Mercurial > octave-antonio
annotate scripts/specfun/legendre.m @ 7508:f501b22c0394
doc fixes for legendre
author | John W. Eaton <jwe@octave.org> |
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date | Wed, 20 Feb 2008 21:58:11 -0500 |
parents | bc6573d2fa40 |
children | f88b720bf254 |
rev | line source |
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7017 | 1 ## Copyright (C) 2000, 2006, 2007 Kai Habel |
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2 ## Copyright (C) 2008 Marco Caliari |
5827 | 3 ## |
4 ## This file is part of Octave. | |
5 ## | |
6 ## Octave is free software; you can redistribute it and/or modify it | |
7 ## under the terms of the GNU General Public License as published by | |
7016 | 8 ## the Free Software Foundation; either version 3 of the License, or (at |
9 ## your option) any later version. | |
5827 | 10 ## |
11 ## Octave is distributed in the hope that it will be useful, but | |
12 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
14 ## General Public License for more details. | |
15 ## | |
16 ## You should have received a copy of the GNU General Public License | |
7016 | 17 ## along with Octave; see the file COPYING. If not, see |
18 ## <http://www.gnu.org/licenses/>. | |
5827 | 19 |
20 ## -*- texinfo -*- | |
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21 ## @deftypefn {Function File} {@var{l} =} legendre (@var{n}, @var{x}) |
7508 | 22 ## @deftypefnx {Function File} {@var{l} =} legendre (@var{n}, @var{x}, @var{normalization}) |
23 ## Compute the Legendre function of degree @var{n} and order @var{m}. | |
24 ## The optional argument my be one of @code{"unnorm"}, @code{"sch"}, or | |
25 ## @code{"norm"}. The default is @code{"unnorm"}. | |
26 ## The value of @var{n} must be a non-negative scalar integer. | |
27 ## | |
28 ## If the optional argument @var{normalization} is missing or is | |
29 ## @code{"unnorm"}, compute the Legendre function of degree @var{n} and | |
30 ## order @var{m} and return all values for @var{m} = 0 @dots{} @var{n}. | |
31 ## The return value has one dimension more than @var{x}. | |
5827 | 32 ## |
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33 ## The Legendre Function of degree @var{n} and order @var{m}: |
5827 | 34 ## |
35 ## @example | |
36 ## @group | |
37 ## m m 2 m/2 d^m | |
38 ## P(x) = (-1) * (1-x ) * ---- P (x) | |
39 ## n dx^m n | |
40 ## @end group | |
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41 ## @end example |
5827 | 42 ## |
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43 ## @noindent |
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44 ## with Legendre polynomial of degree @var{n}: |
5827 | 45 ## |
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46 ## @example |
5827 | 47 ## @group |
48 ## 1 d^n 2 n | |
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49 ## P (x) = ------ [----(x - 1) ] |
5827 | 50 ## n 2^n n! dx^n |
51 ## @end group | |
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52 ## @end example |
5827 | 53 ## |
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54 ## @noindent |
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55 ## @code{legendre (3, [-1.0, -0.9, -0.8])} returns the matrix: |
5827 | 56 ## |
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57 ## @example |
5827 | 58 ## @group |
59 ## x | -1.0 | -0.9 | -0.8 | |
60 ## ------------------------------------ | |
61 ## m=0 | -1.00000 | -0.47250 | -0.08000 | |
62 ## m=1 | 0.00000 | -1.99420 | -1.98000 | |
63 ## m=2 | 0.00000 | -2.56500 | -4.32000 | |
64 ## m=3 | 0.00000 | -1.24229 | -3.24000 | |
65 ## @end group | |
66 ## @end example | |
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67 ## |
7508 | 68 ## If the optional argument @code{normalization} is @code{"sch"}, |
69 ## compute the Schmidt semi-normalized associated Legendre function. | |
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70 ## The Schmidt semi-normalized associated Legendre function is related |
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71 ## to the unnormalized Legendre functions by the following: |
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72 ## |
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73 ## For Legendre functions of degree n and order 0: |
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74 ## |
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75 ## @example |
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76 ## @group |
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77 ## 0 0 |
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78 ## SP (x) = P (x) |
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79 ## n n |
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80 ## @end group |
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81 ## @end example |
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82 ## |
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83 ## For Legendre functions of degree n and order m: |
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84 ## |
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85 ## @example |
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86 ## @group |
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87 ## m m m 2(n-m)! 0.5 |
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88 ## SP (x) = P (x) * (-1) * [-------] |
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89 ## n n (n+m)! |
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90 ## @end group |
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91 ## @end example |
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92 ## |
7508 | 93 ## If the optional argument @var{normalization} is @code{"norm"}, |
94 ## compute the fully normalized associated Legendre function. | |
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95 ## The fully normalized associated Legendre function is related |
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96 ## to the unnormalized Legendre functions by the following: |
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97 ## |
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98 ## For Legendre functions of degree @var{n} and order @var{m} |
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99 ## |
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100 ## @example |
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101 ## @group |
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102 ## m m m (n+0.5)(n-m)! 0.5 |
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103 ## NP (x) = P (x) * (-1) * [-------------] |
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104 ## n n (n+m)! |
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105 ## @end group |
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106 ## @end example |
5827 | 107 ## @end deftypefn |
108 | |
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109 ## Author: Marco Caliari <marco.caliari@univr.it> |
5827 | 110 |
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111 function retval = legendre (n, x, normalization) |
5827 | 112 |
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113 if (nargin < 2 || nargin > 3) |
5827 | 114 print_usage (); |
115 endif | |
116 | |
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117 if (nargin == 3) |
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118 normalization = lower (normalization); |
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119 else |
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120 normalization = "unnorm"; |
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121 endif |
5827 | 122 |
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123 if (! isscalar (n) || n < 0 || n != fix (n)) |
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124 error ("legendre: n must be a non-negative scalar integer"); |
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125 endif |
5827 | 126 |
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127 if (! isvector (x) || any (x < -1 || x > 1)) |
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128 error ("legendre: x must be vector in range -1 <= x <= 1"); |
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129 endif |
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130 |
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131 switch (normalization) |
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132 case "norm" |
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133 scale = sqrt (n+0.5); |
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134 case "sch" |
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135 scale = sqrt (2); |
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136 case "unnorm" |
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137 scale = 1; |
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138 otherwise |
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139 error ("legendre: expecting normalization option to be \"norm\", \"sch\", or \"unnorm\""); |
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140 endswitch |
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142 ## Based on the recurrence relation below |
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143 ## m m m |
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144 ## (n-m+1) * P (x) = (2*n+1)*x*P (x) - (n+1)*P (x) |
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145 ## n+1 n n-1 |
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146 ## http://en.wikipedia.org/wiki/Associated_Legendre_function |
5827 | 147 |
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148 for m = 1:n |
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149 lpm1 = scale; |
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150 lpm2 = (2*m-1) .* x .* scale; |
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151 lpm3 = lpm2; |
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152 for k = m+1:n |
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153 lpm3 = ((2*k-1) .* x .* lpm2 - (k+m-2) .* lpm1)/(k-m+1); |
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154 lpm1 = lpm2; |
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155 lpm2 = lpm3; |
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156 endfor |
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157 retval(m,:) = lpm3; |
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158 if (strcmp (normalization, "unnorm")) |
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159 scale = -scale * (2*m-1); |
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160 else |
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161 ## normalization == "sch" or normalization == "norm" |
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162 scale = scale / sqrt ((n-m+1)*(n+m))*(2*m-1); |
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163 endif |
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164 scale = scale .* sqrt(1-x.^2); |
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165 endfor |
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167 retval(n+1,:) = scale; |
5827 | 168 |
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169 if (strcmp (normalization, "sch")) |
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170 retval(1,:) = retval(1,:) / sqrt (2); |
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171 endif |
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172 |
5827 | 173 endfunction |
174 | |
175 %!test | |
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176 %! result = legendre (3, [-1.0 -0.9 -0.8]); |
5827 | 177 %! expected = [ |
178 %! -1.00000 -0.47250 -0.08000 | |
179 %! 0.00000 -1.99420 -1.98000 | |
180 %! 0.00000 -2.56500 -4.32000 | |
181 %! 0.00000 -1.24229 -3.24000 | |
182 %! ]; | |
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183 %! assert (result, expected, 1e-5); |
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Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
184 |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
185 %!test |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
186 %! result = legendre (3, [-1.0 -0.9 -0.8], "sch"); |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
187 %! expected = [ |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
188 %! -1.00000 -0.47250 -0.08000 |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
189 %! 0.00000 0.81413 0.80833 |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
190 %! -0.00000 -0.33114 -0.55771 |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
191 %! 0.00000 0.06547 0.17076 |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
192 %! ]; |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
193 %! assert (result, expected, 1e-5); |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
194 |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
195 %!test |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
196 %! 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
|
197 %! expected = [ |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
198 %! -1.87083 -0.88397 -0.14967 |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
199 %! 0.00000 1.07699 1.06932 |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
200 %! -0.00000 -0.43806 -0.73778 |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
201 %! 0.00000 0.08661 0.22590 |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
202 %! ]; |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
203 %! assert (result, expected, 1e-5); |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
204 |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
205 %!test |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
206 %! result = legendre (151, 0); |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
207 %! ## 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
|
208 %! 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
|
209 |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
210 %!test |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
211 %! result = legendre (150, 0); |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
212 %! ## This agrees with Matlab's result. |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
213 %! 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
|
214 |
bc6573d2fa40
legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents:
7017
diff
changeset
|
215 |