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A note on computing the inverse of a triangular toeplitz matrix

APPLIED MATHEMATICS AND COMPUTATION 2014年 236卷 512-523页

作者： Belhaj, Skander Dridi, Marwa Manouba Univ ISAMM Tunis 2010 Tunisia Univ Tunis El Manar ENIT LAMSIN Tunis 1002 Tunisia

Using trigonometric polynomial interpolation, a fast and effective numerical algorithm for computing the inverse of a triangular toeplitz matrix with real numbers has been recently proposed (Lin et al., 2004) [7]. The complexity of the algorithm is two fast Fourier transforms (FFTs) and one fast cosine transform (DCT) of 2n-vectors. In this paper, we present an algorithm with two fast Fourier transforms (FFTs) of 2n-vectors for calculating the inverse of a triangular toeplitz matrix with real and/or complex numbers. A theoretical accuracy and error analysis is also considered. Numerical examples are given to illustrate the effectiveness of our method. (C) 2014 Elsevier Inc. All rights reserved.

关键词： Trigonometric polynomial interpolation triangular toeplitz matrix Fast Fourier transforms

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ON THE DECAY OF THE ELEMENTS OF INVERSE triangular toeplitz MATRICES

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SIAM JOURNAL ON matrix ANALYSIS AND APPLICATIONS 2014年第4期35卷 1288-1302页

作者： Ford, Neville J. Savostyanov, Dmitry V. Zamarashkin, Nickolai L. Univ Chester Chester CH1 4BJ Cheshire England Russian Acad Sci Inst Numer Math Moscow 119333 Russia

We consider half-infinite triangular toeplitz matrices with slow decay of the elements and prove under a monotonicity condition that the elements of the inverse matrix, as well as the elements of the fundamental matrix, decay to zero. We provide a quantitative description of the decay of the fundamental matrix in terms of p-norms. The results add to the classical results of Jaffard and Vecchio and are illustrated by numerical examples.

关键词： triangular toeplitz matrix Volterra equation Abel kernel slow decay stability

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Fast inversion of triangular toeplitz matrices

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THEORETICAL COMPUTER SCIENCE 2004年第2-3期315卷 511-523页

作者： Lin, FR Ching, WK Ng, MK Shantou Univ Dept Math Shantou 515063 Guangdong Peoples R China Univ Hong Kong Dept Math Hong Kong Hong Kong Peoples R China

In this paper, we present an approximate inversion method for triangular toeplitz matrices based on trigonometric polynomial interpolation. To obtain an approximate inverse of high accuracy for a triangular toeplitz: matrix of size n, our algorithm requires two fast Fourier transforms (FFTs) and one fast cosine transform of 2n-vectors. We then revise the approximate method proposed by Bini (SIAM J. Comput. 13 (1984) 268). The complexity of the revised Bini algorithm is two FFTs of 2n-vectors. (C) 2004 Published by Elsevier B.V.

关键词： triangular toeplitz matrix interpolation fast Fourier transform fast cosine transform

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Computing the block factorization of complex Hankel matrices

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COMPUTING 2010年第3-4期87卷 169-186页

作者： Belhaj, Skander Univ Franche Comte CNRS UMR 6623 Math LabUFR Sci & Tech F-25030 Besancon France Ecole Natl Ingn Tunis Lab LAMSIN Tunis 1002 Tunisia

In this paper, we present an algorithm for finding an approximate block diagonalization of complex Hankel matrices. Our method is based on inversion techniques of an upper triangular toeplitz matrix, specifically, by simple forward substitution. We also consider an approximate block diagonalization of complex Hankel matrices via Schur complementation. An application of our algorithm by calculating the approximate polynomial quotient and remainder appearing in the Euclidean algorithm is also given. We have implemented our algorithms in Matlab. Numerical examples are included. They show the effectiveness of our strategy.

关键词： Hankel matrix Block diagonalization triangular toeplitz matrix Schur complementation Euclidean algorithm

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Superfast solution of linear convolutional Volterra equations using QTT approximation

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2014年 260卷 434-448页

作者： Roberts, Jason A. Savostyanov, Dmitry V. Tyrtyshnikov, Eugene E. Univ Chester Chester CH1 4BJ Cheshire England Russian Acad Sci Inst Numer Math Moscow 119333 Russia Moscow MV Lomonosov State Univ Moscow 119991 Russia

We address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed by the divide and conquer and modified Bini's algorithms, for which we present the versions with the QTT approximation. We also present an efficient formula for the shift of vectors given in QTT format, which is used in the divide and conquer algorithm. As a result, we reduce the complexity of inversion from the fast Fourier level O (n log n) to the speed of superfast Fourier transform, i.e., O (log(2) n). The results of the paper are illustrated by numerical examples. (C) 2013 Elsevier B.V. All rights reserved.

关键词： Fractional calculus triangular toeplitz matrix Divide and conquer Tensor train format Fast convolution Superfast Fourier transform

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A short note on the weighted sub-partition mean of integers

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OPERATIONS RESEARCH LETTERS 2009年第5期37卷 356-358页

作者： Szabo, Peter Tech Univ Kosice Dept Aerodynam & Simulat Kosice 04021 Slovakia

In this note we study weighted sub-partitions (i(1),...,i(l)) of positive integers on a number n with the greatest sub-partition mean Sigma(l)(k=1) w(i(k))/l, where w : {1,..., n} -> R(+) is a weight function. We show that this problem is closely related with the problem of computing the eigenvalue of a toeplitz matrix in a specific form. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Partition of positive integers Weighted sub-partition mean triangular toeplitz matrix Eigenvalue

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FAST METHODS FOR COMPUTING THE VALUES OF THE DERIVATIVES OF A RATIONAL-FUNCTION

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 1992年第3期39卷 387-391页

作者： ZHU, HY SHANDONG POLYTECH UNIV DEPT MATHJINANPEOPLES R CHINA

We give the relation between computing the values of the derivatives of a rational function and solving a triangular toeplitz system. Fast methods for solving the triangular toeplitz system are shown in this paper.

关键词： VALUES OF THE DERIVATIVES OF A RATIONAL FUNCTION triangular toeplitz matrix COMPLEXITY

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A fast method to block-diagonalize a Hankel matrix

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NUMERICAL ALGORITHMS 2008年第1期47卷 15-34页

作者： Belhaj, Skander Univ Franche Comte UFR Sci & Tech Math Lab CNRS UMR 6623 F-25030 Besancon France

In this paper, we consider an approximate block diagonalization algorithm of an nxn real Hankel matrix in which the successive transformation matrices are upper triangular toeplitz matrices, and propose a new fast approach to compute the factorization in O(n(2)) operations. This method consists on using the revised Bini method (Lin et al., Theor Comp Sci 315: 511-523, 2004). To motivate our approach, we also propose an approximate factorization variant of the customary fast method based on Schur complementation adapted to the nxn real Hankel matrix. All algorithms have been implemented in Matlab and numerical results are included to illustrate the effectiveness of our approach.

关键词： Hankel matrix block diagonalization triangular toeplitz matrix Schur complementation

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On regularity of block triangular fuzzy matrices

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Journal of Applied Mathematics and Computing 2004年第1-2期16卷 207-220页

作者： Meenakshi, Ar. Department of Mathematics Annamalai University Annamalainagar-608002 India

Necessary and sufficient conditions are given for the regularity of block triangular fuzzy matrices. This leads to characterization of idempotency of a class of triangular toeplitz matrices. As an application, the existence of group inverse of a block triangular fuzzy matrix is discussed. Equivalent conditions for a regular block triangular fuzzy matrix to be expressed as a sum of regular block fuzzy matrices is derived. Further, fuzzy relations equations consistency is studied.

关键词： Block triangular matrix Fuzzy matrix Fuzzy relational equation triangular toeplitz matrix

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ON THE COMPLETION OF PARTIALLY GIVEN triangular toeplitz MATRICES TO CONTRACTIONS

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SIAM JOURNAL ON matrix ANALYSIS AND APPLICATIONS 1993年第2期14卷 545-552页

作者： NAEVDAL, G NORWEGIAN INST TECHNOL DIV MATH SCIN-7034 TRONDHEIMNORWAY

In this paper a complete description of the patterns of partial triangular toeplitz contractions, which are completable to triangular toeplitz contractions, is given.

关键词： matrix COMPLETION CONTRACTION triangular toeplitz matrix PARTIAL matrix

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