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    Adaptive linear equation solvers in codes for large stiff systems of odes
    (University of Waterloo, 1991-07) Jackson, K. R.; Seward, W. L.
    Iterative linear equation solvers have been shown to be effective in codes for large systems of stiff initial-value problems for ordinary differential equations (ODEs). While preconditioned iterative methods are required in general for efficiency and robustness, unpreconditioned methods may be cheaper over some ranges of the interval of integration. In this paper, we develop a strategy for switching between unpreconditioned and preconditioned iterative methods depending on the amount of work being done in the iterative solver and properties of the matrix being solved. This strategy is combined with a "type-insensitive" approach to the choice of formula used in the ODE code to develop a method that makes a smooth transition between nonstiff and stiff when the type-insensitive approach is used. If there is a region of the integration that is "mildly" stiff, switchign between unpreconditioned and preconditioned iterative methods also increases the efficiency of the code significantly.
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    Numeration systems, linear recurrences, and regular sets
    (University of Waterloo, 1991-07) Shallit, J. O.
    A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2, . . . expresses a non-negative integer n as a sum of n = Zij=0 ajuj. In this case we say the string aiai-1 . . . a1a0 is a representation for n. If ged (u0,u1, . . . ) = g, then every sufficiently large multiple of g has some representation. If the lexicographic ordering on the representations is the same as the usual ordering of the integers, we say the numeration system is order-preserving. In particular, if u0=1, then the greedy representation, obtained via the greedy algorithm, is order-preserving. We prove that, subject to some technical assumptions, if the set of all representations in an order-preserving numeration system is regular, then the sequence u=(uj)j>0 satisfies a linear recurrence. The converse, however, is not true. The proof uses two lemmas about regular sets that may be of independent interest. The first shows that if L is regular, then the set of lexicographically greatest strings of every length in L is also regular. The second shows that the number of strings of length n is a regular language L is bounded by a constant (independent of n) iff L is the finite union of sets of the form xy*z.
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    Some facts about continued fractions that should be better known
    (University of Waterloo, 1991-07) Shallit, J. O.
    In this report I will give proofs of some simple theorems concerning continued fractions that are known to the cognoscenti, but for which proofs in the literature seem to be missing, incomplete, or hard to locate. In particular, I will give two proofs of the following "folk theorem": if 5 is an irrational number whose continued fraction has bounded partial quotients, then any non-trivial linear fractional transformation of 5 also has bounded partial quotients. The second proof is of interest because it uses the connection between continued fractions and finite automata first enunciated by G. N. Raney [R]. I will assume that the reader knows basic facts about continued fractions, at the level of [HW, Chapter X].
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    Numerical solution of a turning point problem
    (University of Waterloo, 1991-07) Tang, W.-P.
    The turning point problem is known to have some extremely small eigenvalues. No successful numerical solution to this problem has been reported. In this paper, a numerical procedure is proposed. All four boundary layers are well defined and the numerical singularity is successfully removed.
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    An implementationan evaluation of a hierarchical nonlinear planner
    (University of Waterloo, 1991-05) Woods, Steven G.
    Planning is the process of generating sequences of actions in order to provide a method for agents to modify the state of the world in which they exist. It is well known that the application of abstraction can greatly reduce the search involved in creating plans. In this talk, an empirical evaluation of several different approaches to reducing search in AbTweak, an abstract nonlinear planning system, are presented. To solve a complex problem using abstraction, one would like to protect parts of the abstract solution that have already been completed during the abstract plan refinement. However, it has not been well understood how this protection can be done profitably; some subgoal protection may indeed result in a decrease in efficiency. The Monotonic Property has been proposed as a form of goal protection in abstract planning. Different versions of this property are investigated with respect to their effect on planning performance in several domains. Furthermore, a series of empirical tests of the utility of the properties are presented, along with an evaluation of different search strategies for abstract planning. In addition, we present a novel abstract planning control strategy known as LeftWedge. This strategy adds a depth-first flavour to a complete search strategy by taking advantage of the fact that less abstract solutions are generally closer to a concrete level solution than more abstract ones. The relative benefit of this approach under various criticality assignments is demonstrated through comparisons with breadth-first strategy. Furthermore, two subgoal selection strategies are presented and compared empirically.