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annotate scripts/specfun/legendre.m @ 9051:1bf0ce0930be

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Grammar check TexInfo in all .m files Cleanup documentation sources to follow a few consistent rules. Spellcheck was NOT done. (but will be in another changeset)
author Rik <rdrider0-list@yahoo.com>
date Fri, 27 Mar 2009 22:31:03 -0700
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1 ## Copyright (C) 2000, 2006, 2007, 2009 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{} N. The optional argument, @var{normalization},
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25 ## may be one of @code{"unnorm"}, @code{"sch"}, or @code{"norm"}.
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26 ## The default is @code{"unnorm"}. The value of @var{n} must be a
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27 ## non-negative scalar integer.
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28 ##
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29 ## If the optional argument @var{normalization} is missing or is
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30 ## @code{"unnorm"}, compute the Legendre function of degree @var{n} and
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31 ## order @var{m} and return all values for @var{m} = 0 @dots{} @var{n}.
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32 ## The return value has one dimension more than @var{x}.
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33 ##
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34 ## The Legendre Function of degree @var{n} and order @var{m}:
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35 ##
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36 ## @example
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37 ## @group
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38 ## m m 2 m/2 d^m
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39 ## P(x) = (-1) * (1-x ) * ---- P (x)
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40 ## n dx^m n
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41 ## @end group
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42 ## @end example
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43 ##
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44 ## @noindent
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45 ## with Legendre polynomial of degree @var{n}:
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46 ##
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47 ## @example
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48 ## @group
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49 ## 1 d^n 2 n
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50 ## P (x) = ------ [----(x - 1) ]
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51 ## n 2^n n! dx^n
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52 ## @end group
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53 ## @end example
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54 ##
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55 ## @noindent
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56 ## @code{legendre (3, [-1.0, -0.9, -0.8])} returns the matrix:
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57 ##
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58 ## @example
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59 ## @group
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60 ## x | -1.0 | -0.9 | -0.8
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61 ## ------------------------------------
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62 ## m=0 | -1.00000 | -0.47250 | -0.08000
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63 ## m=1 | 0.00000 | -1.99420 | -1.98000
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64 ## m=2 | 0.00000 | -2.56500 | -4.32000
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65 ## m=3 | 0.00000 | -1.24229 | -3.24000
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66 ## @end group
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67 ## @end example
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68 ##
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69 ## If the optional argument @code{normalization} is @code{"sch"},
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70 ## compute the Schmidt semi-normalized associated Legendre function.
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71 ## The Schmidt semi-normalized associated Legendre function is related
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72 ## to the unnormalized Legendre functions by the following:
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73 ##
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74 ## For Legendre functions of degree n and order 0:
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75 ##
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76 ## @example
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77 ## @group
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78 ## 0 0
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79 ## SP (x) = P (x)
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80 ## n n
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81 ## @end group
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82 ## @end example
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83 ##
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84 ## For Legendre functions of degree n and order m:
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85 ##
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86 ## @example
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87 ## @group
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88 ## m m m 2(n-m)! 0.5
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89 ## SP (x) = P (x) * (-1) * [-------]
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90 ## n n (n+m)!
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91 ## @end group
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92 ## @end example
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93 ##
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94 ## If the optional argument @var{normalization} is @code{"norm"},
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95 ## compute the fully normalized associated Legendre function.
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96 ## The fully normalized associated Legendre function is related
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97 ## to the unnormalized Legendre functions by the following:
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98 ##
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99 ## For Legendre functions of degree @var{n} and order @var{m}
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100 ##
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101 ## @example
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102 ## @group
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103 ## m m m (n+0.5)(n-m)! 0.5
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104 ## NP (x) = P (x) * (-1) * [-------------]
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105 ## n n (n+m)!
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106 ## @end group
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107 ## @end example
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108 ## @end deftypefn
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109
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110 ## Author: Marco Caliari <marco.caliari@univr.it>
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111
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112 function retval = legendre (n, x, normalization)
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113
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114 persistent warned_overflow = false;
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115
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116 if (nargin < 2 || nargin > 3)
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117 print_usage ();
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118 endif
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119
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120 if (nargin == 3)
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121 normalization = lower (normalization);
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122 else
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123 normalization = "unnorm";
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124 endif
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125
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126 if (! isscalar (n) || n < 0 || n != fix (n))
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127 error ("legendre: n must be a non-negative scalar integer");
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128 endif
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129
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130 if (! isvector (x) || any (x < -1 || x > 1))
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131 error ("legendre: x must be vector in range -1 <= x <= 1");
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132 endif
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133
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134 switch (normalization)
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135 case "norm"
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136 scale = sqrt (n+0.5);
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137 case "sch"
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138 scale = sqrt (2);
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139 case "unnorm"
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140 scale = 1;
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141 otherwise
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142 error ("legendre: expecting normalization option to be \"norm\", \"sch\", or \"unnorm\"");
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143 endswitch
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144
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145 scale = scale * ones (1, numel (x));
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146
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147 ## Based on the recurrence relation below
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148 ## m m m
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149 ## (n-m+1) * P (x) = (2*n+1)*x*P (x) - (n+1)*P (x)
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150 ## n+1 n n-1
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151 ## http://en.wikipedia.org/wiki/Associated_Legendre_function
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152
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153 overflow = false;
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154 for m = 1:n
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Ben Abbott <bpabbott@mac.com>
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155 lpm1 = scale;
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156 lpm2 = (2*m-1) .* x .* scale;
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157 lpm3 = lpm2;
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Ben Abbott <bpabbott@mac.com>
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158 for k = m+1:n
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159 lpm3a = (2*k-1) .* x .* lpm2;
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160 lpm3b = (k+m-2) .* lpm1;
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161 lpm3 = (lpm3a - lpm3b)/(k-m+1);
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162 lpm1 = lpm2;
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Ben Abbott <bpabbott@mac.com>
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163 lpm2 = lpm3;
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Ben Abbott <bpabbott@mac.com>
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164 if (! warned_overflow)
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Ben Abbott <bpabbott@mac.com>
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165 if (any (abs (lpm3a) > realmax)
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
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166 || any (abs (lpm3b) > realmax)
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
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167 || any (abs (lpm3) > realmax))
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
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168 overflow = true;
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
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169 endif
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
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170 endif
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171 endfor
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Ben Abbott <bpabbott@mac.com>
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diff changeset
172 retval(m,:) = lpm3;
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Ben Abbott <bpabbott@mac.com>
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173 if (strcmp (normalization, "unnorm"))
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Ben Abbott <bpabbott@mac.com>
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diff changeset
174 scale = -scale * (2*m-1);
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Ben Abbott <bpabbott@mac.com>
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175 else
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Ben Abbott <bpabbott@mac.com>
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176 ## normalization == "sch" or normalization == "norm"
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177 scale = scale / sqrt ((n-m+1)*(n+m))*(2*m-1);
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178 endif
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179 scale = scale .* sqrt(1-x.^2);
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180 endfor
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181
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182 retval(n+1,:) = scale;
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183
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184 if (strcmp (normalization, "sch"))
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185 retval(1,:) = retval(1,:) / sqrt (2);
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186 endif
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187
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Ben Abbott <bpabbott@mac.com>
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188 if (overflow && ! warned_overflow)
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John W. Eaton <jwe@octave.org>
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189 warning ("legendre: overflow - results may be unstable for high orders");
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190 warned_overflow = true;
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Ben Abbott <bpabbott@mac.com>
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191 endif
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Ben Abbott <bpabbott@mac.com>
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diff changeset
192
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193 endfunction
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194
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195 %!test
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196 %! result = legendre (3, [-1.0 -0.9 -0.8]);
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197 %! expected = [
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
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198 %! -1.00000 -0.47250 -0.08000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
jwe
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199 %! 0.00000 -1.99420 -1.98000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
jwe
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200 %! 0.00000 -2.56500 -4.32000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
jwe
parents:
diff changeset
201 %! 0.00000 -1.24229 -3.24000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
jwe
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202 %! ];
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203 %! assert (result, expected, 1e-5);
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Ben Abbott <bpabbott@mac.com>
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204
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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205 %!test
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206 %! result = legendre (3, [-1.0 -0.9 -0.8], "sch");
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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207 %! expected = [
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Ben Abbott <bpabbott@mac.com>
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208 %! -1.00000 -0.47250 -0.08000
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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209 %! 0.00000 0.81413 0.80833
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Ben Abbott <bpabbott@mac.com>
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210 %! -0.00000 -0.33114 -0.55771
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Ben Abbott <bpabbott@mac.com>
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211 %! 0.00000 0.06547 0.17076
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Ben Abbott <bpabbott@mac.com>
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212 %! ];
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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213 %! assert (result, expected, 1e-5);
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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diff changeset
214
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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diff changeset
215 %!test
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
216 %! result = legendre (3, [-1.0 -0.9 -0.8], "norm");
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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217 %! expected = [
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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218 %! -1.87083 -0.88397 -0.14967
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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219 %! 0.00000 1.07699 1.06932
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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220 %! -0.00000 -0.43806 -0.73778
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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diff changeset
221 %! 0.00000 0.08661 0.22590
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Ben Abbott <bpabbott@mac.com>
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222 %! ];
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Ben Abbott <bpabbott@mac.com>
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223 %! assert (result, expected, 1e-5);
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Ben Abbott <bpabbott@mac.com>
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diff changeset
224
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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225 %!test
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Ben Abbott <bpabbott@mac.com>
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diff changeset
226 %! result = legendre (151, 0);
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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227 %! ## Don't compare to "-Inf" since it would fail on 64 bit systems.
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Ben Abbott <bpabbott@mac.com>
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228 %! assert (result(end) < -1.7976e308 && all (isfinite (result(1:end-1))));
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Ben Abbott <bpabbott@mac.com>
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diff changeset
229
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
230 %!test
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
231 %! result = legendre (150, 0);
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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diff changeset
232 %! ## This agrees with Matlab's result.
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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diff changeset
233 %! assert (result(end), 3.7532741115719e+306, 0.0000000000001e+306)
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
234
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235 %!test
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
236 %! result = legendre (0, 0:0.1:1);
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Jaroslav Hajek <highegg@gmail.com>
parents: 8920
diff changeset
237 %! assert (result, full(ones(1,11)))