An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems

Summary. We discuss an inverse-free, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix $A$, or a pair of left and right deflating subspaces of a regular matrix pencil $A - \lambda B$. This algorithm is based on earlier ones of Bulgakov, Godunov and Malyshev, but improves on them in several ways. This algorithm only uses easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition, but not matrix inversion. Similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which requires matrix inversion and is faster but can be less stable than the new algorithm.

[1]  A. Sameh.  On Jacobi and Jacobi-I ike Algorithms for a Parallel Computer , 2010 .

[2]  G. Stewart.  Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems , 1973 .

[3]  L. Ahlfors.  Complex Analysis , 1979 .

[4]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[5]  J. L. Howland.  The sign matrix and the separation of matrix eigenvalues , 1983 .

[6]  Gene H. Golub,et al.  Matrix computations , 1983 .

[7]  A. Laub,et al.  Generalized eigenproblem algorithms and software for algebraic Riccati equations , 1984, Proceedings of the IEEE.

[8]  G. Stewart.  A Jacobi-Like Algorithm for Computing the Schur Decomposition of a Nonhermitian Matrix , 1985 .

[9]  Judith Gardiner,et al.  A generalization of the matrix sign function solution for algebraic Riccati equations , 1985, 1985 24th IEEE Conference on Decision and Control.

[10]  N. Higham.  Computing the polar decomposition with applications , 1986 .

[11]  S. Godunov.  Problem of the dichotomy of the spectrum of a matrix , 1986 .

[12]  T. Chan.  Rank revealing QR factorizations , 1987 .

[13]  R. Byers.  Solving the algebraic Riccati equation with the matrix sign function , 1987 .

[14]  Patricia J. Eberlein,et al.  On the Schur Decomposition of a Matrix for Parallel Computation , 1985, IEEE Transactions on Computers.

[15]  S. Godunov,et al.  Circular dichotomy of the spectrum of a matrix , 1988 .

[16]  R. Byers.  A Bisection Method for Measuring the Distance of a Stable Matrix to the Unstable Matrices , 1988 .

[17]  A. Malyshev.  Computing invariant subspaces of a regular linear pencil of matrices , 1989 .

[18]  Stephen P. Boyd,et al.  A bisection method for computing the H∞ norm of a transfer matrix and related problems , 1989, Math. Control. Signals Syst..

[19]  S. Boyd,et al.  A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L ∞ -norm , 1990 .

[20]  Jack J. Dongarra,et al.  A set of level 3 basic linear algebra subprograms , 1990, TOMS.

[21]  Gautam M. Shroff.  A parallel algorithm for the eigenvalues and eigenvectors of a general complex matrix , 1990 .

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

[23]  J. Dongarra,et al.  Generalized QR factorization and its applications , 1992 .

[24]  J. Demmel.  Trading Off Parallelism and Numerical Stability , 1992 .

[25]  L. Auslander,et al.  On parallelizable eigensolvers , 1992 .

[26]  A. Malyshev.  Parallel Algorithm for Solving Some Spectral Problems of Linear Algebra , 1993 .

[27]  J. Demmel,et al.  On swapping diagonal blocks in real Schur form , 1993 .

[28]  James Demmel,et al.  Parallel numerical linear algebra , 1993, Acta Numerica.

[29]  V. Mehrmann,et al.  A MULTISHIFT ALGORITHM FOR THE NUMERICAL SOLUTION OF ALGEBRAIC RICCATI EQUATIONS , 1993 .

[30]  James Demmel,et al.  Design of a Parallel Nonsymmetric Eigenroutine Toolbox, Part I , 1993, PPSC.

[31]  G. Stewart.  Updating a Rank-Revealing ULV Decomposition , 1993, SIAM J. Matrix Anal. Appl..

[32]  Xiaobai Sun,et al.  The PRISM project: infrastructure and algorithms for parallel eigensolvers , 1993, Proceedings of Scalable Parallel Libraries Conference.

[33]  James Demmel,et al.  Faster Numerical Algorithms via Exception Handling , 1994, IEEE Trans. Computers.

[34]  Stanley C. Eisenstat,et al.  A Divide-and-Conquer Algorithm for the Bidiagonal SVD , 1995, SIAM J. Matrix Anal. Appl..

[35]  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..

[36]  James Demmel,et al.  The Spectral Decomposition of Nonsymmetric Matrices on Distributed Memory Parallel Computers , 1997, SIAM J. Sci. Comput..