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Oblique Projection Methods.
In an oblique projection method we are given two subspaces and and
seek an approximation and an element
of that satisfy the PetrovGalerkin condition,

(12) 
The subspace will be referred to as the right subspace and
as the left subspace.
A procedure similar to the RayleighRitz procedure can be devised, and
this can be conveniently described in matrix form by
expressing the approximate eigenvector in matrix form with respect to
some basis and formulating the PetrovGalerkin
conditions eq:4.21 for the basis of . This time we will need
two bases, one which we denote by for the subspace and the other,
denoted by , for the subspace . We assume that these two bases are
biorthogonal, i.e., that
or
where is the identity matrix.
Then, writing as before, the above PetrovGalerkin condition
yields the same approximate problem as eq:4.18 except that the matrix
is now defined by
In order for a biorthogonal pair to exist, the following
additional assumption for and must hold.
For any two bases and , of and , respectively,

(13) 
Obviously this condition
does not depend on the particular bases selected and it is
equivalent to requiring that no vector of be orthogonal to .
The approximate problem obtained for oblique projection methods has
the potential of being much worse conditioned than with orthogonal
projection methods. Therefore, one may wonder whether there is any
need for using oblique projection methods.
However, methods based on oblique projectors can offer
some advantages. In particular, they may be able to compute good
approximations to left as well as right eigenvectors
simultaneously. As will be seen later, there are also methods based on
oblique projection techniques which require far less storage than
similar orthogonal projections methods.
The disadvantages of working with the nonorthogonal and can be
reduced by combining this technique with a few steps of a (more
expensive) orthogonal projection method.
Next: Harmonic Ritz Values.
Up: Basic Ideas Y. Saad
Previous: Orthogonal Projection Methods.
Contents
Index
Susan Blackford
20001120