Abstract not found

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.

. The matrix sign function has several applications in system theory and matrix computations. However, the numericalbehavior of the matrix sign function, and its associated divideand -conquer algorithm for computing invariant subspaces, are still not completely understood. In this paper, we present a new perturbation theory for the matrix sign function, the conditioning of its computation, the numerical stability of the divide-and-conquer algorithm, and iterative refinement schemes. Numerical examples are also presented. An extension of the matrix sign function based algorithm to compute left and right deflating subspaces for a regular pair of matrices is also described. Key words. matrix sign function, Newton's method, eigenvalue problem, invariant subspace, deflating subspaces AMS subject classifications. 65F15, 65F35, 65F30, 15A18 1. Introduction. Since the matrix sign function was introduced in early 1970s, it has been the subject of numerous studies and used in many applications...

. Recently, a number of primal-dual interior-point methods for semidefinite programming have been developed. To reduce the number of floating point operations, each iteration of these methods typically performs block Gaussian elimination with block pivots that are close to singular near the optimal solution. As a result, these methods often exhibit complex numerical properties in practice. We consider numerical issues related to some of these methods. Our error analysis indicates that these methods could be numerically stable if certain coefficient matrices associated with the iterations are well-conditioned, but are unstable otherwise. With this result, we explain why one particular method, the one introduced by Alizadeh, Haeberly and Overton is in general more stable than others. We also explain why the so called least squares variation, introduced for some of these methods, does not yield more numerical accuracy in general. Finally, we present results from our numerical experiments ...

Search directions for primal-dual path-following methods for semidefinite programming (SDP) are proposed. These directions have the properties that (1) under certain nondegeneracy and strict complementarity assumptions, the Jacobian matrix of the associated symmetrized Newton equation has bounded condition number along the central path in the limit as the barrier parameter tends to zero; (2) the Schur complement matrix of the symmetrized Newton equation is symmetric and the cost for computing this matrix is 2mn 3 + 0:5m 2 n 2 ops, where n and m are the dimension of the matrix and vector variables of the SDP, respectively. These two properties imply that a path-following method using the proposed directions can achieve the high accuracy typically attained by methods employing the direction proposed by Alizadeh, Haeberly, and Overton (currently the best search direction in terms of accuracy), but each iteration requires at most half the amount of flops (to leading order).

Abstract. Many of the currently popular \block algorithms " are scalar algorithms in which the operations have been grouped and reordered into matrix operations. One genuine block algorithm in practical use is block LU factorization, and this has recently been shown by Demmel and Higham to be unstable in general. It is shown here that block LU factorization is stable if A is block diagonally dominant by columns. Moreover, for a general matrix the level of instability in block LU factorization can be bounded in terms of the condition number (A) and the growth factor for Gaussian elimination without pivoting. A consequence is that block LU factorization is stable for a matrix A that is symmetric positive de nite or point diagonally dominant byrows or columns as long as A is well-conditioned.

We study classical control problems like pole assignment, stabilization, linear quadratic control and H1 control from a numerical analysis point of view. We present several examples that show the difficulties with classical approaches and suggest reformulations of the problems in a more general framework. We also discuss some new algorithmic approaches. AMS subject classification. 65F15, 93B40, 93B36, 93C60. Authors: Volker Mehrmann and Hongguo Xu Fakultat fur Mathematik, TU Chemnitz, D-09107 Chemnitz, FRG. Supported by Deutsche Forschungsgemeinschaft, within Sonderforschungsbereich SFB393, `Numerische Simulation auf massiv parallelen Rechnern'. 1 Introduction In the last 40 years linear systems theory (control theory) has evolved into a mature field that has found a stable position on the borderline between applied mathematics, engineering and computer science. The major success is not only due to the fact that beautiful mathematical theories (like linear algebra, ring theory, rep...

Robust and reliable algorithms are presented for computing the stable projection with respect to a specified region \Gamma in the complex plane of a transfer matrix given by its generalized state space realization. The algorithms are based on a block-diagonalization of E \Gamma A (in generalized Schur form) with optimally conditioned transformation matrices. A first direct elimination approach reduces to solving a generalized Sylvester equation. In a second approach an equivalence transformation is constructed from unitary bases of pairs of deflating subspaces obtained from two reorderings of the eigenvalues. The sensitivity of the problem and the stability of the proposed algorithms are discussed and compared with a more classical approach. Results from numerical experiments that evaluate the algorithms and confirm the theory are also reported. Keywords: Transfer matrix, additive decomposition, generalized state space realization, generalized Sylvester equation, block-dia...

apier nach 1 ISO 9706 Contents 1 Introduction 7 1.1 What is a PIM? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Different types of PIMs . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Organization and summary of our results . . . . . . . . . . . . . 9 2 Background 13 2.1 Krylov spaces and the Arnoldi process . . . . . . . . . . . . . . . 13 2.2 Exterior mapping functions and Faber polynomials . . . . . . . . 14 2.3 Inclusion sets and asymptotic analysis . . . . . . . . . . . . . . . 15 3 Inclusion sets generated by the conformal 'bratwurst' maps 19 3.1 Derivation of the maps . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Definition and properties of the 'bratwurst' shape sets . . . . . . 23 3.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . 25 4 The hybrid ABF method for non-hermitian linear systems 29 4.1 Faber polynomials for the inclusion sets

Strong trends in chemical engineering and plant operation have made the control of processes increasingly difficult and have driven the process industry's demand for improved control techniques. Improved control leads to savings in resources, smaller downtimes, improved safety, and reduced pollution. Though the need for improved process control is clear, advanced control methodologies have had only limited acceptance and application in industrial practice. The reason for this gap between control theory and practice is that existing control methodologies do not adequately address all of the following control system requirements and problems associated with control design: ffl The controller must be insensitive to plant/model mismatch, and perform well under unmeasured or poorly modeled disturbances. ffl The controlled system must perform well under state or actuator constraints. ffl The controlled system must be safe, reliable, and easy to maintain. ffl Controllers are commonly require...