, (i-1):i) * rot'; |
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81 aa (:, (i-1):i) = aa (:, (i-1):i) * rot'; |
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82 z (:, (i-1):i) = z (:, (i-1):i) * rot'; |
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83 |
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84 endfor |
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85 endfor |
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86 |
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87 bb (2, 1) = 0.0; |
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88 for i = 3:na |
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89 bb (i, 1:(i-1)) = zeros (1, i-1); |
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90 aa (i, 1:(i-2)) = zeros (1, i-2); |
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91 endfor |
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92 |
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93 endfunction |
