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A modification of the Dewilde-van der Veen method for inversion of finite structured matrices

linear ALGEBRA AND ITS APPLICATIONS 2002年 343卷 419-450页

作者： Eidelman, Y Gohberg, I Tel Aviv Univ Raymond & Beverly Sackler Fac Exact Sci Sch Math Sci IL-69978 Tel Aviv Israel

We study a class of block structured matrices R = {R-ij}(i,j=1)(N) with a property that the solution of the corresponding system Rx = y of linear algebraic equations may be performed for O(N) arithmetic operations. In this paper for finite invertible matrices we analyze in detail factorization and inversion algorithms. These algorithms are related to those suggested by P.M. Dewilde and A.J. van der Veen (Time-varying Systems and Computations, Kluwer Academic Publishers, New York, 1998) for a class of finite and infinite matrices with a small Hankel rank. The algorithms presented here are more transparent and are a modification of the algorithms from the above reference, The approach and the proofs are essentially different from those in the above-mentioned reference. The paper contains also analysis of complexity and results of numerical experiments. (C) 2002 Elsevier Science Inc. All rights reserved.

关键词： structured matrices linear complexity algorithms inversion algorithms factorization algorithms solution of linear equations

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Out-of-band quasiseparable matrices

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linear ALGEBRA AND ITS APPLICATIONS 2008年第1期429卷 266-289页

作者： Eidelman, Y. Gohberg, I. Tel Aviv Univ Raymond & Beverly Sackler Fac Exact Sci Sch Math Sci IL-69978 Ramat Aviv Israel

We study structured matrices which consist of a band part and quasiseparable parts below and upper the band. We extend algorithms known for quasiseparable matrices, i.e. for the case when the band consists of the main diagonal only, to a wider class of matrices. The matrices which we consider may be treated as an usual quasiseparable matrices with larger orders of generators. Hence one can apply the methods developed for usual quasiseparable matrices and obtain various linear complexity O(N) algorithms. However in this case the coefficients in N in the complexity estimates turns out to be quite large. In this paper we use the structure more accurately by division of the matrix into three parts in which the middle part is the band instead of diagonal as it is used for usual quasiseparable matrices. This approach allows to use better the structure of the matrix in order to improve the coefficients in N in the complexity estimates for the algorithms. This method works for algorithms which keep invariant the structure. (C) 2008 Elsevier Inc. All rights reserved.

关键词： structured matrices quasiseparable matrices linear complexity algorithms

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Structural and computational properties of possibly singular semiseparable matrices

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linear ALGEBRA AND ITS APPLICATIONS 2002年第1-3期340卷 183-198页

作者： Fasino, D Gemignani, L Univ Pisa Dipartimento Matemat I-56127 Pisa Italy Univ Udine Dipartimento Matemat & Informat I-33100 Udine Italy

A classical result of structured numerical linear algebra states that the inverse of a nonsingular semiseparable matrix is a tridiagonal matrix. Such a property of a semiseparable matrix has been proved to be useful for devising linear complexity solvers, for establishing recurrence relations among its columns or rows and, moreover, for efficiently evaluating its characteristic polynomial. In this paper, we provide sparse structured representations of a semiseparable matrix A which hold independently of the fact that A is singular or not. These relations are found by pointing out the band structure of the inverse of the sum of A plus a certain sparse perturbation of minimal rank. Further, they can be used to determine in a computationally efficient way both a reflexive generalized inverse of A and its characteristic polynomial. (C) 2002 Elsevier Science Inc. All rights reserved.

关键词： semiseparable matrices block tridiagonal matrices linear complexity algorithms

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