Lanczos-type product methods

The abbreviation LTPM stands for the English designation Lanczos-type product methods , which represents a class of methods based on the (asymmetrical) Lanczos method for solving linear systems of equations with large, sparse matrices . LTPM represent a subclass of the Krylow space method, which provides methods that are particularly applicable to asymmetrical matrices, manage with a Krylow space (thus do not need the transpose ) and are based on short recursions. ${\displaystyle Ax=b}$ ${\displaystyle A}$

The methods contained in this class calculate iterates whose residuals can be described as the product of the residual polynomials of the BiCG method with other polynomials of the same degree times the first residual vector . These second polynomials are also used to classify the LTPM. ${\displaystyle r=b-Ax_{0}}$

An approximately chronological incomplete list of the LTPM is as follows:

1. IDR , Peter Sonneveld, approx. 1980,
2. CGS , Peter Sonneveld, 1984/1989,
3. BiCGSTAB , Henk van der Vorst , 1992,
4. BiCGSTAB2 , Martin H. Gutknecht, 1993
5. TFQMR , Roland W. Freund,
6. BiCGSTAB (l) , Diederik Fokkema, Gerard LG Sleijpen,
7. CGS2 , Diederik Fokkema, Gerard LG Sleijpen, Henk A. van der Vorst,
8. shifted CGS , Diederik Fokkema, Gerard LG Sleijpen, Henk A. van der Vorst,
9. QMRCGSTAB , Tony F. Chan, E. Gallopoulos, Valeria Simoncini, T. Szeto, Charles H. Tong, 1994
10. GPBiCG , Zhang

What all the procedures have in common is that they collapse exactly when BiCG collapses.

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