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The Standard Form Package for Linear Systems

The rif algorithm is a generalization of the Standard Form Algorithm [57] for simplifying linear systems to a coordinate-dependent ordered canonical form.

The Long Guide to the Standard Form Package is 90 page manual [67] which documents our algorithms and provides illustrative examples of their use. In [67, §8] we describe the Standard Form Algorithm's strategy for reducing large systems of PDEs to standard form. Upon input into the Standard Form Algorithm a system of PDEs is divided into 4 classes: one-term PDEs (e.g. $\eta_{xxy} = 0$), `easy' PDEs, PDEs which are nonlinear in their highest derivative, and unclassified PDEs. These classes are updated at each iteration. User defined strategy parameters control the flow of PDEs between these lists by setting criteria for `easy' or desirable PDEs. Extreme settings include the greedy and conservative settings. This divide and conquer strategy, which only applies the algorithm to an easy piece of the system at a time, has been key to our success in treating large systems. User controlled interface information helps trace expression swell. To facilitate interactive use there is a storage option which allows a current copy of the system to be stored to disk at each major iteration of the Standard Form Algorithm.

In [67, §9] we give the example of reduction of the determining system for infinitesimal symmetries of the Magneto-Hydrodynamic (MHD) equations to standard form. The original system contains more than 2000 PDEs in 12 independent and 12 dependent variables. The reduced system contains 133 equations, of which 104 are one-term PDEs, It was automatically integrated using the Taylor expansion algorithm. An implemented strategy is given for systems of PDE with unspecified functions or parameters. These yield classification problems - problems where a tree of cases corresponding to different values of the parameters or forms of the functions have to be treated separately. A nontrivial example of group classifying a family of nonlinear telegraph equations is given in [67, §9] and in [58]. In the manual, interfaces between our packages and the following packages are described: the Maple Exterior Forms package Liesymm; the Macsyma package symmgrp.max and Hickman's Maple package Symmetry. All these packages automatically produce determining systems for infinitesimal symmetries of PDEs. We are developing such interfaces to avoid unnecessarily duplicating the work of others.

 


Next: The Reduced Involutive Form Up: Differential Elimination Completion Algorithms Previous: Differential Elimination Completion Algorithms
Greg Reid 2003-11-24