The Unsymmetric Lanczos Algorithms and Their Relations to Pad E Approximation Continued Fractions and the Qd Algorithm

First several algorithms based on the unsymmetric Lanczos process are surveyed the biorthog onalization BO algorithm for constructing a tridiagonal matrix T similar to a given matrix A whose ex treme spectrum is sought typically the BOBC algorithm which generates directly the LU factors of T the Biores Lanczos Orthores Biomin Lanczos Orthomin or biconjugate gradient BCG and the Biodir Lanczos Orthodir algorithms for solving a nonsymmetric system of linear equations The possibilities of breakdown in these algorithms are discussed and brought into relation Then the connections to formal orthogonal polynomials Pad e approximation continued fractions and the qd algorithm are reviewed They allow us to deapen our understanding of breakdowns Next three types of bi conjugate gradient squared CGS algorithms are presented Biores Biomin standard CGS and Biodir Finally fast Hankel solvers related to the Lanczos process are described