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1 ## Copyright (C) 2006, 2007 David Bateman and Marco Caliari |
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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 (at |
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8 ## 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 GNU |
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13 ## 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 ## <http://www.gnu.org/licenses/>. |
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18 |
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19 ## -*- texinfo -*- |
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20 ## @deftypefn {Function File} {[@var{n}, @var{c}] =} normest (@var{a}, @var{tol}) |
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21 ## Estimate the 2-norm of the matrix @var{a} using a power series |
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22 ## analysis. This is typically used for large matrices, where the cost |
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23 ## of calculating the @code{norm (@var{a})} is prohibitive and an approximation |
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24 ## to the 2-norm is acceptable. |
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25 ## |
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26 ## @var{tol} is the tolerance to which the 2-norm is calculated. By default |
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27 ## @var{tol} is 1e-6. @var{c} returns the number of iterations needed for |
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28 ## @code{normest} to converge. |
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29 ## @end deftypefn |
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30 |
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31 function [e1, c] = normest (A, tol) |
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32 if (nargin < 2) |
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33 tol = 1e-6; |
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34 endif |
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35 if (tol < eps) |
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36 tol = eps |
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37 endif |
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38 if (ndims(A) != 2) |
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39 error ("normest: A must be a matrix"); |
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40 endif |
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41 maxA = max (max (abs (A))); |
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42 c = 0; |
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43 if (maxA == 0) |
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44 e1 = 0 |
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45 else |
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46 [m, n] = size (A); |
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47 B = A / maxA; |
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48 Bt = B'; |
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49 if (m > n) |
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50 tmp = B; |
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51 B = Bt; |
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52 Bt = tmp; |
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53 endif |
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54 e0 = 0; |
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55 x = randn (min (m, n), 1); |
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56 e1 = norm (x); |
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57 x = x / e1; |
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58 e1 = sqrt (e1); |
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59 if (issparse (A)) |
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60 while (abs (e1 - e0) > tol * e1) |
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61 e0 = e1; |
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62 x = B * (Bt * x); |
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63 e1 = norm (x); |
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64 x = x / e1; |
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65 e1 = sqrt (e1); |
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66 c = c + 1; |
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67 endwhile |
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68 else |
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69 B = B * Bt; |
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70 while (abs (e1 - e0) > tol * e1) |
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71 e0 = e1; |
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72 x = B * x; |
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73 e1 = norm (x); |
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74 x = x / e1; |
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75 e1 = sqrt (e1); |
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76 c = c + 1; |
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77 endwhile |
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78 endif |
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79 e1 = e1 * maxA; |
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80 endif |
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81 endfunction |
