Hi team, https://github.com/scipy/scipy/pull/20320 by Irwin Zaid updates the legendre functions of the first and second kind, lpn <https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.lpn.html> and lqn <https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.lqn.html> respectively, and the associated legendre functions of the first and second kind, lpmn <https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.lpmn.html> /clpmn <https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.clpmn.html#scipy.special.clpmn> and lqmn <https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.lqmn.html> respectively, to take array arguments for the parameter z. Currently lxn(n, z) returns a tuple of arrays, each of shape (n+1, ), evaluating a legendre function at the point z for degrees 0 through n. lxmn(m, n, z) returns a tuple of arrays, each of shape (m+1, n+1) evaluating an associated legendre function at the point z for orders from 0 to m and degrees from 0 to n. #20320 updates these functions to allow z to be an ndarray, such that the output shape of lxn(n, z) will be (n+1, ) + z.shape and the output shape of lxmn(m, n, z) will be (m+1, n+1) + z.shape.
I think this is a highly valuable change, allowing efficient evaluation of these functions over multiple points. To accomplish this, Irwin worked directly with the NumPy C API to create generalized universal functions (gufuncs <https://numpy.org/doc/stable/reference/c-api/generalized-ufuncs.html>) that allow the inner loops of these functions to be run in native code. His approach extends a C++ metaprogramming approach he implemented to streamline working with the NumPy C API to create ufuncs in https://github.com/scipy/scipy/pull/20260. Our hope is to make the use of gufuncs more widespread in special, and to make them available in CuPy's special as well, along the lines of what was discussed in https://github.com/scipy/scipy/issues/19404. Feel free to share your thoughts either here or in #20320. Cheers, Albert ps. The count of lines changed in #20320 looks imposing, but it's just because kokkos/mdspan <https://github.com/kokkos/mdspan>, the reference implementation of the C++23 addition mdspan, <https://en.cppreference.com/w/cpp/container/mdspan> has been pasted into the under development special function C++ library Irwin and I are working on.
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