R. Byers. Numerical stability and instability in matrix sign function based algorithms. In C.I. Byrnes and A. Lindquist, editors, Computational and Combinatorial Methods in Systems Theory, pages 185--199, Elsevier, North Holland, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is highly parallel. Auslander and Tsao [2] and Lederman, Tsao, and Turnbull [33] use multiply based parallel algorithms based on matrix polynomials to split the spectrum. Bai and Demmel [4] use similar matrix multiply techniques using the matrix sign function to split the spectrum (see also [6, 10, 5, 7]. Dongarra and Sidani [17] introduced tearing methods based on doing rank one updates to an unsymmetric Hessenberg matrix, resulting in two smaller problems, which are solved independently and then glued back together with a Newton iteration. This tends to suffer from stability problems since ....

Byers, R., Numerical Stability and Instability in Matrix Sign Function Based Algorithms, Computational and Combinatorial Methods in Systems Theory, C. Byrnes and A. Lindquist, editors, pp. 185--200, NorthHolland, 1986.

.... :5( 1= i times nearly yields zero) Fortunately, this seems unlikely in practice because the iteration (2.1) moves eigenvalues near the imaginary axis away from it. A more complete discussion of the conditioning of the matrix sign function computation will appear in part 2 of this paper. In [53, 11], some condition number estimation procedures for the matrix sign function, based on Fr echet derivatives, are discussed. 3 Higher Level Tools We can use properties of the matrix sign function to produce two higher level tools for our toolbox: 1. counting the number of eigenvalues in a region of ....

R. Byers. Numerical stability and instability in matrix sign function based algorithms. In C. Byrnes and A. Lindquist, editors, Computational and Combinatorial Methods in Systems Theory, pages 185--200. North-Holland, 1986.

....2.1 The SDC algorithm with Newton iteration The first SDC algorithm uses the matrix sign function, which was introduced by Roberts [46] for solving the algebraic Riccati equation. However, it was soon extended to solving the spectral decomposition problem [8] More recent studies may be found in [11, 42, 6]. The matrix sign function, sign(A) of a matrix A with no eigenvalues on the imaginary axis can be defined via the Jordan canonical form of A (2.1) where the eigenvalues of J are in the open right half plane D, and the eigenvalues of J Gamma are in the open left half plane D. Then sign(A) ....

....Newton iteration with global convergence still need to compute the inverse of a matrix explicitly in one form or another. Dealing with ill conditioned matrices and instability in the Newton iteration for computing the matrix sign function and the subsequent spectral decomposition is discussed in [11, 6, 4] and the references therein. 2.2 The SDC algorithm with inverse free iteration The above algorithm needs an explicit matrix inverse. This could cause numerical instability when the matrix is ill conditioned. The following algorithm, originally due to Godunov, Bulgakov and Malyshev [30, 10, 44] ....

R. Byers. Numerical stability and instability in matrix sign function based algorithms. In C. Byrnes and A. Lindquist, editors, Computational and Combinatorial Methods in Systems Theory, pages 185--200. North-Holland, 1986.

....is highly parallel. Auslander and Tsao [2] and Lederman, Tsao, and Turnbull [36] use multiply based parallel algorithms based on matrix polynomials to split the spectrum. Bai and Demmel [4] use similar matrix multiply techniques using the matrix sign function to split the spectrum (see also [6, 10, 5, 7]. Dongarra and Sidani [17] introduced tearing methods based on doing rank one updates to an unsymmetric Hessenberg matrix, resulting in two smaller problems, which are solved independently and then glued back together with a Newton iteration. This tends to suffer from stability problems since the ....

Byers, R., Numerical Stability and Instability in Matrix Sign Function Based Algorithms, Computational and Combinatorial Methods in Systems Theory, C. Byrnes and A. Lindquist, editors, pp. 185--200, NorthHolland, 1986.

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R. Byers. Numerical stability and instability in matrix sign function based algorithms. In C.I. Byrnes and A. Lindquist, editors, Computational and Combinatorial Methods in Systems Theory, pages 185--199, Elsevier, North Holland, 1986.

....deflating subspaces, solving (generalized) algebraic Riccati equations and (generalized) Lyapunov equations. Matrix sign function algorithms have attracted much recent attention; the survey [23] lists over 100 references. The rounding error analysis and perturbation theory are becoming understood [2, 12, 13, 14, 19, 32]. Because they are rich in matrix matrix operations, matrix sign function algorithms are well suited to computers with advanced architectures [3, 1, 2, 9, 20, 21] Its presence in the ScaLAPACK library [10] is an indication of its maturity and acceptance. The first matrix sign function algorithm ....

R. Byers, Numerical stability and instability in matrix sign function based algorithms, in Computational and Combinatorial Methods in Systems Theory, C. I. Byrnes and A. Lindquist, eds., North-Holland, New York, 1986, pp. 185--200.

.... (1) by Roberts in a 1971 technical report [34] which was not published until 1980 [35] Kato [23, Page 67] reports that the resolvent integral (2) goes back to 1946 [12] and 1949 [21, 22] There is some concern about the numerical stability of numerical methods based upon the matrix sign function [2, 8, 19]. In this paper, we demonstrate that evaluating the matrix sign function is a more ill conditioned computational problem than the problem of finding bases of the invariant subspaces V and V Gamma . See Example 1 in Section 3. Nevertheless, we also give perturbation and error analyses, which ....

....computational problem than the problem of finding bases of the invariant subspaces V and V Gamma . See Example 1 in Section 3. Nevertheless, we also give perturbation and error analyses, which show that (at least for Newton s method for the computation of the matrix sign function [8, 9]) in most circumstances the accuracy is competitive with conventional methods for computing invariant subspaces. Our analysis improves some of the perturbation bounds in [2, 8, 18, 24] In Section 2 we establish some notation and clarify the relationship between the matrix sign function and the ....

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R. Byers. Numerical stability and instability in matrix sign function based algorithms. In C.I. Byrnes and A. Lindquist, editors, Computational and Combinatorial Methods in Systems Theory, pages 185--199, Elsevier, North Holland, 1986.

....problems like finding solutions of the algebraic Riccati equation and stable Lyapunov equation [2, 6, 10, 15, 19, 23, 26] The matrix sign function is attracting much attention. The survey [19] lists over 100 references. The rounding error analysis and perturbation theory are becoming understood [2, 9, 10, 11, 13, 27]. Because it is rich in matrixmatrix operations, the matrix sign function is well suited to computers with advanced architectures [1, 2, 3, 15, 23] It has been demonstrated on matrices of order greater than 10,000 [1] A sign of its maturity and acceptance is its presence in the ScaLAPACK library ....

R. Byers, Numerical stability and instability in matrix sign function based algorithms, in Computational and Combinatorial Methods in Systems Theory, C. I. Byrnes and A. Lindquist, eds., NorthHolland, New York, 1986, pp. 185--200.

....other matrix inverses. The assumption that A Gamma E has no eigenvalue on the unit circle implies that A Gamma E is nonsingular. Nevertheless, A Gamma E may be ill conditioned for inversion. Moreover, in rare cases, even the wellstudied scaled Newton iteration exhibits numerical instabilities [11, 9, 27]. An alternative way to compute disk functions is the inverse free, implicit squaring algorithm [3, 8, 20, 33, 34, 35] which is related to the AB algorithm [28, 30, 31] UR Gamma VR = lim k 1 (U k V k ) Gamma1 (U k Gamma V k ) where U 0 = A V 0 = E U k 1 = Q H k;12 U k V k 1 = Q H ....

R. Byers. Numerical stability and instability in matrix sign function based algorithms. In C.I. Byrnes and A. Lindquist, editors, Computational and Combinatorial Methods in Systems Theory, pages 185--199, Elsevier, North Holland, 1986.

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R. Byers, Numerical stability and instability in matrix sign function based algorithms, Comp. and Comb. Methods in Systems Theory, C.I. Byrnes, A. Lindquist, edts., North Holland, New York (1986), 185--200.

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Byers, R., Numerical Stability and Instability in Matrix Sign Function Based Algorithms, Computational and Combinatorial Methods in Systems Theory, C. Byrnes and A. Lindquist, editors, pp. 185--200, North-Holland, 1986.

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R. Byers. Numerical stability and instability in matrix sign function based algorithms. In Computational and Combinatorial Methods in Systems Theory. Elsevier Science Publishers B.V., North Holland, 1990. Ed. C. Byrnes and A. Lindquist.

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#11# Byers, R., Numerical Stability and Instability in Matrix Sign Function Based Algorithms, Computational and Combinatorial Methods in Systems Theory, C. Byrnes and A. Lindquist, editors, pp. 185#200, NorthHolland,

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