On the inverses of general tridiagonal matrices

In this work, the sign distribution for all inverse elements of general tridiagonal H-matrices is presented. In addition, some computable upper and lower bounds for the entries of the inverses of diagonally dominant tridiagonal matrices are obtained. Based on the sign distribution, these bounds greatly improve some well-known results due to Ostrowski (1952) 23, Shivakumar and Ji (1996) 26, Nabben (1999) 21,22 and recently given by Peluso and Politi (2001) 24, Peluso and Popolizio (2008) 25 and so forth. It is also stated that the inverse of a general tridiagonal matrix may be described by 2n-2 parameters ({θk}k=2n and {φk}k=1n-1) instead of 2n+2 ones as given by El-Mikkawy (2004) 3, El-Mikkawy and Karawia (2006) 4 and Huang and McColl (1997) 10. According to these results, a new symbolic algorithm for finding the inverse of a tridiagonal matrix without imposing any restrictive conditions is presented, which improves some recent results. Finally, several applications to the preconditioning technology, the numerical solution of differential equations and the birth–death processes together with numerical tests are given.

[1]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[2]  Ting-Zhu Huang,et al.  Lower bounds for the minimum eigenvalue of Hadamard product of an M-matrix and its inverse , 2007 .

[3]  Ana M. Urbano,et al.  The N-matrix completion problem under digraphs assumptions☆ , 2004 .

[4]  D. Kershaw.  Inequalities on the elements of the inverse of a certain tridiagonal matrix , 1970 .

[5]  R.M.M. Mattheij,et al.  Estimates for the inverse of tridiagonal matrices arising in boundary-value problems☆ , 1986 .

[6]  Moawwad E. A. El-Mikkawy.  On the inverse of a general tridiagonal matrix , 2004, Appl. Math. Comput..

[7]  New lower bounds on eigenvalue of the Hadamard product of an M-matrix and its inverse , 2009 .

[8]  Richard S. Varga,et al.  Matrix Iterative Analysis , 2000, The Mathematical Gazette.

[9]  Jerry W. Lewis.  Inversion of tridiagonal matrices , 1982 .

[10]  Ting-Zhu Huang,et al.  Estimates for the inverse elements of tridiagonal matrices , 2006, Appl. Math. Lett..

[11]  Ranjan K. Mallik,et al.  The inverse of a tridiagonal matrix , 2001 .

[12]  F. Gantmakher,et al.  Sur les matrices complètement non négatives et oscillatoires , 1937 .

[13]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[14]  Ting-Zhu Huang,et al.  On some new approximate factorization methods for block tridiagonal matrices suitable for vector and parallel processors , 2009, Math. Comput. Simul..

[15]  W. Mccoll,et al.  Analytical inversion of general tridiagonal matrices , 1997 .

[16]  P. Shivakumar,et al.  Upper and lower bounds for inverse elements of finite and infinite tridiagonal matrices , 1996 .

[17]  T. Politi,et al.  A NOTE ON ESTIMATES OF DIAGONAL ELEMENTS OF THE INVERSE OF DIAGONALLY DOMINANT TRIDIAGONAL MATRICES , 2008 .

[18]  Y. Ikebe.  On inverses of Hessenberg matrices , 1979 .

[19]  Qiang Ye,et al.  On Two-Sided Bounds Related to Weakly Diagonally Dominant M-Matrices with Application to Digital Circuit Dynamics , 1996, SIAM J. Matrix Anal. Appl..

[20]  T. Politi,et al.  Some improvements for two-sided bounds on the inverse of diagonally dominant tridiagonal matrices , 2001 .

[21]  Reinhard Nabben,et al.  Decay Rates of the Inverse of Nonsymmetric Tridiagonal and Band Matrices , 1999, SIAM J. Matrix Anal. Appl..

[22]  William F. Moss,et al.  Decay rates for inverses of band matrices , 1984 .

[23]  Moawwad E. A. El-Mikkawy,et al.  Inversion of general tridiagonal matrices , 2006, Appl. Math. Lett..

[24]  Wai-Ki Ching.  Iterative Methods for Queuing and Manufacturing Systems , 2001 .

[25]  A. Ostrowski.  Note on bounds for determinants with dominant principal diagonal , 1952 .

[26]  R. Nabben.  Two-sided bounds on the inverses of diagonally dominant tridiagonal matrices , 1999 .

[27]  Winfried K. Grassmann,et al.  Real eigenvalues of certain tridiagonal matrix polynomials, with queueing applications , 2002 .

[28]  Mei Han An,et al.  accuracy and stability of numerical algorithms , 1991 .

[29]  G. Golub,et al.  Bounds for the Entries of Matrix Functions with Applications to Preconditioning , 1999 .