Showing papers in "International Journal of Computer Mathematics in 1986"
TL;DR: In this paper, an explicit 8-point group over-relaxation scheme is presented for the numerical solution of the sparse linear systems derived from the finite difference discretisation of a self-adjoint elliptic partial differential equation in 3-space dimensions.
Abstract: In this paper, a new explicit 8-point group over-relaxation scheme is presented for the numerical solution of the sparse linear systems derived from the finite difference discretisation of a self-adjoint elliptic partial differential equation in 3-space dimensions. A comparison with the usual point S.O.R. scheme for the Laplace model problem shows the new technique to be competitive.
40 citations
TL;DR: Some work of van Bokhoven is extended to a class of nonsymmetric P-matrices, and several new iterative algorithms for the linear complementarity problem are developed and compared.
Abstract: Direct complementary pivot algorithms for the linear complementarity problem with P-matrices are known to have exponential computational complexity. The analog of Gauss-Seidel and SOR iteration for linear complementarity problems with P-matrices has not been extensively developed. This paper extends some work of van Bokhoven to a class of nonsymmetric P-matrices, and develops and compares several new iterative algorithms for the linear complementarity problem. Numerical results for several hundred test problems are presented. Such indirect iterative algorithms may prove useful for large sparse complementarity problems.
36 citations
TL;DR: The notion of Hanoi Graph is introduced the first time for the Towers of H Vietnam puzzle which greatly helps investigate the problem and some new conclusions are presented.
Abstract: The notion of Hanoi Graph is introduced the first time for the Towers of Hanoi puzzle which greatly helps investigate the problem. And some new conclusions are presented in this paper.
26 citations
TL;DR: The family of f-disjunctive languages is a natural generalization of the disjunctive language and both families are anti-AFL as discussed by the authors, and it is shown that they form a strict hierarchy.
Abstract: A language L is f-disjunctive if every class of its syntactic congruence is finite. The family of f-disjunctive languages is a natural generalization of the family of disjunctive languages and both families are anti-AFL. Every f-disjunctive language is dense and every class (≠ 1) of its syntactic congruence is an infix code. Five different classes of f-disjunctive languages are considered and it is shown that they form a strict hierarchy. Also some properties of the syntactic monoid of a f-disjunctive language are established, in particular its decomposition as a subdirect product of nil monoids.
19 citations
TL;DR: In this article, the convergence rate of SAP for two-dimensional model problems was shown to be convergent with respect to the number of points and the dimension of the model and the model.
Abstract: This is the second of a series of papers about the convergence rate of SAP (Schwarz Alternating Procedure) for solving Mathematical Physics problems. In this paper, the convergence rate is obtained for two-dimensional model problems.
16 citations
TL;DR: An algorithm for simulating a pushdown stack of size S{n) on an ITA of depth logS(n) in real-time is presented.
Abstract: An iterative tree array (ITA) is a binary tree-connected systolic network in which each cell is a finite-state machine and the input is provided serially to the root. We present an algorithm for simulating a pushdown stack of size S{n) on an ITA of depth logS(n) in real-time. Some interesting applications are the following: 1) Every linear iterative array operating in (simultaneous) time T(n) and space S(n) can be simulated by an ITA in time T(n) and depth log S(n). 2) S(n)-space bounded on-line TM's are equivalent to log S(n)-depth bounded ITA's. 3) log n depth is a necessary and sufficient condition for an ITA to recognize every context-free language. 4) log log n depth is a necessary condition for an ITA to recognize a nonregular set. 5) Every on-line nondeterministic TM with log n-bounded nondeterminism operating in linear time and space can be simulated by an ITA with O(log n) depth in linear time.
14 citations
TL;DR: In this article, four time linearization techniques and two operator-splitting algorithms have been employed to study the propagation of a one-dimensional wave governed by a reaction-diffusion equation.
Abstract: Four time linearization techniques and two operator-splitting algorithms have been employed to study the propagation of a one-dimensional wave governed by a reaction-diffusion equation. Comparisons amongst the methods are shown in terms of the L 2-norm error and computed wave speeds. The calculations have been performed with different numerical grids in order to determine the effects of the temporal and spatial step sizes on the accuracy. It is shown that a time linearization procedure with a second-order accurate temporal approximation and a fourth-order accurate spatial discretization yields the most accurate results. The numerical calculations are compared with those reported in Parts 1 and 2. It is concluded that the most accurate time linearization method described in this paper offers a great promise for the computation of multi-dimensional reaction-diffusion equations.
14 citations
TL;DR: Backward differentiation methods based on trigonometric polynomials for the initial value problems whose solutions are known to be periodic are constructed in this article, assuming that the frequency wcan be estimated in advance.
Abstract: Backward differentiation methods based on trigonometric polynomials for the initial value problems whose solutions are known to be periodic are constructed. It is assumed that the frequency wcan be estimated in advance. The resulting methods depend on a parameter v = hw, where his the step size, and reduce to classical backward methods if v→0. Neta and Ford [6] constructed Nystrom and generalized Milne-Simpson type methods. Those methods require the Jacobian matrix to have purely imaginary eigenvalues. The methods we construct here will not suffer of this deficiency.
14 citations
TL;DR: In this paper, a new numerical method for solving non-linear boundary value problems with the boundary values specified at multiple points is presented, where boundary conditions are specified only at the end.
Abstract: A new numerical method for solving non-linear boundary value problems with the boundary values specified at multiple points is presented. The present paper is an extension of an earlier work where boundary conditions were specified only at the end. The method proceeds with first linearizing the problem by an initial guess for the nonlinear terms. Next the method of weighted residuals is applied to compute all boundary quantities for the approximate solution corresponding to the linearized version. This converts the boundary value problem to an initial value problem which is solved by a Runge-Kutta scheme. The resulting solution is used as an improved guess for the next iteration. The process is repeated until convergence to a prescribed tolerance is achieved. Illustrative applications from bending of sandwich beams and outflow of an incompressible fluid from a narrow two dimensional slit are included.
12 citations
TL;DR: In this article, it is shown that if only the transitive reduction of the given relation is required and not all the implications by transitivity, one can restrict oneself to the direct dominances in the corresponding point set N; here a dominating b directly means that a dominates b and there is no intermediate c in N. To estimate the advantage of this restriction, information about the number of dominant and directly dominant pairs in a set of n points is required.
Abstract: Several transitive relations of geometrical objects (like inclusion of intervals on a line or polygons in the plain), which are important in VLSI design applications, can be translated into the dominance relation a dominates b iff (a ≠ b and a j ≧ b j for j = 1,…d) of points a = (a 1,...,a d ),b = (b 1,…b d ) in R d by representing the objects as points in a suitable way. If only the transitive reduction (see [7]) of the given relation is required and not all the implications by transitivity, one can restrict oneself to the direct dominances in the corresponding point set N; here a dominates b directly means that a dominates b and there is no—with respect to dominance—intermediate c in N (see [5]). To estimate the advantage of this restriction, information about the numbers of dominant and directly dominant pairs in a set of n points is required, both numbers essentially depending upon how the points are distributed in R d . In this paper we assume the n points in question to be identically and independen...
9 citations
TL;DR: In this paper, two methods are given for the stable evaluation of weighted Cauchy principal value integrals, one is applicable when we have the points and weights of an interpolatory product integration rule and the second method can be used with any suitable integration ruie but requires that the weight function be non-negative.
Abstract: Two methods are given for the stable evaluation of weighted Cauchy principal value integrals. The first method is applicable when we have the points and weights of an interpolatory product integration rule. The second method can be used with any suitable integration ruie but requires that the weight function be non-negative.
TL;DR: The propagation of a one-dimensional, confined, laminar flame is studied by means of two operator-splitting methods, four linear block implicit schemes and the standard implicit and Crank-Nicolson techniques, and it is shown that the implicit method underestimates the flame speed and pressure.
Abstract: The propagation of a one-dimensional, confined, laminar flame is studied by means of two operator-splitting methods, four linear block implicit schemes and the standard implicit and Crank-Nicolson techniques. It is shown that the implicit method underestimates the flame speed and pressure because of first-order temporal truncation errors. Second-order accurate, in both space and time, methods yield nearly the same flame location and pressure as the operator-splitting techniques and the Crank-Nicolson scheme. Block linear methods which are first-order accurate in time and second-order accurate in space show temperature oscillations in the burned gas region near the flame front. These methods are found to be unstable when a delta formulation is used and the flame approaches the combustor wall. The instabilities are attributed to the large magnitude of the source terms and the rate of change of these terms with respect to time. A reduction in the time step brings the results of first-order accurate block met...
TL;DR: Their explicit nature and their obvious parallelism makes them most suitable for the solution of the resulting linear systems arising from the discretisation of a class of elliptic PDEs on a Parallel Computer of SIMD (Single Instruction Stream-Multiple Data Stream) type.
Abstract: In this paper the Alternating Group Explicit (AGE) methods, modified analogues of the classical Alternating Direction Implicit (ADI) methods, are introduced. Their explicit nature and their obvious parallelism, in contrast with the inherent parallelism of the ADI methods, makes them most suitable for the solution of the resulting linear systems arising from the discretisation of a class of elliptic PDEs on a Parallel Computer of SIMD (Single Instruction Stream-Multiple Data Stream) type. The determination of optimum sets of acceleration parameters in 2-dimensional problems is discussed and numerical experiments are successfully completed.
TL;DR: For any two positive integers n 1 and n 2, the maximum function of stable marriages satisfies this partial answer to a problem of D. E. Knuth as discussed by the authors, and the algebraic operation (sum and product) of two marriages as well as two preferences by which we are able to study the maximum problem of stable marriage.
Abstract: In this paper, we introduce the algebraic operation (sum and product) of two marriages as well as two preferences by which we are able to study the maximum problem of stable marriages. For any two positive integers n 1 and n 2, the maximum function of stable marriages satisfies This presents a partial answer to a problem of D. E. Knuth.
TL;DR: Preconditioned conjugate gradient methods for solving symmetric linear systems resulting from high order discretization techniques for elliptic partial differential equations are investigated.
Abstract: Preconditioned conjugate gradient methods for solving symmetric linear systems resulting from high order discretization techniques for elliptic partial differential equations are investigated The preconditionings are based on an incomplete LU factorization to another matrix that arises from the application of a lower order approximation to the same elliptic equation The use of R similarity transformation to estimate the extreme eigenvalues and the condition numbers of the linear systems is described The efficiency and effectiveness of the preconditioned algorithms are demonstrated by the computational experiments
TL;DR: This paper considers two machine models equivalent in power to Turing machines and shows their equivalence to terminal weighted regular grammars, thus proving that time varying generalized finite automata have the same power as Turing machines.
Abstract: In this paper, we consider two machine models equivalent in power to Turing machines. Time varying finite automata are defined and it is shown that time varying nondeterministic finite automata are equivalent to time varying deterministic finite automata. But, we find that, when e-moves are introduced, the power is increased to that of Turing machines. Equivalence between time varying regular grammars [6] and time varying nondeterministic finite automata with e-moves is shown. We also consider time varying generalized finite automata and show their equivalence to terminal weighted regular grammars [5], thus proving that time varying generalized finite automata have the same power as Turing machines.
TL;DR: Explicit mappings from the first 22p integers into the ordered vertices of the pth Hilbert polygon and vice versa are derived in this article, and algorithms to achieve the transformations directly are given.
Abstract: Explicit mappings from the first 22p integers into the ordered vertices of the pth Hilbert polygon and vice versa are derived. Generalisations of this result are discussed. Algorithms to achieve the transformations directly are given.
TL;DR: A loopless algorithm is formulated that moves each disc in a constant time, independent of the number of discs, and is found to be the fastest algorithm for solving the Towers of Hanoi problem.
Abstract: A loopless algorithm for the Towers of Hanoi problem is formulated. This algorithm moves each disc in a constant time, independent of the number of discs. An empirical test reveals that this loopless algorithm is the fastest algorithm for solving the Towers of Hanoi problem.
TL;DR: The relationship between context-free languages and homomorphic images of Szilard languages is considered and it is shown that there is a proper hierarchy of language classes.
Abstract: In this paper we study homomorphic images of Szilard languages (of context-free grammars). First, the relationship between context-free languages and homomorphic images of Szilard languages is considered. Secondly, it is shown that there is a proper hierarchy of language classes. , is the class of languages obtained from Szilard languages by homomorphisms h such that the length of h(ρ) is at most i for each label ρ
TL;DR: In this paper, an alternative method is proposed whereby the Cauchy principal value integral is expressed first as the Hilbert transform of the density function, then as an expression which involves sine and cosine transforms of f(t), which can be evaluated with the fast Fourier transform (FFT).
Abstract: The numerical evaluation of Cauchy principal value integrals is largely based on quadrature rules of primitive or Gauss-type which provide accurate solutions for selected points of the abscissae. An alternative method is proposed whereby the Cauchy principal value integral is expressed first as the Hilbert transform of the density functionf(t), then as an expression which involves sine and cosine transforms of f(t) which can be evaluated with the fast Fourier transform (FFT). An example is considered which compares the FFT method with Ivanov's and Laguerre quadrature methods, and it is shown that although the quadrature methods obtain accurate solutions, they are relatively inefficient if solutions are sought at numerous locations along the abscissae.
TL;DR: In this article, implicit iterative methods are presented for the efficient numerical solution of non-linear elliptic boundary value problems in 2D and 3D space dimensions, where isomorphic iterative schemes in conjunction with preconditioning techniques are used for solving nonlinear equations in two and three-space dimensions.
Abstract: New implicit iterative methods are presented for the efficient numerical solution of non-linear elliptic boundary-value problems. Isomorphic iterative schemes in conjunction with preconditioning techniques are used for solving non-linear elliptic equations in two and three-space dimensions. The application of the derived methods on characteristic 2D and 3D non-linear boundary-value problems is discussed and numerical results are given.
TL;DR: Parallel algorithms for finding a fundamental set of cycles of a graph, for locating the bridges of a connected graph and for strongly orienting a bridgeless connected graph are proposed in this paper.
Abstract: Parallel algorithms for finding a fundamental set of cycles of a graph, for locating the bridges of a connected graph and for strongly orienting a bridgeless connected graph are proposed in this paper. Each of these algorithms runs in time and requires 0(n(m – n+ 1)) processors on a shared memory model of a single instruction-stream multiple data-stream computer, where m and n refer respectively to the number of arcs and the number nodes of the underlying graph. The running time of the algorithms is reduced to provided 0(n 3) processors are used, where d refers to the diameter of the graph.
TL;DR: A parallel breadth first search algorithm for general graphs and a parallel depth first search algorithms for acyclic digraphs which run in time O(logd-logn) using 0(n 2[n/log n]) processors are developed.
Abstract: Two graph search problems are considered in a parallel computational environment. In this paper, we use the shortest path algorithm as a useful parallel computational technique. Based on the parallel shortest path algorithm presented by Dekel [8], we develop a parallel breadth first search algorithm for general graphs and a parallel depth first search algorithm for acyclic digraphs which run in time O(logd-logn) using 0(n 2[n/log n]) processors. For a breadth first search, the resulting algorithm is the same with that of Ghosh and Bhattacharjee [9]. But the difference between them is that the algorithm presented in [9] is based on the parallel breadth first search algorithm for trees while our algorithm is based on the shortest path algorithm. Furthermore, the algorithms presented in this paper have a unified algorithm structure.
TL;DR: A general procedure for deriving second-order linearly implicit methods and a typical method derived by this procedure is found to compare favourably with Gear's method when only moderate accuracy is required.
Abstract: The second-order linearly implicit method of Steihaug and Wolfbrandt is found to give poor numerical accuracy if unsuitable error estimates are used. The criteria for satisfactory error estimates are discussed. A general procedure for deriving second-order linearly implicit methods is described. A typical method derived by this procedure is found to compare favourably with Gear's method when only moderate accuracy is required.
TL;DR: This paper argues that at least for sets of orthogonal objects divide-and-conquer is competitive, if a suitable representation of the objects is used, and sketches three (new) time-optimal dividing algorithms to solve the line segment intersection problem, the measure problem and the contour problem.
Abstract: In the last few years line-sweep has become the standard method to solve problems that involve computing some property of a set of planar objects. In this paper we argue that at least for sets of orthogonal objects divide-and-conquer is competitive, if a suitable representation of the objects is used. We support this claim by sketching three (new) time-optimal divide-and-conquer algorithms to solve the line segment intersection problem, the measure problem and the contour problem, respectively. It turns out that divide-and-conquer requires simpler supporting data structures while line-sweep permits an easier reduction to a one-dimensional problem.
TL;DR: An algorithm recently discovered by J. M. Morris which requires neither stack nor tag fields is derived, which leads to a new representation for threaded binary trees requiring no tag fields.
Abstract: Starting from a stack-based binary tree traversal algorithm for preorder and/or inorder, we derive an algorithm recently discovered by J. M. Morris which requires neither stack nor tag fields. This algorithm may also be derived from the familiar threaded binary tree traversal algorithm. By demonstrating how searching may proceed in parallel with traversal, we show that the algorithm is “almost read-only”. This leads to a new representation for threaded binary trees requiring no tag fields. We show how to perform the usual operations efficiently for this representation, including strictly read-only traversal. In addition, we analyse the performance of variants of the traversal algorithm for binary trees represented with/without threads and with/without tag fields.
TL;DR: It is shown that the problem of determining the minimum number of node/edge deletions that break a network into three or more components is a computationally hard problem, by proving that its decision-version is NP-Complete.
Abstract: A simple variant of the node connectivity problem arises in the domain of network reliability. Here, break-up of a network into two components can be tolerated, but not into three or more components. We show that the problem of determining the minimum number of node/edge deletions that break a network into three or more components is a computationally hard problem, by proving that its decision-version is NP-Complete. In the process, we show that a closely related problem, which can be of interest in itself, is also NP-Complete.
TL;DR: An algorithm using second derivatives for solving unconstrained optimization problems is presented and the method performs quite well and the numerical results are presented in Section 4.
Abstract: An algorithm using second derivatives for solving unconstrained optimization problems is presented. In this brief note the descent direction of the algorithm is based on a modification of the Newton direction, while the Armijo rule for choosing the stepsize is used. The rate of convergence of the algorithm is shown to be superlinear. Our computational experience shows that the method performs quite well and our numerical results are presented in Section 4.
TL;DR: This paper briefly describes the implementation of the odd-even merge algorithm on a parallel MIMD computer and discusses its computational complexity.
Abstract: This paper briefly describes the implementation of the odd-even merge algorithm on a parallel MIMD computer and discusses its computational complexity.
TL;DR: In this paper, a new method for the solution of real nonlinear algebraic or transcendental equations with one simple root along a finite interval is proposed, based on a modification of Picard's method.
Abstract: A new method for the solution of a single real nonlinear algebraic or transcendental equation with one simple root along a finite interval is proposed. This method is based on a modification of Picard's method for the determination of the number of roots of a nonlinear equation along a finite interval and a relevant formula for the determination of such a root. Alternatively, it can be considered to be based on the method of integration by parts. The present method leads to a very simple non-iterative approximate formula, based on the classical Gauss quadrature rule for the computation of the sought root. The convergence of the method, for increasing values of the number of nodes n, is proved and numerical results for two transcendental equations are presented.