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Re: QR vs. Gram Schmidt

  • To: mathgroup at
  • Subject: [mg20742] Re: QR vs. Gram Schmidt
  • From: Jens-Peer Kuska <kuska at>
  • Date: Wed, 10 Nov 1999 00:17:52 -0500
  • Organization: Universitaet Leipzig
  • References: <7vrbdh$>
  • Sender: owner-wri-mathgroup at

Hi Greg,

I would like to see the QR decomposition of the vectors
{1,x,x^2,x^3, ...}

with InnerProduct[f_,g_]:=Integrate[f[x]*g[x] *Exp[-x],{x,0,Infinity}]


garnold wrote:
> Here's something interesting... QR & Gram Schmidt are fundamentally the same
> thing.  They are exactly the same for a #rows>=#cols and trivially the same
> for #rows<#cols.
> So... why is GramSchmidt a separate package that must be loaded in?  QR is
> MUCH faster at least in the Mathematica 4.0 implementation.  Of course this
> is expected since QR is in the kernel.
> So... I have 2 related questions:
> (1) Is one algorithm theoretically faster than the other?
> (2) Does anybody have a good reference for understanding the QR algorithm
> when #rows > # cols? (Mathematica claims it uses Hausdorf transformations,
> but I don't understand this since Q is no longer square).
> Thanks!
> Greg

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