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A completed theory of the unsymmetric Lanczos process and related algorithms. II. (English) Zbl 0809.65028

In the first part [ibid. 13, No. 2, 594-639 (1992; Zbl 0760.65039)], the author extended the “unsymmetric” Lanczos bi-orthogonalization algorithm to the non-generic case with the help of a connection with Padé approximants, thus being able to cure the non-generic breakdown. The present paper extends this program for the biconjugate gradient or BIOMIN and for the related BIODIR method in an analogous way.

Again the breakdowns of these methods can be cured by using also non- regular formal orthogonal polynomials. The sequence of formal orthogonal polynomials corresponds to a diagonal in the Padé table. New recurrences are derived for sequences of formal orthogonal polynomials belonging to two adjacent diagonals of the Padé table. Finally, the cure for exact breakdown is extended to the case of near-breakdown.

MSC:
65F10Iterative methods for linear systems
65F15Eigenvalues, eigenvectors (numerical linear algebra)
41A21Padé approximation