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I/O Efficient Algorithms for Block Hessenberg Reduction Using Panel Approach

  • Sraban Kumar Mohanty
  • Gopalan Sajith
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7678)

Abstract

Reduction to Hessenberg form is a major performance bottleneck in the computation of the eigenvalues of a nonsymmetric matrix; which takes O(N 3) flops. All the known blocked and unblocked direct Hessenberg reduction algorithms have an I/O complexity of O(N 3/B). To improve the performance by incorporating matrix-matrix operations in the computation, usually the Hessenberg reduction is computed in two steps: the first reducing the matrix to a banded Hessenberg form, and the second further reducing it to Hessenberg form. We propose and analyse the first step of the reduction, i.e., reduction of a nonsymmetric matrix to banded Hessenberg form of bandwidth t for varying values of N and M (the size of the internal memory), on external memory model introduced by Aggarwal and Vitter for the I/O complexity and show that the reduction can be performed in \(O(N^3/\min\{t,\sqrt{M}\}B)\) I/Os.

Keywords

Large Matrix Computation External Memory Algorithms Out-of-Core Algorithms Matrix Computations Hessenberg Reduction I/O Efficient Eigenvalue Problem 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sraban Kumar Mohanty
    • 1
  • Gopalan Sajith
    • 2
  1. 1.Computer Science & Engineering DisciplinePDPM Indian Institute of Information Technology, Design and Manufacturing JabalpurJabalpurIndia
  2. 2.Computer Science & Engineering DepartmentIndian Institute of Technology GuwahatiGuwahatiIndia

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