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The stability of LU-decompositions of block tridiagonal matrices

Published online by Cambridge University Press:  17 April 2009

R. M. M. Mattheij
R. M. M. Mattheij
Affiliation:
Mathematisch Instituut, Katholieke Universiteit, Nijmegen, The Netherlands; Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12181, USA.
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An investigation is made of the stability of block LU-decomposition of matrices A arising from boundary value problems of differential equations, in particular of ordinary differential equations with separated boundary conditions. It is shown that for such matrices the pivotal growth can be bounded by constants of the order of ‖A‖ and, if solution space is dichotomic, often by constants of order one. Furthermore a method to estimate the growth of the pivotal blocks is given. A number of examples support the analysis.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 29 , Issue 2 , April 1984 , pp. 177 - 205
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Ascher, U., Pruess, U. S. and Russell, R. D., “On spline basis selection for solving differential equations”, SIAM J. Numer. Anal. 20 (1983), 121–142.CrossRefGoogle Scholar
[2]de Boor, Carl, “Dichotomies for band matrices” SIAM J. Numer. Anal. 17 (1980), 894–907.CrossRefGoogle Scholar
[3]de Boor, Carl and Weiss, Richard, “SOLVEBLOK: a package for solving almost block diagonal linear systems”, ACM Trans. Math. Software 6 (1980), 80–87.CrossRefGoogle Scholar
[4]Keller, Herbert B., “Accurate difference methods for nonlinear two-point boundary value problems”, SIAM J. Numer. Anal. 11 (1974), 305–320.CrossRefGoogle Scholar
[5]Keller, Herbert B., Numerical solution of two point boundary value problems(Regional Conference Series in Applied Mathematics, 24, Society of Industrial and Applied Mathematics,Philadelphia, Pennsylvania,1976).Google Scholar
[6]Mattheij, R.M.M., “Accurate estimates of solutions of second order recursions”, Lin. Algebra Appl. 12 (1975), 29–54.CrossRefGoogle Scholar
[7]Mattheij, R.M.M., “Characterizations of dominant and dominated solutions of linear recursions”, Numer. Math. 35 (1980), 421–442.CrossRefGoogle Scholar
[8]Mattheij, R.M.M., Stable computation of solutions of unstable linear initial value recursions” BIT 22 (1982), 79–93.CrossRefGoogle Scholar
[9]Mattheij, R.M.M., “The conditioning of linear boundary value porblems”, SIAM J. Numer. Anal. 19 (1982), 963–978.CrossRefGoogle Scholar
[10]Russell, Robert D., “A comparison of collocation and finite differences for two–point boundary value problems”, SIAM J. Numer. Anal. 14 (1977), 19–39.CrossRefGoogle Scholar
[11]Varah, J.M., “On the solution of block-tridiagonal systems arising from certain finite-difference equations”, Math. Comp. 26 (1972), 859–868.CrossRefGoogle Scholar
[12]Varah, James M., “A comparison of some numerical methods for two-point boundary value problems”, Math. Comp. 28 (1974), 743–755.CrossRefGoogle Scholar
[13]Varah, J.M., “Alternate row and column elimination for solving certain linear systems”, SIAM J. Numer. Anal. 13 (1976), 71–75.CrossRefGoogle Scholar
[14]Wilkinson, J.H., The algebraic eigenvalue problem (Clarendon Press, Oxford, 1965).Google Scholar