Parallel Reduction of a Block Hessenberg-Triangular Matrix Pair to Hessenberg-Triangular Form—Algorithm Design and Performance Results
نویسنده
چکیده
The design, implementation and performance of a parallel algorithm for reduction of a matrix pair in block upper Hessenberg-Triangular form (Hr, T ) to upper Hessenberg-triangular form (H, T ) is presented. This reduction is the second stage in a two-stage reduction of a regular matrix pair (A, B) to upper Hessenberg-Triangular from. The desired upper Hessenberg-triangular form is computed using two-sided Givens rotations. The parallel implementation is analyzed with regard to scalability properties and the selection of near to optimal algorithm parameters. Performance results for the ScaLAPACK-style implementation show that the parallel algorithm can be used to solve large scale problems effectively.
منابع مشابه
Reduction of a Regular Matrix Pair (A, B) to Block Hessenberg Triangular Form
An algorithm for reduction of a regular matrix pair (A; B) to block Hessenberg-triangular form is presented. This condensed form Q T (A; B)Z = (H; T), where H and T are block upper Hessenberg and upper triangular, respectively, and Q and Z orthogonal, may serve as a rst step in the solution of the generalized eigenvalue problem Ax = Bx. It is shown how an elementwise algorithm can be reorganize...
متن کاملBlocked Algorithms for Reduction of a Regular Matrix Pair to Generalized Schur Form
This contribution considers the problem of transforming a regular matrix pair (A;B) to generalized Schur form. The focus is on blocked algorithms for the reduction process that typically includes two major steps. The rst is a two-stage reduction of a regular matrix pair (A;B) to condensed form (H;T ) using orthogonal transformations Q and Z such that H = QAZ is upper Hessenberg and T = QBZ is u...
متن کاملAlgorithms for Hessenberg-Triangular Reduction of Fiedler Linearization of Matrix Polynomials
Smallto medium-sized polynomial eigenvalue problems can be solved by linearizing the matrix polynomial and solving the resulting generalized eigenvalue problem using the QZ algorithm. The QZ algorithm, in turn, requires an initial reduction of a matrix pair to Hessenberg– triangular form. In this paper, we discuss the design and evaluation of high-performance parallel algorithms and software fo...
متن کاملEfficient Algorithm for Simultaneous Reduction to the m-Hessenberg–Triangular–Triangular Form
This paper proposes an efficient algorithm for simultaneous reduction of three matrices. The algorithm is a blocked version of the algorithm described by Miminis and Page (1982) which reduces A to the m-Hessenberg form, and B and E to the triangular form. The m-Hessenberg– triangular–triangular form of matrices A, B and E is specially suitable for solving multiple shifted systems. Such shifted ...
متن کاملDistributed One-Stage Hessenberg-Triangular Reduction with Wavefront Scheduling
A novel parallel formulation of Hessenberg-triangular reduction of a regular matrix pair on distributed memory computers is presented. The formulation is based on a sequential cache-blocked algorithm by Kågström, Kressner, E.S. Quintana-Ortí, and G. QuintanaOrtí (2008). A static scheduling algorithm is proposed that addresses the problem of underutilized processes caused by two-sided updates of...
متن کامل