annotate scripts/linear-algebra/cond.m @ 27898:4d6d21839dfd

remove Author: jwe and Adapted-by: jwe lines from source files These lines have little meaning now in the age of modern version control systems. Files affected: url-transfer.cc, url-transfer.h, acot.m, acoth.m, acsc.m, acsch.m, asec.m, asech.m, cot.m, coth.m, csc.m, csch.m, sec.m, sech.m, cart2pol.m, cart2sph.m, deal.m, fliplr.m, flipud.m, int2str.m, logspace.m, num2str.m, pol2cart.m, rot90.m, sortrows.m, sph2cart.m, waitbar.m, doc.m, hsv2rgb.m, rgb2hsv.m, beep.m, isstr.m, setstr.m, strmatch.m, cond.m, rank.m, trace.m, bug_report.m, cast.m, delete.m, dir.m, list_primes.m, ls.m, ls_command.m, menu.m, pack.m, recycle.m, substruct.m, version.m, glpk.m, axis.m, box.m, grid.m, __axis_label__.m, text.m, title.m, view.m, xlabel.m, ylabel.m, zlabel.m, bar.m, barh.m, comet3.m, hist.m, line.m, loglog.m, mesh.m, patch.m, plot.m, polar.m, __bar__.m, __line__.m, __plt__.m, __stem__.m, semilogx.m, semilogy.m, sombrero.m, stairs.m, stem.m, surface.m, __default_plot_options__.m, __gnuplot_drawnow__.m, __next_line_color__.m, __plt_get_axis_arg__.m, __pltopt__.m, clf.m, closereq.m, gca.m, gnuplot_binary.in.m, isfigure.m, meshgrid.m, __gnuplot_draw_axes__.m, __gnuplot_draw_figure__.m, shg.m, addpref.m, getpref.m, ispref.m, loadprefs.m, prefsfile.m, saveprefs.m, rmpref.m, setpref.m, ismember.m, setdiff.m, union.m, hankel.m, hilb.m, vander.m, median.m, std.m, cstrcat.m, isletter.m, str2num.m, strcat.m, oruntests.m, rundemos.m, asctime.m, clock.m, ctime.m, date.m, etime.m, is_leap_year.m.
author John W. Eaton <jwe@octave.org>
date Thu, 02 Jan 2020 15:43:01 -0500
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1 ## Copyright (C) 1993-2019 John W. Eaton
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2 ##
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3 ## This file is part of Octave.
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4 ##
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5 ## Octave is free software: you can redistribute it and/or modify it
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6 ## under the terms of the GNU General Public License as published by
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7 ## the Free Software Foundation, either version 3 of the License, or
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8 ## (at your option) any later version.
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9 ##
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10 ## Octave is distributed in the hope that it will be useful, but
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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13 ## GNU General Public License for more details.
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14 ##
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15 ## You should have received a copy of the GNU General Public License
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16 ## along with Octave; see the file COPYING. If not, see
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17 ## <https://www.gnu.org/licenses/>.
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19 ## -*- texinfo -*-
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20 ## @deftypefn {} {} cond (@var{A})
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21 ## @deftypefnx {} {} cond (@var{A}, @var{p})
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22 ## Compute the @var{p}-norm condition number of a matrix with respect to
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23 ## inversion.
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24 ##
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25 ## @code{cond (@var{A})} is defined as
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26 ## @tex
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27 ## $ {\parallel A \parallel_p * \parallel A^{-1} \parallel_p .} $
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28 ## @end tex
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29 ## @ifnottex
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30 ## @code{norm (@var{A}, @var{p}) * norm (inv (@var{A}), @var{p})}.
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31 ## @end ifnottex
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32 ##
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33 ## By default, @code{@var{p} = 2} is used which implies a (relatively slow)
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34 ## singular value decomposition. Other possible selections are
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35 ## @code{@var{p} = 1, Inf, "fro"} which are generally faster. See @code{norm}
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36 ## for a full discussion of possible @var{p} values.
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37 ##
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38 ## The condition number of a matrix quantifies the sensitivity of the matrix
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39 ## inversion operation when small changes are made to matrix elements. Ideally
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40 ## the condition number will be close to 1. When the number is large this
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41 ## indicates small changes (such as underflow or round-off error) will produce
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42 ## large changes in the resulting output. In such cases the solution results
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43 ## from numerical computing are not likely to be accurate.
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44 ## @seealso{condest, rcond, condeig, norm, svd}
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45 ## @end deftypefn
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47 function retval = cond (A, p = 2)
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48
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49 if (nargin < 1 || nargin > 2)
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50 print_usage ();
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51 endif
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52
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53 if (ndims (A) > 2)
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54 error ("cond: A must be a 2-D matrix");
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55 endif
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56
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57 if (p == 2)
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58 if (isempty (A))
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59 retval = 0.0;
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60 elseif (any (! isfinite (A(:))))
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61 error ("cond: A must not contain Inf or NaN values");
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62 else
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63 sigma = svd (A);
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64 sigma_1 = sigma(1);
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65 sigma_n = sigma(end);
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66 if (sigma_1 == 0 || sigma_n == 0)
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67 retval = Inf;
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68 else
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69 retval = sigma_1 / sigma_n;
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70 endif
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71 endif
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72 else
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73 retval = norm (A, p) * norm (inv (A), p);
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74 endif
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75
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76 endfunction
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78
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79 %!test
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80 %! y = [7, 2, 3; 1, 3, 4; 6, 4, 5];
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81 %! tol = 1e-6;
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82 %! type = {1, 2, "fro", "inf", inf};
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83 %! for n = 1:numel (type)
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84 %! rcondition(n) = 1 / cond (y, type{n});
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85 %! endfor
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86 %! assert (rcondition, [0.017460, 0.019597, 0.018714, 0.012022, 0.012022], tol);
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87
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88 %!assert (cond ([1, 2; 2, 1]), 3, sqrt (eps))
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89 %!assert (cond ([1, 2, 3; 4, 5, 6; 7, 8, 9]) > 1.0e+16)
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91 %!error cond ()
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92 %!error cond (1, 2, 3)
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93 %!error <A must be a 2-D matrix> cond (ones (1,3,3))
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94 %!error <A must not contain Inf or NaN value> cond ([1, 2;Inf 4])
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95 %!error <A must not contain Inf or NaN value> cond ([1, 2;NaN 4])