Publisher's Synopsis
Excerpt from Quasi-Tridiagonal Matrices and Type-Insensitive Differences Equations
The processes described here are, in part, extensions of those described by Karlqvist[4] and Cornock[2]. Furthermore it is shown that the finite difference equations obtained from symmetric positive systems, as defined by K. O. Friedrichs[5], also fall into the class of matrices of the form These include, in addition to pure elliptic or hyperbolic equations, a certain class of boundary problems for equations of mixed type such as the Tricomi equation. Where iterative methods for such problems seem to present difficulties, even when Q is positive definite and symmetric, the direct methods for solving are shown below to be feasible for this larger class of problems.
A criterion is given for the process to apply which is similar to that found for the ldu theorem[3]. It is also shown that when the Mn have the same order p, and the Dn are easily invertible, then the process may be reduced to multiplication of matrices and the inversion of one matrix of order p. This last fact was noted by Corhook[2] for the Poisson and bi-harmonic case.
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