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Thread: Lanczos Biorthogonalization

  1. #1
    Nov 2006

    Lanczos Biorthogonalization


    Can someone possibly explain or direct me to a source that explains (though not Lanczos's original work) Lanczos biorthogonalization as a means for building a basis for a Krylov subspace? The Arnoldi/Gram-Schmidt approach is very easy to understand. However, I can't really follow the justification behind building a basis using the Lanczos technique. Why is it okay to biorthogonalize? What makes that doable? And why is the result a basis? Clearly, it's not an orthonormal basis like that one obtains from the Arnoldi technique. But I'm not clear on the justification for building the basis this way in the first place.

    Several texts I've read, including Y. Saad's book basically gloss over it, saying that you build two simultaneous bases, one for the Krylov space $\displaystyle (Ar_0|v)$ and another for $\displaystyle (A^T r_0|w)$, such that $\displaystyle (v_i, w_j)=\delta_{ij}$.

    Thanks for any help
    Last edited by fade; Aug 27th 2008 at 09:59 AM.
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