ENH: special: add log trick to roots_jacobi and roots_sh_jacobi

[itamarfaran]

Reference issue

closes #12425: roots_sh_jacobi: Error handling for OverflowError

What does this implement/fix?

add option that allows returning of the log values in orthogonal._gen_roots_and_weights when symmetrize=False

Additional information

needed to imitate statmod::gauss.quad.prob function in R

[sethtroisi]

I failing tests are

FAILED scipy/special/tests/test_orthogonal.py::test_roots_jacobi - AssertionE...
FAILED scipy/special/tests/test_orthogonal.py::test_roots_sh_jacobi - Asserti...

So that needs to be looked into

[itamarfaran]

@sethtroisi the tests pass now for log_weights=False. however, when testing for log_weights=True, I'm getting the following failure in test_orthogonal.py#L331:

I'm getting an absolute difference of .023, which is a relative difference of about 1.69e-15 (mu is in the scale of 10^13). So the test is a bit to harsh for taking exponents and logs. what should I do about that?

[itamarfaran]

@person142 any new with this pr :)?

[person142]

Ach let me try to find some time to review... first thought though is that a flag determining whether we return weights or log weights is non-ideal API-wise; generally outputs of functions shouldn't change completely depending on flags. (Admittedly yes the API already does that, but we can no longer correct that because of back compat. Would be nice to not make the situation worse though.)

[itamarfaran]

@person142 would it be better to open a completely new module with 'roots_log_ ...' functions instead?

[person142]

would it be better to open a completely new module with 'roots_log_ ...' functions instead?

Yes, new functions would be the way to go I think. It would match well with the rest of the API-e.g. loggamma versus gamma, log_ndtr versus ndtr and so on. But before making you put more effort into this, let me read #12425 more carefully first... might not have time for that until next week though.