Algorithm underflows when computing Wigner 3j symbols with large quantum numbers

[joeydumont]

When the quantum numbers become large, the relative sizes of the Wigner symbols can span tens of orders of magnitude. This frequently leads to overflow in the recurrence relations, which we deal with by dividing by a large number. However, this leads to underflows in the algorithm, resulting in the algorithm returning 0s instead of valid Wigner symbols. See

Wigner 3j symbol                 Mathematica       WignerSymbols
[ 751  856 1200  464 -828  364] -9.41731061215e-58 -0.0
[ 841  379 1011 -631  313  318] -2.44096504011e-41 -0.0
[ 570 1007 1392  327 -933  606] -1.74376347733e-98 0.0
[ 970  727 1202  533 -663  130] -6.93009562166e-12 0.0
[ 905  919 1670  869 -594 -275] -3.48516309858e-195 -0.0
[ 895  574 1392  793 -365 -428] -1.41868655509e-146 -0.0

for examples of this.

[joeydumont]

The Fortran algorithm does not have this problem.

[jeffzhen]

Hi Joey, Is this difference caused by different implementation in C++ and Fortran, or because somehow these languages handle floating point numbers differently? Presumably one could multiply a large number to deal with underflows very much the same way as you dealt with the overflow?

Thanks a lot! Jeff