Volume 47, pp. 127-152, 2017.

An optimal Q-OR Krylov subspace method for solving linear systems

Gérard Meurant


Today the most popular iterative methods for solving nonsymmetric linear systems are Krylov methods. In this paper we show how to construct a non-orthogonal basis of the Krylov subspace such that the quasi-orthogonal residual (Q-OR) Krylov method using this basis yields the same residual norms as GMRES up to the final stagnation phase, provided GMRES is not stagnating. In many examples this new Krylov method gives a better maximum attainable accuracy than GMRES with a modified Gram-Schmidt (MGS) implementation. Even though the number of floating point operations per iteration is larger than for GMRES, the optimal Q-OR method offers more potential for parallelism than GMRES with MGS.

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Key words

linear systems, Krylov methods, Q-OR algorithm

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