Notes and References

The symmetric indefinite Lanczos procedure was first presented in [357].

The symmetric indefinite Lanczos procedure shares many of the same characteristics of the non-Hermitian Lanczos procedure. These two algorithms are intimately related. The symmetric indefinite Lanczos algorithm may be viewed as a special case of the non-Hermitian Lanczos procedure where the starting vectors have been chosen appropriately to take advantage of the symmetry of the underlying problem [,173,91,92,357,,]. On the other hand, when $H^{-1} B$ is diagonalizable, the non-Hermitian Lanczos procedure may be viewed as a modified version of the indefinite symmetric Lanczos procedure.

Many of the challenges of working with an algorithm such as the symmetric indefinite Lanczos algorithm, particularly the breakdown phenomenon, may be explained in the context of metric geometry. A ``metric'' in this sense refers to the indefinite inner product $\langle\cdot,\cdot\rangle_B$ determined by the symmetric matrix $B$. Detailed descriptions of the mathematical properties of vector spaces equipped with an indefinite inner product may be found in [20,417].

Other iterative methods, e.g. subspace iteration, have been adapted to take advantage of the symmetry of a symmetric indefinite pencil[].

For very large problems which may involve a significant amount of data transfer to and from secondary storage, it may be advantageous to implement a block version of the symmetric indefinite Lanczos procedure. A block version also permits the use of adaptive blocking to treat breakdowns [29,298].