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.

**[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