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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
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] Mark Lotkin, The numerical integration of heat conduction equations, J. Math. and Phys. 37 (1958), 178–187. MR 0099118

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