Next: Invert .
Up: Generalized Non-Hermitian Eigenvalue Problems
Previous: Direct Methods
A common approach for the numerical solution of the large sparse generalized
eigenvalue problem (8.1) is to transform the problem first
to an equivalent standard eigenvalue problem and then to apply an appropriate
iterative method as described in Chapter 7.
In this section, we will
discuss three approaches for the transformation to a standard eigenproblem.
The first approach (invert ) is recommended
only if the matrix has
a very simple structure and when linear
systems of equations with matrices and can be solved
The second approach (split-invert )
is suitable when the matrix is Hermitian
positive definite and when the Cholesky decomposition of can be
computed efficiently a priori. For numerical stability,
these two approaches require that the matrix
The third approach is the
shift-and-invert spectral transformation (SI), and this is the most common
approach. We recommend using it whenever possible.
Transformation to Standard Problems