| Title:
|
Derivation of BiCG from the conditions defining Lanczos' method for solving a system of linear equations (English) |
| Author:
|
Tichý, Petr |
| Author:
|
Zítko, Jan |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
43 |
| Issue:
|
5 |
| Year:
|
1998 |
| Pages:
|
381-388 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Lanczos’ method for solving the system of linear algebraic equations $Ax=b$ consists in constructing a sequence of vectors $x_k$ in such a way that $r_k=b-Ax_k \in r_0+A{\mathcal K}_{k}(A,r_0)$ and $r_k \perp {\mathcal K}_{k}(A^T,\widetilde{r}_0)$. This sequence of vectors can be computed by the BiCG (BiOMin) algorithm. In this paper is shown how to obtain the recurrences of BiCG (BiOMin) directly from this conditions. (English) |
| Keyword:
|
biorthogonalization |
| Keyword:
|
linear equations |
| Keyword:
|
biconjugate gradient method |
| MSC:
|
65F10 |
| MSC:
|
65F25 |
| idZBL:
|
Zbl 0938.65061 |
| idMR:
|
MR1644128 |
| DOI:
|
10.1023/A:1022238402573 |
| . |
| Date available:
|
2009-09-22T17:58:47Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134394 |
| . |
| Reference:
|
[Brezn–94] C. Brezinski, M. Redivo-Zaglia: Treatment of near-breakdown in the CGS algorithm.Numerical Algorithms 7 (1994), 33–73. MR 1283334, 10.1007/BF02141260 |
| Reference:
|
[Fletch–76] R. Fletcher: Conjugate gradient methods for indefinite systems.Numerical Analysis, Dundee, 1975, G. A. Watson (ed.), Vol. 506 of Lecture Notes in Mathematics, Springer, Berlin, 1976. Zbl 0326.65033, MR 0461857 |
| Reference:
|
[Gutkn–97] M. H. Gutknecht: Lanczos-type Solvers for Nonsymmetric Linear Systems of Equations.Technical Report TR-97-04, Swiss Center for Scientific Computing ETH-Zentrum, Switzerland, 1997. Zbl 0888.65030, MR 1489258 |
| Reference:
|
[Lancz–50] C. Lanczos: An iteration method for the solution of eigenvalue problem of linear differential and integral operators.J. Res. Nat. Bureau Standards 45 (1950). MR 0042791, 10.6028/jres.045.026 |
| Reference:
|
[Lancz–52] C. Lanczos: Solution of systems of linear equations by minimized iterations.J. Res. Nat. Bureau Standards 49 (1952). MR 0051583, 10.6028/jres.049.006 |
| Reference:
|
[Leary–80] D. P. O‘Leary: The block conjugate gradient algorithm.Linear Algebra Appl. 99 (1980), 293–322. MR 0562766 |
| Reference:
|
[Tichý–97] P. Tichý: Behaviour of BiCG and CGS algorithms.Mgr. thesis, Department of Numerical Mathematics, Faculty of Mathematics and Physics Praha, 1997. |
| Reference:
|
[Weiss–95] R. Weiss: A theoretical overview of Krylov subspace methods.Applied Numerical Mathematics 19 (1995), 207–233. Zbl 0854.65031, MR 1374350, 10.1016/0168-9274(95)00084-4 |
| . |
