Re: normalized ALF (Assotiated Legendre Function)

[Ben Abbott]

On Feb 11, 2008, at 10:20 AM, Marco Caliari wrote:

Please, try the enclosed legendre.m. It is based on the recurrence relation found in http://en.wikipedia.org/wiki/Associated_Legendre_function. I think it should be much more stable.

I compared Matlab's script and Octave's and your proposal.

  result_matlab = legendre (80, [-1:0.1:1]);

result_octave0 respects the original and result_octave1 respects your algorithm.

I calculated an error with respect to the matlab version (I'm not sure Matlab's is to be trusted as correct in all cases).

d0 = abs (result_matlab - result_octave0) / abs (result_matlab); d1 = abs (result_matlab - result_octave1) / abs (result_matlab); er1 = max (d1, [], 2); er0 = max (d0, [], 2);

[er0(:), er1(:)] produces the following

ans =

   2.3424e-119   2.0877e-144

   1.1666e-120   2.8601e-144

   3.1132e-119   1.3575e-140

   8.1518e-118   1.0167e-139

   2.1137e-116   8.8310e-137

   5.4094e-115   1.6560e-135

   1.3778e-113   5.7097e-133

   3.5365e-112   1.3705e-131

   9.1806e-111   3.3573e-129

   2.5238e-109   1.3295e-127

   8.9744e-108   1.8397e-125

   3.8727e-106   1.2779e-123

   1.9273e-104   9.8135e-122

   1.0123e-102   9.4790e-120

   5.4782e-101   4.6351e-118

    2.9220e-99   6.1185e-116

    1.5443e-97   1.6927e-114

    7.9263e-96   5.0573e-112

    3.9915e-94   4.8575e-111

    1.9686e-92   2.5731e-108

    9.4747e-91   1.0210e-106

    4.4378e-89   1.2955e-104

    2.0255e-87   9.2168e-103

    8.9000e-86   5.0162e-101

    3.7992e-84    7.4455e-99

    1.5630e-82    6.0867e-98

    6.3553e-81    4.0858e-95

    2.4881e-79    1.5141e-93

    9.3534e-78    1.6912e-91

    3.4021e-76    1.8648e-89

    1.2025e-74    5.4167e-88

    4.1090e-73    1.0303e-85

    1.3681e-71    4.2479e-84

    4.2235e-70    3.6317e-82

    1.2357e-68    4.2982e-80

    3.4224e-67    7.9412e-79

    9.0728e-66    2.0485e-76

    2.1559e-64    1.1802e-74

    4.2186e-63    5.7797e-73

    4.8458e-62    7.6818e-71

    1.0928e-61    4.0895e-69

    4.4609e-59    2.5363e-67

    3.6901e-57    2.3634e-65

    2.5598e-55    3.8612e-64

    8.8479e-54    8.5079e-62

    3.4999e-52    7.1329e-60

    1.0974e-50    1.5563e-58

    3.8537e-49    2.3469e-56

    8.7346e-48    2.4322e-54

    2.2104e-46    6.8924e-53

    7.5648e-45    4.6632e-51

    1.6279e-43    6.4798e-49

    3.0916e-42    2.7505e-47

    7.0616e-42    7.3142e-46

    8.1111e-40    7.6535e-44

    7.9301e-39    6.1412e-42

    5.7348e-37    3.2699e-40

    6.0936e-36    1.4796e-38

    4.3591e-34    9.0850e-37

    1.2343e-32    5.8704e-35

    3.4634e-31    3.9178e-33

    1.0903e-29    2.0041e-31

    8.0223e-29    9.2888e-30

    2.4566e-27    3.6487e-28

    4.2821e-26    1.7018e-26

    1.9256e-25    6.5910e-25

    8.1766e-25    2.4639e-23

    4.2551e-23    6.8962e-22

    1.1459e-22    1.7888e-20

    3.2583e-20    3.9508e-19

    1.8758e-18    6.5279e-18

    3.1305e-17    1.1644e-16

    9.8748e-16    1.9183e-15

    1.5795e-16    3.3544e-14

    2.9368e-13    4.7596e-13

    2.8870e-12    7.9144e-12

    5.9192e-11    9.2655e-11

    6.6729e-10    9.5730e-10

    4.7224e-09    7.8467e-09

    2.2427e-08    3.8634e-08

    5.3147e-08    1.0190e-07

I'm inclined to agree that the recursion form should work better. I'm suspicious that Matlab's version is reliable for such high order legendre polynomials.

Anyone, is there a reliable method to verify the correct answers?

Ben