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subplot.m: Add suppport for "align" and "replace" options.
author Ben Abbott <bpabbott@mac.com>
date Thu, 02 Dec 2010 18:46:10 -0500
parents be55736a0783
children e7ed20f87e82
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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 (!isscalar (n) || n < 0 || n != fix (n))
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121 error ("legendre: N must be a non-negative scalar integer");
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122 endif
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123
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124 if (!isvector (x) || !isreal (x) || any (x < -1 | x > 1))
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125 error ("legendre: X must be real-valued vector in the range -1 <= X <= 1");
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126 endif
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127
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128 if (nargin == 3)
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129 normalization = lower (normalization);
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130 else
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131 normalization = "unnorm";
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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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1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
145 if (rows (x) != 1)
1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
146 x = x';
1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
147 endif
8255
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
148 scale = scale * ones (1, numel (x));
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
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diff changeset
149
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Ben Abbott <bpabbott@mac.com>
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diff changeset
150 ## Based on the recurrence relation below
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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diff changeset
151 ## m m m
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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diff changeset
152 ## (n-m+1) * P (x) = (2*n+1)*x*P (x) - (n+1)*P (x)
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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diff changeset
153 ## n+1 n n-1
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
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diff changeset
154 ## http://en.wikipedia.org/wiki/Associated_Legendre_function
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jwe
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diff changeset
155
8255
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Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
156 overflow = false;
7507
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Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
157 for m = 1:n
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
158 lpm1 = scale;
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
159 lpm2 = (2*m-1) .* x .* scale;
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
160 lpm3 = lpm2;
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
161 for k = m+1:n
8255
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Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
162 lpm3a = (2*k-1) .* x .* lpm2;
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
163 lpm3b = (k+m-2) .* lpm1;
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
164 lpm3 = (lpm3a - lpm3b)/(k-m+1);
7507
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
165 lpm1 = lpm2;
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
166 lpm2 = lpm3;
8255
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
167 if (! warned_overflow)
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
168 if (any (abs (lpm3a) > realmax)
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
169 || any (abs (lpm3b) > realmax)
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
170 || any (abs (lpm3) > realmax))
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
171 overflow = true;
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
172 endif
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
173 endif
7507
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Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
174 endfor
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
175 retval(m,:) = lpm3;
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
176 if (strcmp (normalization, "unnorm"))
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
177 scale = -scale * (2*m-1);
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
178 else
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
179 ## normalization == "sch" or normalization == "norm"
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
180 scale = scale / sqrt ((n-m+1)*(n+m))*(2*m-1);
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
181 endif
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
182 scale = scale .* sqrt(1-x.^2);
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
183 endfor
5827
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jwe
parents:
diff changeset
184
7507
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Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
185 retval(n+1,:) = scale;
5827
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
jwe
parents:
diff changeset
186
7507
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Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
187 if (strcmp (normalization, "sch"))
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
188 retval(1,:) = retval(1,:) / sqrt (2);
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
189 endif
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
190
8255
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
191 if (overflow && ! warned_overflow)
8664
e07e93c04080 style fixes
John W. Eaton <jwe@octave.org>
parents: 8255
diff changeset
192 warning ("legendre: overflow - results may be unstable for high orders");
8255
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
193 warned_overflow = true;
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
194 endif
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
195
5827
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jwe
parents:
diff changeset
196 endfunction
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
jwe
parents:
diff changeset
197
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
jwe
parents:
diff changeset
198 %!test
7507
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Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
199 %! result = legendre (3, [-1.0 -0.9 -0.8]);
5827
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
jwe
parents:
diff changeset
200 %! expected = [
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
jwe
parents:
diff changeset
201 %! -1.00000 -0.47250 -0.08000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
jwe
parents:
diff changeset
202 %! 0.00000 -1.99420 -1.98000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
jwe
parents:
diff changeset
203 %! 0.00000 -2.56500 -4.32000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
jwe
parents:
diff changeset
204 %! 0.00000 -1.24229 -3.24000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe]
jwe
parents:
diff changeset
205 %! ];
7507
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Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
206 %! assert (result, expected, 1e-5);
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
207
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
208 %!test
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
209 %! 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
210 %! expected = [
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
211 %! -1.00000 -0.47250 -0.08000
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
212 %! 0.00000 0.81413 0.80833
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
213 %! -0.00000 -0.33114 -0.55771
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
214 %! 0.00000 0.06547 0.17076
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
215 %! ];
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
216 %! assert (result, expected, 1e-5);
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
217
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
218 %!test
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
219 %! 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
220 %! expected = [
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
221 %! -1.87083 -0.88397 -0.14967
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
222 %! 0.00000 1.07699 1.06932
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
223 %! -0.00000 -0.43806 -0.73778
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
224 %! 0.00000 0.08661 0.22590
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
225 %! ];
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
226 %! assert (result, expected, 1e-5);
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
227
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
228 %!test
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
229 %! result = legendre (151, 0);
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
230 %! ## 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
231 %! 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
232
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
233 %!test
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
234 %! result = legendre (150, 0);
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
235 %! ## This agrees with Matlab's result.
bc6573d2fa40 legendre.m: Added normalization and improved stability.
Ben Abbott <bpabbott@mac.com>
parents: 7017
diff changeset
236 %! 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
237
8255
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
238 %!test
3f1199ad212f legendre.m: Warn once on under/overflow.
Ben Abbott <bpabbott@mac.com>
parents: 7517
diff changeset
239 %! result = legendre (0, 0:0.1:1);
8975
2e9af3363669 fix tests with ones
Jaroslav Hajek <highegg@gmail.com>
parents: 8920
diff changeset
240 %! assert (result, full(ones(1,11)))
10672
1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
241
1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
242 %% Check correct invocation
1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
243 %!error legendre ();
1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
244 %!error legendre (1);
1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
245 %!error legendre (1,2,3,4);
1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
246 %!error legendre ([1, 2], [-1, 0, 1]);
1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
247 %!error legendre (-1, [-1, 0, 1]);
1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
248 %!error legendre (1.1, [-1, 0, 1]);
1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
249 %!error legendre (1, [-1+i, 0, 1]);
1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
250 %!error legendre (1, [-2, 0, 1]);
1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
251 %!error legendre (1, [-1, 0, 2]);
1cd7c39a96ee legendre.m: Orient row vector correctly (bug #29997).
Rik <octave@nomad.inbox5.com>
parents: 9051
diff changeset
252 %!error legendre (1, [-1, 0, 1], "badnorm");