[libcxx-commits] [libcxx] [libc++][math] Mathematical Special Functions: Hermite Polynomial (PR #89982)

[via libcxx-commits]

================
@@ -765,6 +766,53 @@ _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 _Tp __constexpr_scalbn(_Tp _
   return __builtin_scalbn(__x, __exp);
 }
 
+#if _LIBCPP_STD_VER >= 17
+
+/// \return the Hermite polynomial \f$ H_n(x) \f$
+/// \note The implementation is based on the recurrence formula
+/// \f[
+///   H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}
+/// \f]
+/// Press, William H., et al. Numerical recipes 3rd edition: The art of
+/// scientific computing. Cambridge university press, 2007, p. 183.
+template <class _Real>
+_LIBCPP_HIDE_FROM_ABI _Real __hermite(unsigned __n, _Real __x) {
+  if (std::isnan(__x))
+    return std::numeric_limits<_Real>::quiet_NaN();
+
+  _Real __H_0{1};
+  if (__n == 0)
+    return __H_0;
+
+  _Real __H_n_prev = __H_0;
+  _Real __H_n      = 2 * __x;
+  for (unsigned __i = 1; __i < __n; ++__i) {
+    _Real __H_n_next = 2 * (__x * __H_n - __i * __H_n_prev);
+    __H_n_prev       = __H_n;
+    __H_n            = __H_n_next;
+  }
+  return __H_n;
----------------
lntue wrote:

+1 to everything in your comment.
- For the first point a simple non-trivial (might be not that useful) example is when `x = std::numeric_limits<_Real>::max()`, following the iterations, you will get that:
```
  hermite(0, x) = 1.0
  hermite(1, x) = +Inf
  hermite(2, x) = +Inf
  hermite(3, x) = NaN
  ...
```
and there are definitely more intricate cases.
- Another issue that we might want to address later is precision loss.  I haven't looked into the stability of the iterations, but we might have catastrophic cancellation when `__x * __H_n ~ __i * __H_n_prev`.  It is interesting to find some non-trivial examples, but we can defer dealing with this case to later PR.

https://github.com/llvm/llvm-project/pull/89982