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  • 1.
    Kågström, Bo
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Kressner, Daniel
    Shao, Meiyue
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    On aggressive early deflation in parallel variants of the QR algorithm2012Ingår i: Applied parallel and scientific computing, PT I, Berlin, Heidelberg: Springer, 2012, s. 1-10Konferensbidrag (Refereegranskat)
    Abstract [en]

    The QR algorithm computes the Schur form of a matrix and is by far the most popular approach for solving dense nonsymmetric eigenvalue problems. Multishift and aggressive early deflation (AED) techniques have led to significantly more efficient sequential implementations of the QR algorithm during the last decade. More recently, these techniques have been incorporated in a novel parallel QR algorithm on hybrid distributed memory HPC systems. While leading to significant performance improvements, it has turned out that AED may become a computational bottleneck as the number of processors increases. In this paper, we discuss a two-level approach for performing AED in a parallel environment, where the lower level consists of a novel combination of AED with the pipelined QR algorithm implemented in the ScaLAPACK routine PDLAHQR. Numerical experiments demonstrate that this new implementation further improves the performance of the parallel QR algorithm.

  • 2.
    Kågström, Bo
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Kressner, Daniel
    Shao, Meiyue
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    On aggressive early deflation in parallel variants of the QR algorithm2011Konferensbidrag (Refereegranskat)
    Abstract [en]

    The QR algorithm computes the Schur form of a matrix and is by far the most popular approach for solving dense nonsymmetric eigenvalue problems. Multishift and aggressive early deflation (AED) techniques have led to significantly more efficient sequential implementations of the QR algorithm during the last decade. More recently, these techniques have been incorporated in a novel parallel QR algorithm on hybrid distributed memory HPC systems. While leading to significant performance improvements, it has turned out that AED may become a computational bottleneck as the number of processors increases. In this paper, we discuss a two-level approach for performing AED in a parallel environment, where the lower level consists of a novel combination of AED with the pipelined QR algorithm implemented in the ScaLAPACK routine PDLAHQR. Numerical experiments demonstrate that this new iplementation further improves the performance of the parallel QR algorithm.

  • 3.
    Shao, Meiyue
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Högpresterande beräkningscentrum norr (HPC2N). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Gao, Weiguo
    Xue, Jungong
    Aggressively truncated Taylor series method for accurate computation of exponentials of essentially nonnegative matrices2014Ingår i: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 35, nr 2, s. 317-338Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Small relative perturbations to the entries of an essentially nonnegative matrix introduce small relative errors to entries of its exponential. It is thus desirable to compute the exponential with high componentwise relative accuracy. Taylor series approximation coupled with scaling and squaring is used to compute the exponential of an essentially nonnegative matrix. An a priori componentwise relative error bound of truncation is established, from which one can choose the degree of Taylor series expansion and the scale factor so that the exponential is computed with desired componentwise relative accuracy. To reduce the computational cost, the degree of the Taylor series expansion is chosen small, while the scale factor is chosen sufficiently large to achieve the desired accuracy. The rounding errors in the squaring stage are not serious as squaring is forward stable for nonnegative matrices. We also establish a posteriori componentwise error bounds and derive a novel interval algorithm for the matrix exponential of an essentially nonnegative matrix. Rounding error analysis and numerical experiments demonstrate the efficiency and accuracy of the proposed methods.

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