Showing papers in "Bit Numerical Mathematics in 1985"
414 citations
TL;DR: A new formulation of the notion of duality that allows the unified treatment of a number of geometric problems is used, to solve two long-standing problems of computational geometry and to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane.
Abstract: This paper uses a new formulation of the notion of duality that allows the unified treatment of a number of geometric problems. In particular, we are able to apply our approach to solve two long-standing problems of computational geometry: one is to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane; the other is to produce an optimal algorithm for the half-plane range query problem. This problem is to preprocessn points in the plane, so that given a test half-plane, one can efficiently determine all points lying in the half-plane. We describe an optimalO(k + logn) time algorithm for answering such queries, wherek is the number of points to be reported. The algorithm requiresO(n) space andO(n logn) preprocessing time. Both of these results represent significant improvements over the best methods previously known. In addition, we give a number of new combinatorial results related to the computation of line arrangements.
286 citations
TL;DR: A complete analysis of approximate counting is provided which establishes good convergence properties of the algorithm and allows to quantify precisely complexity-accuracy tradeoffs.
Abstract: Approximate counting is an algorithm proposed by R. Morris which makes it possible to keep approximate counts of large numbers in small counters. The algorithm is useful for gathering statistics of a large number of events as well as for applications related to data compression (Todd et al.). We provide here a complete analysis of approximate counting which establishes good convergence properties of the algorithm and allows to quantify precisely complexity-accuracy tradeoffs.
182 citations
TL;DR: This work surveys preconditioned iterative methods with the emphasis on solving large sparse systems such as arise by discretization of boundary value problems for partial differential equations and reports in particular on the state of the art of preconditionsing methods for vectorizable and/or parallel computers.
Abstract: We survey preconditioned iterative methods with the emphasis on solving large sparse systems such as arise by discretization of boundary value problems for partial differential equations. We discuss shortly various acceleration methods but the main emphasis is on efficient preconditioning techniques. Numerical simulations on practical problems have indicated that an efficient preconditioner is the most important part of an iterative algorithm. We report in particular on the state of the art of preconditioning methods for vectorizable and/or parallel computers.
181 citations
TL;DR: The design concepts underlying the implementation of the grid file are presented and applications of thegrid file system are described.
Abstract: The grid file is an adaptable, symmetric multikey file structure. It stores highly dynamic sets of multidimensional data in such a way that different types of queries can be performed using few disk accesses. We present the design concepts underlying our implementation of the grid file and describe applications of thegrid file system.
71 citations
TL;DR: It is shown how the same modifications which have been applied to segment trees can be applied to the data structure of Swart and Ladner as well, leading to anO((n+k)logn) time hidden line elimination algorithm that improves the fastest previous line-sweep algorithm for the problem by a factorO(logn).
Abstract: Fast hidden line elimination algorithms can be obtained by minor modifications to algorithms developed for reporting intersections of polygons. We show how the same modifications which have been applied to segment trees can be applied to the data structure of Swart and Ladner as well, leading to anO((n+k)logn) time hidden line elimination algorithm (n is the number of boundary edges of the input polygons andk is the number of intersections of the edges on the projection plane). The algorithm improves the fastest previous line-sweep algorithm for the problem by a factorO(logn).
57 citations
TL;DR: In this paper, the existence and convergence conditions for a class of interval iteration algorithms for the enclosure of a zero of a system of nonlinear equations are given, in particular a quadratically convergent method which throughout the iteration uses the same interval enclosure of the derivative.
Abstract: Easily verifiable existence and convergence conditions are given for a class of interval iteration algorithms for the enclosure of a zero of a system of nonlinear equations. In particular, a quadratically convergent method is obtained which throughout the iteration uses the same interval enclosure of the derivative.
52 citations
TL;DR: Upper bounds for the global discretization error of the implicit midpoint rule and the trapezoidal rule for the case of arbitrary variable stepsizes are presented and second order optimal B-convergence for both methods is proved.
Abstract: We present upper bounds for the global discretization error of the implicit midpoint rule and the trapezoidal rule for the case of arbitrary variable stepsizes. Specializing our results for the case of constant stepsizes they easily prove second order optimal B-convergence for both methods. 1980 AMS Subject Classification: 65L05, 65L20.
45 citations
TL;DR: In this paper, a discrete Gronwall inequality is employed to prove that, away from the origin, the error in product integration and collocation schemes for these equations is of order 2-α.
Abstract: In general, second kind Volterra integral equations with weakly singular kernels of the formk(t,s)(t −s)−α posses solutions which have discontinuous derivatives att=0. A discrete Gronwall inequality is employed to prove that, away from the origin, the error in product integration and collocation schemes for these equations is of order 2-α.
43 citations
TL;DR: This paper examines the subclass mechanism offered in the Simula language, and discusses how the flexibility of this mechanism would be enhanced if more than one prefix were allowed for classes (multiple inheritance).
Abstract: This paper examines the subclass mechanism offered in the Simula language, and discusses how the flexibility of this mechanism would be enhanced if more than one prefix were allowed for classes (multiple inheritance). A strategy is presented for implementing this with efficiency comparable to that of current Simula.
38 citations
TL;DR: TwoGrid file algorithms were suggested in [12] to provide multi-key access to records in a dynamically growing file are specified and the average sizes of the corresponding directories are derived and an asymptotic analysis is provided.
Abstract: Grid file algorithms were suggested in [12] to provide multi-key access to records in a dynamically growing file. We specify here two algorithms and derive the average sizes of the corresponding directories. We provide an asymptotic analysis. The growth of the indexes appears to be non-linear for uniform distributions:O(vc) orO(vξ), wherec=1+b−1, ξ=1+(s-1)/(sb+1),s is the number of attributes being used,v the file size, andb the page capacity of the system. Finally we give corresponding results for biased distributions and compare transient phases.
Journal Article•
TL;DR: In this paper, a discrete Gronwall inequality is employed to prove that, away from the origin, the error in product integration and collocation schemes for these equations is of order 2-a.
Abstract: In general, second kind Volterra integral equations with weakly singular kernels of the form k(t, s)(t-s) -~ posses solutions which have discontinuous derivatives at t = 0. A discrete Gronwall inequality is employed to prove that, away from the origin, the error in product integration and collocation schemes for these equations is of order 2-a.
TL;DR: In this article, the authors present a talk given in 1981 at the Zurich symposium to commemorate the tenth anniversary of the death of the eminent Swiss numerical analyst, Heinz Rutishauser.
Abstract: BIT has played and plays a great role in the development of concepts concerning numerical (in)stability in initial value problems forODE's and related questions. This development is here seen through the looking-glass of the author, who experienced much of its pains and pleasures. The article is based on a talk given in 1981 at the Zurich symposium to commemorate the tenth anniversary of the death of the eminent Swiss numerical analyst, Heinz Rutishauser. The presentation is mainly chronological with a few digressions. Part I ends at the beginning of the stiff epoch.
TL;DR: In this paper, it was shown that forn a non-negative integer, there does not exist an explicit Runge-Kutta method with 10 +n stages and order 8 +n.
Abstract: It is shown that forn a non-negative integer, there does not exist an explicit Runge-Kutta method with 10 +n stages and order 8 +n. It follows that for order 8, the minimum number of stages is 11.
TL;DR: It is shown that if the subnetwork is required to be biconnected or respectively edge-biconconnected, and the underlying network is outerplanar, the GSP can be solved in linear time.
Abstract: The generalized Steiner problem (GSP) is concerned with the determination of a minimum cost subnetwork of a given network where some (not necessarily all) vertices satisfy certain pairwise (vertex or edge) connectivity requirements.
TL;DR: Experimental results show that the length of the text can be reduced more than 50% with no a priori knowledge of the nature of thetext.
Abstract: A new technique for compression of character strings is presented. The technique is based on the use of a dictionary forest which is built simultaneously with the encoding and decoding. Codes representing substrings are addresses in the dictionary forest. Experimental results show that the length of the text can be reduced more than 50% with no a priori knowledge of the nature of the text.
Journal Article•
TL;DR: It is shown that the traveling salesman problem, where cities are bit strings with Hamming distances, is NP-complete.
Abstract: It is shown that the traveling salesman problem, where cities are bit strings with Hamming distances, is NP-complete.
TL;DR: Evidence is presented showing that the McVitie and Wilson algorithm to solve the stable marriage problem has a sequential component that is quite large on the average, and an approximate solution with a few unstable pairings can be found much faster than an exact solution.
Abstract: Evidence is presented showing that the McVitie and Wilson algorithm to solve the stable marriage problem has a sequential component that is quite large on the average. Hence parallel implementations of the algorithm are likely to achieve only mediocre average case speedup. A corollary result is that an approximate solution with a few unstable pairings can be found much faster than an exact solution.
TL;DR: An error analysis of the G-algorithm is presented which shows that it is as stable as any of the standard orthogonal decomposition methods for solving least squares problems.
Abstract: The G-algorithm was proposed by Bareiss [1] as a method for solving the weighted linear least squares problem. It is a square root free algorithm similar to the fast Givens method except that it triangularizes a rectangular matrix a column at a time instead of one element at a time.
TL;DR: A theoretical analysis of the performance of a parallel form of the Quicksort algorithm is presented and shown to be in qualitative agreement with experiments carried out on the Loughborough University NEPTUNE parallel system.
Abstract: In this paper a theoretical analysis of the performance of a parallel form of the Quicksort algorithm is presented and shown to be in qualitative agreement with experiments carried out on the Loughborough University NEPTUNE parallel system.
TL;DR: A two-stage implicit fifth order multipoint iterative method for solving equations and an implicit third order and a semiexplicit fourth order multipointed method are derived.
Abstract: We have derived a two-stage implicit fifth order multipoint iterative method for solving equations. Further, we have also obtained an implicit third order and a semiexplicit fourth order multipoint method. Comparison of computational results are made with other well-known methods on a number of difficult problems. The implicit multipoint methods are accurate and robust.
TL;DR: This paper gives a negative result on BSI-stability for the Lobatto IIIC-methods with more than two stages and gives a necessary condition forBSI-Stability.
Abstract: This paper gives a negative result onBSI-stability for the Lobatto IIIC-methods with more than two stages. We also give a necessary condition forBSI-stability.
TL;DR: It is shown that the Double Distributive Partitioning algorithm runs, for all practical purposes, inO(n) time for many distributions of keys.
Abstract: A new sorting algorithm, Double Distributive Partitioning, is introduced and compared against Sedgewick's quicksort. It is shown that the Double Distributive Partitioning algorithm runs, for all practical purposes, inO(n) time for many distributions of keys. Furthermore, the combined number of comparisons, additions, and assignments required to sort by the new method on these distributions is always less than quicksort.
TL;DR: This work presents the iterative solutions of the Towers of Hanoi problems (standard, cyclic, and generalized) using the program transformation methodology of Burstall-Darlington, and derives algorithms with minimal time × space requirements.
Abstract: We present the iterative solutions of the Towers of Hanoi problems (standard, cyclic, and generalized) using the program transformation methodology of Burstall-Darlington. We derive algorithms with minimal time × space requirements. Their correctness proofs are trivial, as usual when applying the program transformation technique.
TL;DR: In this article, the authors considered singular integral equations of the second kind and proved that the method converges also in this case, but stronger conditions than for the first kind equations must be imposed.
Abstract: For solving singular integral equations of the first kind Erdogan proposed a method of Galerkin type, and convergence was proved by Linz. In this paper we consider equations of the second kind, and it is found that the method converges also in this case. However, stronger conditions than for the first kind equations must be imposed. The computational aspect of the convergence problem is also considered.
TL;DR: Szilard languages and label languages are studied as examples of languages generable by permutative grammars, particularly, sufficient conditions for apermutative grammar to generate a context-free language.
Abstract: A grammar is said to be permutative if it has permutation productions of the formAB ρBA in addition to context-free productions. Szilard languages and label languages are studied as examples of languages generable by permutative grammars. Particularly, sufficient conditions for a permutative grammar to generate a context-free language are studied.
TL;DR: These algorithms are based on the modified Gill-Møller algorithm for summation of very many terms, iterative refinement of a linear system with a special algorithm for the computation of residuals in single precision and on a property of floating point subtraction of nearby numbers.
Abstract: A typical approach for finding the approximate solution of a continuous problem is through discretization with meshsizeh such that the truncation error goes to zero withh. The discretization problem is solved in floating point arithmetic. Rounding-errors spoil the theoretical convergence and the error may even tend to infinity.
TL;DR: In this paper, the authors give an overview over how problems connected to degeneracy have been attacked in the literature in connection with extreme point enumeration in a convex polyhedron.
Abstract: We give an overview over how problems connected to degeneracy have been attacked in the literature in connection with extreme point enumeration in a convex polyhedron. We treat both the question of unique pivots and how to attach only one basis to each extreme point.