https://gcc.gnu.org/bugzilla/show_bug.cgi?id=83140
Bug ID: 83140
Summary: assoc_legendre returns negated value when m is odd
Product: gcc
Version: 7.1.0
Status: UNCONFIRMED
Severity: normal
Priority: P3
Component: libstdc++
Assignee: unassigned at gcc dot gnu.org
Reporter: kwsm.gcc at gmail dot com
Target Milestone: ---
In the standard, Legendre polynomials and associated Legendre functions are
defined as
P_l = (1 / 2^l l!) d^l/dx^l (x^2 - 1)^l,
and
P_l^m = (1 - x^2)^(m/2) d^m/dx^m P_l,
respectively.
Then, for example, P_1^1 should be √(1 - x^2), but GCC returns -√(1 - x^2): the
return value of std::assoc_legendre(1, 1, 0.5) is -0.866.
GCC's implementation uses recurrence of associated Legendre function
P_m^m = (-1)^m (2m - 1)!! (1 - x^2)^(m/2),
but this corresponds to
P_l^m = (-1)^m (1 - x^2)^(m/2) d^m/dx^m P_l.
Namely, std::assoc_legendre returns negated value when m is odd.
