Characterizing the inverses of block tridiagonal, block Toeplitz matrices
Abstract
We consider the inversion of block tridiagonal, block Toeplitz matrices and comment on the behaviour of these inverses as one moves away from the diagonal. Using matrix Möbius transformations, we first present an O(1) representation (with respect to the number of block rows and block columns) for the inverse matrix and subsequently use this representation to characterize the inverse matrix. There are four symmetrydistinct cases where the blocks of the inverse matrix (i) decay to zero on both sides of the diagonal, (ii) oscillate on both sides, (iii) decay on one side and oscillate on the other and (iv) decay on one side and grow on the other. This characterization exposes the necessary conditions for the inverse matrix to be numerically banded and may also aid in the design of preconditioners and fast algorithms. Finally, we present numerical examples of these matrix types.
 Authors:

 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Center for Nanophase Materials Science (CNMS)
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). National Center for Computational Sciences
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computer Science & Mathematics Division
 Publication Date:
 Research Org.:
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF)
 Sponsoring Org.:
 USDOE Office of Science (SC), Workforce Development for Teachers and Scientists (WDTS) (SC27)
 OSTI Identifier:
 1567555
 DOE Contract Number:
 AC0500OR22725
 Resource Type:
 Journal Article
 Journal Name:
 Computational Science and Discovery
 Additional Journal Information:
 Journal Volume: 8; Journal Issue: 1; Journal ID: ISSN 17494699
 Country of Publication:
 United States
 Language:
 English
 Subject:
 matrix inversion algorithms; matrix Möbius transformations; block tridiagonal matrices; block Toeplitz matrices
Citation Formats
Boffi, Nicholas M., Hill, Judith C., and Reuter, Matthew G. Characterizing the inverses of block tridiagonal, block Toeplitz matrices. United States: N. p., 2015.
Web. doi:10.1088/17494680/8/1/015001.
Boffi, Nicholas M., Hill, Judith C., & Reuter, Matthew G. Characterizing the inverses of block tridiagonal, block Toeplitz matrices. United States. doi:10.1088/17494680/8/1/015001.
Boffi, Nicholas M., Hill, Judith C., and Reuter, Matthew G. Thu .
"Characterizing the inverses of block tridiagonal, block Toeplitz matrices". United States. doi:10.1088/17494680/8/1/015001.
@article{osti_1567555,
title = {Characterizing the inverses of block tridiagonal, block Toeplitz matrices},
author = {Boffi, Nicholas M. and Hill, Judith C. and Reuter, Matthew G.},
abstractNote = {We consider the inversion of block tridiagonal, block Toeplitz matrices and comment on the behaviour of these inverses as one moves away from the diagonal. Using matrix Möbius transformations, we first present an O(1) representation (with respect to the number of block rows and block columns) for the inverse matrix and subsequently use this representation to characterize the inverse matrix. There are four symmetrydistinct cases where the blocks of the inverse matrix (i) decay to zero on both sides of the diagonal, (ii) oscillate on both sides, (iii) decay on one side and oscillate on the other and (iv) decay on one side and grow on the other. This characterization exposes the necessary conditions for the inverse matrix to be numerically banded and may also aid in the design of preconditioners and fast algorithms. Finally, we present numerical examples of these matrix types.},
doi = {10.1088/17494680/8/1/015001},
journal = {Computational Science and Discovery},
issn = {17494699},
number = 1,
volume = 8,
place = {United States},
year = {2015},
month = {1}
}
Works referenced in this record:
Marching Algorithms for Elliptic Boundary Value Problems. I: The Constant Coefficient Case
journal, October 1977
 Bank, Randolph E.; Rose, Donald J.
 SIAM Journal on Numerical Analysis, Vol. 14, Issue 5
Block Preconditioning for the Conjugate Gradient Method
journal, January 1985
 Concus, P.; Golub, G. H.; Meurant, G.
 SIAM Journal on Scientific and Statistical Computing, Vol. 6, Issue 1
Estimates for the inverse of tridiagonal matrices arising in boundaryvalue problems
journal, January 1986
 Mattheij, R. M. M.; Smooke, M. D.
 Linear Algebra and its Applications, Vol. 73
A Review on the Inverse of Symmetric Tridiagonal and Block Tridiagonal Matrices
journal, July 1992
 Meurant, Gérard
 SIAM Journal on Matrix Analysis and Applications, Vol. 13, Issue 3
Some algorithms for solving special tridiagonal block Toeplitz linear systems
journal, July 2003
 Minchev, Borislav V.
 Journal of Computational and Applied Mathematics, Vol. 156, Issue 1
Exact solution for the resolvent matrix of a generalized tridiagonal Hamiltonian
journal, November 1979
 Dy, K. S.; Wu, ShiYu; Spratlin, T.
 Physical Review B, Vol. 20, Issue 10
Simple scheme for surfaceband calculations. II. The Green's function
journal, May 1981
 Lee, D. H.; Joannopoulos, J. D.
 Physical Review B, Vol. 23, Issue 10
Edge effects in finite elongated graphene nanoribbons
journal, December 2007
 Hod, Oded; Peralta, Juan E.; Scuseria, Gustavo E.
 Physical Review B, Vol. 76, Issue 23
Probing the surfacetobulk transition: A closedform constantscaling algorithm for computing subsurface Green functions
journal, February 2011
 Reuter, Matthew G.; Seideman, Tamar; Ratner, Mark A.
 Physical Review B, Vol. 83, Issue 8
The role of dimensionality in the decay of surface effects
journal, February 2013
 Reuter, Matthew G.; Boffi, Nicholas M.; Ratner, Mark A.
 The Journal of Chemical Physics, Vol. 138, Issue 8
Theoretical analysis of electron transport through organic molecules
journal, January 2004
 Tomfohr, John; Sankey, Otto F.
 The Journal of Chemical Physics, Vol. 120, Issue 3
Block tridiagonal matrix inversion and fast transmission calculations
journal, March 2008
 Petersen, Dan Erik; Sørensen, Hans Henrik B.; Hansen, Per Christian
 Journal of Computational Physics, Vol. 227, Issue 6
Distributed nonequilibrium Green’s function algorithms for the simulation of nanoelectronic devices with scattering
journal, August 2011
 Cauley, Stephen; Luisier, Mathieu; Balakrishnan, Venkataramanan
 Journal of Applied Physics, Vol. 110, Issue 4
Simple Analytic Description of Collection Efficiency in Organic Photovoltaics
journal, February 2013
 Savoie, Brett M.; Movaghar, Bijan; Marks, Tobin J.
 The Journal of Physical Chemistry Letters, Vol. 4, Issue 5
On a new class of structured matrices
journal, September 1999
 Eidelman, Y.; Gohberg, I.
 Integral Equations and Operator Theory, Vol. 34, Issue 3
On generators of quasiseparable finite block matrices
journal, December 2005
 Eidelman, Y.; Gohberg, I.
 Calcolo, Vol. 42, Issue 34
Algorithm 15 inversion of a blockwise tridiagonal matrix
journal, September 1971
 Wallach, Y.; Barlevi, A.
 Computing, Vol. 7, Issue 34
On inverses of Hessenberg matrices
journal, April 1979
 Ikebe, Yasuhiko
 Linear Algebra and its Applications, Vol. 24
Inversion of band matrices
journal, April 1979
 Yamamoto, Tetsuro; Ikebe, Yasuhiko
 Linear Algebra and its Applications, Vol. 24
Inversion of tridiagonal matrices
journal, October 1982
 Lewis, Jerry W.
 Numerische Mathematik, Vol. 38, Issue 3
Inverses of quasitridiagonal matrices
journal, January 1984
 Rizvi, S. A. H.
 Linear Algebra and its Applications, Vol. 56
Parallel solution of block tridiagonal linear systems
journal, June 1988
 Bevilacqua, Roberto; Codenotti, Bruno; Romani, Francesco
 Linear Algebra and its Applications, Vol. 104
A method to compute the inverse of an nblock tridiagonal quasiHermitian matrix
journal, October 1991
 Godfrin, E. M.
 Journal of Physics: Condensed Matter, Vol. 3, Issue 40
On periodic blocktridiagonal matrices
journal, April 1992
 Rózsa, Pál; Romani, Francesco
 Linear Algebra and its Applications, Vol. 167
Upper and lower bounds for inverse elements of finite and infinite tridiagonal matrices
journal, November 1996
 Shivakumar, P. N.; Ji, Chuanxiang
 Linear Algebra and its Applications, Vol. 247
Analytical inversion of general tridiagonal matrices
journal, November 1997
 Huang, Y.; McColl, W. F.
 Journal of Physics A: Mathematical and General, Vol. 30, Issue 22
Twosided bounds on the inverses of diagonally dominant tridiagonal matrices
journal, January 1999
 Nabben, Reinhard
 Linear Algebra and its Applications, Vol. 287, Issue 13
The inverse of a tridiagonal matrix
journal, March 2001
 Mallik, Ranjan K.
 Linear Algebra and its Applications, Vol. 325, Issue 13
Firstprinciples electronic transport calculations in finite elongated systems: A divide and conquer approach
journal, September 2006
 Hod, Oded; Peralta, Juan E.; Scuseria, Gustavo E.
 The Journal of Chemical Physics, Vol. 125, Issue 11
An efficient, blockbyblock algorithm for inverting a block tridiagonal, nearly block Toeplitz matrix
journal, January 2012
 Reuter, Matthew G.; Hill, Judith C.
 Computational Science & Discovery, Vol. 5, Issue 1
On computingINV block preconditionings for the conjugate gradient method
journal, December 1986
 Concus, P.; Meurant, G.
 BIT, Vol. 26, Issue 4
Inequalities on the elements of the inverse of a certain tridiagonal matrix
journal, January 1970
 Kershaw, D.
 Mathematics of Computation, Vol. 24, Issue 109
Inverses of Band Matrices and Local Convergence of Spline Projections
journal, September 1977
 Demko, Stephen
 SIAM Journal on Numerical Analysis, Vol. 14, Issue 4
Decay rates of inverses of banded Mmatrices that are near to Toeplitz matrices
journal, October 1988
 Eijkhout, Victor; Polman, Ben
 Linear Algebra and its Applications, Vol. 109
Decay Rates of the Inverse of Nonsymmetric Tridiagonal and Band Matrices
journal, January 1999
 Nabben, Reinhard
 SIAM Journal on Matrix Analysis and Applications, Vol. 20, Issue 3
Some improvements for twosided bounds on the inverse of diagonally dominant tridiagonal matrices
journal, June 2001
 Peluso, R.; Politi, T.
 Linear Algebra and its Applications, Vol. 330, Issue 13
Estimates for the inverse elements of tridiagonal matrices
journal, June 2006
 Liu, XiaoQin; Huang, TingZhu; Fu, YingDing
 Applied Mathematics Letters, Vol. 19, Issue 6
Matrix möbius transformations
journal, January 1981
 Schwarz, Binyamin; Zaks, Abraham
 Communications in Algebra, Vol. 9, Issue 19
Closedform solutions to surface Green's functions
journal, February 1997
 Umerski, A.
 Physical Review B, Vol. 55, Issue 8
Theory of tunneling magnetoresistance of an epitaxial Fe/MgO/Fe(001) junction
journal, May 2001
 Mathon, J.; Umerski, A.
 Physical Review B, Vol. 63, Issue 22
LevelScheme: A level scheme drawing and scientific figure preparation system for Mathematica
journal, September 2005
 Caprio, M. A.
 Computer Physics Communications, Vol. 171, Issue 2