The sign matrix and the separation of matrix eigenvalues

Howland, James Lucien

doi:10.1016/0024-3795(83)90104-0

Cited by 39 publications

(34 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The iteration is globally and ultimately quadratically convergent with lim j →∞ A j = sign(A) [45,38]. The iteration could fail to converge if A has pure imaginary eigenvalues (or, in finite precision, if A is "close" to having pure imaginary eigenvalues.)…”

Section: Inverse-free Iteration Vs the Matrix Sign Functionmentioning

confidence: 99%

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

Bai

Demmel

1997

Numerische Mathematik

116

164

View full text Add to dashboard Cite

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 − λB . 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. Mathematics Subject Classification (1991): 65F15

show abstract

Section: Inverse-free Iteration Vs the Matrix Sign Functionmentioning

confidence: 99%

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

Bai

Demmel

1997

Numerische Mathematik

116

164

View full text Add to dashboard Cite

show abstract

“…Roberts in [1] for the first time extended this definition for matrices, which has several important applications in scientific computing, for example see [2][3][4] and the references cited therein. For example, the off-diagonal decay of the matrix function of sign is also a well-developed area of study in statistics and statistical physics [5].…”

Section: Preliminariesmentioning

confidence: 99%

Constructing a High-Order Globally Convergent Iterative Method for Calculating the Matrix Sign Function

Jebreen

2018

Mathematical Problems in Engineering

View full text Add to dashboard Cite

This work is concerned with the construction of a new matrix iteration in the form of an iterative method which is globally convergent for finding the sign of a square matrix having no eigenvalues on the axis of imaginary. Toward this goal, a new method is built via an application of a new four-step nonlinear equation solver on a particulate matrix equation. It is discussed that the proposed scheme has global convergence with eighth order of convergence. To illustrate the effectiveness of the theoretical results, several computational experiments are worked out. PreliminariesThe sign function for the scalar case is expressed bywherein ∈ C is not located on the imaginary axis. Roberts in [1] for the first time extended this definition for matrices, which has several important applications in scientific computing, for example see [2][3][4] and the references cited therein. For example, the off-diagonal decay of the matrix function of sign is also a well-developed area of study in statistics and statistical physics [5].To proceed formally, let us consider that ∈ C × is a square matrix possessing no eigenvalues on the axis of imaginary. We consideras a form of Jordan canonical written such that wherein + = . Noting that sign( ) is definable once is nonsingular. This procedure takes into account a clear application of the form of Jordan canonical and of the matrix . Herein none of the matrices and are unique. However, it is possible to investigate that sign( ) as provided in (4) does not rely on the special selection of or .Here, a simpler interpretation for the sign matrix in the case of Hermitian case (namely, all eigenvalues are real) can be given byis a diagonalization of the square matrix . The significance of calculating and finding in (4) is because of the point that the function of sign plays a central role in matrix functions theory, specially for principal matrix

show abstract

“…It can be shown that the iteration is globally and ultimately quadratically convergent with lim j→∞ A j = sign(A), provided A has no pure imaginary eigenvalues [31,23]. The iteration fails otherwise.…”

Section: An Sdc Algorithm With Newton Iterationmentioning

confidence: 99%

“…However, it was soon extended to solving the spectral decomposition problem [5]. More recent studies may be found in [28,3,23].…”

Section: An Sdc Algorithm With Newton Iterationmentioning

confidence: 99%

The Spectral Decomposition of Nonsymmetric Matrices on Distributed Memory Parallel Computers

Bai¹,

Demmel²,

Dongarra³

et al. 1997

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

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.

show abstract

The sign matrix and the separation of matrix eigenvalues

Cited by 39 publications

References 6 publications

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

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

Constructing a High-Order Globally Convergent Iterative Method for Calculating the Matrix Sign Function

The Spectral Decomposition of Nonsymmetric Matrices on Distributed Memory Parallel Computers

Contact Info

Product

Resources

About