This substantial paper provides a general theory of the unsymmetric Lanczos algorithm. The whole theory is here developed from the point of view of orthogonal polynomials and Padé approximants. A rather comprehensive introduction to this subject is therefore provided. The recurrence relations for formal orthogonal polynomials are then interpreted in a matrix setting, leading naturally to the Lanczos process.
Using the connection to Padé approximants, the block structure theorem for those is used to overcome the difficulties associated with nongeneric break-down. Connections are also established to other algorithms from the same group as Lanczos’s, such as BIORES or BIORTHORES, and bi-conjugate gradients.