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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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Quasi-Tridiagonal Matrices and Type-Insensitive Differences Equations
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Paperback. Condizione: New. Print on Demand. This book details direct methods for solving quasi-tridiagonal matrices and their applications to solving type-insensitive difference equations. These matrices arise in solving finite difference equations of problems for pure elliptic equations and for symmetric positive systems. The author outlines a partitioned decomposition of the matrix into a product of lower and upper triangular matrices, which serves as the foundation of the direct methods proposed. A criterion is given for applying these methods, which may be preferable to iterative methods in certain situations. The author also discusses other methods for special types of matrices involving the inversion of relatively small matrices, extending the applicability of the approach. Through illustrative examples, the book demonstrates the effectiveness of these direct methods for solving boundary problems, such as those involving the Tricomi equation. This book provides valuable insights into solving linear equations arising from finite difference approximations, offering a comprehensive treatment of direct methods and their applications in solving complex problems in applied mathematics and beyond. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book. print-on-demand item. Codice articolo 9781330199152_0
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