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 *A*y = **b** where *A* is a tridiagonal matrix whose elements on the principal diagonal are = 2 + 2*r* and whose elements off the principal diagonal are = -r.

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

*r*.

**[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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DOI:
https://doi.org/10.1090/S0025-5718-1960-0136054-2

Article copyright:
© Copyright 1960
American Mathematical Society