Journal of Mathematical Chemistry

, Volume 50, Issue 1, pp 289–299 | Cite as

Theory and algorithm of the inversion method for pentadiagonal matrices

Original Paper


A recently developed inversion method for pentadiagonal matrices is reconsidered in this work. The mathematical structure of the previously suggested method is fully developed. In the process of establishing the mathematical structure, certain determinantial relations specific to pentadiagonal matrices were also established. This led to a rather general necessary and sufficient condition for the method to work. All the so called forward and backward leading principal submatrices need to be non-singular. While this condition sounds restrictive it really is not so. These are in fact the conditions for forward and backward Gauss Eliminations without any pivoting requirement. Additionally, the method is more effective computational complexity wise then recently published competitive methods.


Direct methods for linear systems and matrix inversion Difference equations Matrices, determinants 


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  1. 1.
    Znojil M.: Perturbation method with triangular propogators and anharmonicities of intermediate strength. J. Math. Chem. 28, 140–167 (2000)CrossRefGoogle Scholar
  2. 2.
    Kanal M.E., Baykara N.A., Demiralp M.A.: A novel approach to the numerical inversion algorithm for adjacent pentadiagonal matrices. Tools Math. Model. 9, 270–280 (2003)Google Scholar
  3. 3.
    Huang G.H., McColl W.F.: Analytical inversion of general tridiagonal matrices. J. Phys. A Math. Gen. 30, 7919–7993 (1997)CrossRefGoogle Scholar
  4. 4.
    M.E. Kanal, Parallel algorithm on inversion for adjacent pentadiagonal matrices with MPI. J. Supercomput. (2010). doi: 10.1007/s11227-010-0487-y
  5. 5.
    Zhao X.-L., Huang T.-Z.: On the inverse of a general pentadiagonal matrix. Appl. Math. Comput. 202, 639–646 (2008)CrossRefGoogle Scholar
  6. 6.
    Hadj X.-A.D.A., Elouafi M.: A fast numerical algorithm for the inverse of a tridiagonal and pentadiagonal matrix. Appl. Math. Comput. 202, 441–445 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Informatics InstituteIstanbul Technical UniversityMaslak, IstanbulTurkey
  2. 2.Mathematics DepartmentMarmara UniversityGoztepe, IstanbulTurkey

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