Geodesic
Using successive approximations for improving the convergence of GMRES method
Applications of Mathematics, Tome 43 (1998) no. 5, pp. 321-350 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper, our attention is concentrated on the GMRES method for the solution of the system $(I-T)x=b$ of linear algebraic equations with a nonsymmetric matrix. We perform $m$ pre-iterations $y_{l+1}=Ty_l+b $ before starting GMRES and put $y_m $ for the initial approximation in GMRES. We derive an upper estimate for the norm of the error vector in dependence on the $m$th powers of eigenvalues of the matrix $T$. Further we study under what eigenvalues lay-out this upper estimate is the best one. The estimate shows and numerical experiments verify that it is advisable to perform pre-iterations before starting GMRES as they require fewer arithmetic operations than GMRES. Towards the end of the paper we present a numerical experiment for a system obtained by the finite difference approximation of convection-diffusion equations.
In this paper, our attention is concentrated on the GMRES method for the solution of the system $(I-T)x=b$ of linear algebraic equations with a nonsymmetric matrix. We perform $m$ pre-iterations $y_{l+1}=Ty_l+b $ before starting GMRES and put $y_m $ for the initial approximation in GMRES. We derive an upper estimate for the norm of the error vector in dependence on the $m$th powers of eigenvalues of the matrix $T$. Further we study under what eigenvalues lay-out this upper estimate is the best one. The estimate shows and numerical experiments verify that it is advisable to perform pre-iterations before starting GMRES as they require fewer arithmetic operations than GMRES. Towards the end of the paper we present a numerical experiment for a system obtained by the finite difference approximation of convection-diffusion equations.
DOI : 10.1023/A:1022291601664
Classification : 65F10, 65N22, 65N35
Keywords: GMRES; iterative method; numerical experiments; solution of discretized equations 
@article{10_1023_A:1022291601664,
     author = {Z{\'\i}tko, Jan},
     title = {Using successive approximations for improving the convergence of {GMRES} method},
     journal = {Applications of Mathematics},
     pages = {321--350},
     year = {1998},
     volume = {43},
     number = {5},
     doi = {10.1023/A:1022291601664},
     mrnumber = {1644136},
     zbl = {0938.65060},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1023/A:1022291601664/}
}
TY  - JOUR
AU  - Zítko, Jan
TI  - Using successive approximations for improving the convergence of GMRES method
JO  - Applications of Mathematics
PY  - 1998
SP  - 321
EP  - 350
VL  - 43
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.1023/A:1022291601664/
DO  - 10.1023/A:1022291601664
LA  - en
ID  - 10_1023_A:1022291601664
ER  - 
%0 Journal Article
%A Zítko, Jan
%T Using successive approximations for improving the convergence of GMRES method
%J Applications of Mathematics
%D 1998
%P 321-350
%V 43
%N 5
%U http://geodesic.mathdoc.fr/articles/10.1023/A:1022291601664/
%R 10.1023/A:1022291601664
%G en
%F 10_1023_A:1022291601664
Zítko, Jan. Using successive approximations for improving the convergence of GMRES method. Applications of Mathematics, Tome 43 (1998) no. 5, pp. 321-350. doi: 10.1023/A:1022291601664

Cité par Sources :