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doc: Update more docstrings to have one sentence summary as first line. Reviewed specfun, special-matrix, testfun, and time script directories. * scripts/specfun/expint.m, scripts/specfun/isprime.m, scripts/specfun/legendre.m, scripts/specfun/primes.m, scripts/specfun/reallog.m, scripts/specfun/realsqrt.m, scripts/special-matrix/gallery.m, scripts/special-matrix/hadamard.m, scripts/special-matrix/hankel.m, scripts/special-matrix/hilb.m, scripts/special-matrix/invhilb.m, scripts/special-matrix/magic.m, scripts/special-matrix/pascal.m, scripts/special-matrix/rosser.m, scripts/special-matrix/toeplitz.m, scripts/special-matrix/vander.m, scripts/special-matrix/wilkinson.m, scripts/testfun/assert.m, scripts/testfun/demo.m, scripts/testfun/example.m, scripts/testfun/fail.m, scripts/testfun/rundemos.m, scripts/testfun/runtests.m, scripts/testfun/speed.m, scripts/time/asctime.m, scripts/time/calendar.m, scripts/time/clock.m, scripts/time/ctime.m, scripts/time/datenum.m, scripts/time/datestr.m, scripts/time/datevec.m, scripts/time/etime.m, scripts/time/is_leap_year.m, scripts/time/now.m, scripts/time/weekday.m: Update more docstrings to have one sentence summary as first line.
author Rik <rik@octave.org>
date Sun, 03 May 2015 17:00:11 -0700
parents 9fc020886ae9
children
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1 ## Copyright (C) 2000-2015 Kai Habel
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2 ## Copyright (C) 2008 Marco Caliari
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3 ##
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4 ## This file is part of Octave.
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5 ##
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6 ## Octave is free software; you can redistribute it and/or modify it
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7 ## under the terms of the GNU General Public License as published by
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8 ## the Free Software Foundation; either version 3 of the License, or (at
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9 ## your option) any later version.
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10 ##
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11 ## Octave is distributed in the hope that it will be useful, but
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12 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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14 ## General Public License for more details.
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15 ##
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16 ## You should have received a copy of the GNU General Public License
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17 ## along with Octave; see the file COPYING. If not, see
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18 ## <http://www.gnu.org/licenses/>.
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19
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20 ## -*- texinfo -*-
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21 ## @deftypefn {Function File} {@var{l} =} legendre (@var{n}, @var{x})
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22 ## @deftypefnx {Function File} {@var{l} =} legendre (@var{n}, @var{x}, @var{normalization})
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23 ## Compute the Legendre function of degree @var{n} and order
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24 ## @var{m} = 0 @dots{} @var{n}.
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25 ##
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26 ## The value @var{n} must be a real non-negative integer.
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27 ##
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28 ## @var{x} is a vector with real-valued elements in the range [-1, 1].
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29 ##
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30 ## The optional argument @var{normalization} may be one of @qcode{"unnorm"},
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31 ## @qcode{"sch"}, or @qcode{"norm"}. The default if no normalization is given
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32 ## is @qcode{"unnorm"}.
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33 ##
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34 ## When the optional argument @var{normalization} is @qcode{"unnorm"}, compute
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35 ## the Legendre function of degree @var{n} and order @var{m} and return all
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36 ## values for @var{m} = 0 @dots{} @var{n}. The return value has one dimension
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37 ## more than @var{x}.
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38 ##
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39 ## The Legendre Function of degree @var{n} and order @var{m}:
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40 ##
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41 ## @tex
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42 ## $$
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43 ## P^m_n(x) = (-1)^m (1-x^2)^{m/2}{d^m\over {dx^m}}P_n (x)
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44 ## $$
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45 ## @end tex
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46 ## @ifnottex
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47 ##
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48 ## @example
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49 ## @group
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50 ## m m 2 m/2 d^m
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51 ## P(x) = (-1) * (1-x ) * ---- P(x)
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52 ## n dx^m n
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53 ## @end group
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54 ## @end example
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55 ##
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56 ## @end ifnottex
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57 ##
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58 ## @noindent
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59 ## with Legendre polynomial of degree @var{n}:
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60 ##
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61 ## @tex
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62 ## $$
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63 ## P(x) = {1\over{2^n n!}}\biggl({d^n\over{dx^n}}(x^2 - 1)^n\biggr)
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64 ## $$
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65 ## @end tex
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66 ## @ifnottex
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67 ##
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68 ## @example
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69 ## @group
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70 ## 1 d^n 2 n
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71 ## P(x) = ------ [----(x - 1) ]
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72 ## n 2^n n! dx^n
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73 ## @end group
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74 ## @end example
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75 ##
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76 ## @end ifnottex
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77 ##
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78 ## @noindent
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79 ## @code{legendre (3, [-1.0, -0.9, -0.8])} returns the matrix:
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80 ##
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81 ## @example
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82 ## @group
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83 ## x | -1.0 | -0.9 | -0.8
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84 ## ------------------------------------
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85 ## m=0 | -1.00000 | -0.47250 | -0.08000
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86 ## m=1 | 0.00000 | -1.99420 | -1.98000
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87 ## m=2 | 0.00000 | -2.56500 | -4.32000
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88 ## m=3 | 0.00000 | -1.24229 | -3.24000
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89 ## @end group
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90 ## @end example
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91 ##
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92 ## When the optional argument @code{normalization} is @qcode{"sch"}, compute
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93 ## the Schmidt semi-normalized associated Legendre function. The Schmidt
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94 ## semi-normalized associated Legendre function is related to the unnormalized
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95 ## Legendre functions by the following:
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96 ##
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97 ## For Legendre functions of degree @var{n} and order 0:
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98 ##
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99 ## @tex
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100 ## $$
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101 ## SP^0_n (x) = P^0_n (x)
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102 ## $$
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103 ## @end tex
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104 ## @ifnottex
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105 ##
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106 ## @example
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107 ## @group
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108 ## 0 0
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109 ## SP(x) = P(x)
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110 ## n n
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111 ## @end group
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112 ## @end example
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113 ##
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114 ## @end ifnottex
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115 ##
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116 ## For Legendre functions of degree n and order m:
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117 ##
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118 ## @tex
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119 ## $$
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120 ## SP^m_n (x) = P^m_n (x)(-1)^m\biggl({2(n-m)!\over{(n+m)!}}\biggl)^{0.5}
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121 ## $$
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122 ## @end tex
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123 ## @ifnottex
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124 ##
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125 ## @example
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126 ## @group
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127 ## m m m 2(n-m)! 0.5
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128 ## SP(x) = P(x) * (-1) * [-------]
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129 ## n n (n+m)!
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130 ## @end group
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131 ## @end example
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132 ##
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133 ## @end ifnottex
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134 ##
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135 ## When the optional argument @var{normalization} is @qcode{"norm"}, compute
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136 ## the fully normalized associated Legendre function. The fully normalized
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137 ## associated Legendre function is related to the unnormalized Legendre
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138 ## functions by the following:
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139 ##
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140 ## For Legendre functions of degree @var{n} and order @var{m}
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141 ##
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142 ## @tex
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143 ## $$
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144 ## NP^m_n (x) = P^m_n (x)(-1)^m\biggl({(n+0.5)(n-m)!\over{(n+m)!}}\biggl)^{0.5}
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145 ## $$
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146 ## @end tex
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147 ## @ifnottex
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148 ##
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149 ## @example
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150 ## @group
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151 ## m m m (n+0.5)(n-m)! 0.5
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152 ## NP(x) = P(x) * (-1) * [-------------]
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153 ## n n (n+m)!
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154 ## @end group
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155 ## @end example
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156 ##
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157 ## @end ifnottex
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158 ## @end deftypefn
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159
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160 ## Author: Marco Caliari <marco.caliari@univr.it>
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161
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162 function retval = legendre (n, x, normalization)
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163
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164 persistent warned_overflow = false;
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165
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166 if (nargin < 2 || nargin > 3)
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167 print_usage ();
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168 endif
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169
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170 if (! isreal (n) || ! isscalar (n) || n < 0 || n != fix (n))
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171 error ("legendre: N must be a real non-negative integer");
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172 elseif (! isreal (x) || any (x(:) < -1 | x(:) > 1))
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173 error ("legendre: X must be real-valued vector in the range -1 <= X <= 1");
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174 endif
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175
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176 if (nargin == 3)
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177 normalization = lower (normalization);
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178 else
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179 normalization = "unnorm";
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180 endif
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181
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182 unnorm = false;
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183 switch (normalization)
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184 case "unnorm"
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185 scale = 1;
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186 unnorm = true;
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187 case "norm"
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188 scale = sqrt (n+0.5);
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189 case "sch"
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190 scale = sqrt (2);
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191 otherwise
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192 error ('legendre: NORMALIZATION option must be "unnorm", "norm", or "sch"');
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193 endswitch
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194
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195 scale = scale * ones (size (x));
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196
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197 ## Based on the recurrence relation below
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198 ## m m m
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199 ## (n-m+1) * P (x) = (2*n+1)*x*P (x) - (n+1)*P (x)
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200 ## n+1 n n-1
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201 ## http://en.wikipedia.org/wiki/Associated_Legendre_function
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202
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203 overflow = false;
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204 retval = zeros ([n+1, size(x)]);
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205 for m = 1:n
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206 lpm1 = scale;
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207 lpm2 = (2*m-1) .* x .* scale;
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208 lpm3 = lpm2;
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209 for k = m+1:n
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210 lpm3a = (2*k-1) .* x .* lpm2;
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211 lpm3b = (k+m-2) .* lpm1;
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212 lpm3 = (lpm3a - lpm3b) / (k-m+1);
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213 lpm1 = lpm2;
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214 lpm2 = lpm3;
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215 if (! warned_overflow)
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216 if ( any (abs (lpm3a) > realmax)
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217 || any (abs (lpm3b) > realmax)
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218 || any (abs (lpm3) > realmax))
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219 overflow = true;
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220 endif
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221 endif
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222 endfor
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223 retval(m,:) = lpm3(:);
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224 if (unnorm)
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225 scale *= -(2*m-1);
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226 else # normalization = "sch" or "norm"
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227 scale *= (2*m-1) / sqrt ((n-m+1)*(n+m));
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228 endif
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229 scale .*= sqrt (1-x.^2);
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230 endfor
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231
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232 retval(n+1,:) = scale(:);
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233
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234 if (isvector (x))
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235 ## vector case is special
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236 retval = reshape (retval, n + 1, length (x));
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237 endif
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238
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239 if (strcmp (normalization, "sch"))
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240 retval(1,:) ./= sqrt (2);
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241 endif
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242
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243 if (overflow && ! warned_overflow)
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244 warning ("legendre: overflow - results may be unstable for high orders");
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245 warned_overflow = true;
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246 endif
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247
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248 endfunction
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249
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250
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251 %!test
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252 %! result = legendre (3, [-1.0 -0.9 -0.8]);
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253 %! expected = [
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254 %! -1.00000 -0.47250 -0.08000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
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255 %! 0.00000 -1.99420 -1.98000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
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256 %! 0.00000 -2.56500 -4.32000
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257 %! 0.00000 -1.24229 -3.24000
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258 %! ];
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259 %! assert (result, expected, 1e-5);
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260
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261 %!test
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262 %! result = legendre (3, [-1.0 -0.9 -0.8], "sch");
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263 %! expected = [
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264 %! -1.00000 -0.47250 -0.08000
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265 %! 0.00000 0.81413 0.80833
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266 %! -0.00000 -0.33114 -0.55771
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267 %! 0.00000 0.06547 0.17076
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268 %! ];
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269 %! assert (result, expected, 1e-5);
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Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
270
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
271 %!test
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
272 %! result = legendre (3, [-1.0 -0.9 -0.8], "norm");
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
273 %! expected = [
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
274 %! -1.87083 -0.88397 -0.14967
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
275 %! 0.00000 1.07699 1.06932
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
276 %! -0.00000 -0.43806 -0.73778
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
277 %! 0.00000 0.08661 0.22590
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
278 %! ];
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
279 %! assert (result, expected, 1e-5);
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
280
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
281 %!test
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
282 %! result = legendre (151, 0);
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
283 %! ## Don't compare to "-Inf" since it would fail on 64 bit systems.
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
284 %! assert (result(end) < -1.7976e308 && all (isfinite (result(1:end-1))));
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
285
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
286 %!test
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
287 %! result = legendre (150, 0);
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
288 %! ## This agrees with Matlab's result.
12893
72ffa81a68d4 legendre.m: Allow ND-array inputs (Bug #33526).
Marco Caliari <marco.caliari@univr.it>
parents: 11587
diff changeset
289 %! assert (result(end), 3.7532741115719e+306, 0.0000000000001e+306);
7507
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
290
8255
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
291 %!test
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
292 %! result = legendre (0, 0:0.1:1);
14363
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
Rik <octave@nomad.inbox5.com>
parents: 14327
diff changeset
293 %! assert (result, full (ones (1,11)));
12893
72ffa81a68d4 legendre.m: Allow ND-array inputs (Bug #33526).
Marco Caliari <marco.caliari@univr.it>
parents: 11587
diff changeset
294
72ffa81a68d4 legendre.m: Allow ND-array inputs (Bug #33526).
Marco Caliari <marco.caliari@univr.it>
parents: 11587
diff changeset
295 %!test
19099
55b613e5183d legendre.m: Overhaul function.
Rik <rik@octave.org>
parents: 17744
diff changeset
296 %! ## Test matrix input
12893
72ffa81a68d4 legendre.m: Allow ND-array inputs (Bug #33526).
Marco Caliari <marco.caliari@univr.it>
parents: 11587
diff changeset
297 %! result = legendre (3, [-1,0,1;1,0,-1]);
72ffa81a68d4 legendre.m: Allow ND-array inputs (Bug #33526).
Marco Caliari <marco.caliari@univr.it>
parents: 11587
diff changeset
298 %! expected(:,:,1) = [-1,1;0,0;0,0;0,0];
72ffa81a68d4 legendre.m: Allow ND-array inputs (Bug #33526).
Marco Caliari <marco.caliari@univr.it>
parents: 11587
diff changeset
299 %! expected(:,:,2) = [0,0;1.5,1.5;0,0;-15,-15];
72ffa81a68d4 legendre.m: Allow ND-array inputs (Bug #33526).
Marco Caliari <marco.caliari@univr.it>
parents: 11587
diff changeset
300 %! expected(:,:,3) = [1,-1;0,0;0,0;0,0];
72ffa81a68d4 legendre.m: Allow ND-array inputs (Bug #33526).
Marco Caliari <marco.caliari@univr.it>
parents: 11587
diff changeset
301 %! assert (result, expected);
72ffa81a68d4 legendre.m: Allow ND-array inputs (Bug #33526).
Marco Caliari <marco.caliari@univr.it>
parents: 11587
diff changeset
302
72ffa81a68d4 legendre.m: Allow ND-array inputs (Bug #33526).
Marco Caliari <marco.caliari@univr.it>
parents: 11587
diff changeset
303 %!test
72ffa81a68d4 legendre.m: Allow ND-array inputs (Bug #33526).
Marco Caliari <marco.caliari@univr.it>
parents: 11587
diff changeset
304 %! result = legendre (3, [-1,0,1;1,0,-1]');
72ffa81a68d4 legendre.m: Allow ND-array inputs (Bug #33526).
Marco Caliari <marco.caliari@univr.it>
parents: 11587
diff changeset
305 %! expected(:,:,1) = [-1,0,1;0,1.5,0;0,0,0;0,-15,0];
72ffa81a68d4 legendre.m: Allow ND-array inputs (Bug #33526).
Marco Caliari <marco.caliari@univr.it>
parents: 11587
diff changeset
306 %! expected(:,:,2) = [1,0,-1;0,1.5,0;0,0,0;0,-15,0];
72ffa81a68d4 legendre.m: Allow ND-array inputs (Bug #33526).
Marco Caliari <marco.caliari@univr.it>
parents: 11587
diff changeset
307 %! assert (result, expected);
10672
1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
308
19833
9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
Rik <rik@octave.org>
parents: 19697
diff changeset
309 ## Test input validation
14363
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
Rik <octave@nomad.inbox5.com>
parents: 14327
diff changeset
310 %!error legendre ()
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
Rik <octave@nomad.inbox5.com>
parents: 14327
diff changeset
311 %!error legendre (1)
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
Rik <octave@nomad.inbox5.com>
parents: 14327
diff changeset
312 %!error legendre (1,2,3,4)
19099
55b613e5183d legendre.m: Overhaul function.
Rik <rik@octave.org>
parents: 17744
diff changeset
313 %!error <must be a real non-negative integer> legendre (i, [-1, 0, 1])
55b613e5183d legendre.m: Overhaul function.
Rik <rik@octave.org>
parents: 17744
diff changeset
314 %!error <must be a real non-negative integer> legendre ([1, 2], [-1, 0, 1])
55b613e5183d legendre.m: Overhaul function.
Rik <rik@octave.org>
parents: 17744
diff changeset
315 %!error <must be a real non-negative integer> legendre (-1, [-1, 0, 1])
55b613e5183d legendre.m: Overhaul function.
Rik <rik@octave.org>
parents: 17744
diff changeset
316 %!error <must be a real non-negative integer> legendre (1.1, [-1, 0, 1])
55b613e5183d legendre.m: Overhaul function.
Rik <rik@octave.org>
parents: 17744
diff changeset
317 %!error <must be real-valued vector> legendre (1, [-1+i, 0, 1])
55b613e5183d legendre.m: Overhaul function.
Rik <rik@octave.org>
parents: 17744
diff changeset
318 %!error <in the range -1 .= X .= 1> legendre (1, [-2, 0, 1])
55b613e5183d legendre.m: Overhaul function.
Rik <rik@octave.org>
parents: 17744
diff changeset
319 %!error <in the range -1 .= X .= 1> legendre (1, [-1, 0, 2])
55b613e5183d legendre.m: Overhaul function.
Rik <rik@octave.org>
parents: 17744
diff changeset
320 %!error <NORMALIZATION option must be> legendre (1, [-1, 0, 1], "badnorm")
14363
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
Rik <octave@nomad.inbox5.com>
parents: 14327
diff changeset
321