The dense nonsymmetric eigenproblem is one of the hardest linear algebra problems to solve effectively on massively parallel machines. Rather than trying to design a "black box" eigenroutine in the spirit of EISPACK or LAPACK, we propose building a toolbox for this problem. The tools are meant to be used in different combinations on different problems and architectures. In this paper, we will describe these tools which include basic block matrix computations, the matrix sign function, 2-dimensional bisection, and spectral divide and conquer using the matrix sign function to find selected eigenvalues. We also outline how we deal with ill-conditioning and potential instability. Numerical examples are included. A future paper will discuss error analysis in detail and extensions to the generalized eigenproblem.

We discuss two inverse free, highly parallel, spectral divide and conquer algorithms: one for computing an invariant subspace of a nonsymmetric matrix and another one for computing left and right de ating subspaces of a regular matrix pencil A, B. These two closely related algorithms are based on earlier ones of Bulgakov, Godunov and Malyshev, but improve on them in several ways. These algorithms only use easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition. The existing parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which is faster but can be less stable than the new algorithm. Appears also as

The sign function of a square matrix was introduced by Roberts in 1971. We show that it is useful to regard S = sign(A) as being part of a matrix sign decomposition A = SN , where N = (A 2 ) 1=2 . This decomposition leads to the new representation sign(A) = A(A 2 ) \Gamma1=2 . Most results for the matrix sign decomposition have a counterpart for the polar decomposition A = UH, and vice versa. To illustrate this, we derive best approximation properties of the factors U , H and S, determine bounds for kA \Gamma Sk and kA \Gamma Uk, and describe integral formulas for S and U . We also derive explicit expressions for the condition numbers of the factors S and N . An important equation expresses the sign of a block 2 \Theta 2 matrix involving A in terms of the polar factor U of A. We apply this equation to a family of iterations for computing S by Pandey, Kenney and Laub, to obtain a new family of iterations for computing U . The iterations have some attractive properties, includin...

Abstract. The implementation and performance of a class of divide-and-conquer algorithms for computing the spectral decomposition of nonsymmetric matrices on distributed memory parallel computers are studied in this paper. After presenting a general framework, we focus on a spectral divide-and-conquer (SDC) algorithm with Newton iteration. Although the algorithm requires several times as many floating point operations as the best serial QR algorithm, it can be simply constructed from a small set of highly parallelizable matrix building blocks within Level 3 basic linear algebra subroutines (BLAS). Efficient implementations of these building blocks are available on a wide range of machines. In some ill-conditioned cases, the algorithm may lose numerical stability, but this can easily be detected and compensated for. The algorithm reached 31 % efficiency with respect to the underlying PUMMA matrix multiplication and 82 % efficiency with respect to the underlying ScaLAPACK matrix inversion on a 256 processor Intel Touchstone Delta system, and 41 % efficiency with respect to the matrix multiplication in CMSSL on a 32 node Thinking Machines CM-5 with vector units. Our performance model predicts the performance reasonably accurately. To take advantage of the geometric nature of SDC algorithms, we have designed a graphical user interface to let the user choose the spectral decomposition according to specified regions in the complex plane.

. In this paper, preliminary research results on a new algorithm for finding all the eigenvalues and eigenvectors of a real diagonalizable matrix with real eigenvalues are presented. The basic mathematical theory behind this approach is reviewed and is followed by a discussion of the numerical considerations of the actual implementation. The numerical algorithm has been tested on thousands of matrices on both a Cray-2 and an IBM RS/6000 Model 580 workstation. The results of these tests are presented. Finally, issues concerning the parallel implementation of the algorithm are discussed. The algorithm's heavy reliance on matrix--matrix multiplication, coupled with the divide and conquer nature of this algorithm, should yield a highly parallelizable algorithm. Key words. eigenvalues, divide and conquer algorithm, invariant subspaces, parallel algorithm AMS subject classification. 65F15 PII. S1064827592228833 1. Introduction. Computation of all the eigenvalues and eigenvectors of a dens...

Linear quadratic optimal control problems and the computation of Kalman filters require numerical solutions of discrete and continuous algebraic Riccati equations.

In this paper, we discuss work in progress on a complete eigensolver based on the Invariant Subspace Decomposition Algorithm for dense symmetric matrices (SYISDA). We describe a recently developed acceleration technique that substantially reduces the overall work required by this algorithm and review the algorithmic highlights of a distributed-memory implementation of this approach. These include a fast matrix-matrix multiplication algorithm, a new approach to parallel band reduction and tridiagonalization, and a harness for coordinating the divide-and-conquer parallelism in the problem. We present performance results for the dominant kernel, dense matrix multiplication, as well as for the overall SYISDA implementation on the Intel Touchstone Delta and the Intel Paragon. 1. Introduction Computation of eigenvalues and eigenvectors is an essential kernel in many applications, and several promising parallel algorithms have been investigated [26, 3, 28, 22, 25, 6]. The work presented in t...

This paper uses a forward and backward error analysis to try to identify some classes of matrices for . P which the matrix sign function is a numerically stable algorithm for extracting invariant subspaces roper scaling is essential to numerical stability as well as to rapid convergence. g a Roberts [21] and Beavers and Denman [7] introduced the matrix sign function as a means of solvin lgebraic Riccati equations and Lyapunov equations. The matrix sign function has since attracted the ] a attention of control engineers and some applied mathematicians ([1] to [21]). Balzer [3], Barraud [5 nd Byers [9] have suggested strategies for accelerating convergence. Denman and Beavers [11] - - 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002 extended matrix sign function algorithms to a list of invariant subspace related calculations. Howland e a [16] used the matrix sign function to count eigenvalues in boxes in the complex plane. Some of th lgorithms have been refined and extended by Attarzadeh [2], Bierman [8], Byers [9]. Gardiner and d d Laub [14] have extended the use of the matrix sign function to generalized Riccati equations an iscrete Riccati equations. Higham [15] has used matrix sign function techniques to calculate polar decompositions

The goal of the PRISM project is the development of infrastructure and algorithms for the parallel solution of eigenvalue problems. We are currently investigating a complete eigensolver based on the Invariant Subspace Decomposition Algorithm for dense symmetric matrices (SYISDA). After briefly reviewing SYISDA, we discuss the algorithmic highlights of a distributed-memory implementation of this approach. These include a fast matrix-matrix multiplication algorithm, a new approach to parallel band reduction and tridiagonalization, and a harness for coordinating the divide-and-conquer parallelism in the problem. We also present performance results of these kernels as well as the overall SYISDA implementation on the Intel Touchstone Delta prototype. 1. Introduction Computation of eigenvalues and eigenvectors is an essential kernel in many applications, and several promising parallel algorithms have been investigated [29, 24, 3, 27, 21]. The work presented in this paper is part of the PRI...

We survey recent developments in dense numerical linear algebra, covering linear systems, least squares problems and eigenproblems. Topics considered include the design and analysis of block, partitioned and parallel algorithms, condition number estimation, componentwise error analysis, and the computation of practical error bounds. Frequent reference is made to LAPACK, the state of the art package of Fortran software designed to solve linear algebra problems efficiently and accurately on high-performance computers.