Martin H. Gutknecht

An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices

The nonsymmetric Lanczos method can be used to compute eigenvalues of large sparse non-Hermitian matrices or to solve large sparse non-Hermitian linear systems. However, the original Lanczos algorithm is susceptible to possible breakdowns and potential instabilities. An implementation of a look-ahead version of the Lanczos algorithm is presented that, except for the very special situation of an incurable breakdown, overcomes these problems by skipping over those steps in which a breakdown or near-breakdown would occur in the standard process. The proposed algorithm can handle look-ahead steps of any length and requires the same number of matrix–vector products and inner products as the standard Lanczos process without look-ahead.

Variants of BICGSTAB for matrices with complex spectrum

Recently Van der Vorst [SIAM J. Sci. Statist. Comput.,13 (1992), pp. 631–644] proposed for solving nonsymmetric linear systems

Lanczos-type solvers for nonsymmetric linear systems of equations

Az = b

The Carathéodory–Fejér Method for Real Rational Approximation

a biconjugate gradient (BICG)-based Krylov space method called BICGSTAB that, like the biconjugate gradient squared (BICGS) method of Sonneveld, does not require matrix–vector multiplications with the transposed matrix

Accuracy of Two Three-term and Three Two-term Recurrences for Krylov Space Solvers

A^T

Numerical conformal mapping methods based on function conjugation

, and that has typically a much smoother convergence behavior than BICG and BICGS. Its nth residual polynomial is the product of the one of BICG (i.e., the nth Lanczos polynomial) with a polynomial of the same degree with real zeros. Therefore, nonreal eigenvalues of A are not approximated well by the second polynomial factor. Here, the author presents for real nonsymmetric matrices a method BICGSTAB2 in which the second factor may have complex conjugate zeros. Moreover, versions suitable for complex matrices are given for both methods.

Stable row recurrences for the Padé table and generically superfast lookahead solvers for non-Hermitian Toeplitz systems

Among the iterative methods for solving large linear systems with a sparse (or, possibly, structured) nonsymmetric matrix, those that are based on the Lanczos process feature short recurrences for the generation of the Krylov space. This means low cost and low memory requirement. This review article introduces the reader not only to the basic forms of the Lanczos process and some of the related theory, but also describes in detail a number of solvers that are based on it, including those that are considered to be the most efficient ones. Possible breakdowns of the algorithms and ways to cure them by look-ahead are also discussed.

A Framework for Deflated and Augmented Krylov Subspace Methods

A “Caratheodory–Fejer method” is presented for near-best real rational approximation on intervals, based on the eigenvalue (or singular value) analysis of a Hankel matrix of Chebyshev coefficients. In approximation of a smooth function F, the CF approximant

Solving Theodorsen's integral equation for conformal maps with the fast fourier transform and various nonlinear iterative methods

R^{cf}

The CF Table

frequently differs from the best approximation