An efficient method for computing the inverse of arrowhead matrices

Abstract In this paper we propose a simple and effective method to find the inverse of arrowhead matrices which often appear in wide areas of applied science and engineering such as wireless communications systems, molecular physics, oscillators vibrationally coupled with Fermi liquid, and eigenvalue problems. A modified Sherman–Morrison inverse matrix method is proposed for computing the inverse of an arrowhead matrix. The effectiveness of the proposed method is illustrated and numerical results are presented along with comparative results.

[1]  G. Stewart,et al.  Computing the eigenvalues and eigenvectors of symmetric arrowhead matrices , 1990 .

[2]  Sabine Van Huffel,et al.  Efficient reduction algorithms for bordered band matrices , 1995, Numer. Linear Algebra Appl..

[3]  Gene H. Golub,et al.  QR-Like Algorithms for Symmetric Arrow Matrices , 1992, SIAM J. Matrix Anal. Appl..

[4]  C. Fonseca,et al.  Explicit inverses of some tridiagonal matrices , 2001 .

[5]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[6]  R. Plemmons,et al.  IMPROVING JACOBI AND GAUSS-SEIDEL ITERATIONS , 1987 .

[7]  H. Saberi Najafi,et al.  Computational algorithms for computing the inverse of a square matrix, quasi-inverse of a non-square matrix and block matrices , 2006, Appl. Math. Comput..

[8]  P. Robinson,et al.  Variational bounds on the entries of the inverse of a matrix , 1992 .

[9]  George A. Gravvanis,et al.  High Performance Inverse Preconditioning , 2009 .

[10]  Maria Morandi Cecchi,et al.  The modified bordering method to evaluate eigenvalues and eigenvectors of normal matrices , 1996, Numerical Algorithms.

[11]  Hubert Pickmann,et al.  TWO INVERSE EIGENPROBLEMS FOR SYMMETRIC DOUBLY ARROW MATRICES , 2009 .

[12]  Victor Y. Pan,et al.  Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations , 2005, Numerische Mathematik.

[13]  I. Duff,et al.  Direct Methods for Sparse Matrices , 1987 .

[14]  Li Chen,et al.  Efficient Recursive Multichannel Blind Image Restoration , 2007, EURASIP J. Adv. Signal Process..

[15]  Lorenz S. Cederbaum,et al.  The eigenvalue problem for ‘‘arrow’’ matrices , 1984 .

[16]  George A. Gravvanis,et al.  Solving Symmetric Arrowhead and Special Tridiagonal Linear Systems by Fast Approximate Inverse Preconditioning , 2002, J. Math. Model. Algorithms.

[18]  George A. Gravvanis,et al.  An approximate inverse matrix technique for arrowhead matrices , 1998, Int. J. Comput. Math..

[19]  Uwe Schäfer.  The Feasibility of the Interval Gaussian Algorithm for Arrowhead Matrices , 2001, Reliab. Comput..

[20]  B. Parlett,et al.  The use of a refined error bound when updating eigenvalues of tridiagonals , 1985 .

[21]  Joshua Jortner,et al.  Intramolecular Radiationless Transitions , 1968 .

[22]  Jesse L. Barlow,et al.  Accurate eigenvalue decomposition of real symmetric arrowhead matrices and applications , 2013, 1302.7203.

[23]  S. A. Edalatpanah,et al.  Comparison analysis for improving preconditioned SOR-type iterative method , 2013 .

[24]  Suely Oliveira.  A new parallel chasing algorithm for transforming arrowhead matrices to tridiagonal form , 1998, Math. Comput..

[25]  Jan Brandts,et al.  Computable eigenvalue bounds for rank-k perturbations , 2010 .

[26]  Jack J. Dongarra,et al.  Solving linear systems on vector and shared memory computers , 1990 .

[27]  Aurél Galántai,et al.  Parallel ABS projection methods for linear and nonlinear systems with block arrowhead structure , 1999 .

[28]  S. A. Edalatpanah,et al.  Iterative methods with analytical preconditioning technique to linear complementarity problems: application to obstacle problems , 2013, RAIRO Oper. Res..

[29]  James Demmel,et al.  On structure-exploiting trust-region regularized nonlinear least squares algorithms for neural-network learning , 2003, Neural Networks.

[30]  Bruce W. Suter,et al.  A Hub Matrix Theory and Applications to Wireless Communications , 2007, EURASIP J. Adv. Signal Process..

[31]  J. W. Gadzuk.  Localized vibrational modes in Fermi liquids. General theory , 1981 .

[32]  David E. Stewart,et al.  A graph-theoretic model of symmetric givens operations and its implications , 1997 .

[33]  N. Mastronardi,et al.  Fast and Stable Reduction of Diagonal Plus Semi-Separable Matrices to Tridiagonal and Bidiagonal Form , 2001 .

[34]  Efstratios Gallopoulos,et al.  Computing symmetric nonnegative rank factorizations , 2012 .

[35]  George A. Gravvanis,et al.  High performance finite element approximate inverse preconditioning , 2008, Appl. Math. Comput..

[36]  George A. Gravvanis,et al.  On the Solution of Boundary Value Problems by Using Fast Generalized Approximate Inverse Banded Matrix Techniques , 2003, The Journal of Supercomputing.

[37]  Gérard Meurant,et al.  A Review on the Inverse of Symmetric Tridiagonal and Block Tridiagonal Matrices , 1992, SIAM J. Matrix Anal. Appl..