ENH: Improve the computation of polynomials from roots.

The original method was overly sensitive to roundoff. Of the two
approaches considered, gauss integration or binary subdivision of
the roots, the latter is more compatible with using other number
representations such as mpmath. No method is going to be suitable
for large numbers of arbitrary zeros but the current method is a
significant improvement.

1 files changed, 11 insertions, 4 deletions

diff --git a/numpy/polynomial/legendre.py b/numpy/polynomial/legendre.py
index 8fd64985b..b76e678c2 100644
--- a/numpy/polynomial/legendre.py
+++ b/numpy/polynomial/legendre.py
@@ -314,10 +314,17 @@ def legfromroots(roots) :
         return np.ones(1)
     else :
         [roots] = pu.as_series([roots], trim=False)
-        prd = np.array([1], dtype=roots.dtype)
-        for r in roots:
-            prd = legsub(legmulx(prd), r*prd)
-        return prd
+        roots.sort()
+        n = len(roots)
+        p = [legline(-r, 1) for r in roots]
+        while n > 1:
+            m = n//2
+            tmp = [legmul(p[i], p[i+m]) for i in range(m)]
+            if n%2:
+                tmp[0] = legmul(tmp[0], p[-1])
+            p = tmp
+            n = m
+        return p[0]
 
 
 def legadd(c1, c2):