Oblique Projection Methods.

In an oblique projection method we are given two subspaces $\LL$ and $\KK$ and seek an approximation $\tlu \in \KK$ and an element $\tilde \lambda$ of ${\cal C}$ that satisfy the Petrov-Galerkin condition,

$$v^{\ast} (A - \tilde \lambda I) \tlu = 0 \quad \forall \ v \in \LL \ .$$ (12)

The subspace $\KK$ will be referred to as the right subspace and $\LL$ as the left subspace. A procedure similar to the Rayleigh-Ritz procedure can be devised, and this can be conveniently described in matrix form by expressing the approximate eigenvector $\tlu$ in matrix form with respect to some basis and formulating the Petrov-Galerkin conditions eq:4.21 for the basis of $\LL$. This time we will need two bases, one which we denote by $V$ for the subspace $\KK$ and the other, denoted by $W$, for the subspace $\LL$. We assume that these two bases are biorthogonal, i.e., that $w^{\ast}_i v_j = \delta_{ij}$ or

$$W^{\ast} V\ =\ I ,$$

where $I$ is the identity matrix. Then, writing $\tlu = V y$ as before, the above Petrov-Galerkin condition yields the same approximate problem as eq:4.18 except that the matrix $B_m$ is now defined by

$$B_m \ =\ W ^{\ast} A V .$$

In order for a biorthogonal pair $V, W$ to exist, the following additional assumption for $\LL$ and $\KK$ must hold.

For any two bases $V$ and $W$, of $\KK$ and $\LL$, respectively,

$$\det ( W ^{\ast} V ) \ne 0 .$$ (13)

Obviously this condition does not depend on the particular bases selected and it is equivalent to requiring that no vector of $\KK$ be orthogonal to $\LL$.

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 $V$ and $W$ can be reduced by combining this technique with a few steps of a (more expensive) orthogonal projection method.