The return value specified in "8.1.2 associated Legendre polynomials"
of ISO/IEC JTC 1/SC 22/WG 21 N3060 (which is identical to the
expression in the doxygen comment of the patched function) is well-
defined for m>l: it is always zero because $ P_l(x) $ is a polynomial
of degree l.
The standard does not enforce an exception in this case because none of
the requirements in 8.1 (5) on page 11 of ISO/IEC JTC 1/SC 22/WG 21
N3060 are met.
Note: the implementation of st::assoc_legendre in Visual Studio 2017
(tested with Visual Studio 15.9.7) silently returns zero.
Index: libstdc++-v3/include/tr1/legendre_function.tcc
===================================================================
--- libstdc++-v3/include/tr1/legendre_function.tcc (revision 269352)
+++ libstdc++-v3/include/tr1/legendre_function.tcc (working copy)
@@ -67,13 +67,13 @@
/**
* @brief Return the Legendre polynomial by recursion on degree
* @f$ l @f$.
- *
+ *
* The Legendre function of @f$ l @f$ and @f$ x @f$,
* @f$ P_l(x) @f$, is defined by:
* @f[
* P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l}
* @f]
- *
+ *
* @param l The degree of the Legendre polynomial. @f$l >= 0@f$.
* @param x The argument of the Legendre polynomial. @f$|x| <= 1@f$.
*/
@@ -120,17 +120,17 @@
/**
* @brief Return the associated Legendre function by recursion
* on @f$ l @f$.
- *
+ *
* The associated Legendre function is derived from the Legendre function
* @f$ P_l(x) @f$ by the Rodrigues formula:
* @f[
* P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x)
* @f]
- *
+ *
* @param l The degree of the associated Legendre function.
* @f$ l >= 0 @f$.
* @param m The order of the associated Legendre function.
- * @f$ m <= l @f$.
+ * @f$ m >= 0 @f$.
* @param x The argument of the associated Legendre function.
* @f$ |x| <= 1 @f$.
* @param phase The phase of the associated Legendre function.
@@ -146,8 +146,7 @@
std::__throw_domain_error(__N("Argument out of range"
" in __assoc_legendre_p."));
else if (__m > __l)
- std::__throw_domain_error(__N("Degree out of range"
- " in __assoc_legendre_p."));
+ return _Tp(0);
else if (__isnan(__x))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (__m == 0)
@@ -192,7 +191,7 @@
/**
* @brief Return the spherical associated Legendre function.
- *
+ *
* The spherical associated Legendre function of @f$ l @f$, @f$ m @f$,
* and @f$ \theta @f$ is defined as @f$ Y_l^m(\theta,0) @f$ where
* @f[
@@ -202,7 +201,7 @@
* @f]
* is the spherical harmonic function and @f$ P_l^m(x) @f$ is the
* associated Legendre function.
- *
+ *
* This function differs from the associated Legendre function by
* argument (@f$x = \cos(\theta)@f$) and by a normalization factor
* but this factor is rather large for large @f$ l @f$ and @f$ m @f$
@@ -210,7 +209,7 @@
* and @f$ m @f$.
* @note Unlike the case for __assoc_legendre_p the Condon-Shortley
* phase factor @f$ (-1)^m @f$ is present here.
- *
+ *
* @param l The degree of the spherical associated Legendre function.
* @f$ l >= 0 @f$.
* @param m The order of the spherical associated Legendre function.
