Another Proof for Modified Gram-schmidt with Reorthogonalization
نویسندگان
چکیده
In this note, we consider the modified Gram-Schmidt algorithm with reorthogonalization applied on a numerical nonsingular matrix, we explain why the resulting set of vectors is orthogonal up to the machine precision level. To establish this result, we show that a certain L-criterion is necessarily verified after the second reorthogonalization step, then we prove that this L-criterion implies the desired level of orthogonality. If the L-criterion is verified after the first orthogonalization step, then there is no need to reorthogonalize. From this simple observation, we deduce that the L-criterion is an interesting selective reorthogonalization criterion for modified Gram-Schmidt algorithm. AMS Subject Classification : 65F25, 65G50, 15A23.
منابع مشابه
A Robust Criterion for the Modified Gram-Schmidt Algorithm with Selective Reorthogonalization
A new criterion for selective reorthogonalization in the modified Gram–Schmidt algorithm is proposed. We study its behavior in the presence of rounding errors. We give some counterexample matrices which prove that the standard criteria might fail. Through numerical experiments, we illustrate that our new criterion seems to be suitable also for the classical Gram– Schmidt algorithm with selectiv...
متن کاملRobust Selective Gram-schmidt Reorthogonalization
A new criterion for selective reorthogonalization in the Gram-Schmidt procedure is given. We establish its comportment in presence of rounding errors when the criterion is used with modified Gram-Schmidt algorithm and show counter-example matrices which prove that standard criteria are not always valid. Experimentally, our criterion is fine also for the classical Gram-Schmidt algorithm with reo...
متن کاملPerformance Evaluation of Some Inverse Iteration Algorithms on PowerXCell 8i Processor
In this paper, we compare with the inverse iteration algorithms on PowerXCell 8i processor, which has been known as a heterogeneous environment. When some of all the eigenvalues are close together or there are clusters of eigenvalues, reorthogonalization must be adopted to all the eigenvectors associated with such eigenvalues. Reorthogonalization algorithms need a lot of computational cost. The...
متن کاملA reorthogonalization procedure for modified Gram–Schmidt algorithm based on a rank-k update
The modified Gram–Schmidt algorithm is a well–known and widely used procedure to orthogonalize the column vectors of a given matrix. When applied to ill–conditioned matrices in floating point arithmetic, the orthogonality among the computed vectors may be lost. In this work, we propose an a posteriori reorthogonalization technique based on a rank–k update of the computed vectors. The level of o...
متن کاملPerformance Evaluation of Golub-Kahan-Lanczos Algorithm with Reorthogonalization by Classical Gram-Schmidt Algorithm and OpenMP
The Golub-Kahan-Lanczos algorithm with reorthogonalization (GKLR algorithm) is an algorithm for computing a subset of singular triplets for large-scale sparse matrices. The reorthogonalization tends to become a bottleneck of elapsed time, as the iteration number of the GKLR algorithm increases. In this paper, OpenMP-based parallel implementation of the classical Gram-Schmidt algorithm with reor...
متن کامل