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This research evaluates the applicability to microcomputers of various methods for determining the solution of large systems of simultaneous linear algebraic equations. Such systems of equations characterize physical systems often encountered in Civil Engineering and other engineering disciplines. Many methods of solution involving either in-core or out-of-core storage of data have been developed for use with large digital computers. These methods are reviewed and their applicability to microcomputers is evaluated. A comparison of several schemes is made regarding core size required, time of execution, and precision of results. The out-of-core solution schemes for banded matrices are found to be most applicable to microcomputers with large out-of-core storage capacity.

In this dissertation we study a pioneering model of analog computation called General Purpose Analog Computer (GPAC), introduced by Shannon in 1941. The GPAC is capable of manipulating real-valued data streams. Its power is characterized by the class of differentially algebraic functions, which includes the solutions of initial value problems for ordinary differential equations. We address two limitations of this model. The first is its fundamental inability to reason about functions of more than one independent variable (the `time' variable). In particular, the Shannon GPAC cannot be used to specify solutions of partial differential equations. The second concerns the notion of approximability, a desirable property in computation over continuous spaces that is however absent in the GPAC. To overcome these limitations, we extend the class of data types by taking channels carrying information on a general complete metric space X; for example the class of continuous functions of one real variable. We consider the original modules in Shannon's construction (constants, adders, multipliers, integrators) and add two new modules: a differential module which computes spatial derivatives; and a continuous limit module which computes limits. We then build networks using X-stream channels and the abovementioned modules. This leads us to a framework in which the speci cations of such analog systems are given by fi xed points of certain operators on continuous data streams, as considered by Tucker and Zucker. We study the properties of these analog systems and their associated operators. We present a characterization which generalizes Shannon's results. We show that some non-differentially algebraic functions such as the gamma function are generable by our model. Finally, we attempt to relate our model of computation to the notion of tracking computability as studied by Tucker and Zucker. / Thesis / Doctor of Philosophy (PhD)

Systems of differential-algebraic equations (DAEs) arise in many areas including electrical circuit simulation, chemical engineering, and robotics. The difficulty of solving a DAE is characterized by its index. For index-1 DAEs, there are several solvers, while a high-index DAE is numerically much more difficult to solve. The DAETS solver by Nedialkov and Pryce integrates numerically high-index DAEs. This solver is based on explicit Taylor series method and is efficient on non-stiff to mildly stiff problems, but can have severe stepsize restrictions on highly stiff problems. Hermite-Obreschkoff (HO) methods can be viewed as a generalization of Taylor series methods. The former have smaller error than the latter and can be A- or L- stable. In this thesis, we develop an implicit HO method for numerical solution of stiff high-index DAEs. Our method reduces a given DAE to a system of generally nonlinear equations and a constrained optimization problem. We employ Pryce’s structural analysis to determine the constraints of the problem and to organize the computations of higher-order Taylor coefficients (TCs) and their gradients. Then, we use automatic differentiation to compute these TCs and gradients, which are needed for evaluating the resulting system and its Jacobian. We design an adaptive variable-stepsize and variable-order algorithm and implement it in C++ using literate programming. The theory and implementation are interwoven in this thesis, which can be verified for correctness by a human expert. We report numerical results on stiff DAEs illustrating the accuracy and performance of our method, and in particular, its ability to take large steps on stiff problems. / Thesis / Doctor of Philosophy (PhD)

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No description available.