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1 ## Copyright (C) 1996, 1997 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 2, or (at your option) |
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8 ## 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, write to the Free |
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17 ## Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |
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18 ## 02110-1301, USA. |
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19 |
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20 ## -*- texinfo -*- |
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21 ## @deftypefn {Function File} {} norm (@var{a}, @var{p}) |
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22 ## Compute the p-norm of the matrix @var{a}. If the second argument is |
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23 ## missing, @code{p = 2} is assumed. |
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24 ## |
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25 ## If @var{a} is a matrix: |
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26 ## |
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27 ## @table @asis |
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28 ## @item @var{p} = @code{1} |
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29 ## 1-norm, the largest column sum of the absolute values of @var{a}. |
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30 ## |
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31 ## @item @var{p} = @code{2} |
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32 ## Largest singular value of @var{a}. |
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33 ## |
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34 ## @item @var{p} = @code{Inf} |
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35 ## @cindex infinity norm |
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36 ## Infinity norm, the largest row sum of the absolute values of @var{a}. |
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37 ## |
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38 ## @item @var{p} = @code{"fro"} |
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39 ## @cindex Frobenius norm |
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40 ## Frobenius norm of @var{a}, @code{sqrt (sum (diag (@var{a}' * @var{a})))}. |
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41 ## @end table |
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42 ## |
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43 ## If @var{a} is a vector or a scalar: |
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44 ## |
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45 ## @table @asis |
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46 ## @item @var{p} = @code{Inf} |
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47 ## @code{max (abs (@var{a}))}. |
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48 ## |
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49 ## @item @var{p} = @code{-Inf} |
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50 ## @code{min (abs (@var{a}))}. |
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51 ## |
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52 ## @item other |
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53 ## p-norm of @var{a}, @code{(sum (abs (@var{a}) .^ @var{p})) ^ (1/@var{p})}. |
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54 ## @end table |
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55 ## @end deftypefn |
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56 ## |
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57 ## @seealso{cond and svd} |
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58 |
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59 ## Author: jwe |
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60 |
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61 function retval = norm (x, p) |
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62 |
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63 if (nargin < 1 || nargin > 2) |
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64 usage ("norm (x, p)"); |
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65 endif |
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66 |
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67 if (isempty (x)) |
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68 retval = []; |
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69 return; |
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70 endif |
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71 |
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72 if (ndims (x) > 2) |
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73 error ("norm: Only valid on 2-D objects") |
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74 endif |
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75 |
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76 ## Do we have a vector or matrix as the first argument? |
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77 |
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78 if (rows (x) == 1 || columns (x) == 1) |
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79 |
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80 if (nargin == 2) |
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81 if (ischar (p)) |
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82 if (strcmp (p, "fro")) |
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83 retval = sqrt (sum (abs (x) .^ 2)); |
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84 elseif (strcmp (p, "inf")) |
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85 retval = max (abs (x)); |
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86 else |
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87 error ("norm: unrecognized norm"); |
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88 endif |
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89 else |
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90 if (p == Inf) |
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91 retval = max (abs (x)); |
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92 elseif (p == -Inf) |
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93 retval = min (abs (x)); |
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94 else |
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95 retval = sum (abs (x) .^ p) ^ (1/p); |
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96 endif |
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97 endif |
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98 elseif (nargin == 1) |
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99 retval = sqrt (sum (abs (x) .^ 2)); |
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100 endif |
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101 |
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102 else |
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103 |
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104 if (nargin == 2) |
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105 if (ischar (p)) |
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106 if (strcmp (p, "fro")) |
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107 retval = sqrt (sum (sum (abs (x) .^ 2))); |
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108 elseif (strcmp (p, "inf")) |
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109 retval = max (sum (abs (x'))); |
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110 else |
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111 error ("norm: unrecognized norm"); |
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112 endif |
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113 else |
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114 if (p == 1) |
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115 retval = max (sum (abs (x))); |
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116 elseif (p == 2) |
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117 s = svd (x); |
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118 retval = s (1); |
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119 elseif (p == Inf) |
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120 retval = max (sum (abs (x'))); |
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121 endif |
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122 endif |
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123 elseif (nargin == 1) |
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124 s = svd (x); |
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125 retval = s (1); |
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126 endif |
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127 |
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128 endif |
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129 |
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130 endfunction |
