On the matrix sign function method for the computation of invariant subspaces

Byers, Ralph; He, Chunyang; Mehrmann, Volker

doi:10.1109/cacsd.1996.555200

Cited by 24 publications

(37 citation statements)

References 21 publications

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“…We consider important matrix-valued functions f (A) as, e.g., the inverse A −1 and the square root √ A [3,10,[28][29][30]41]. In particular, we are interested in evaluations of f (A) for matrices A arising from partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

Approximate iterations for structured matrices

2008

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Important matrix-valued functions f (A) are, e.g., the inverse A −1 , the square root √ A and the sign function. Their evaluation for large matrices arising from pdes is not an easy task and needs techniques exploiting appropriate structures of the matrices A and f (A) (often f (A) possesses this structure only approximately). However, intermediate matrices arising during the evaluation may lose the structure of the initial matrix. This would make the computations inefficient and even infeasible. However, the main result of this paper is that an iterative fixed-point like process for the evaluation of f (A) can be transformed, under certain general assumptions, into another process which preserves the convergence rate and benefits from the underlying structure. It is shown how this result applies to matrices in a tensor format with a bounded tensor rank and to the structure of the hierarchical matrix technique. We demonstrate our results by verifying all requirements in the case of the iterative computation of A −1 and √ A.

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Section: Introductionmentioning

confidence: 99%

Approximate iterations for structured matrices

2008

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“…The connection of matrix iterative expressions with the function of sign is not that straightforward, but in practice, such methods could be constructed by considering a suitable root-finding method to the nonlinear matrix equation (8). Noting that sign( ) is a solution of this equation (refer to [12] and the references cited therein).…”

Section: The Literaturementioning

confidence: 99%

“…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 2 Mathematical Problems in Engineering roots and the polar decomposition; for more one may refer to [7][8][9].…”

Section: Preliminariesmentioning

confidence: 99%

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

Jebreen

2018

Mathematical Problems in Engineering

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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

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“…With respect to rounding and truncation errors, the sign function has been found to often work just as well as invariant subspace techniques, but it can be shown that occasionally ill-conditioned sign iterations will cause numerical instability ( [52], [53]), and with intermediate low order approximations, such numerical difficulties are even more prominent. Hence we cannot 'solve' the Lyapunov and Riccati equations; there will always be non-zero residuals, , , the norms of which are not necessarily a good measure of the backwards error [54], [55], and the suggested criteria are not easily calculable using SSS methods.…”

Section: B Numerical Issues Relating To the Approximationsmentioning

confidence: 99%

Distributed Control: A Sequentially Semi-Separable Approach for Spatially Heterogeneous Linear Systems

Rice

Verhaegen

2009

IEEE Trans. Automat. Contr.

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Abstract-We consider the problem of designing controllers for spatially-varying interconnected systems distributed in one spatial dimension. The matrix structure of such systems can be exploited to allow fast analysis and design of centralized controllers with simple distributed implementations. Iterative algorithms are provided for stability analysis, analysis and sub-optimal controller synthesis. For practical implementation of the algorithms, approximations can be used, and the computational efficiency and accuracy are demonstrated on an example.

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On the matrix sign function method for the computation of invariant subspaces

Cited by 24 publications

References 21 publications

Approximate iterations for structured matrices

Approximate iterations for structured matrices

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

Distributed Control: A Sequentially Semi-Separable Approach for Spatially Heterogeneous Linear Systems

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