skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: 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 symmetry-distinct 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:
 [1];  [2];  [3]
  1. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Center for Nanophase Materials Science (CNMS)
  2. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). National Center for Computational Sciences
  3. 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) (SC-27)
OSTI Identifier:
1567555
DOE Contract Number:  
AC05-00OR22725
Resource Type:
Journal Article
Journal Name:
Computational Science and Discovery
Additional Journal Information:
Journal Volume: 8; Journal Issue: 1; Journal ID: ISSN 1749-4699
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:10.1088/1749-4680/8/1/015001.
Boffi, Nicholas M., Hill, Judith C., & Reuter, Matthew G. Characterizing the inverses of block tridiagonal, block Toeplitz matrices. United States. doi:10.1088/1749-4680/8/1/015001.
Boffi, Nicholas M., Hill, Judith C., and Reuter, Matthew G. Thu . "Characterizing the inverses of block tridiagonal, block Toeplitz matrices". United States. doi:10.1088/1749-4680/8/1/015001.
@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 symmetry-distinct 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 = {10.1088/1749-4680/8/1/015001},
journal = {Computational Science and Discovery},
issn = {1749-4699},
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
  • DOI: 10.1137/0714055

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
  • DOI: 10.1137/0906018

Estimates for the inverse of tridiagonal matrices arising in boundary-value problems
journal, January 1986


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
  • DOI: 10.1137/0613045

Some algorithms for solving special tridiagonal block Toeplitz linear systems
journal, July 2003


Exact solution for the resolvent matrix of a generalized tridiagonal Hamiltonian
journal, November 1979


Simple scheme for surface-band calculations. II. The Green's function
journal, May 1981


Edge effects in finite elongated graphene nanoribbons
journal, December 2007


Probing the surface-to-bulk transition: A closed-form constant-scaling algorithm for computing subsurface Green functions
journal, February 2011


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
  • DOI: 10.1063/1.4792643

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
  • DOI: 10.1063/1.1625911

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
  • DOI: 10.1016/j.jcp.2007.11.035

Distributed non-equilibrium 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
  • DOI: 10.1063/1.3624612

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
  • DOI: 10.1021/jz302148z

On a new class of structured matrices
journal, September 1999

  • Eidelman, Y.; Gohberg, I.
  • Integral Equations and Operator Theory, Vol. 34, Issue 3
  • DOI: 10.1007/BF01300581

On generators of quasiseparable finite block matrices
journal, December 2005


Algorithm 15 inversion of a blockwise tridiagonal matrix
journal, September 1971


On inverses of Hessenberg matrices
journal, April 1979


Inversion of band matrices
journal, April 1979


Inversion of tridiagonal matrices
journal, October 1982


Inverses of quasi-tridiagonal matrices
journal, January 1984


Parallel solution of block tridiagonal linear systems
journal, June 1988


A method to compute the inverse of an n-block tridiagonal quasi-Hermitian matrix
journal, October 1991


On periodic block-tridiagonal matrices
journal, April 1992


Upper and lower bounds for inverse elements of finite and infinite tridiagonal matrices
journal, November 1996


Analytical inversion of general tridiagonal matrices
journal, November 1997


Two-sided bounds on the inverses of diagonally dominant tridiagonal matrices
journal, January 1999


The inverse of a tridiagonal matrix
journal, March 2001


First-principles 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
  • DOI: 10.1063/1.2349482

An efficient, block-by-block algorithm for inverting a block tridiagonal, nearly block Toeplitz matrix
journal, January 2012


On computingINV block preconditionings for the conjugate gradient method
journal, December 1986


Inequalities on the elements of the inverse of a certain tridiagonal matrix
journal, January 1970


Inverses of Band Matrices and Local Convergence of Spline Projections
journal, September 1977

  • Demko, Stephen
  • SIAM Journal on Numerical Analysis, Vol. 14, Issue 4
  • DOI: 10.1137/0714041

Decay rates of inverses of banded M-matrices that are near to Toeplitz matrices
journal, October 1988


Decay Rates of the Inverse of Nonsymmetric Tridiagonal and Band Matrices
journal, January 1999


Some improvements for two-sided bounds on the inverse of diagonally dominant tridiagonal matrices
journal, June 2001


Estimates for the inverse elements of tridiagonal matrices
journal, June 2006


Matrix möbius transformations
journal, January 1981


Closed-form solutions to surface Green's functions
journal, February 1997


Theory of tunneling magnetoresistance of an epitaxial Fe/MgO/Fe(001) junction
journal, May 2001


LevelScheme: A level scheme drawing and scientific figure preparation system for Mathematica
journal, September 2005