5.2.1 Sequential algorithm

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5.2.1 Sequential algorithm

For solving a system of linear equations with a non-symmetric matrix

( $K \neq K^T$ ) one can use the General Conjugate Residual Method. If

is additionally not positive definite, i.e., $(K\underline{v},\underline{v})_{L_2} \not > 0$ , then a variety of methods like GMRES, QMR, Bi-cg, Bi-cgstab are available.
We will investigate the GMRES method based on the Arnoldi iteration which is a generalization of the Lanczos method.
$\begin{algorithmus} % latex2html id marker 14942 [H] \caption{Sequential GMRES w... ...}^0 + \sum\limits_{i=1}^k z_i\cdot \underline{w}^j \end{array}$\end{algorithmus}$
The number of stored vectors $\underline{w}^k$ and entries in matrix $H=\{h_{i,j}\}_{i,j=\overline{1,k} \makebox[0pt]{}}$ increases with the number of iterations

and may cause a memory overflow. To prevent this unexpected termination of the iteration the REPEAT-loop will be stopped after

iterations and restarted with $\underline{u}^m$ as initial solution. This shortening of the cycle leads to an unstable behavior of the method.

For preconditioning techniques see section 5.8.

Gundolf Haase 2000-03-20