[Scipy-tickets] [SciPy] #1466: More accurate roots/weights for Gauss-Hermite quadrature

SciPy Trac scipy-tickets@scipy....
Mon Jul 4 15:03:31 CDT 2011

#1466: More accurate roots/weights for Gauss-Hermite quadrature
 Reporter:  Bogdan         |       Owner:  pv         
     Type:  enhancement    |      Status:  new        
 Priority:  normal         |   Milestone:  Unscheduled
Component:  scipy.special  |     Version:  devel      
 Keywords:                 |  

Comment(by charris):

 I've played with this a bit more.

 1) Numpy's Hermite zero finder dies up around deg 70. I need to symmetrize
 the companion matrix

 2) Scipy needs to use eigh instead of eig, but this doesn't help much. The
 eigenvectors are also off.

 3) Using zeros from Scipy, improving them with Newton, and using the w =
 a/(H_{n}'(r)*H_{n-1}), where a is chosen so that the w sum up to sqrt(pi),
 works well.

 Note that the routine form NR also uses Newton. I've attached a plot with
 the weights from the three (eig, eigh, improved), where the improved
 behavior for degree 100 is easy to see.

Ticket URL: <http://projects.scipy.org/scipy/ticket/1466#comment:3>
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