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On the solution of block-tridiagonal systems arising from certain finite-difference equations


Author: J. M. Varah
Journal: Math. Comp. 26 (1972), 859-868
MSC: Primary 65F05
MathSciNet review: 0323087
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Abstract: We consider the solution of the linear systems arising from certain implicit finite-difference approximations to systems of linear differential equations. In particular, we consider those schemes which lead to matrices of block-tridiagonal form. There are two common methods for solving such equations: using a block-tridiagonal factorization (blocksolve), or treating the matrix as a band matrix (bandsolve).

First, we discuss conditions for ensuring the numerical stability of the block-tridiagonal factorization for general matrices of this form. Then, we compare the two methods for general block-tridiagonal matrices (including matrices arising from the Crank-Nicolson scheme for systems of parabolic equations) and for a more specialized block-tridiagonal matrix which arises from schemes of H. B. Keller for systems of two-point boundary value problems and parabolic equations.


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  • [1] David G. Feingold and Richard S. Varga, Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem, Pacific J. Math. 12 (1962), 1241–1250. MR 0151473
  • [2] Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964. MR 0175290
  • [3] Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0201039
  • [4] Herbert B. Keller, Accurate difference methods for linear ordinary differential systems subject to linear constraints, SIAM J. Numer. Anal. 6 (1969), 8–30. MR 0253562
  • [5] Herbert B. Keller, A new difference scheme for parabolic problems, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 327–350. MR 0277129
  • [6] Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
  • [7] Burton Wendroff, Theoretical numerical analysis, Academic Press, New York-London, 1966. MR 0196896
  • [8] J. H. Wilkinson, Error analysis of direct methods of matrix inversion, J. Assoc. Comput. Mach. 8 (1961), 281–330. MR 0176602
  • [9] Alan George, Nested dissection of a regular finite element mesh, SIAM J. Numer. Anal. 10 (1973), 345–363. Collection of articles dedicated to the memory of George E. Forsythe. MR 0388756

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1972-0323087-4
Keywords: Solution of difference equations, band matrices, block-tridiagonal matrices
Article copyright: © Copyright 1972 American Mathematical Society