TABLE OF CONTENTS The previous chapter considered a number of Krylov subspace methods that relied on some form of orthogonalization of the Krylov vectors in order to compute an approximate solution. This chapter will describe a class of Krylov subspace methods that are instead based on a biorthogonalization algorithm due to Lanczos. These are projection methods that are intrinsically nonorthogonal. They have some appealing properties, but are harder to analyze theoretically.
The Lanczos biorthogonalization algorithm is an extension to nonsymmetric matrices of the symmetric Lanczos algorithm seen in the previous chapter. One such extension, the Arnoldi procedure, has already been seen. However, the nonsymmetric Lanczos algorithm is quite different in concept from Arnoldi's method because it relies on biorthogonal sequences instead of orthogonal sequences.
The algorithm proposed by Lanczos for nonsymmetric matrices builds a pair of biorthogonal bases for the two subspaces
and
The algorithm that achieves this is the following.
Choose two vectors v 1, w 1 such that ( v 1, w 1) = 1
Set ? 1 = ? 1 ? 0, w 0 = v 0 ? 0
For j = 1, 2, , m, Do
a j = ( Av j, w j)
If ? j + 1 = 0 Stop
EndDo
Note that there are numerous ways to choose the scalars ? j+1, ? j+1 in lines...
