Showing papers in "International Journal of Computer Mathematics in 1991"
TL;DR: This paper introduces a four point explicit decoupled group (EDG) iterative method as a new Poisson solver and is shown to be very much faster compared to existing explicit group (EG) methods.
Abstract: The aim of this paper is to introduce a four point explicit decoupled group (EDG) iterative method as a new Poisson solver. The method is shown to be very much faster compared to existing explicit group (EG) methods due to D. J. Evans and M. J. Biggins (1982) and W. Yousif and D. J. Evans (1985). Some numerical experiments are included to confirm our recommendation.
209 citations
TL;DR: Zero-stable Block methods of orders 3/4 are proposed for Second Order Initial Value Problems given and there is anticipated speed up of computations as a result of admissible parallelism across the method and cheap error estimators.
Abstract: Zero-stable Block methods of orders 3/4 are proposed for Second Order Initial Value Problems given. The matrix coefficients of the schemes are chosen as to ensure zero-stability and consistency (hence convergence of the resultant schemes). There is anticipated speed up of computa-tions as a result of admissible parallelism across the method (or time) and cheap error estimators.
182 citations
TL;DR: A general model of SA is given, which has both dynamic generation probability and acceptance probability and proves its convergence, and a method of extending SA's neighbourhood is proposed, which uses a discrete approximation to some continuous probability function as the generation function in SA.
Abstract: Simulated Annealing (SA) is a powerful stochastic search method applicable to a wide range of problems for which little prior knowledge is available. It can produce very high quality solutions for hard combinatorial optimization problems. However, the computation time required by SA is very large. Various methods have been proposed to reduce the computation time, but they mainly deal with the careful tuning of SA'ol parameters. This paper first analyzes the impact of SA'neighbourhood on SA'performance and shows that SA with a larger neighbourhood is better than SA with a smaller one. The paper also gives a general model of SA, which has both dynamic generation probability and acceptance probability, and proves its convergence. All variants of SA can be unified under such a generalization. Finally, a method of extending SA's neighbourhood is proposed, which uses a discrete approximation to some continuous probability function as the generation function in SA, and several important corollaries of the genera...
82 citations
TL;DR: A two step method with phase-lag of order infinity is developed for the numerical integration of second order periodic initial-value problem and is generally more accurate than other two-step methods.
Abstract: A two-step method with phase-lag of order infinity is developed for the numerical integration of second order periodic initial-value problem. The method has algebraic order six. Extensive numerical testing indicates that the new method is generally more accurate than other two-step methods.
58 citations
TL;DR: An exact algorithm for the maximum weighted independent set problem based on an unconstrained quadratic 0–1 formulation and branch and bound techniques is proposed and implemented and computational results are presented.
Abstract: We propose and implement an exact algorithm for the maximum weighted independent set problem based on an unconstrained quadratic 0–1 formulation and branch and bound techniques. The use of a non-greedy branch and bound method using depth first search is presented and justified along with various heuristics to improve performance. We also present computational results using randomly generated graphs with up to 500 vertices.
42 citations
TL;DR: AB mod N where N is odd is shown to have significant advantages over other algorithms which make it suitable for use in hardware for public key encryption and could run at approximately twice the speed of the best currently available.
Abstract: This paper describes a method for quickly computing AB mod N where N is odd. It is shown to have significant advantages over other algorithms which make it suitable for use in hardware for public key encryption. Such hardware could run at approximately twice the speed of the best currently available.
39 citations
TL;DR: The fourth order parabolic equation governing the transverse vibrations of a homogeneous beam is solved numerically using the AGE method and the results obtained confirm the superiority of the method over existing methods.
Abstract: The fourth order parabolic equation governing the transverse vibrations of a homogeneous beam is solved numerically using the AGE method. The results obtained confirm the superiority of the method over existing methods.
38 citations
TL;DR: A new algorithm for computing the medial axis of a simple polygon is presented that is simple to implement and it does not require the complex data-structures required for the faster methods.
Abstract: A new algorithm for computing the medial axis of a simple polygon is presented. Although the algorithm runs in O(kN) time where k is the hierarchy of the Voronoi diagram of the polygon ranging from O(N) to O(logN) it is simple to implement and it does not require the complex data-structures required for the faster methods. This is an important factor in many applications of the medial axis.
36 citations
TL;DR: A new 4th order Runge-Kutta method for solving initial value problems is derived which provides an estimate of the truncation error without any extra function evaluations and is suitable to be used as an error control strategy.
Abstract: A new 4th order Runge-Kutta method for solving initial value problems is derived by replacing the arithmetic means in the formula where etc., by their geometric means i.e. etc. to yield initially a low order accuracy formula. However by re-comparing the Taylor series expansions of k 1 k 2 k 3 and k 4 in terms of the functional derivatives and the α ij parameters, a fourth order accuracy formula is obtained which is confirmed by numerical experiments. Then, a new fourth order Runge-Kutta method for solving linear initial value problems of the form y′ = Ay is derived which provides an estimate of the truncation error without any extra function evaluations. The idea follows from the fact that two numerical solutions of similar order can be obtained by using the arithmetic mean (AM) and the geometric mean (GM) averaging of the functional values. The numerical results given confirm that this new method is suitable to be used as an error control strategy.
33 citations
TL;DR: It is shown that the high-radix methods with optimal choice of the radix provide significant reductions in the number of multiplications required for modular exponentiation, and bit recoding techniques similar to those used in binary multiplication algorithms can be used to further reduce the total number ofmultiplications.
Abstract: Algorithms that make use of high-radix and bit recoding techniques to perform modular exponentiation are proposed. It is shown that the high-radix methods with optimal choice of the radix provide significant reductions in the number of multiplications required for modular exponentiation. It is then shown that bit recoding techniques similar to those used in binary multiplication algorithms can be used to further reduce the total number of multiplications. The algorithms presented are analyzed by counting the maximum and the average number of multiplications required.
30 citations
TL;DR: The ASE-I method is extended to the case of the two-space dimensional problem, and in this case the method is called the Alternating Block Explicit-Implicit (ABE-I) method.
Abstract: We have developed the Alternating Segment Explicit-Implicit (ASE-I) method for the one-space dimensional diffusion equation in Zhang Bao-lin [1]. Mathematically, the ASE-I method is unconditionally stable and is good from the view point of parallel computing. In this paper the method is extended to the case of the two-space dimensional problem, and in this case the method is called the Alternating Block Explicit-Implicit (ABE-I) method.
TL;DR: This theory provides a framework in which known results about codes can be expressed elegantly and in which several new results are derived and can be generalized to relations of arbitrary finite arity in a very natural fashion.
Abstract: Many classes of codes can be characterized as families of antichains with respect to partial orders on the free monoid or, in more general terms, as families of independent sets with respect to some binary relations. In this paper we investigate the general properties of this connection between families of sets and binary relations. This theory provides a framework in which known results about codes can be expressed elegantly and in which several new results are derived. Moreover, this theory can be generalized to relations of arbitrary finite arity in a very natural fashion. This allows us, for instance, to prove new hierarchy results. More importantly, however, this theory provides a new and profound insight into the mechanisms by which classes of codes are defined.
TL;DR: An expansion procedure using the Chebyshev polynomials is proposed by using El-Gendi method, which yields more accurate results than those computed by D. Hatziavrmidis as indicated from solving the Orr-Sommerfeld equation for both the plane poiseuille flow and the Blasius velocity profile.
Abstract: An expansion procedure using the Chebyshev polynomials is proposed by using El-Gendi method [1], which yields more accurate results than those computed by D. Hatziavrmidis [2] as indicated from solving the Orr-Sommerfeld equation for both the plane poiseuille flow and the Blasius velocity profile. This method is accomplished by starting with Chebyshev approximation for the highest-order derivative and generating approximations to the lower-order derivatives through integration of the highest-order derivative.
TL;DR: A family of numerical methods is developed for the numerical solution of a linear third-order dispersive partial differential equation in one space variable with time-dependent boundary conditions and evolves from first- and second-order rational approximants to an exponential function in a recurrence relation.
Abstract: A family of numerical methods is developed for the numerical solution of a linear third-order dispersive partial differential equation in one space variable with time-dependent boundary conditions. The methods are seen to evolve from first- and second-order rational approximants to an exponential function in a recurrence relation. Global extrapolation procedures in time-only and in both space and time are discussed.
TL;DR: Several results on properties of the stable partitions are established, a simple proof of a recent theorem of Tan is presented, and a maximum stable matching problem is solved.
Abstract: Recently Tan [7] defined a new structure for the stable roommates problem, called a “stable partition” which is a generalization of the notion of the stable matching. He proved that every instance of that problem contains at least one such structure, and obtained a succinct certificate of the non-existence of a stable matching. In this paper. we establish several results on properties of the stable partitions, present a simple proof of a recent theorem of Tan, and solve a maximum stable matching problem.
TL;DR: It is shown that there is no advantage in taking a very large radix, although a small increase above 2 is beneficial, and the effect of varying the radix on the efficiency of hardware implementations is considered.
Abstract: Details are given here of how to generalise Brickell's fast modular multiplication algorithm to when the number representations have a general 2-power radix. Correct action depends upon the satisfaction of a complicated inequality and speed upon the use of a redundant number system to enable parallel digit operations. The effect of varying the radix on the efficiency of hardware implementations is considered. Improved efficiency has repercussions in public key cryptography where the RSA encryption scheme may use this type of algorithm for its modular exponentiations. However, it is shown that there is no advantage in taking a very large radix, although a small increase above 2 is beneficial.
TL;DR: Nonlinear trapezoidal formulae based on a variety of means other than the arithmetic and geometric means, i.e. harmonic, logarithmic, etc., for solving first order differential equations with specified initial values are presented.
Abstract: Nonlinear trapezoidal formulae based on a variety of means other than the arithmetic and geometric means, ie harmonic, logarithmic, etc, for solving first order differential equations with specified initial values are presented The accuracy and stability properties of the methods are investigated and some conclusions drawn from a numerical example
TL;DR: The single-term Walsh series (STWS) method is used to find the numerical solution of a singular nonlinear differential equation from fluid dynamics.
Abstract: The single-term Walsh series (STWS) method is used to find the numerical solution of a singular nonlinear differential equation from fluid dynamics. The STWS method is better than the Runge-Kutta method for this problem.
TL;DR: The structures of context-free languages contained in Q are determined and various problems related to context- free languages consisting of non-primitive words are solved.
Abstract: Let Q set of all primitive words over an alphabet In this paper, the structures of context-free languages contained in are determined. Observing these structures, we solve various problems related to context-free languages consisting of non-primitive words.
TL;DR: Canonical polynomials are constructed as new basis for collocation solution in the smooth region which is superposed with an exponential function in the boundary layer region for the linear problem.
Abstract: This paper concerns the numerical solutions of two-point singularly perturbed boundary value problems for second order ordinary differential equations. For the linear problem, Canonical polynomials are constructed as new basis for collocation solution in the smooth region which is superposed with an exponential function in the boundary layer region. Numerical examples are given which show that the exponential fitting leads to a numerical asymptotic procedure. Extension to nonlinear problems is demonstrated by an example.
TL;DR: This paper presents an efficient election algorithm which works on all chordal ring networks, and shows that the algorithm has an O(n) message complexity both on a regular chordalRing with only O( n log log n) links and on an irregular chordal Ring withonly O(N) links.
Abstract: Distributed election arises in all situations where a single processor is needed to control a certain function. In this paper, we address this problem in networks with a “sense of direction”which is the capability to distinguish between its adjacent communication links. We present an efficient election algorithm which works on all chordal ring networks. We show that the algorithm has an O(n) message complexity both on a regular chordal ring with only O(n log log n) links and on an irregular chordal ring with only O(n) links. This is an improvement over the algorithms presented in [1,9] where O(n 2) and O(n log n) links are needed respectively to achieve O(n) message complexity in distributed election with a sense of direction. Since at least O(n) messages and 0(n) links are needed to perform an election in a network with n processors, our algorithm is optimal when irregular chordal ring networks are considered.
TL;DR: A class of singular perturbation problems is considered, in which a numerical-asymptotic method is constructed in which asymptotic solution techniques are combined with standard numerical methods.
Abstract: A class of singular perturbation problems is considered. In order to solve them a numerical-asymptotic method is constructed in which asymptotic solution techniques are combined with standard numerical methods. The proposed method is distinguished by the following facts: First, we construct the division point which divides the initial interval into two subintervals, so that the layer belongs only to one of them. The division point is constructed in such a way that it provides the best possible accuracy of such a combined method. The inner solution problem is solved as a two point boundary value problem, where the terminal boundary conditions are supplied by the solution of reduced problem. As the outer solution the solution of reduced problem is taken. A numerical example is included.
TL;DR: Good multigrid performance with discrete problems arising from the biharmonic (plate) equation is demonstrated and an approximate inverse of a matrix found by solving a Frobenius matrix norm minimization problem is used in the multigrids of the approximate inverse based FAPIN algorithm.
Abstract: The approximate inverse based multigrid algorithm FAPIN for the solution of large sparse linear systems of equations is examined. This algorithm has proven successful in the numerical solution of several second order boundary value problems. Here we are concerned with its application to fourth order problems. In particular, we demonstrate good multigrid performance with discrete problems arising from the biharmonic (plate) equation. The work presented also represents new experience with FAPIN using bicubic Hermite basis functions. Central to our development is the concept of an approximate inverse of a matrix. In particular, we use a least squares approximate inverse found by solving a Frobenius matrix norm minimization problem. This approximate inverse is used in the multigrid smoothers of our algorithm FAPIN. The algorithms presented are well suited for implementation on hypercube multiprocessors.
TL;DR: This paper considers the relocation problem with processing times and deadlines in which each job is additionally associated with a processing length and an individual deadline and proposes two polynomial algorithms, one runs inO(h∗log h)time and the other in O(h2∗ log h) time, to solve some further restricted problems.
Abstract: The relocation problem originates from the public housing project so as to minimize the budgets. Given a set of jobs each demands nt units of resources for processing and returns a, units of resources. The relocation problem is to determine the minimum resource requirements demanded by the successful completion of this set of jobs. In the literature, an O(h∗log h) algorithm has been proposed by Kaplan and Amir, where h is the number of jobs. In this paper, we consider the relocation problem with processing times and deadlines in which each job is additionally associated with a processing length and an individual deadline. Studies of NP-completeness of some versions of this problem are included. Also, we propose two polynomial algorithms, one runs inO(h∗log h)time and the other in O(h2∗log h)time, to solve some further restricted problems.
TL;DR: A very simple iterative solution for the Generalized Tower of Hanoi Problem is presented, taking a minimal number of disc moves.
Abstract: The Generalized Tower of Hanoi Problem concerns the transformation of an arbitrary initial configuration of n discs distributed among three pegs to an arbitrary final configuration, subject to the well-known Tower of Hanoi rules. A very simple iterative solution for this problem is presented, taking a minimal number of disc moves.
TL;DR: A systolic array for matrix-vector multiplication {mvm), with the additional feature that its output can be re-used as input for a consecutive mvm, makes the array especially useful for iterative syStolic algorithms based on successive mvm computations.
Abstract: This paper presents a systolic array for matrix-vector multiplication {mvm), with the additional feature that its output can be re-used as input for a consecutive mvm. This fact makes the array especially useful for iterative systolic algorithms based on successive mvm computations. Some of these applications are presented in this paper. The array is simulated in Occam.
TL;DR: A parallel algorithm for generating all the n! permutations is presented and the basic idea used is the iterative method and the exchange of two consecutive components in an existing permutation.
Abstract: Given n items, a parallel algorithm for generating all the n! permutations is presented. The computational model used is a linear array which consists of n identical processing elements with a simple structure. One permutation is produced at each other time step. The elapsed time to produce a permutation is independent of the integer n. The basic idea used is the iterative method and the exchange of two consecutive components in an existing permutation. The design procedures of this algorithm are considered in detail. The ranking and unranking functions of the required permutations are also discussed.
TL;DR: An efficient algorithm for generating the P-sequences characterising all shapes of binary trees with n nodes is presented, based on the connection between the P -sequences and the ballot sequences.
Abstract: An efficient algorithm for generating the P-sequences characterising all shapes of binary trees with n nodes is presented. This is based on the connection between the P-sequences and the ballot sequences. A P-sequence results from a ballot sequence if the integers 0, 1, 2,…,n−1 in ballot sequence is replaced by integers n, n−1, n−2, …, 1 in their corresponding positions.
TL;DR: An implicit eighth-order finite difference method for the general second order non-linear differential equation Yn=f(t,y,y) subject to the initial conditions y(to) =yo, y( to)=yO is presented.
Abstract: We present an implicit eighth-order finite difference method for the general second order non-linear differential equation Yn=f(t,y,y) subject to the initial conditions y(to) =yo, y(to)=yO. The method is based on mixed interpolation containing a parameter which can be used to improve the accuracy. The same technique gives rise to an explicit sixth-order integration scheme. Numerical solutions of problems are given to illustrate both methods.
TL;DR: High-order finite difference methods of O(k 2 +kh 2 + h 4) using 9-spatial grid points for integrating the system of two-dimensional nonlinear parabolic partial differential equations subject to Dirichlet boundary conditions are given.
Abstract: In this paper, high-order finite difference methods of O(k 2 +kh 2 + h 4) using 9-spatial grid points for integrating the system of two-dimensional nonlinear parabolic partial differential equations subject to Dirichlet boundary conditions are given. The method having two variables is tested on 2-D unsteady Navier-Stokes equations. Numerical examples given here show that the methods developed here retain their order and accuracy.