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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 ## @seealso{cond, svd} |
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56 ## @end deftypefn |
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57 |
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58 ## Author: jwe |
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59 |
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60 function retval = norm (x, p) |
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61 |
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62 if (nargin < 1 || nargin > 2) |
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63 print_usage (); |
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64 endif |
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65 |
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66 if (isempty (x)) |
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67 retval = []; |
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68 return; |
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69 endif |
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70 |
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71 if (ndims (x) > 2) |
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72 error ("norm: only valid on 2-D objects") |
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73 endif |
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74 |
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75 if (nargin == 1) |
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76 p = 2; |
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77 endif |
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78 |
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79 ## Do we have a vector or matrix as the first argument? |
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80 |
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81 if (is_vector (x)) |
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82 if (isinteger (x) || issparse (x)) |
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83 if (ischar (p)) |
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84 if (strcmp (p, "fro")) |
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85 retval = sqrt (sum (abs (x) .^ 2)); |
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86 elseif (strcmp (p, "inf")) |
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87 retval = max (abs (x)); |
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88 else |
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89 error ("norm: unrecognized norm"); |
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90 endif |
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91 else |
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92 if (p == Inf) |
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93 retval = max (abs (x)); |
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94 elseif (p == -Inf) |
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95 retval = min (abs (x)); |
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96 else |
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97 retval = sum (abs (x) .^ p) ^ (1/p); |
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98 endif |
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99 endif |
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100 else |
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101 retval = __vnorm__ (x, p); |
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102 endif |
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103 else |
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104 if (ischar (p)) |
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105 if (strcmp (p, "fro")) |
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106 retval = sqrt (sum (sum (abs (x) .^ 2))); |
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107 elseif (strcmp (p, "inf")) |
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108 retval = max (sum (abs (x'))); |
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109 else |
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110 error ("norm: unrecognized vector norm"); |
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111 endif |
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112 else |
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113 if (p == 1) |
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114 retval = max (sum (abs (x))); |
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115 elseif (p == 2) |
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116 s = svd (x); |
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117 retval = s (1); |
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118 elseif (p == Inf) |
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119 retval = max (sum (abs (x'))); |
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120 else |
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121 error ("norm: unrecognized matrix norm"); |
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122 endif |
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123 endif |
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124 endif |
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125 |
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126 endfunction |
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127 |
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128 %!shared x |
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129 %! x = [1, -3, 4, 5, -7]; |
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130 %!assert(norm(x,1), 20); |
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131 %!assert(norm(x,2), 10); |
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132 %!assert(norm(x,3), 8.24257059961711, -4*eps); |
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133 %!assert(norm(x,Inf), 7); |
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134 %!assert(norm(x,-Inf), 1); |
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135 %!assert(norm(x,"inf"), 7); |
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136 %!assert(norm(x,"fro"), 10); |
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137 %!assert(norm(x), 10); |
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138 %!assert(norm([1e200, 1]), 1e200); |
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139 %!shared m |
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140 %! m = magic (4); |
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141 %!assert(norm(m,1), 34); |
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142 %!assert(norm(m,2), 34); |
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143 %!assert(norm(m,Inf), 34); |
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144 %!assert(norm(m,"inf"), 34); |
