Newton's Iteration for Inversion of Cauchy-Like and Other Structured Matrices

Pan, Victor Y.; Zheng, Ai-Long; Huang, Xiaohan; Dias, Olen

doi:10.1006/jcom.1997.0431

Cited by 9 publications

(10 citation statements)

References 21 publications

(30 reference statements)

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“…The very idea of iterations with truncation has been already advocated in several papers, chiefly for Toeplitz-like matrices [8,9,[31][32][33], rank-structured matrices [4,5,12,13,21,27,32] (see also [22][23][24][25][26]) and wavelet-based sparsification [1,6,12,13]. However, the proofs provided so far only for some particular cases of structures have appeared as different individual proofs.…”

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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“…It is usually an essential part of many solutions, e.g., as preliminary steps for optimization, single processing, electromagnetic systems, robotic control, statistics and physics. Some effective direct solution algorithms exploiting displacement representation can be found in [2,3]. Alternative iterative methods were proposed in [1].…”

Section: Introductionmentioning

confidence: 99%

NEW ELECTRONIC EDUCATION SERVICES USING THE DISTRIBUTED E-LEARNING PLATFORM (DisPeL)

Rahnev¹,

Pavlov²,

Golev³

et al. 2014

Int. Elec. J. of Pure and Appl. Math.

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In this paper we study the solution of a system of equations Ax=b with singular and nearly singular, symmetric positive definite coefficient matrix A. Our algorithm based on, the Divide and Conquer strategy leading to the Divide-andConquer Algorithm (D&C algorithm) with, Cholesky's factorization algorithm. The Cholesky's factorization will be used to convert the matrix into a product of the form LL T , where L is a lower triangular matrix. The algorithm will be implemented on MATLAB and simulated as a user-subroutine. The user-subroutine is considering MATLAB features for reducing the round-off error especially for sensitive systems. Numerical examples will be given of a non-singular matrix and another for illconditioned matrix. The effect of round-off error will be analyzed. Results will be compared with previous ones, where LU factorization is used.

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“…The Matlab SVD subroutine was used to obtain the singular value decomposition. In general, the complexity of the SVD on an n n × square matrix is ( ) 3 O n arithmetic operations. However, it is actually applied to structured matrices with smaller rank, so it is not really expensive.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…The algorithms are based on the inversion formulas for Toeplitz matrices. The Cauchy-like case was studied in [3]. Since those matrices can be represented by their short generators, which allow faster com-putations based on the displacement operators tool, we can employ Newton's iteration to compute the inverse of the input structured matrices.…”

Section: Introductionmentioning

confidence: 99%

New Approach for the Inversion of Structured Matrices via Newton’s Iteration

Tabanjeh¹

2015

ALAMT

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X to the inverse of a general nonsingular matrix. In this paper, we will extend and apply this method to n n × structured matrices M , in which matrix multiplication has a lower computational cost. These matrices can be represented by their short generators which allow faster computations based on the displacement operators tool. However, the length of the generators is tend to grow and the iterations do not preserve matrix structure. So, the main goal is to control the growth of the length of the short displacement generators so that we can operate with matrices of low rank and carry out the computations much faster. In order to achieve our goal, we will compress the computed approximations to the inverse to yield a superfast algorithm. We will describe two different compression techniques based on the SVD and substitution and we will analyze these approaches. Our main algorithm can be applied to more general classes of structured matrices.

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Newton's Iteration for Inversion of Cauchy-Like and Other Structured Matrices

Cited by 9 publications

References 21 publications

Approximate iterations for structured matrices

Approximate iterations for structured matrices

NEW ELECTRONIC EDUCATION SERVICES USING THE DISTRIBUTED E-LEARNING PLATFORM (DisPeL)

New Approach for the Inversion of Structured Matrices via Newton’s Iteration

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