Re: normalized ALF (Assotiated Legendre Function) *patch*

[Ben Abbott]

kruvalig wrote:

In matlab enxist function legendre, and it can compute fully normalized   
assotiated function, and normalized by Schmidt. Does exist anything the   
same in octave. 

I now that legendre-function in octave can compute un-normalized function. 

And i interested in fully normalized ALF 

                 (2-delta(0,m))(2n+1)(n-m)! 
P_nm(t) = sqrt(----------------------------) P_nm(t) 
                         (n+m)! 

I inadvertently deleted the mailing by kruvalig :-( ... so I've forged a reply. I'll try to contact kruvalig as well.

In that I've added the normalization options present in Matabs legendre.m to Octave, I've moved this to the maintainers list.

2008-02-09  Ben Abbott <address@hidden>

* specfun/cond.m: Added Schmidt semi-normalized and fully normalized

 legendre functions.

I've attached a patch.

Ben

--- /Users/bpabbott/Development/cvs/octave/scripts/specfun/legendre.m 2007-10-12 17:27:26.000000000 -0400

+++ legendre.m 2008-02-09 17:18:09.000000000 -0500

@@ -18,6 +18,7 @@

 

 ## -*- texinfo -*-

 ## @deftypefn {Function File} address@hidden =} legendre (@var{n}, @var{X})

+## @deftypefnx {Function File} address@hidden =} legendre (@var{n}, @var{X}, "unnorm")

 ##

 ## Legendre Function of degree n and order m

 ## where all values for m = address@hidden are returned.

@@ -53,19 +54,75 @@

 ## m=3 |  0.00000 | -1.24229 | -3.24000 

 ## @end group

 ## @end example

+##

+## @deftypefnx {Function File} address@hidden =} legendre (@var{n}, @var{X}, "sch")

+##

+## Computes the Schmidt semi-normalized associated Legendre function.

+## The Schmidt semi-normalized associated Legendre function is related

+## to the unnormalized Legendre functions by

+##

+## @example

+## For Legendre functions of degree n and order 0

+##

+## @group

+##   0       0

+## SP (x) = P (x)

+##   n       n

+## @end group

+##

+## For Legendre functions of degree n and order m

+##

+## @group

+##   m       m          m    2(n-m)! 0.5

+## SP (x) = P (x) * (-1)  * [-------]

+##   n       n               (n+m)!

+## @end group

+## @end example

+##

+## @deftypefnx {Function File} address@hidden =} legendre (@var{n}, @var{X}, "norm")

+##

+## Computes the fully normalized associated Legendre function.

+## The fully normalized associated Legendre function is related

+## to the unnormalized Legendre functions by

+##

+## @example

+## For Legendre functions of degree n and order m

+##

+## @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 deftypefn

 

-## FIXME Add Schmidt semi-normalized and fully normalized legendre functions

-

 ## Author: Kai Habel <address@hidden>

 

-function L = legendre (n, x)

+function L = legendre (n, x, normalization)

 

   warning ("legendre is unstable for higher orders");

 

-  if (nargin != 2)

+  if (nargin < 2 || nargin > 3)

     print_usage ();

   endif

+  if nargin == 3

+    switch lower(normalization)

+      case "sch"

+        m = 1:n;

+        scale = (-1).^m .* sqrt (2 * factorial (n-m) ./ factorial (n+m));

+        scale = [1, scale];

+      case "norm"

+        m = 0:n;

+        scale = (-1).^m .* sqrt ((n+0.5) .* factorial (n-m) ./ factorial (n+m));

+      case "unnorm"

+        scale = ones (1, n+1);

+      otherwise

+        print_usage ();

+    endswitch

+  else

+    scale = ones (1, n+1);

+  endif

 

     if (! isscalar (n) || n < 0 || n > 255 || n != fix (n))

       error ("n must be a integer between 0 and 255]");

@@ -119,6 +176,11 @@

       M = kron ((0:n)', ones (1, l));

       L = X .^ (M / 2) .* (-1) .^ M .* Y;

     endif

+

+    if nargin == 3

+      scale = scale(:) * ones (1, numel (x));

+      L = L .* reshape (scale, size (L));

+    endif

 endfunction

 

 %!test

@@ -130,3 +192,25 @@

 %!     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);

+

+

legendre.diff
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