Parallel and Blocked Algorithms for Reduction of a Regular Matrix Pair to Hessenberg-Triangular and Generalized Schur Forms

A parallel three-stage algorithm for reduction of a regular matrix pair (A, B) to generalized Schur from (S, T) is presented. The first two stages transform (A, B) to upper Hessenberg-triangular form (H, T) using orthogonal equivalence transformations. The third stage iteratively reduces the matrix in (H, T) form to generalized Schur form. Algorithm and implementation issues regarding the single-/double-shift QZ algorithm are discussed. We also describe multishift strategies to enhance the performance in blocked as well as in parallell variants of the QZ method.

[1]  Isak Jonsson,et al.  Recursive blocked algorithms for solving triangular systems—Part II: two-sided and generalized Sylvester and Lyapunov matrix equations , 2002, TOMS.

[2]  Bruno Lang,et al.  Using Level 3 BLAS in Rotation-Based Algorithms , 1998, SIAM J. Sci. Comput..

[3]  Bo Kågström,et al.  LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs , 1994, TOMS.

[4]  Krister Dackland,et al.  A ScaLAPACK-Style Algorithm for Reducing a Regular Matrix Pair to Block Hessenberg-Triangular Form , 1998, PARA.

[5]  Isak Jonsson,et al.  Recursive blocked algorithms for solving triangular systems—Part I: one-sided and coupled Sylvester-type matrix equations , 2002, TOMS.

[6]  Karen S. Braman,et al.  The Multishift QR Algorithm. Part II: Aggressive Early Deflation , 2001, SIAM J. Matrix Anal. Appl..

[7]  James Demmel,et al.  On a Block Implementation of Hessenberg Multishift QR Iteration , 1989, Int. J. High Speed Comput..

[8]  Krister Dackland,et al.  Blocked algorithms and software for reduction of a regular matrix pair to generalized Schur form , 1999, TOMS.

[9]  Gene H. Golub,et al.  Linear algebra for large scale and real-time applications , 1993 .

[10]  Karen S. Braman,et al.  The Multishift QR Algorithm. Part I: Maintaining Well-Focused Shifts and Level 3 Performance , 2001, SIAM J. Matrix Anal. Appl..

[11]  Bo Kågström,et al.  Distributed and Shared Memory Block Algorithms for the Triangular Sylvester Equation with øperatornamesep - 1 Estimators , 1992, SIAM J. Matrix Anal. Appl..

[12]  Robert A. van de Geijn,et al.  Deferred Shifting Schemes for Parallel QR Methods , 1993, SIAM J. Matrix Anal. Appl..

[13]  Jack J. Dongarra,et al.  A Parallel Implementation of the Nonsymmetric QR Algorithm for Distributed Memory Architectures , 2002, SIAM J. Sci. Comput..

[14]  Krister Dackland,et al.  A Hierarchical Approach for Performance Analysis of ScaLAPACK-Based Routines Using the Distributed Linear Algebra Machine , 1996, PARA.

[15]  David S. Watkins,et al.  Shifting Strategies for the Parallel QR Algorithm , 1994, SIAM J. Sci. Comput..

[16]  Peter Poromaa.  Parallel Algorithms for Triangular Sylvester Equations: Design, Scheduling and Saclability Issues , 1998, PARA.

[17]  Erik Elmroth,et al.  A Web Computing Environment for the SLICOT Library , 2001 .

[18]  G. Stewart,et al.  An Algorithm for Generalized Matrix Eigenvalue Problems. , 1973 .

[19]  Jack Dongarra,et al.  ScaLAPACK Users' Guide , 1987 .

[20]  Bo Kågström,et al.  A Perturbation Analysis of the Generalized Sylvester Equation $( AR - LB,DR - LE ) = ( C,F )$ , 1994, SIAM J. Matrix Anal. Appl..

[21]  Robert A. van de Geijn,et al.  Parallelizing the QR Algorithm for the Unsymmetric Algebraic Eigenvalue Problem: Myths and Reality , 1996, SIAM J. Sci. Comput..

[22]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[23]  Krister Dackland,et al.  Parallel Two-Stage Reduction of a Regular Matrix Pair to Hessenberg-Triangular Form , 2000, PARA.

[24]  David S. Watkins.  Bulge Exchanges in Algorithms of QR Type , 1998 .

[25]  David S. Watkins,et al.  Theory of Decomposition and Bulge-Chasing Algorithms for the Generalized Eigenvalue Problem , 1994 .

[26]  B. Kågström,et al.  A Direct Method for Reordering Eigenvalues in the Generalized Real Schur form of a Regular Matrix Pair (A, B) , 1993 .