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/laguerre.py b/numpy/polynomial/laguerre.py
index fc9afec00..866aab5c4 100644
--- a/numpy/polynomial/laguerre.py
+++ b/numpy/polynomial/laguerre.py
@@ -283,10 +283,17 @@ def lagfromroots(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 = lagsub(lagmulx(prd), r*prd)
-        return prd
+        roots.sort()
+        n = len(roots)
+        p = [lagline(-r, 1) for r in roots]
+        while n > 1:
+            m = n//2
+            tmp = [lagmul(p[i], p[i+m]) for i in range(m)]
+            if n%2:
+                tmp[0] = lagmul(tmp[0], p[-1])
+            p = tmp
+            n = m
+        return p[0]
 
 
 def lagadd(c1, c2):