Showing papers in "International Journal of Computer Mathematics in 1997"
TL;DR: A class of synchronous as well as asynchronous matrix multisplitting methods for solving the linear complementarity problem is set up, and its convergence is proved when the coefficient matrix is an H-matrix with positive diagonal elements.
Abstract: In this paper, a class of synchronous as well as asynchronous matrix multisplitting methods for solving the linear complementarity problem is set up, and its convergence is proved when the coefficient matrix is an H-matrix with positive diagonal elements
115 citations
TL;DR: This paper describes a method for using the existence of a formal specification as the basis for the development of a detailed functional testing strategy and demonstrates that this method can provide a more convincing approach to the problem of detecting all faults.
Abstract: Although a great deal of research has been done in the area of formal methods and their practical use for the specification and verification of software systems, testing issues are very seldom mentioned by those within the formal methods community. Almost all the methods currently used for testing software are experience based rather than theoretically founded methods. In particular, very few methods allow us to make any statement about the type or number of faults that remain undetected after testing is completed. This paper describes a method for using the existence of a formal specification as the basis for the development of a detailed functional testing strategy. By considering testing from a theoretical point of view we demonstrate that this method can provide a more convincing approach to the problem of detecting all faults. The formal method used, X-machines, is a blend of finite state machines, data structures and processing functions and provides a simple and intuitive way for specifying compute...
102 citations
TL;DR: A finite difference method for the singular two point boundary value problems has been developed and second order convergence of the method has been established under quite general conditions on the functions p(x) and f(x,y).
Abstract: A finite difference method for the singular two point boundary value problems has been developed and second order convergence of the method has been established under quite general conditions on the functions p(x) and f(x,y). In case p(x)=x α, the assumptions on the function p(x) allow α to be 1, 2 or any number greater than 2. Order of the method has been verified by one example and two problems of physiology have also been solved by this method.
41 citations
TL;DR: An efficient L 0-stable parallel algorithm is developed for the two-dimensional diffusion equation with non-local time-dependent boundary conditions and is tested on two model problems from the literature for which discontinuities between initial and boundary conditions exist.
Abstract: An efficient L 0-stable parallel algorithm is developed for the two-dimensional diffusion equation with non-local time-dependent boundary conditions The algorithm is based on subdiagonal Pade approximation to the matrix exponentials arising from the use of the method of lines and may be implemented on a parallel architecture using two processors running concurrently with each processor employing the use of tridiagonal solvers at every time-step The algorithm is tested on two model problems from the literature for which discontinuities between initial and boundary conditions exist The CPU times together with the associated error estimates are compared
27 citations
TL;DR: A tight Ω(n lg n) lower bound is proved in the more powerful algebraic decision tree model and the apparently simpler problem of finding the length of a longest increasing subsequence is solved.
Abstract: The longest increasing subsequence problem is as follows: Given a sequence of n real numbers, find a longest increasing subsequence of . There is a well-known O(n lg n)-time comparison tree algorithm for solving this problem. Also, a tight Ω(n lg n) lower bound in the comparison tree model is known. We prove a tight Ω(n lg n) lower bound in the more powerful algebraic decision tree model. The above lower bounds also apply to the apparently simpler problem of finding the length of a longest increasing subsequence.
20 citations
TL;DR: An equational theory of networks, called BNA (Basic Network Algebra), is presented, which is essentially a part of the algebra of flownomials, captures the basic algebraic properties of networks.
Abstract: Network algebra is proposed as a uniform algebraic framework for the description and analysis of dataflow networks. An equational theory of networks, called BNA (Basic Network Algebra), is presented. BNA, which is essentially a part of the algebra of flownomials, captures the basic algebraic properties of networks. For asynchronous dataflow networks, additional constants and axioms are given; and a corresponding process algebra model is introduced. This process algebra model is compared with previous models for asynchronous dataflow.
18 citations
TL;DR: The original system to solve nonlinear Volterra integro-differential equations with two point boundary conditions is modified to allow the implementation of a special fast algorithm for solving tridiagonal systems.
Abstract: Using finite difference methods to solve nonlinear Volterra integro-differential equations with two point boundary conditions give rise to a symmetric banded coefficient matrix. A typical method for solving systems of this form involves the LU method. In this paper the original system is modified to allow the implementation of a special fast algorithm for solving tridiagonal systems. Numerical examples are given to compare an efficient form of the LU method with the new approach.
18 citations
TL;DR: The study formally shows that the partial solutions of partial differential equations for initial/boundary conditions using the Adomian decomposition method are always identical for all styles of boundary conditions and it is proved that thepartial solution in the t-direction requires less computational work if compared with other partial solutions developed in any space variable direction.
Abstract: This work considers the partial solutions of partial differential equations for initial/boundary conditions using the Adomian decomposition method. The study formally shows that the partial solutions are always identical for all styles of boundary conditions. We also prove that the partial solution in the t-direction requires less computational work if compared with other partial solutions developed in any space variable direction. In addition, several mathematical models that govern the heat distribution and the wave propagation phenomenas have been tested, and the results obtained have shown that the t-solution minimizes the size of calculations if compared with the traditional techniques.
17 citations
TL;DR: This paper proposes two efficient parallel algorithms to speed up the computation of the modular multi-exponentiation ∏n i=1Mi Ei (modN), which is an important but time-consuming arithmetic operation used in many scientific researches and applications, especially in the contemporary cryptosystems.
Abstract: Some methods of fast modular exponentiation have been proposed in the past years. However, there are only a few parallel mechanisms for evaluating the modular multi-exponentiation. In this paper, we propose two efficient parallel algorithms to speed up the computation of the modular multi-exponentiation ∏n i=1Mi Ei (modN), which is an important but time-consuming arithmetic operation used in many scientific researches and applications, especially in the contemporary cryptosystems. We also show that our two proposed methods are faster than the best known sequential method (the Shamir's method) and parallel method (the Chiou's method). Furthermore, our methods can be implemented easily in the multicomputer systems
15 citations
TL;DR: A class of parallel blockwise matrix multisplitting block relaxation methods is established for the large sparse block system of linear equations and its convergence theory is set up thorouthly when the coefficient matrix is a block H-matrix.
Abstract: In this paper, a class of parallel blockwise matrix multisplitting block relaxation methods, including the blockwise matrix multisplitting block symmetric accelerated overrelaxation method, the blockwise matrix multisplitting block unsymmetric and symmetric successive overrelaxation methods and the blockwise matrix multisplitting block unsymmetric and symmetric Gauss-Seidel methods, etc., is established for the large sparse block system of linear equations, and its convergence theory is set up thorouthly when the coefficient matrix is a block H-matrix. Also, the new methods are further extended by relaxing different block elements of the iterations with different relaxation parameters and, therefore, general frameworks of parallel blockwise matrix multisplitting block relaxation methods for solving the block system of linear equations are naturally obtained.
14 citations
TL;DR: The family of recursively enumerable languages is characterized by scattered context Grammars with four nonterminals if these grammars start their derivations from a word rather than a symbol.
Abstract: The family of recursively enumerable languages is characterized by scattered context grammars with four nonterminals. Moreover, this family is characterized by scattered context grammars with three nonterminals if these grammars start their derivations from a word rather than a symbol. Three open problem areas are suggested
TL;DR: An alternative method based on expansion is presented here and this method is compared with other approaches, and the algorithms are applied to problems with large oscillatory factors.
Abstract: Many current problems in applied mathematics require the numerical integration of irregular oscillatory integrals. Few methods have been specifically found for these problems. An alternative method based on expansion is presented here and this method is compared with other approaches. Tests are carried out on a representative set of examples, and the algorithms are applied to problems with large oscillatory factors.
TL;DR: Canonical polynomials constructed in [1] are modified and used as a new basis for a collocation solution via the perturbed collocation method with and without exponentially fitting.
Abstract: This paper concerns the numerical methods based on collocation approximation of fourth-order boundary value problems for the linear problem. Canonical polynomials constructed in [1] are modified and used as a new basis for a collocation solution via the perturbed collocation method with and without exponentially fitting. Nonlinear cases are treated by the Newton's linearization scheme of order 4. The Newton's scheme from the Taylor's series expansion of order 4 is given by: is used throughout this paper. Numerical examples are given to illustrate the effectiveness of the methods discussed in this paper.
TL;DR: This work proposes an iterative scheme based on a splitting of the matrix of the objective function and called splitting algorithm (SA), which transforms the original problem into a sequence of subproblems easier to solve, and uses a special version of SA called diagonalization algorithm (DA).
Abstract: Many problems arising in data analysis can be formulated as a large sparse strictly convex quadratic programming problems with equality and inequality linear constraints. In order to solve these problems, we propose an iterative scheme based on a splitting of the matrix of the objective function and called splitting algorithm (SA). This algorithm transforms the original problem into a sequence of subproblems easier to solve, for which there exists a large number of efficient methods in literature. Each subproblem can be solved as a linear complementarity problem or as a constrained least distance problem. We give conditions for SA convergence and we present an application on a large scale sparse problem arising in constrained bivariate interpolation. In this application we use a special version of SA called diagonalization algorithm (DA). An extensive experimentation on CRAY C90 permits to evaluate the DA performance
TL;DR: Borsch-Supan's method and its modification with Weierstrass' correction and the comparison of the considered two methods on MIMD parallel computers (synchronous and asynchronous implementation) are studied.
Abstract: In this paper we consider Borsch-Supan's method and its modification with Weierstrass' correction. These methods are suitable for the simultaneous approximation of all simple zeros of polynomials and have the convergence order three and four, respectively. In the first part we give an initial condition for the safe convergence of the method with correction. This condition depends only on attainable data and has a practical importance. In the second part the comparison of the considered two methods on MIMD parallel computers (synchronous and asynchronous implementation) are studied.
TL;DR: A trust region method, based on approximation of f(·) and f1 (·) of higher order, is presented and a convergence analysis for the method is considered.
Abstract: In this paper a trust region method, based on approximation of f(·) and f1 (·) of higher order, is presented. A convergence analysis for the method is considered too. Numerical results are reported.
TL;DR: This analysis has represented the bin packing problem as an optimization problem and considered the objective function as fitness function for GA and the numerical results indicate that the GA is robust and it yields better results when compared with the results obtained from other heuristics.
Abstract: In this paper we have analysed the bin packing problem by applying Genetic Algorithm (GA). In this analysis we have represented the problem as an optimization problem and considered the objective function as fitness function for GA. The numerical results obtained from the GA indicate that the GA is robust and it yields better results when compared with the results obtained from other heuristics such as First Fit Decreasing (FFD) and Best Fit Decreasing (BFD)
TL;DR: A procedure for determining whether a box intersects a tetrahedron is developed, and it gives a guaranteed answer (modulo rounding errors).
Abstract: A procedure for determining whether a box intersects a tetrahedron is developed. The procedure uses interval arithmetic tools and barycentric coordinates, and it gives a guaranteed answer (modulo rounding errors). The basic geometric idea uses the separating plane approach. The procedure is robust and easy to implement. Numerical examples are given for each of the possible results (i.e., inclusion, overlap and exclusion).
TL;DR: This paper demonstrates that both transforms (i.e. MAT and CDT) are interchangeable and proposes the corresponding algorithms, which are used in computer vision and image processing.
Abstract: The distance transform (DT) and the medial axis transform (MAT) are two image computation tools used to extract the information about the shape and the position of the foreground pixels relative to each other. The DT converts a binary image into an image, where each pixel has a value to represent the distance from it to its nearest foreground pixel. The MAT of an image is a set of maximal squares that represents the foreground pixels of an image. Extensively applications of these two transforms are used in the fields of computer vision and image processing, such as expanding shrinking, thinning and computing shape factor etc. There are many different distance transforms based on different distance metrics. The chessboard distance transform (CDT) is one kind of distance transform (DT) which converts an image based on the chessboard distance metrics. In this paper, we first demonstrate that both transforms (i.e. MAT and. CDT) are interchangeable. Then the corresponding algorithms are also proposed. That is,...
TL;DR: A variable-step variable-order algorithm for stiff ODEs based on previously derived stabilized extended one-step methods is established and tested on certain initial-value problems for systems of ODE's contained in the test set proposed by CWI.
Abstract: A variable-step variable-order algorithm for stiff ODEs based on previously derived stabilized extended one-step methods is established. The developed code is tested on certain initial-value problems for systems of ODEs contained in the test set proposed by CWI.
TL;DR: This investigation demonstrates that the family of recursively enumerable languages is characterized by six-nonterminal multi-sequential grammars.
Abstract: The present paper investigates the descriptional complexity of multi-sequential grammars with respect to the number of nonterminals. This investigation demonstrates that the family of recursively enumerable languages is characterized by six-nonterminal multi-sequential grammars.
TL;DR: The BLAGE preconditioner is shown to be extremely effective in achieving optimal convergence rates for the class of unsymmetric fourth order difference schemes considered in this paper.
Abstract: We consider the block alternating group explicit method (BLAGE) as a preconditioner for a class of unsymmetric linear systems arising from fourth order finite difference schemes. An advantage of the preconditioner proposed is that the preconditioning operations can be divided into several subproblems which can then be run in parallel. The BLAGE preconditioner is shown to be extremely effective in achieving optimal convergence rates for the class of unsymmetric fourth order difference schemes considered in this paper
TL;DR: It is proved that the iterative algorithm converges to a smooth approximate solution provided the boundary value problem is well posed and the algorithm is applied appropriately.
Abstract: In this paper, by using the ideas employed in the analysis of interpolatory subdivision algorithms for the generation of smooth curves, an iterative scheme for solving nonlinear two point boundary value problems is formulated. This method is basically a collocation method for nonlinear second order two point boundary value problems. It is proved that the iterative algorithm converges to a smooth approximate solution provided the boundary value problem is well posed and the algorithm is applied appropriately. Error estimates in the case of uniform partitions are also investigated. Some numerical examples are included to show the convergence of the proposed algorithm.
TL;DR: This work presents a new class of trapezoidal formulas obtained by suitably modifying the classical arithmetic mean formula and the recently proposed geometric and harmonic mean formulas, and shows that each is second order and L-stable.
Abstract: We present a new class of trapezoidal formulas obtained by suitably modifying the classical arithmetic mean formula and the recently proposed geometric and harmonic mean formulas. Interestingly, when applied to the test equation, a modified trapezoidal formula leads to a single linear 1-step recurrence. We show that each of these modified trapezoidal formulas is second order and L-stable. For the integration of autonomous problems, both the modified geometric and harmonic mean trapezoidal formulas are well-defined around f=0, at least for sufficiently small step-lengths
TL;DR: The global convergence theorem of the MPSD algorithm is established under the some conditions and the numerical examples given show that the new algorithm is feasible and efficient.
Abstract: In this paper, we propose the parallel multisplitting PSD (MPSD) method for solving system of weakly nonlinear equations Ax + ф(x) = b. The global convergence theorem of the MPSD algorithm is established under the some conditions. Finally, the numerical examples are given, they show that our new algorithm is feasible and efficient.
TL;DR: New block methods of order two and three for the numerical solution of initial value problems are derived such that low powers of the blocksize appear in the principal local truncation errors.
Abstract: New block methods of order two and three for the numerical solution of initial value problems are derived. The matrix coefficients of these methods are chosen such that low powers of the blocksize appear in the principal local truncation errors.
TL;DR: A new four-stage fifth-order explicit method of Runge-Kutta type is presented based on a composite derivative strategy and stability and error analyses confirm that the method is stable of fifth order.
Abstract: In this paper we present a new four-stage fifth-order explicit method of Runge-Kutta type. The new method is based on a composite derivative strategy and stability and error analyses confirm that the method is stable of fifth order. Two examples are given which illustrate the effectiveness of the method compared to the RK-Nystrom six-stage fifth order method.
TL;DR: The evaluation of an integral of the product of Laguerre polynomials which appeared recently in this Journal by Mavromatis is shown to be a particular case of a general result of Erdelyi.
Abstract: The evaluation of an integral of the product of Laguerre polynomials which appeared recently in this Journal by Mavromatis [36, 1990, p. 257] is shown to be a particular case of a general result of Erdelyi. An error in the special case derived earlier and other possible explicit evaluations on account of special forms of the 3 F 2(1) function are discussed.
TL;DR: New high order Hermite interpolants for two well known embedded Runge-Kutta methods of orders 7 and 8 are presented, which are constructed using values from two successive integration steps, and require only one or four extra function evaluations per step respectively.
Abstract: Computer assisted derivation and improved techniques have led to effective explicit Runge-Kutta methods of higher order. These methods become inefficient when the step size must be reduced often to produce approximations at specified points. Considerable effort has been devoted to providing Runge-Kutta methods with an interpolation capability, so that approximations can be produced inexpensively at intermediate points of a successful step. New high order Hermite interpolants for two well known embedded Runge-Kutta methods of orders 7 and 8 are presented. These interpolants are constructed using values from two successive integration steps, are locally of O(h 8) or O(h 9), and require only one or four extra function evaluations per step respectively.
TL;DR: This paper presents a new algorithm, dynamic adaptation of the learning rate to accelerate steepest descent, to partition the iteration number domain into n intervals and a suitable value for theLearning rate is assigned for each respective iteration interval.
Abstract: Standard backpropagation, as with many gradient based optimization methods converges slowly as neural networks training problems become larger and more complex. In this paper, we present a new algorithm, dynamic adaptation of the learning rate to accelerate steepest descent. The underlying idea is to partition the iteration number domain into n intervals and a suitable value for the learning rate is assigned for each respective iteration interval. We present a derivation of the new algorithm and test the algorithm on several classification problems. As compared to standard backpropagation, the convergence rate can be improved immensely with only a minimal increase in the complexity of each iteration.