Bugfix/Workaround sign error Asso. Legendre Function at z=-1

[mberz]

This might fix the bug reported in scipy/scipy#23101.

It seems that

xsf/include/xsf/legendre.h

Line 313 in 15ffa02

res[0] = T(1);

should depend on the sign of z.

This is an edge case for abs(z)=1, imag(z)=0 (see

xsf/include/xsf/legendre.h

Line 402 in 15ffa02

assoc_legendre_p_pm1(norm, j, m, z, branch_cut, res[1]);

)

I've implemented a test which catches the original issue in scipy/scipy#24017.

I am not sure if this is a proper solution to the problem or only a workaround. I did not find references in the code of the Legendre functions which were used for the original implementation.

[lucascolley]

LGTM, thanks @mberz ! I've confirmed that the test fails on main and passes on this branch:

scipy on 🎋 bug/assoc_legendre is 📦 v1.17.0.dev0 via 🐍 via 🧚 v0.59.0 took 4s
❯ pixi run test -t scipy.special.tests.test_legendre "--" -k "assoc_legendre_bug"
✨ Pixi task (build in build): spin build --setup-args=-Dblas=blas --setup-args=-Dlapack=lapack --setup-args=-Duse-g77-abi=true: (Build SciPy (default settings))                                                                           $ meson compile -j 8 -C build
INFO: autodetecting backend as ninja
INFO: calculating backend command to run: /Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy/.pixi/envs/build/bin/ninja -C /Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy/build -j 8
ninja: Entering directory `/Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy/build'
[2/338] Generating subprojects/highs/src/HConfig.h with a custom command
$ meson install --only-changed -C build --destdir ../build-install --tags=runtime,python-runtime,tests,devel
                                                                                                                      ✨ Pixi task (test in test): spin test --no-build -t scipy.special.tests.test_legendre -- -k assoc_legendre_bug: (Test SciPy (default settings))                                                                                            $ export PYTHONPATH="/Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy/build-install/usr/lib/python3.14/site-packages"
$ /Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy/.pixi/envs/test/bin/python3.14 -P -c 'import scipy'
$ cd /Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy/build-install/usr/lib/python3.14/site-packages
$ /Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy/.pixi/envs/test/bin/python3.14 -P -m pytest -m 'not slow' -k assoc_legendre_bug --pyargs scipy.special.tests.test_legendre
================================================ test session starts =================================================
platform darwin -- Python 3.14.0, pytest-9.0.0, pluggy-1.6.0
rootdir: /Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy
configfile: pytest.ini
plugins: hypothesis-6.147.0, xdist-3.8.0, timeout-2.4.0
collected 417 items / 416 deselected / 1 selected

scipy/special/tests/test_legendre.py F                                                                         [100%]

====================================================== FAILURES ======================================================
_____________________________________ TestAssocLegendreP.test_assoc_legendre_bug _____________________________________

self = <scipy.special.tests.test_legendre.TestAssocLegendreP object at 0x11a0bf750>

    def test_assoc_legendre_bug(self):
        """
        This test detects the bug reported in
        https://github.com/scipy/scipy/issues/23101
        """
        z = np.array([-1, -.5, 0, .5, 1])
        expected = assoc_legendre_p_1_0(z)
        result = assoc_legendre_p(1, 0, z)
>       assert_allclose(np.squeeze(result), expected)
E       AssertionError:
E       Not equal to tolerance rtol=1e-07, atol=0
E
E       Mismatched elements: 1 / 5 (20%)
E       Max absolute difference among violations: 2.
E       Max relative difference among violations: 2.
E        ACTUAL: array([ 1. , -0.5,  0. ,  0.5,  1. ])
E        DESIRED: array([-1. , -0.5,  0. ,  0.5,  1. ])

expected   = array([-1. , -0.5,  0. ,  0.5,  1. ])
result     = array([[ 1. , -0.5,  0. ,  0.5,  1. ]])
self       = <scipy.special.tests.test_legendre.TestAssocLegendreP object at 0x11a0bf750>
z          = array([-1. , -0.5,  0. ,  0.5,  1. ])

scipy/special/tests/test_legendre.py:79: AssertionError
============================================== short test summary info ===============================================
FAILED scipy/special/tests/test_legendre.py::TestAssocLegendreP::test_assoc_legendre_bug - AssertionError:
========================================= 1 failed, 416 deselected in 0.52s ==========================================

scipy on 🎋 bug/assoc_legendre is 📦 v1.17.0.dev0 via 🐍 via 🧚 v0.59.0 took 5s
❯ cd subprojects/xsf/

xsf on 🎋 HEAD (33768a0) is 📦 v0.1.3 via △ v4.1.2 via 🧚 v0.59.0
❯ gpc
? Select a pull request 76	OPEN Bugfix/Workaround sign error Asso. Legendre Function at z=-1 [mberz:bugfix/legendre_sign_error]
remote: Enumerating objects: 11, done.
remote: Counting objects: 100% (11/11), done.
remote: Compressing objects: 100% (3/3), done.
remote: Total 11 (delta 5), reused 10 (delta 5), pack-reused 0 (from 0)
Unpacking objects: 100% (11/11), 1.97 KiB | 155.00 KiB/s, done.
From https://github.com/scipy/xsf
 * [new ref]         refs/pull/76/head -> bugfix/legendre_sign_error
Previous HEAD position was 33768a0 REL: bump version to 0.1.3 (#67)
Switched to branch 'bugfix/legendre_sign_error'

xsf on 🎋 bugfix/legendre_sign_error is 📦 v0.1.3 via △ v4.1.2 via 🧚 v0.59.0 took 6s
❯ cd ../..

scipy on 🎋 bug/assoc_legendre [🔨] is 📦 v1.17.0.dev0 via 🐍 via 🧚 v0.59.0
❯ pixi run test -t scipy.special.tests.test_legendre "--" -k "assoc_legendre_bug"
✨ Pixi task (build in build): spin build --setup-args=-Dblas=blas --setup-args=-Dlapack=lapack --setup-args=-Duse-g77-abi=true: (Build SciPy (default settings))                                                                                                                                                         $ meson compile -j 8 -C build
INFO: autodetecting backend as ninja
INFO: calculating backend command to run: /Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy/.pixi/envs/build/bin/ninja -C /Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy/build -j 8
ninja: Entering directory `/Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy/build'
[12/12] Linking target scipy/special/_special_ufuncs.cpython-314-darwin.so
$ meson install --only-changed -C build --destdir ../build-install --tags=runtime,python-runtime,tests,devel
                                                                                                                                                             ✨ Pixi task (test in test): spin test --no-build -t scipy.special.tests.test_legendre -- -k assoc_legendre_bug: (Test SciPy (default settings))             $ export PYTHONPATH="/Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy/build-install/usr/lib/python3.14/site-packages"
$ /Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy/.pixi/envs/test/bin/python3.14 -P -c 'import scipy'
$ cd /Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy/build-install/usr/lib/python3.14/site-packages
$ /Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy/.pixi/envs/test/bin/python3.14 -P -m pytest -m 'not slow' -k assoc_legendre_bug --pyargs scipy.special.tests.test_legendre
==================================================================== test session starts ====================================================================
platform darwin -- Python 3.14.0, pytest-9.0.0, pluggy-1.6.0
rootdir: /Users/lucascolley/ghq/github.com/rgommers/pixi-dev-scipystack/scipy/scipy
configfile: pytest.ini
plugins: hypothesis-6.147.0, xdist-3.8.0, timeout-2.4.0
collected 417 items / 416 deselected / 1 selected

scipy/special/tests/test_legendre.py .                                                                                                                [100%]

============================================================= 1 passed, 416 deselected in 0.39s =============================================================