Summary
Stiefel's theory of optimal relaxation methods is applied to study the behavior of the error, measured by (the square of) an energy-type norm, as the number of iteration steps tends to infinity. It is shown how certain features of the initial residual vector affect the rate of convergence. Of particular interest are cases in which the higher-order components of the initial residual vector, in the coordinate system of principal axes, are more and more attenuated.
Zusammenfassung
Stiefels Theorie der optimalen Relaxationsverfahren wird angewandt, um das Verhalten des Fehlers zu studieren, wenn die Zahl der Iterationsschritte nach Unendlich geht, wobei der Fehler durch das Quadrat einer Energie-Norm gemessen wird. Es wird gezeigt, wie gewisse Eigenschaften des Anfangsresiduumvektors die Konvergenzgeschwindigkeit beeinflussen. Von besonderem Interesse sind Fälle, in welchen die höheren Komponenten des Anfangsresiduumvektors in den Koordinaten der Hauptachsen mehr und mehr abgeschwächt sind.
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The work of the first author was supported, in part, by the National Science Foundation under Grant MCS-7927158. The work of the second author was supported, in part, by the National Science Foundation under Grant MCS-7610225.
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Gautschi, W., Lynch, R.E. Error behavior in optimal relaxation methods. Z. angew. Math. Phys. 33, 24–35 (1982). https://doi.org/10.1007/BF00948310
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DOI: https://doi.org/10.1007/BF00948310