Synchronous and Asynchronous Interval Newton-Schwarz Methods for a Class of Large Systems of Nonlinear Equations

We introduce a class of parallel interval arithmetic iteration methods for nonlinear systems of equations, especially of the type Ax+(x) = f, diagonal, in R ^N . These methods combine enclosure and global convergence properties of Newton-like interval methods with the computational efficiency of parallel block iteration methods: algebraic forms of Schwarz-type methods which generalize both the well-known additive algebraic Schwarz Alternating Procedure and multisplitting methods. We discuss both synchronous and asynchronous variants. Besides enclosure and convergence properties, we present numerical results from a CRAY T3E.     

Rational Interpolation from Stochastic Data: A New Froissart's Phenomenon

The analytic structure of Rational Interpolants (R.I.) f (z) built from randomly perturbed data is explored; the interpolation nodes x _j , j = 1,...,M, are real points where the function f reaches these prescribed data . It is assumed that the data are randomly perturbed values of a rational function _(n) ^(m) (m / n is the degree of the numerator/denominator). Much attention is paid to the R.I. familyf _(n+1) ^(m–1), in the small stochasticity régime. The main result is that the additional zero and pole are located nearby the root of the same random polynomial, called the Froissart Polynomial (F.P.). With gaussian hypothesis on the noise, the random real root of F.P. is distributed according to a Cauchy-Lorentz law, with parameters such that the integrated probability over the interpolation interval x ₁, x _M is always larger than 1/2; in two cases studied in detail, it reaches 2/3 in one case and almost 3/4 in the other. For the families f _(n+k) ^(m+k), numerical explorations point to similar phenomena; inspection shows that, in the mean, the localization occurs in the complex and/or real vicinity of the interpolation interval.     

On closure properties of bounded two-sided error complexity classes

We show that if a complexity classC is closed downward under polynomial-time majority truth-table reductions ( _mtt ^p ), then practically every other polynomial closure property it enjoys is inherited by the corresponding bounded two-sided error class BP[C]. For instance, the Arthur-Merlin game class AM [B1] enjoys practically every closure property of NP. Our main lemma shows that, for any relativizable classD which meets two fairly transparent technical conditions, we haveC ^BP[C] BP[D ^C]. Among our applications, we simplify the proof by Toda [Tol], [To2] that the polynomial hierarchy PH is contained in BP[P]. We also show that relative to a random oracleR, PH ^R is properly contained in P ^R .The first author was supported in part by NSF Grant CCR-9011248 and the second author was supported in part by NSF Grant CCR-89011154.     

Pseudo-Additive Measures and Triangular-Norm-Based Conditioning

Conditioning in the framework of fuzzy measures (monotone normalized set functions vanishing in the empty set) is introduced. For every set B with non-null measure m(B) a conditional measure m _B , based on a triangular norm T, is introduced. Universal conditioning preserving the lower semi-continuity is shown to be necessarily based on some strict triangular norm. Then also each conditional measure m _B related to a pseudo-additive measure m is pseudo-additive. However, the pseudo-addition _B operating on the measures m _B is in general different from the pseudo-addition operating on the measure m. Specific cases of universal conditioning preserving the pseudo-addition are characterized. Classical probabilistic conditioning is shown to be a special case.     

Predicate Synthesis for Correcting Faulty Conjectures: The Proof Planning Paradigm

The synthesis of programs as well as other synthetic tasks often end up with an unprovable, partially false conjecture. A successful subsequent synthesis attempt depends on determining why the conjecture is faulty and how it can be corrected. Hence, it is highly desirable to have an automated means for detecting and correcting faulty conjectures.We introduce a method for patching faulty conjectures. The method is based on abduction and performs its task during an attempt to prove a given conjecture. On input X. G(X), the method builds a definition for a corrective predicate, P(X), such that X. P(X) G(X) is a theorem. The synthesis of a corrective predicate is guided by the constructive principle of formulae as types, relating inference with computation.We take the construction of a corrective predicate as a program transformation task. The method consists of a collection of construction commands. A construction command is a small program that makes use of one or more program editing commands, geared towards building recursive, equational procedures.A synthesised corrective predicate is guaranteed to be correct, turning a faulty conjecture into a theorem. Whether conditional or not, it will be well-defined. If recursive, it will also be terminating.Our method is amenable to mechanisation, but careful search guidance is required for making a productive use of the failure of a proof. A failed proof attempt quickly yields a huge, possibly infinite, deduction tree, giving rise to exponentially many abductive explanations. We suggest that a proof planning approach can structure the task of correcting a formula in such a way as to allow significant automation, while dramatically restricting the search space.     

Thiele Rational Interpolation for the Numerical Computation of the Reversible Randles–Sevcik Function in Electrochemistry

This paper uses Thiele rational interpolation to derive a simple method for computing the Randles–Sevcik function ^1/2(x), with relative error at most 1.9 × 10^–5 for – < x < . We develop a piecewise approximation method for the numerical computation of ^1/2(x) on the union (–, –10) [–10, 10] (10, ). This approximation is particularly convenient to employ in electrochemical applications where four significant digits of accuracy are usually sufficient. Although this paper is primarily concerned with the approximation of the Randles–Sevcik function, some examples are included that illustrate how Thiele rational interpolation can be employed to generate useful approximations to other functions of interest in scientific work.     

On Bounding Solutions of Underdetermined Systems

Sufficient conditions for the existence and uniqueness of a solution x* D (R ⁿ ) of Y(x) = 0 where : R ⁿ R ^m (m n) with C ²(D) where D R ⁿ is an open convex set and Y = (x)⁺ are given, and are compared with similar results due to Zhang, Li and Shen (Reliable Computing 5(1) (1999)). An algorithm for bounding zeros of f (·) is described, and numerical results for several examples are given.     

Schedulers for larger classes of pinwheel instances

The pinwheel is a hard-real-time scheduling problem for scheduling satellite ground stations to service a number of satellites without data loss. Given a multiset of positive integers (instance)A={a₁,..., a_n}, the problem is to find an infinite sequence (schedule) of symbols from {1,2,...,n} such that there is at least one symboli within any interval of a_i symbols (slots). Not all instancesA can be scheduled; for example, no successful schedule exists for instances whose density,(A)= _i ⁱ (l/a_i), is larger than 1. It has been shown that all instances whose densities are less than a 0.5 density threshold can always be scheduled. If a schedule exists, another concern is the design of a fast on-line scheduler (FOLS) which can generate each symbol of the schedule in constant time. Based on the idea of integer reduction, two new FOLSs which can schedule different classes of pinwheel instances, are proposed in this paper. One uses single-integer reduction and the other uses double-integer reduction. They both improve the previous 0.5 result and have density thresholds of 13/20 and2/3, respectively. In particular, if the elements inA are large, the density thresholds will asymptotically approach In 2 and 1/R2, respectively.This research was supported in part by ONR Grant N00014-87-K-0833, and was done while Francis Chin was visiting the Computer Science Program, The University of Texas at Dallas, Richardson, TX 75083, USA.     

Language Simplification through Error-Correcting and Grammatical Inference Techniques

In many language processing tasks, most of the sentences generally convey rather simple meanings. Moreover, these tasks have a limited semantic domain that can be properly covered with a simple lexicon and a restricted syntax. Nevertheless, casual users are by no means expected to comply with any kind of formal syntactic restrictions due to the inherent spontaneous nature of human language. In this work, the use of error-correcting-based learning techniques is proposed to cope with the complex syntactic variability which is generally exhibited by natural language. In our approach, a complex task is modeled in terms of a basic finite state model, F, and a stochastic error model, E. F should account for the basic (syntactic) structures underlying this task, which would convey the meaning. E should account for general vocabulary variations, word disappearance, superfluous words, and so on. Each natural user sentence is thus considered as a corrupted version (according to E) of some simple sentence of L(F). Adequate bootstrapping procedures are presented that incrementally improve the structure of F while estimating the probabilities for the operations of E. These techniques have been applied to a practical task of moderately high syntactic variability, and the results which show the potential of the proposed approach are presented.     

Self-organizing neural networks for the analysis and representation of data: Some financial cases

Many recent papers have dealt with the application of feedforward neural networks in financial data processing. This powerful neural model can implement very complex nonlinear mappings, but when outputs are not available or clustering of patterns is required, the use of unsupervised models such as self-organizing maps is more suitable. The present work shows the capabilities of self-organizing feature maps for the analysis and representation of financial data and for aid in financial decision-making. For this purpose, we analyse the Spanish banking crisis of 1977–1985 and the Spanish economic situation in 1990 and 1991, making use of this unsupervised model. Emphasis is placed on the analysis of the synaptic weights, fundamental for delimiting regions on the map, such as bankrupt or solvent regions, where similar companies are clustered. The time evolution of the companies and other important conclusions can be drawn from the resulting maps.Characters and symbols used and their meaning nx x dimension of the neuron grid, in number of neurons - ny y dimension of the neuron grid, in number of neurons - n dimension of the input vector, number of input variables - (i, j) indices of a neuron on the map - k index of the input variables - w _ijk synaptic weight that connects thek input with the (i, j) neuron on the map - W _ij weight vector of the (i, j) neuron - x _k input vector - X input vector - (t) learning rate - _o starting learning rate - _f final learning rate - R(t) neighbourhood radius - R₀ starting neighbourhood radius - R _f final neighbourhood radius - t iteration counter - t _rf number of iterations until reachingR _f - t _f number of iterations until reaching _f - h(·) lateral interaction function - standard deviation - for every - d (x, y) distance between the vectors x and y     

The existence of solutions,stability, and linearization of volterra systems

LetB be a Banach space ofR _n valued continuous functions on [0, ) withfB. Consider the nonlinear Volterra integral equation (^*)x(t)+ _o ^t K(t,s,x(s))ds. We use the implicit function theorem to give sufficient conditions onB andK (t,s,x) for the existence of a unique solutionxB to (^*) for eachf B with f ^B sufficiently small. Moreover, there is a constantM>0 independent off with Mf^B.Part of this work was done while the author was visiting at Wright State University.     

Computational Divided Differencing and Divided-Difference Arithmetics

Tools for computational differentiation transform a program that computes a numerical function F(x) into a related program that computes F(x) (the derivative of F). This paper describes how techniques similar to those used in computational-differentiation tools can be used to implement other program transformations—in particular, a variety of transformations for computational divided differencing. The specific technical contributions of the paper are as follows:– It presents a program transformation that, given a numerical function F(x) defined by a program, creates a program that computes F[x ₀, x ₁], the first divided difference of F(x), where – It shows how computational first divided differencing generalizes computational differentiation.– It presents a second program transformation that permits the creation of higher-order divided differences of a numerical function defined by a program.– It shows how to extend these techniques to handle functions of several variables.The paper also discusses how computational divided-differencing techniques could lead to faster and/or more robust programs in scientific and graphics applications.Finally, the paper describes how computational divided differencing relates to the numerical-finite-differencing techniques that motivated Robert Paige's work on finite differencing of set-valued expressions in SETL programs.     

On the Quadratic Convergence of the Levenberg-Marquardt Method without Nonsingularity Assumption

Recently, Yamashita and Fukushima [11] established an interesting quadratic convergence result for the Levenberg-Marquardt method without the nonsingularity assumption. This paper extends the result of Yamashita and Fukushima by using _k=||F(x_k)||, where [1,2], instead of _k=||F(x_k)||² as the Levenberg-Marquardt parameter. If ||F(x)|| provides a local error bound for the system of nonlinear equations F(x)=0, it is shown that the sequence {x_k} generated by the new method converges to a solution quadratically, which is stronger than dist(x_k,X^*)0 given by Yamashita and Fukushima. Numerical results show that the method performs well for singular problems.     

On the power of parity polynomial time

This paper proves that the complexity class P, parity polynomial time [PZ], contains the class of languages accepted byNP machines with few accepting paths. Indeed, P contains a broad class of languages accepted by path-restricted nondeterministic machines. In particular, P contains the polynomial accepting path versions ofNP, of the counting hierarchy, and of Mod _m NP form>1. We further prove that the class of nondeterministic path-restricted languages is closed under bounded truth-table reductions.These results were announced at the 6th Symposium on Theoretical Aspects of Computer Science [CH3]. Jin-yi Cai was supported in part by NSF Grant CCR-8709818 and the work was done while he was at Yale University. Lane A. Hemachandra was supported in part by a Hewlett-Packard Corporation equipment grant and the National Science Foundation under Grant CCR-8809174/CCR-8996198 and a Presidential Young Investigator Award. His work was done in part while at Columbia University.     

Multilayer grid embeddings for VLSI

In this paper we propose two new multilayer grid models for VLSI layout, both of which take into account the number of contact cuts used. For the first model in which nodes exist only on one layer, we prove a tight area × (number of contact cuts) = (n ²) tradeoff for embeddingn-node planar graphs of bounded degree in two layers. For the second model in which nodes exist simultaneously on all layers, we give a number of upper bounds on the area needed to embed groups using no contact cuts. We show that anyn-node graph of thickness 2 can be embedded on two layers inO(n ²) area. This bound is tight even if more layers and any number of contact cuts are allowed. We also show that planar graphs of bounded degree can be embedded on two layers inO(n ^3/2(logn)²) area.Some of our embedding algorithms have the additional property that they can respect prespecified grid placements of the nodes of the graph to be embedded. We give an algorithm for embeddingn-node graphs of thicknessk ink layers usingO(n ³) area, using no contact cuts, and respecting prespecified node placements. This area is asymptotically optimal for placement-respecting algorithms, even if more layers are allowed, as long as a fixed fraction of the edges do not use contact cuts. Our results use a new result on embedding graphs in a single-layer grid, namely an embedding ofn-node planar graphs such that each edge makes at most four turns, and all nodes are embedded on the same line.The first author's research was partially supported by NSF Grant No. MCS 820-5167.     

Randomized vs. deterministic decision tree complexity for read-once Boolean functions

We consider the deterministic and the randomized decision tree complexities for Boolean functions, denotedDC(f) andRC(f), respectively. A major open problem is how smallRC(f) can be with respect toDC(f). It is well known thatRC(f)DC(f) ^0.5 for every Boolean functionf (called 0.5-exponent). On the other hand, some Boolean functionf is known to haveRC(f) = (DC(f))^0.753...) (or 0.753...-exponent). It is not known whether there is a Boolean function with exponent smaller than 0.753... Likewise, no lower bound for arbitrary Boolean functions with exponent greater than 0.5 is known.Our result is a 0.51 lower bound on the exponent for everyread-once function. Read-once means that each input variable appears exactly once in the Boolean formula representing the function. To obtain this result we generalize an existing lower bound technique and combine it with restriction arguments. This result provides a lower bound ofn ^0.51 on the number of positions that have to be evaluated by any randomized - pruning algorithm computing the value of any two-person zero-sum game tree withn final positions.     

A faster divide-and-conquer algorithm for constructing delaunay triangulations

An easily implemented modification to the divide-and-conquer algorithm for computing the Delaunay triangulation ofn sites in the plane is presented. The change reduces its (n logn) expected running time toO(n log logn) for a large class of distributions that includes the uniform distribution in the unit square. Experimental evidence presented demonstrates that the modified algorithm performs very well forn2¹⁶, the range of the experiments. It is conjectured that the average number of edges it creates—a good measure of its efficiency—is no more than twice optimal forn less than seven trillion. The improvement is shown to extend to the computation of the Delaunay triangulation in theL _p metric for 1<p.This research was supported by National Science Foundation Grants DCR-8352081 and DCR-8416190.     

A probabilistic analysis of the height of tries and of the complexity of triesort

Summary We consider binary tries formed by using the binary fractional expansions of X ₁, ...,X _n, a sequence of independent random variables with common density f on [0,1]. For H _n, the height of the trie, we show that either E(H_n)21og₂ n or E(H_n)= for all n2 according to whether f ²(x)dx is finite or infinite. Thus, the average height is asymptotically twice the average depth (which is log₂ n when f ²(x)dx>). The asymptotic distribution of H _n is derived as well.If f is square integrable, then the average number of bit comparisons in triesort is nlog₂ n+0(n), and the average number of nodes in the trie is 0(n).Research of the author was supported in part by FCAC Grant EQ-1678     

The privacy of dense symmetric functions

Ann argument function,f, is calledt-private if there exists a distributed protocol for computingf so that no coalition of at mostt processors can infer any additional information from the execution of the protocol. It is known that every function defined over a finite domain is [(n–1)/2]-private. The general question oft-privacy (fort[n/2]) is still unresolved.In this work, we relate the question of [n/2]-privacy for the class of symmetric functions of Boolean argumentsf: {0, 1} ⁿ {0, 1,...,n} to the structure of Hamming weights inf ^–1(b) (b{0, 1, ...,n}). We show that iff is [n/2]-private, then every set of Hamming weightsf ^–1(b) must be an arithmetic progression. For the class ofdense symmetric functions (defined in the sequel), we refine this to the following necessary and sufficient condition for [n/2]-privacy off: Every collection of such arithmetic progressions must yield non-identical remainders, when computed modulo the greatest common divisor of their differences. This condition is used to show that for dense symmetric functions, [n/2]-privacy impliesn-privacy.