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annotate scripts/linear-algebra/cond.m @ 27898:4d6d21839dfd
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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> |
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date | Thu, 02 Jan 2020 15:43:01 -0500 |
parents | 00f796120a6d |
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1 ## Copyright (C) 1993-2019 John W. Eaton |
2313 | 2 ## |
3 ## This file is part of Octave. | |
4 ## | |
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5 ## Octave is free software: you can redistribute it and/or modify it |
2313 | 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. |
2313 | 9 ## |
10 ## Octave is distributed in the hope that it will be useful, but | |
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. |
2313 | 14 ## |
15 ## You should have received a copy of the GNU General Public License | |
7016 | 16 ## along with Octave; see the file COPYING. If not, see |
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17 ## <https://www.gnu.org/licenses/>. |
245 | 18 |
3372 | 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} |
3372 | 45 ## @end deftypefn |
4 | 46 |
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47 function retval = cond (A, p = 2) |
4 | 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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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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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; |
7484 | 68 else |
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69 retval = sigma_1 / sigma_n; |
7484 | 70 endif |
3250 | 71 endif |
4 | 72 else |
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73 retval = norm (A, p) * norm (inv (A), p); |
4 | 74 endif |
75 | |
76 endfunction | |
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78 |
7484 | 79 %!test |
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80 %! y = [7, 2, 3; 1, 3, 4; 6, 4, 5]; |
7484 | 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) |
7484 | 84 %! rcondition(n) = 1 / cond (y, type{n}); |
85 %! endfor | |
86 %! assert (rcondition, [0.017460, 0.019597, 0.018714, 0.012022, 0.012022], tol); | |
7411 | 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]) |