X-Git-Url: https://git.creatis.insa-lyon.fr/pubgit/?p=CreaPhase.git;a=blobdiff_plain;f=octave_packages%2Fm%2Fspecfun%2Flegendre.m;fp=octave_packages%2Fm%2Fspecfun%2Flegendre.m;h=7e35f10fe340b6d95b8b1a508d774cd22ca51b10;hp=0000000000000000000000000000000000000000;hb=1c0469ada9531828709108a4882a751d2816994a;hpb=63de9f36673d49121015e3695f2c336ea92bc278
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+## Copyright (C) 2000-2012 Kai Habel
+## Copyright (C) 2008 Marco Caliari
+##
+## This file is part of Octave.
+##
+## Octave is free software; you can redistribute it and/or modify it
+## under the terms of the GNU General Public License as published by
+## the Free Software Foundation; either version 3 of the License, or (at
+## your option) any later version.
+##
+## Octave is distributed in the hope that it will be useful, but
+## WITHOUT ANY WARRANTY; without even the implied warranty of
+## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+## General Public License for more details.
+##
+## You should have received a copy of the GNU General Public License
+## along with Octave; see the file COPYING. If not, see
+## .
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {@var{l} =} legendre (@var{n}, @var{x})
+## @deftypefnx {Function File} {@var{l} =} legendre (@var{n}, @var{x}, @var{normalization})
+## Compute the Legendre function of degree @var{n} and order
+## @var{m} = 0 @dots{} N@. The optional argument, @var{normalization},
+## may be one of @code{"unnorm"}, @code{"sch"}, or @code{"norm"}.
+## The default is @code{"unnorm"}. The value of @var{n} must be a
+## non-negative scalar integer.
+##
+## If the optional argument @var{normalization} is missing or is
+## @code{"unnorm"}, compute the Legendre function of degree @var{n} and
+## order @var{m} and return all values for @var{m} = 0 @dots{} @var{n}.
+## The return value has one dimension more than @var{x}.
+##
+## The Legendre Function of degree @var{n} and order @var{m}:
+##
+## @tex
+## $$
+## P^m_n(x) = (-1)^m (1-x^2)^{m/2}{d^m\over {dx^m}}P_n (x)
+## $$
+## @end tex
+## @ifnottex
+##
+## @example
+## @group
+## m m 2 m/2 d^m
+## P(x) = (-1) * (1-x ) * ---- P(x)
+## n dx^m n
+## @end group
+## @end example
+##
+## @end ifnottex
+##
+## @noindent
+## with Legendre polynomial of degree @var{n}:
+##
+## @tex
+## $$
+## P(x) = {1\over{2^n n!}}\biggl({d^n\over{dx^n}}(x^2 - 1)^n\biggr)
+## $$
+## @end tex
+## @ifnottex
+##
+## @example
+## @group
+## 1 d^n 2 n
+## P(x) = ------ [----(x - 1) ]
+## n 2^n n! dx^n
+## @end group
+## @end example
+##
+## @end ifnottex
+##
+## @noindent
+## @code{legendre (3, [-1.0, -0.9, -0.8])} returns the matrix:
+##
+## @example
+## @group
+## x | -1.0 | -0.9 | -0.8
+## ------------------------------------
+## m=0 | -1.00000 | -0.47250 | -0.08000
+## m=1 | 0.00000 | -1.99420 | -1.98000
+## m=2 | 0.00000 | -2.56500 | -4.32000
+## m=3 | 0.00000 | -1.24229 | -3.24000
+## @end group
+## @end example
+##
+## If the optional argument @code{normalization} is @code{"sch"},
+## compute the Schmidt semi-normalized associated Legendre function.
+## The Schmidt semi-normalized associated Legendre function is related
+## to the unnormalized Legendre functions by the following:
+##
+## For Legendre functions of degree n and order 0:
+##
+## @tex
+## $$
+## SP^0_n (x) = P^0_n (x)
+## $$
+## @end tex
+## @ifnottex
+##
+## @example
+## @group
+## 0 0
+## SP(x) = P(x)
+## n n
+## @end group
+## @end example
+##
+## @end ifnottex
+##
+## For Legendre functions of degree n and order m:
+##
+## @tex
+## $$
+## SP^m_n (x) = P^m_n (x)(-1)^m\biggl({2(n-m)!\over{(n+m)!}}\biggl)^{0.5}
+## $$
+## @end tex
+## @ifnottex
+##
+## @example
+## @group
+## m m m 2(n-m)! 0.5
+## SP(x) = P(x) * (-1) * [-------]
+## n n (n+m)!
+## @end group
+## @end example
+##
+## @end ifnottex
+##
+## If the optional argument @var{normalization} is @code{"norm"},
+## compute the fully normalized associated Legendre function.
+## The fully normalized associated Legendre function is related
+## to the unnormalized Legendre functions by the following:
+##
+## For Legendre functions of degree @var{n} and order @var{m}
+##
+## @tex
+## $$
+## NP^m_n (x) = P^m_n (x)(-1)^m\biggl({(n+0.5)(n-m)!\over{(n+m)!}}\biggl)^{0.5}
+## $$
+## @end tex
+## @ifnottex
+##
+## @example
+## @group
+## m m m (n+0.5)(n-m)! 0.5
+## NP(x) = P(x) * (-1) * [-------------]
+## n n (n+m)!
+## @end group
+## @end example
+##
+## @end ifnottex
+## @end deftypefn
+
+## Author: Marco Caliari
+
+function retval = legendre (n, x, normalization)
+
+ persistent warned_overflow = false;
+
+ if (nargin < 2 || nargin > 3)
+ print_usage ();
+ endif
+
+ if (!isscalar (n) || n < 0 || n != fix (n))
+ error ("legendre: N must be a non-negative scalar integer");
+ endif
+
+ if (!isreal (x) || any (x(:) < -1 | x(:) > 1))
+ error ("legendre: X must be real-valued vector in the range -1 <= X <= 1");
+ endif
+
+ if (nargin == 3)
+ normalization = lower (normalization);
+ else
+ normalization = "unnorm";
+ endif
+
+ switch (normalization)
+ case "norm"
+ scale = sqrt (n+0.5);
+ case "sch"
+ scale = sqrt (2);
+ case "unnorm"
+ scale = 1;
+ otherwise
+ error ('legendre: expecting NORMALIZATION option to be "norm", "sch", or "unnorm"');
+ endswitch
+
+ scale = scale * ones (size (x));
+
+ ## Based on the recurrence relation below
+ ## m m m
+ ## (n-m+1) * P (x) = (2*n+1)*x*P (x) - (n+1)*P (x)
+ ## n+1 n n-1
+ ## http://en.wikipedia.org/wiki/Associated_Legendre_function
+
+ overflow = false;
+ retval = zeros([n+1, size(x)]);
+ for m = 1:n
+ lpm1 = scale;
+ lpm2 = (2*m-1) .* x .* scale;
+ lpm3 = lpm2;
+ for k = m+1:n
+ lpm3a = (2*k-1) .* x .* lpm2;
+ lpm3b = (k+m-2) .* lpm1;
+ lpm3 = (lpm3a - lpm3b)/(k-m+1);
+ lpm1 = lpm2;
+ lpm2 = lpm3;
+ if (! warned_overflow)
+ if (any (abs (lpm3a) > realmax)
+ || any (abs (lpm3b) > realmax)
+ || any (abs (lpm3) > realmax))
+ overflow = true;
+ endif
+ endif
+ endfor
+ retval(m,:) = lpm3(:);
+ if (strcmp (normalization, "unnorm"))
+ scale = -scale * (2*m-1);
+ else
+ ## normalization == "sch" or normalization == "norm"
+ scale = scale / sqrt ((n-m+1)*(n+m))*(2*m-1);
+ endif
+ scale = scale .* sqrt(1-x.^2);
+ endfor
+
+ retval(n+1,:) = scale(:);
+
+ if (isvector (x))
+ ## vector case is special
+ retval = reshape (retval, n + 1, length (x));
+ endif
+
+ if (strcmp (normalization, "sch"))
+ retval(1,:) = retval(1,:) / sqrt (2);
+ endif
+
+ if (overflow && ! warned_overflow)
+ warning ("legendre: overflow - results may be unstable for high orders");
+ warned_overflow = true;
+ endif
+
+endfunction
+
+
+%!test
+%! result = legendre (3, [-1.0 -0.9 -0.8]);
+%! expected = [
+%! -1.00000 -0.47250 -0.08000
+%! 0.00000 -1.99420 -1.98000
+%! 0.00000 -2.56500 -4.32000
+%! 0.00000 -1.24229 -3.24000
+%! ];
+%! assert (result, expected, 1e-5);
+
+%!test
+%! result = legendre (3, [-1.0 -0.9 -0.8], "sch");
+%! expected = [
+%! -1.00000 -0.47250 -0.08000
+%! 0.00000 0.81413 0.80833
+%! -0.00000 -0.33114 -0.55771
+%! 0.00000 0.06547 0.17076
+%! ];
+%! assert (result, expected, 1e-5);
+
+%!test
+%! result = legendre (3, [-1.0 -0.9 -0.8], "norm");
+%! expected = [
+%! -1.87083 -0.88397 -0.14967
+%! 0.00000 1.07699 1.06932
+%! -0.00000 -0.43806 -0.73778
+%! 0.00000 0.08661 0.22590
+%! ];
+%! assert (result, expected, 1e-5);
+
+%!test
+%! result = legendre (151, 0);
+%! ## Don't compare to "-Inf" since it would fail on 64 bit systems.
+%! assert (result(end) < -1.7976e308 && all (isfinite (result(1:end-1))));
+
+%!test
+%! result = legendre (150, 0);
+%! ## This agrees with Matlab's result.
+%! assert (result(end), 3.7532741115719e+306, 0.0000000000001e+306);
+
+%!test
+%! result = legendre (0, 0:0.1:1);
+%! assert (result, full(ones(1,11)));
+
+%!test
+%! result = legendre (3, [-1,0,1;1,0,-1]);
+%! ## Test matrix input
+%! expected(:,:,1) = [-1,1;0,0;0,0;0,0];
+%! expected(:,:,2) = [0,0;1.5,1.5;0,0;-15,-15];
+%! expected(:,:,3) = [1,-1;0,0;0,0;0,0];
+%! assert (result, expected);
+
+%!test
+%! result = legendre (3, [-1,0,1;1,0,-1]');
+%! expected(:,:,1) = [-1,0,1;0,1.5,0;0,0,0;0,-15,0];
+%! expected(:,:,2) = [1,0,-1;0,1.5,0;0,0,0;0,-15,0];
+%! assert (result, expected);
+
+%% Check correct invocation
+%!error legendre ();
+%!error legendre (1);
+%!error legendre (1,2,3,4);
+%!error legendre ([1, 2], [-1, 0, 1]);
+%!error legendre (-1, [-1, 0, 1]);
+%!error legendre (1.1, [-1, 0, 1]);
+%!error legendre (1, [-1+i, 0, 1]);
+%!error legendre (1, [-2, 0, 1]);
+%!error legendre (1, [-1, 0, 2]);
+%!error legendre (1, [-1, 0, 1], "badnorm");