Showing papers in "Bit Numerical Mathematics in 1982"
TL;DR: The weighted pseudoinverse theory is used to analyse least squares problems with linear and quadratic constraints as mentioned in this paper, and a numerical algorithm for the computation of the weighted pseudo-inverse is briefly described.
Abstract: The weighted pseudoinverse providing the minimum semi-norm solution of the weighted linear least squares problem is studied. It is shown that it has properties analogous to those of the Moore-Penrose pseudoinverse. The relation between the weighted pseudoinverse and generalized singular values is explained. The weighted pseudoinverse theory is used to analyse least squares problems with linear and quadratic constraints. A numerical algorithm for the computation of the weighted pseudoinverse is briefly described.
185 citations
TL;DR: An algorithm for the geometric problem of determining a line (called a stabbing line) which intersects each ofn given line segments in the plane and a purely geometric fact is proved which infers that this description requiresO(n) space to be specified.
Abstract: An algorithm for the geometric problem of determining a line (called a stabbing line) which intersects each ofn given line segments in the plane is presented. As a matter of fact, the algorithm computes a description of all stabbing lines. A purely geometric fact is proved which infers that this description requiresO(n) space to be specified. Our algorithm computes it inO(n logn) time which is optimal in the worst case.
106 citations
TL;DR: The extendible cell method is an application of order preserving extendible hashing to multidimensional point files that derives some of its performance characteristics and shows its expected case optimality for closest point problems.
Abstract: The extendible cell method is an application of order preserving extendible hashing to multidimensional point files We derive some of its performance characteristics and show its expected case optimality for closest point problems
60 citations
TL;DR: In this paper, a unified approach to the derivation of high order formulae for the numerical integration of two-point boundary value problems is described, and a numerical comparison with finite difference methods is made and this comparison indicates the efficiency of the high order methods for problems having a suitably smooth solution.
Abstract: In a recent paper, Cash and Moore have given a fourth order formula for the approximate numerical integration of two-point boundary value problems in O.D.E.s. The formula presented was in effect a “one-off” formula in that it was obtained using a trial and error approach. The purpose of the present paper is to describe a unified approach to the derivation of high order formulae for the numerical integration of two-point boundary value problems. It is shown that the formula derived by Cash and Moore fits naturally into this framework and some new formulae of orders 4, 6 and 8 are derived using this approach. A numerical comparison with certain existing finite difference methods is made and this comparison indicates the efficiency of the high order methods for problems having a suitably smooth solution.
58 citations
TL;DR: Conditions for the parameters of a semi-implicit method imply that the method produces contractive numerical solutions over a large class of nonlinear problems for sufficiently smallh, but the restriction on the stepsize does not depend on the stiffness of the differential equation.
Abstract: The aim of this paper is to analyze the stability properties of semi-implicit methods (such as Rosenbrock methods,W-methods, and semi-implicit extrapolation methods) for nonlinear stiff systems of differential equations. First it is shown that the numerical solution satisfies ‖y1‖ ≦ϕ(hμ)‖y0‖, if the method is applied with stepsizeh to the systemy′ =Ay (μ denotes the logarithmic norm ofA). Properties of the functionϕ(x) are studied. Further, conditions for the parameters of a semi-implicit method are given, which imply that the method produces contractive numerical solutions over a large class of nonlinear problems for sufficiently smallh. The restriction on the stepsize, however, does not depend on the stiffness of the differential equation. Finally, the presented theory is applied to the extrapolation method based on the semi-implicit mid-point rule.
57 citations
TL;DR: It is found that certain arguments given for using particular formal modes of expression in developing and proving programs correct are invalid.
Abstract: The concepts of specification and formalization, as relevant to the development of programs, are introduced and discussed. It is found that certain arguments given for using particular formal modes of expression in developing and proving programs correct are invalid. As illustration a formalized description of Algol 60 is discussed and found deficient. Emphasis on formalization is shown to have harmful effects on program development, such as neglect of informal precision and simple formalizations. A style of specifications using formalizations only to enhance intuitive understandability is recommended.
47 citations
TL;DR: A general method whereby certain primary key hasing schemes can be extended to partial-match retrieval schemes that can handle arbitrarily dynamic files and can be optimized with respect to the number of page faults required to answer a query is proposed.
Abstract: This paper studies file designs for answering partial-match queries for dynamic files A partial-match query is a specification of the value of zero or more fields in a record An answer to a query consists of a listing of all records in the file satisfying the values specified
46 citations
TL;DR: In this article, a unified derivation of affine invariant convergence results for Newton's method is presented, where the perturbation lemma and mean value theorem are used to obtain an optimal radius of convergence.
Abstract: We present a unified derivation of affine invariant convergence results for Newton's method. Initially we derive affine invariant forms of the perturbation lemma and a mean value theorem. With their aid we obtain an optimal radius of convergence for Newton's method, from which further radius of convergence estimates follow. From the Newton-Kantorovitch theorem we derive other estimates of the radius of convergence. We discuss estimation of the parameters in the expressions we have derived.
33 citations
TL;DR: In this article, the authors presented an archetype (scalar) equation for assessing the quality of integrator/interpolator pairs used to solve differential-difference equations.
Abstract: The equationu(t)=pu(t)+qu(t-τ) is presented as an archetype (scalar) equation for assessing the quality of integrator/interpolator pairs used to solve retarded differential-difference equations. The relationships ofP-stability andP[α, β]-stability, defined with respect to this archetype equation, to stability and order of multistep integrators and to passivity and order of Lagrange interpolators are developed. Composite multistep integrators and composite Lagrange interpolators are considered as a means of obtaining high order pairs stable for all step-sizes over a large portion, if not all, of the (p, q)-domain on which the archetype equation is stable.
27 citations
TL;DR: The convex hull of a finite set of points in the plane can be computed in constant time using a polynomial number of processors.
Abstract: The convex hull of a finite set of points in the plane can be computed in constant time using a polynomial number of processors.
22 citations
TL;DR: In this paper, an algorithm for approximating dominated solutions of linear recursions with initial conditions is given, and the stability of this algorithm is investigated and expressions for the truncation and rounding errors are derived.
Abstract: An algorithm is given for approximating dominated solutions of linear recursions, when some initial conditions are given. The stability of this algorithm is investigated and expressions for the truncation and rounding errors are derived. A number of practical questions concerning the algorithm is considered, and several numerical examples sustain the theory.
TL;DR: It is proved that the 0/1-INTERCHANGE scheduling heuristic can be modified, without increasing its time complexity fromO(N logM), so that its worst-case performance bound is reduced from 2 to 4/3 times optimal.
Abstract: Consideration is given to the problem of nonpreemptively scheduling a set ofN independent tasks to a system ofM identical processors, with the objective to minimize the overall finish time. It is proved that the 0/1-INTERCHANGE scheduling heuristic can be modified, without increasing its time complexity fromO(N logM), so that its worst-case performance bound is reduced from 2 to 4/3 times optimal.
TL;DR: It is shown that this approximation converges uniformly, provided that the kernel and the input functions possess a continuous derivative, and the error of the Nyström interpolant is bounded by the Gaussian quadrature errors and thus convergence is fast, especially for smooth functions.
Abstract: Nystrom's interpolation formula is applied to the numerical solution of singular integral equations. For the Gauss-Chebyshev method, it is shown that this approximation converges uniformly, provided that the kernel and the input functions possess a continuous derivative. Moreover, the error of the Nystrom interpolant is bounded from above by the Gaussian quadrature errors and thus convergence is fast, especially for smooth functions. ForC∞ input functions, a sharp upper bound for the error is obtained. Finally numerical examples are considered. It is found that the actual computational error agrees well with the theoretical derived bounds.
TL;DR: In this article, a family of two-step fourth-order methods, which requires two function evaluations per step, is derived fory″=f(x,y), and the existence of a sub-family of these methods which when applied toy″=−k 2y,k real, areP-stable.
Abstract: A family of two-step fourth order methods, which requires two function evaluations per step, is derived fory″=f(x,y). We then show the existence of a sub-family of these methods which when applied toy″=−k2y,k real, areP-stable.
TL;DR: For evaluation schemes based on the Lagrangian form of a polynomial with degreen, a rigorous error analysis is performed, taking into account that data, computation and even the nodes of interpolation might be perturbed by round-off.
Abstract: For evaluation schemes based on the Lagrangian form of a polynomial with degreen, a rigorous error analysis is performed, taking into account that data, computation and even the nodes of interpolation might be perturbed by round-off. The error norm of the scheme is betweenn2 andn2+(3n+7)λ n , where λ n denotes the Lebesgue constant belonging to the nodes. Hence, the error norm is of least possible orderO(n2) if, for instance, the nodes are chosen to be the Chebyshev points or the Fekete points.
TL;DR: It is shown that the minimal error of an algorithm using the values of the righthand side function and/or its derivatives atm points is of orderm−r for functions with boundedrth derivative and bounded support and is infinite for function with non-bounded support.
Abstract: We study the error of algorithms for solving an initial value problem for a scalar autonomous differential equation. We show that the minimal error of an algorithm using the values of the righthand side function and/or its derivatives atm points is of orderm −r for functions with boundedrth derivative and bounded support and is infinite for functions with non-bounded support. The asymptotic optimality of the Taylor algorithm is shown and the complexity of the problem is discussed.
TL;DR: In this paper, it was shown that the number of positive weights of a quadrature formula is related to the rotations of a path in the plane, and necessary and sufficient conditions for all weights to be positive can then be obtained.
Abstract: In this paper we show that the number of positive weights of a quadrature formula is related to the number of rotations of a certain path in the plane. Necessary and sufficient conditions for all weights to be positive can then be obtained. Also, much of classical theory appears in a new light.
TL;DR: A new hybrid of Distributive Partitioning Sorting with Pointer sort versions of both algorithms is described and tested against Quicksort on uniformly distributed items.
Abstract: A new hybrid of Distributive Partitioning Sorting is described and tested against Quicksort on uniformly distributed items. Pointer sort versions of both algorithms are also tested.
TL;DR: In this article, generalized diagonal dominant matrices (GDM matrices) were defined and generalized to general matrices, which improved the known results about the intervals of convergence of the AOR method for linear systems.
Abstract: With the definition of generalized diagonal dominant matrices we improve the known results about the intervals of convergence of the (AOR) method for linear systems. We consider this problem for different kinds of matrices and we get some important results forH-matrices.
TL;DR: A strategy is given for selecting the dimensionr of the linear variety which is used to define the criterion functionalJvrm and which determines the shape of the data clusters detected by the correspondingc-Varieties (FCV) clustering algorithms.
Abstract: A strategy is given for selecting the dimensionr of the linear variety which is used to define the criterion functionalJ
vrm
and which determines the shape of the data clusters detected by the correspondingc-Varieties (FCV) clustering algorithms.
TL;DR: A new method is developed which further improves the number of pairwise comparisons required in the Hwang-Lin algorithm by a factor 52/336m and it is possible that even larger improvements could be achieved.
TL;DR: This work shows how to construct accurate predictors and adaptive steplength estimators for use in predictor-corrector algorithms which follow trajectories through turning points by using different parametrization at every step along the trajectory.
Abstract: The use of the continuation principle in the solution of systems of nonlinear equations frequently leads to the need to follow trajectories through turning points. This can be done by using a different parametrization at every step along the trajectory. We show how to construct accurate predictors and adaptive steplength estimators for use in predictor-corrector algorithms which follow trajectories in this way.
TL;DR: In this paper the concept of event is analyzed within the framework of first-order predicate logic and an axiom system for discrete time points and axiom schemes for events is presented.
Abstract: Most approaches to information modelling are so-called snapshot approaches. This means that they focus on static properties of a universe of discourse only. Some approaches consider the temporal dimension of a universe of discourse. In these approaches the concept of event is central. It is used to denote, e.g., a decision or an action which takes place at a certain time point. In this paper the concept of event is analyzed within the framework of first-order predicate logic. An axiom system for discrete time points and axiom schemes for events is presented. It is shown that the axiom schemes obtained apply to a number of cases with wide applicability. Further, several remaining problems are pointed out.
TL;DR: This paper presents a simple, clear, efficient algorithm to generate a set of fundamental (or distinct) solutions to the eight queens problem.
Abstract: Previous algorithms presented to solve the eight queens problem have generated the set of all solutions Many of these solutions are identical after applying sequences of rotations and reflections In this paper we present a simple, clear, efficient algorithm to generate a set of fundamental (or distinct) solutions to the problem
TL;DR: An analytic approximation to the complex normal probability integral, Φ(x+iy)=(2π)−1/2∝ −t x exp[−(t−iy)2/2]dt, is given together with a study of the error in the approximation.
Abstract: An analytic approximation to the complex normal probability integral, Φ(x+iy)=(2π)−1/2∝ −t x exp[−(t−iy)2/2]dt, is given together with a study of the error in the approximation.
TL;DR: An attempt to switch dynamically between stiff and non-stiff algorithms in the specific area of stiffness detection has traditionally required user interaction.
Abstract: The specific area of stiffness detection has traditionally required user interaction. This paper presents an attempt to switch dynamically between stiff and non-stiff algorithms.
TL;DR: In this paper, it was shown that by introducing one or three interpolated values in each subinterval the local truncation error of finite difference (box, gap and deferred correction) methods can be decreased by two to four orders of magnitude when solving two point boundary value ODEs.
Abstract: It is shown that by introducing one or three interpolated values in each subinterval the local truncation error of finite difference (box, gap and deferred correction) methods can be decreased by two to four orders of magnitude when solving two point boundary value ordinary differential equations.