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Mathematics of Computation

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On the propagation of errors in the inversion of certain tridiagonal matrices


Author: Arnold N. Lowan
Journal: Math. Comp. 14 (1960), 333-338
MSC: Primary 65.35
DOI: https://doi.org/10.1090/S0025-5718-1960-0136054-2
MathSciNet review: 0136054
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Abstract: When the differential equation of heat conduction is replaced by the implicit difference analog, one is led to the solution of Ay = b where A is a tridiagonal matrix whose elements on the principal diagonal are = 2 + 2r and whose elements off the principal diagonal are = -r.

The system of equations may be solved by the following algorithm:

$\displaystyle {\beta _k}\, = \,u\, = \,{r^2}\beta _{k - 1}^{ - 1},\,{\beta _1}\... ...1}};\qquad{y_k}\, = \,{z_k}\, - \,{\gamma _k}{y_{k + 1}},\qquad{y_M}\,=\,{z_M}.$

An upper bound of the round-off errors in the computed values of the $ {y_k}$'s is obtained. An actual test case showed that the theoretical upper bound is about four times larger than the true round-off error. Moreover, the theoretical upper bound does not seem to vary appreciably with r.

References [Enhancements On Off] (What's this?)

  • [1] The implicit scheme was first suggested by J. Crank and P. Nicolson in the paper entitled, ``A practical method for the numerical evaluation of solutions of differential equations of the heat conduction type,'' Camb. Phil. Soc. Proc., v. 43, 1943, p. 50-67. Its stability is discussed by G. G. O'Brien, Morton A. Hyman, and Sidney Kaplan in ``A Study of the numerical solution of partial differential equations,'' Jn. Math. and Phys., v. 29, 1950/51, p. 223-251. It is also discussed in the writer's book The Operator Approach to Problems of Stability and Convergence, Scripta Mathematica, 1957.
  • [2] See for instance M. Lotkin's ``The numerical integration of heat conduction equations,'' Jn. Math. and Phys., v. 37, 1958, p. 178, and R. D. Richtmyer, Difference Methods for Initial Value Problems, Chap. VI., Interscience, N. Y., 1957. MR 0099118 (20:5562)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1960-0136054-2
Article copyright: © Copyright 1960 American Mathematical Society

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