octave-antonio: scripts/specfun/legendre.m annotate

annotate scripts/specfun/legendre.m @ 7507:bc6573d2fa40

legendre.m: Added normalization and improved stability.

author	Ben Abbott <bpabbott@mac.com>
date	Wed, 20 Feb 2008 21:47:39 -0500
parents	a1dbe9d80eee
children	f501b22c0394

rev	line source
7017 a1dbe9d80eee [project @ 2007-10-12 21:27:11 by jwe] jwe parents: 7016 diff changeset	1 ## Copyright (C) 2000, 2006, 2007 Kai Habel
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	2 ## Copyright (C) 2008 Marco Caliari
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	3 ##
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	4 ## This file is part of Octave.
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	5 ##
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	6 ## Octave is free software; you can redistribute it and/or modify it
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	7 ## under the terms of the GNU General Public License as published by
7016 93c65f2a5668 [project @ 2007-10-12 06:40:56 by jwe] jwe parents: 7001 diff changeset	8 ## the Free Software Foundation; either version 3 of the License, or (at
93c65f2a5668 [project @ 2007-10-12 06:40:56 by jwe] jwe parents: 7001 diff changeset	9 ## your option) any later version.
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	10 ##
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	11 ## Octave is distributed in the hope that it will be useful, but
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	12 ## WITHOUT ANY WARRANTY; without even the implied warranty of
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	14 ## General Public License for more details.
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	15 ##
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	16 ## You should have received a copy of the GNU General Public License
7016 93c65f2a5668 [project @ 2007-10-12 06:40:56 by jwe] jwe parents: 7001 diff changeset	17 ## along with Octave; see the file COPYING. If not, see
93c65f2a5668 [project @ 2007-10-12 06:40:56 by jwe] jwe parents: 7001 diff changeset	18 ## <http://www.gnu.org/licenses/>.
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	19
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	20 ## -- texinfo --
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	21 ## @deftypefn {Function File} {@var{l} =} legendre (@var{n}, @var{x})
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	22 ## @deftypefnx {Function File} {@var{l} =} legendre (@var{n}, @var{x}, "unnorm")
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	23 ## Compute the Legendre function of degree @var{n} and order @var{m}
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	24 ## where all values for @var{m} = 0 @dots{} @var{n} are returned.
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	25 ## @var{n} must be a non-negative scalar integer in the range greater
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	26 ## than or equal to 0. The return value has one dimension more than @var{x}.
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	27 ##
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	28 ## The Legendre Function of degree @var{n} and order @var{m}:
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	29 ##
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	30 ## @example
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	31 ## @group
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	32 ## m m 2 m/2 d^m
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	33 ## P(x) = (-1) * (1-x ) * ---- P (x)
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	34 ## n dx^m n
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	35 ## @end group
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	36 ## @end example
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	37 ##
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	38 ## @noindent
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	39 ## with Legendre polynomial of degree @var{n}:
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	40 ##
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	41 ## @example
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	42 ## @group
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	43 ## 1 d^n 2 n
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	44 ## P (x) = ------ [----(x - 1) ]
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	45 ## n 2^n n! dx^n
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	46 ## @end group
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	47 ## @end example
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	48 ##
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	49 ## @noindent
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	50 ## @code{legendre (3, [-1.0, -0.9, -0.8])} returns the matrix:
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	51 ##
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	52 ## @example
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	53 ## @group
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	54 ## x \| -1.0 \| -0.9 \| -0.8
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	55 ## ------------------------------------
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	56 ## m=0 \| -1.00000 \| -0.47250 \| -0.08000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	57 ## m=1 \| 0.00000 \| -1.99420 \| -1.98000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	58 ## m=2 \| 0.00000 \| -2.56500 \| -4.32000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	59 ## m=3 \| 0.00000 \| -1.24229 \| -3.24000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	60 ## @end group
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	61 ## @end example
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	62 ##
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	63 ## @deftypefnx {Function File} {@var{l} =} legendre (@var{n}, @var{x}, "sch")
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	64 ## Compute the Schmidt semi-normalized associated Legendre function.
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	65 ## The Schmidt semi-normalized associated Legendre function is related
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	66 ## to the unnormalized Legendre functions by the following:
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	67 ##
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	68 ## For Legendre functions of degree n and order 0:
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	69 ##
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	70 ## @example
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	71 ## @group
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	72 ## 0 0
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	73 ## SP (x) = P (x)
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	74 ## n n
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	75 ## @end group
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	76 ## @end example
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	77 ##
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	78 ## For Legendre functions of degree n and order m:
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	79 ##
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	80 ## @example
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	81 ## @group
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	82 ## m m m 2(n-m)! 0.5
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	83 ## SP (x) = P (x) * (-1) * [-------]
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	84 ## n n (n+m)!
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	85 ## @end group
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	86 ## @end example
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	87 ##
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	88 ## @deftypefnx {Function File} {@var{l} =} legendre (@var{n}, @var{x}, "norm")
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	89 ## Compute the fully normalized associated Legendre function.
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	90 ## The fully normalized associated Legendre function is related
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	91 ## to the unnormalized Legendre functions by the following:
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	92 ##
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	93 ## For Legendre functions of degree @var{n} and order @var{m}
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	94 ##
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	95 ## @example
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	96 ## @group
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	97 ## m m m (n+0.5)(n-m)! 0.5
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	98 ## NP (x) = P (x) * (-1) * [-------------]
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	99 ## n n (n+m)!
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	100 ## @end group
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	101 ## @end example
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	102 ## @end deftypefn
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	103
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	104 ## Author: Marco Caliari <marco.caliari@univr.it>
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	105
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	106 function retval = legendre (n, x, normalization)
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	107
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	108 if (nargin < 2 \|\| nargin > 3)
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	109 print_usage ();
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	110 endif
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	111
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	112 if (nargin == 3)
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	113 normalization = lower (normalization);
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	114 else
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	115 normalization = "unnorm";
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	116 endif
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	117
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	118 if (! isscalar (n) \|\| n < 0 \|\| n != fix (n))
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	119 error ("legendre: n must be a non-negative scalar integer");
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	120 endif
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	121
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	122 if (! isvector (x) \|\| any (x < -1 \|\| x > 1))
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	123 error ("legendre: x must be vector in range -1 <= x <= 1");
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	124 endif
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	125
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	126 switch (normalization)
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	127 case "norm"
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	128 scale = sqrt (n+0.5);
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	129 case "sch"
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	130 scale = sqrt (2);
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	131 case "unnorm"
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	132 scale = 1;
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	133 otherwise
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	134 error ("legendre: expecting normalization option to be \"norm\", \"sch\", or \"unnorm\"");
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	135 endswitch
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	136
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	137 ## Based on the recurrence relation below
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	138 ## m m m
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	139 ## (n-m+1) * P (x) = (2n+1)xP (x) - (n+1)P (x)
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	140 ## n+1 n n-1
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	141 ## http://en.wikipedia.org/wiki/Associated_Legendre_function
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	142
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	143 for m = 1:n
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	144 lpm1 = scale;
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	145 lpm2 = (2m-1) . x .* scale;
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	146 lpm3 = lpm2;
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	147 for k = m+1:n
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	148 lpm3 = ((2k-1) . x .* lpm2 - (k+m-2) .* lpm1)/(k-m+1);
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	149 lpm1 = lpm2;
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	150 lpm2 = lpm3;
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	151 endfor
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	152 retval(m,:) = lpm3;
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	153 if (strcmp (normalization, "unnorm"))
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	154 scale = -scale * (2*m-1);
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	155 else
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	156 ## normalization == "sch" or normalization == "norm"
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	157 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	158 endif
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	159 scale = scale .* sqrt(1-x.^2);
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	160 endfor
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	161
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	162 retval(n+1,:) = scale;
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	163
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	164 if (strcmp (normalization, "sch"))
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	165 retval(1,:) = retval(1,:) / sqrt (2);
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	166 endif
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	167
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	168 endfunction
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	169
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	170 %!test
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	171 %! result = legendre (3, [-1.0 -0.9 -0.8]);
5827 1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	172 %! expected = [
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	173 %! -1.00000 -0.47250 -0.08000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	174 %! 0.00000 -1.99420 -1.98000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	175 %! 0.00000 -2.56500 -4.32000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	176 %! 0.00000 -1.24229 -3.24000
1fe78adb91bc [project @ 2006-05-22 06:25:14 by jwe] jwe parents: diff changeset	177 %! ];
7507 bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	178 %! assert (result, expected, 1e-5);
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	179
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	180 %!test
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	181 %! 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	182 %! expected = [
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	183 %! -1.00000 -0.47250 -0.08000
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	184 %! 0.00000 0.81413 0.80833
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	185 %! -0.00000 -0.33114 -0.55771
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	186 %! 0.00000 0.06547 0.17076
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	187 %! ];
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	188 %! assert (result, expected, 1e-5);
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	189
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	190 %!test
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	191 %! 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	192 %! expected = [
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	193 %! -1.87083 -0.88397 -0.14967
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	194 %! 0.00000 1.07699 1.06932
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	195 %! -0.00000 -0.43806 -0.73778
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	196 %! 0.00000 0.08661 0.22590
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	197 %! ];
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	198 %! assert (result, expected, 1e-5);
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	199
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	200 %!test
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	201 %! result = legendre (151, 0);
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	202 %! ## 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	203 %! 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	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 (150, 0);
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	207 %! ## This agrees with Matlab's result.
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	208 %! 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	209
bc6573d2fa40 legendre.m: Added normalization and improved stability. Ben Abbott <bpabbott@mac.com> parents: 7017 diff changeset	210

Mercurial > octave-antonio

annotate scripts/specfun/legendre.m @ 7507:bc6573d2fa40