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Showing papers in "International Journal of Computer Mathematics in 1983"


Journal ArticleDOI
TL;DR: Rectangle intersections involving rectilinearly-oriented (hyper-) rectangles in d-dimensional real space are examined and a data structure is developed which is efficient in time and space and allows us to report all d- dimensional rectangles stored which intersect a d- dimension query rectangle.
Abstract: Rectangle intersections involving rectilinearly-oriented (hyper-) rectangles in d-dimensional real space are examined from two points of view. First, a data structure is developed which is efficient in time and space and allows us to report all d-dimensional rectangles stored which intersect a d-dimensional query rectangle. Second, in Part II, a slightly modified version of this new data structure is applied to report all intersecting pairs of rectangles of a given set. This approach yields a solution which is optimal in time and space for planar rectangles and reasonable in higher dimensions.

187 citations


Journal ArticleDOI
TL;DR: In this article, new explicit methods for the finite difference solution of a parabolic PDE are derived using stable asymmetric approximations to the partial differential equation which when coupled in groups of 2 adjacent points on the grid result in implicit equations which can be easily converted to explicit form which in turn offer many advantages.
Abstract: In this paper, new explicit methods for the finite difference solution of a parabolic partial differential equation are derived. The new methods use stable asymmetric approximations to the partial differential equation which when coupled in groups of 2 adjacent points on the grid result in implicit equations which can be easily converted to explicit form which in turn offer many advantages. By judicious use of alternating this strategy on the grid points of the domain results in an algorithm which possesses unconditional stability. The merit of this approach results in more accurate solutions because of truncation error cancellations. The stability, consistency, convergence and truncation error of the new method is discussed and the results of numerical experiments presented.

150 citations


Journal ArticleDOI
TL;DR: In this paper, the main spline relations are presented and incorporated into solution procedures for partial differential equations, and the computational algorithm in every case is a tridiagonal matrix system amenable to efficient inversion methods.
Abstract: This paper presents techniques for the numerical solution of partial differential equations using cubic spline collocation. The main spline relations are presented and incorporated into solution procedures for partial differential equations. The computational algorithm in every case is a tridiagonal matrix system amenable to efficient inversion methods. Truncation errors and stability are briefly discussed. Finally, some examples of their application to parabolic and hyperbolic systems with mixed boundary conditions are presented. The results obtained are encouraging and justify further research in this field.

121 citations


Journal ArticleDOI
TL;DR: In this article, a method of order three for finding multiple zeros of nonlinear functions is developed, which requires two evaluations of the function and one evaluation of the derivative per step.
Abstract: A method of order three for finding multiple zeros of nonlinear functions is developed. The method requires two evaluations of the function and one evaluation of the derivative per step.

78 citations


Journal ArticleDOI
TL;DR: In this paper, an explicit finite difference approximation procedure which is unconditionally stable for the solution of the general multidimensional, non-homogeneous diffusion equation is presented, which possesses the advantages of the implicit methods, i.e., no severe limitation on the size of the time increment.
Abstract: An explicit finite difference approximation procedure which is unconditionally stable for the solution of the general multidimensional, non-homogeneous diffusion equation is presented. This method possesses the advantages of the implicit methods, i.e., no severe limitation on the size of the time increment. Also it has the simplicity of the explicit methods and employs the same “marching” type technique of solution. Results obtained by this method for several different problems were compared with the exact solution and agreed closely with those obtained by other finite-difference methods. For the examples solved the numerical results obtained by the present method are in satisfactory agreement with the exact solution.

41 citations


Journal ArticleDOI
TL;DR: In this paper, a boundary integral procedure for the solution of an important class of crack problems in anisotropic elasticity is outlined, and a specific numerical example is considered in order to assess the effectiveness of the procedure.
Abstract: A boundary integral procedure for the solution of an important class of crack problems in anisotropic elasticity is outlined. A specific numerical example is considered in order to assess the effectiveness of the procedure.

35 citations


Journal ArticleDOI
TL;DR: In this article, a first order stationary iterative scheme for the solution of a linear system of equations is given and an algorithm for finding the optimum parameters is presented and a number of applications and examples is given.
Abstract: In this paper it is assumed that a first order stationary iterative scheme for the solution of a linear system of equations is given. It is also assumed that all the eigenvalues of the iteration matrix of the scheme are known and have real parts which are all either less or greater than one. Under the previous assumptions the solution of the optimization problem by means of an extrapolation scheme of the original one is studied, analyzed and found. Finally an algorithm for finding the optimum parameters is presented and a number of applications and examples is given.

31 citations


Journal ArticleDOI
TL;DR: In this article, the authors proposed a three-step method with convergence rate 10.81525 which is much better than the six-order method of the Neta family of methods.
Abstract: Neta's three step sixth order family of methods for solving nonlinear equations require 3 function and 1 derivative evaluation per iteration. Using exactly the same information another three step method can be obtained with convergence rate 10.81525 which is much better than the sixth order.

30 citations


Journal ArticleDOI
TL;DR: The notion of a Towers of Hanoi contest is introduced and the complexity of some adjudication methods are investigated, and linear time and space adjudication algorithms are presented, that is optimal adjudications algorithms.
Abstract: The notion of a Towers of Hanoi contest is introduced and the complexity of some adjudication methods are investigated. We present linear time and space adjudication algorithms, that is optimal adjudication algorithms.

24 citations


Journal ArticleDOI
TL;DR: An algorithm to recursively compute D(X, Y), which can serve to be a common measure between X and Y, defined in terms of two abstract operators ⊕ and ⊛, and a binary function d( [SDot], [sdot] ) whose arguments are symbols of an alphabet A.
Abstract: Many numerical indices which quantify the similarity and dissimilarity between a pair of stringsX and Y, have been defined in the literature. Some of these include the Length of their Longest Common Subsequence (LLCS(X, Y)), the Length of their Shortest Common Supersequence (LSCS(X, Y)), and their Generalized Levenshtein Distance (GLD(X, Y)). Some non-numerical indices relating the strings are the set of their common subsequences, the set of their common supersequences and the set of their shuffles. In this paper, we consider an abstract measure between X and Y, written as D(X, Y), defined in terms of two abstract operators ⊕ and ⊛, and a binary function d( [sdot], [sdot] ) whose arguments are symbols of an alphabet A Depending on the various concrete operators used for ⊕ and ⊛ and the specific function used for d( [sdot], [sdot] ), all the quantities discussed above can be seen to be particular cases of D(X, Y). We have presented an algorithm to recursively compute D(X, Y), which can serve to be a common...

20 citations


Journal ArticleDOI
TL;DR: A new general method such that the Jacobi, the Gauss-Seidel and the Successive Overrelaxation methods become special cases of it.
Abstract: For the solution of the linear system Ax=b many iterative methods based on a splitting of A exist. Among them the Jacobi, the Gauss-Seidel and the Successive Overrelaxation (SOR) methods as well as their extrapolated counterparts are the most popular. This paper presents a new general method such that the aforementioned methods become special cases of it. Besides its four degrees of freedom, which make it a very flexible method, another of its main characteristics is that it is well-defined even when some elements on the diagonal of A are zero. The first results concerning the new method show that a proper exploitation of its basic properties will make it a very powerful technique.

Journal ArticleDOI
TL;DR: A recursive backtracking algorithm to find a maximum internally stable set of a weighted undirected graph and it is shown that the proposed algorithm is very effective.
Abstract: The problem of determining a maximum internally stable set in a weighted undirected graph belongs to the class of computationally intractable problems known as NP-hard. Furthermore it has a variety of many interesting applications. In this paper we develop a recursive backtracking algorithm to find a maximum internally stable set of a weighted undirected graph. Computational experience on more than 1500 randomly generated graphs ranging from 100 to 250 vertices and from 15% to 80%, densities has shown that the proposed algorithm is very effective.

Journal ArticleDOI
TL;DR: In this article, an improved algorithm for solving boundary-layer type equations is presented, which is rapidly convergent, stable and can find solutions to which other methods do not converge.
Abstract: An improved algorithm is presented for solving boundary-layer type equations. The Falkner-Skan equation is used to illustrate the benefit of formulating an integro- differential equation to avoid imposing finite difference approximations for the boundary conditions. The method is rapidly convergent, stable and can find solutions to which other methods do not converge.

Journal ArticleDOI
TL;DR: In this paper, the first part of a paper consisting of two parts that investigates how various language-theoretical properties of influence the closure properties of selective substitution grammars are established.
Abstract: Let be the class of languages generated by selective substitution grammars, where: 1) arbitrary productions of the form are allowed, where b is a letter and w is a word and 2) the selectors used are from . This is the first part of a paper consisting of two parts that investigates how various language-theoretical properties of influence the closure properties of In this part basic techniques for manipulating selectors of selective substitution grammars are established. Then we investigate how properties of influence the closure of under union and concatenation.

Journal ArticleDOI
TL;DR: In this paper, finite difference methods of orders 2 and 4 are developed and analyzed for the solution of a two-point boundary value problem associated with a fourth-order linear differential equation.
Abstract: In this report, finite difference methods of orders 2 and 4 are developed and analysed for the solution of a two-point boundary value problem associated with a fourth- order linear differential equation. A sufficient condition guaranteeing a unique solution of the boundary value problem is also given. Numerical results for a typical boundary value problem are tabulated and compared with the shooting technique using the fourth-order Runge-Kutta method.

Journal ArticleDOI
TL;DR: In this paper, two new families of higher-order iteration functions for simultaneously improving approximations to all the zeros of a polynomial are presented, and the resulting iteration schemes exhibit improved R-order convergence.
Abstract: Two new families of higher-order iteration functions for simultaneously improving approximations to all the zeros of a polynomial are presented The resulting iteration schemes exhibit improved R-order convergence

Journal ArticleDOI
TL;DR: In this article, simple polynomial changes of variable when used in conjunction with the progressive quadrature method of Patterson are extremely efficient in alleviating a wide class of commonly occurring singularities.
Abstract: It is shown that simple polynomial changes of variable when used in conjunction with the progressive quadrature method of Patterson are extremely efficient in alleviating a wide class of commonly occurring singularities. Accurate values of the singular integrals involved are obtainable with a highly economic number of quadrature grid- points. The results of extensive numerical tests are presented.

Journal ArticleDOI
TL;DR: In this article, a measure D(X Y,…Z) has been defined involving the set of strings in terms of two abstract operators ⊕ and ⊛ and a function δ(·, ·) which has as many arguments as there are strings in the set.
Abstract: In the companion paper [3], we have presented a common basis for many of the similarity and dissimilarity measures involving a pair of strings. In this paper, we extend the results to capture various numerical and nonnumerical measures involving more than two strings. A measure D(X Y,…Z) has been defined involving the set of strings {X Y,…Z} in terms of two abstract operators ⊕ and ⊛ and a function δ(·, ·) which has as many arguments as there are strings in the set {X Y,…Z}. The quantity D(X Y,…Z) represents various numerical and nonnumerical quantities involving {X Y,…Z} such as Length of their Longest Common Subsequence, (LLCS) the Length of their Shortest Common Supersequence, (LSCS) the set of their common subsequences, the set of their common supersequences and the set of their shuffles. The computational properties of D(X Y,…Z) have also been discussed.

Journal ArticleDOI
TL;DR: It is shown that a language is semi-discrete and context-free iff it is a discrete union of languages of the form , iffit is a finite disjoint union of discrete context- free languages.
Abstract: A language L over a finite alphabet is said to be semi-discrete if there exists a positive integer k such that L contains at most k words of any given length. If k=1, the language is said to be discrete. It is shown that a language is semi-discrete and context-free iff it is a discrete union of languages of the form , iff it is a finite disjoint union of discrete context-free languages. Closure properties and decision problems are studied for the class of semi-discrete context-free languages

Journal ArticleDOI
TL;DR: AGFS as discussed by the authors is an anti-AFL language where the growth takes place only at the two ends of a sentential form, at each step of a derivation either a left rule to the left most symbol or a right rule to rightmost symbol of a given form.
Abstract: In this paper we propose a new family of languages called Filamentous Systems with Apical Growth (AGFS), where the growth takes place only at the two ends of a sentential form. At each step of a derivation either a left rule to the left-most symbol or a right rule to the rightmost symbol of a sentential form is applied. It is shown that this family is a proper subset of the family of regular sets and is an anti-AFL. It is compared with the developmental languages such as OL, TOL and parallel-OL languages. We generalize AGFS by considering the effect of E and F on AGFS. We observe that the closure properties are not affected by F whereas the use of E will make the family of AGFL an AFL and equal to the regular sets.

Journal ArticleDOI
TL;DR: In this article, the authors show the uniqueness and analytical expressions for the optimum values of the parameters involved in the second order stationary scheme proposed in [2] and prove that the aforementioned scheme has exactly the same asymptotic rate of convergence and the same structure with (as well as a little less operations per iteration than) the non-stationary scheme.
Abstract: In this paper we show the uniqueness and give analytical expressions for the optimum values of the parameters involved in the second order stationary scheme proposed in [2]. We also prove that the aforementioned scheme has exactly the same asymptotic rate of convergence and the same structure with (as well as a little less operations per iteration than) the non-stationary scheme proposed in [5] (or [6]).

Journal ArticleDOI
TL;DR: In this paper, the authors discuss finite grammar forms and show that successor and nonsuccessor families, as well as normal form (with respect to minimality) results are obtained.
Abstract: This paper discusses finite grammar forms. In particular, results concerning successor and nonsuccessor families, as well as normal form (with respect to minimality) results are obtained. Interconnections with recent work concerning graph families are also pointed out.

Journal ArticleDOI
TL;DR: A method VLSI for design of programmable finite automata that can be programmed to deterministically simulate a non-deterministic finite auto- maton with at most k states is described.
Abstract: We describe a method VLSI for design of programmable finite automata. For any fixed k it can be programmed to deterministically simulate a non-deterministic finite auto- maton with at most k states. For some constant c it runs in time on inputs of length n>b and in time on inputs of length n≧b where b is a constant influencing the size of the realisation, which can be chosen so that ck is much smaller than b. The design is modular, it consists of 0(bk 2) identical processes connected in a uniform manner.

Journal ArticleDOI
TL;DR: In this article, an extrapolation procedure for the evaluation of finite range singular integrals is suggested which is based on the application of the e-algorithm to accelerate sequences of quadrature approximations.
Abstract: An extrapolation procedure for the evaluation of finite range singular integrals is suggested which is based on the application of the e-algorithm to accelerate sequences of quadrature approximations. These sequences are produced by integrating over increasingly small sub-intervals using the powerful pseudo-Gaussian quadrature formulae of Patterson. Extensive numerical tests are carried out on a large number of test integrals and critical comparisons made with existing methods.

Journal ArticleDOI
TL;DR: This paper discusses the implementation of a subgradient projection algorithm due to Sreedharan for the minimization, subject to a finite number of smooth, convex constraints, of an objective function which is the sum of a smooth, strictly convex function and a piecewise smooth conveX function.
Abstract: This paper discusses the implementation of a subgradient projection algorithm due to Sreedharan [13] for the minimization, subject to a finite number of smooth, convex constraints, of an objective function which is the sum of a smooth, strictly convex function and a piecewise smooth convex function. Computational experience with the algorithm on several test problems and comparison of this experience with previously published results is presented.

Journal ArticleDOI
TL;DR: A relaxed Jacobi-type iterative scheme is presented for solving linear algebraic systems and it is preferrable to let the parameter equal to unity and use Conjugate Gradient methods to further increase the convergence rate.
Abstract: A relaxed Jacobi-type iterative scheme is presented for solving linear algebraic systems. The method possesses a high level of parallelism and can be implemented on a multiprocessor system with or without synchronisation. The convergence region of the relaxation parameter is determined under the condition that the Jacobi iteration matrix possesses real eigenvalues. However, when the method is applied to consistently ordered systems, it is preferrable to let the parameter equal to unity and use Conjugate Gradient methods [3] to further increase the convergence rate.

Journal ArticleDOI
TL;DR: In this paper, a numerical method for finding periodic solutions to nonlinear ordinary differential equations is described, where the solution is approximated by a trigonometric series and the series is substituted into the differential equation using the FORMAC computer algebra system for the resulting lengthy algebraic manipulations.
Abstract: This paper describes a numerical method for finding periodic solutions to nonlinear ordinary differential equations. The solution is approximated by a trigonometric series. The series is substituted into the differential equation using the FORMAC computer algebra system for the resulting lengthy algebraic manipulations. This lead to a set of nonlinear algebraic equations for the series coefficients. Modern search methods are used to solve for the coefficients. The method is illustrated by application to Duffing’ equation.

Journal ArticleDOI
TL;DR: In this article, a graphical root-finding procedure for nonlinear algebraic equations is proposed, which employs two graphs of the zero-curves of the real and imaginary parts of the polynomial to give information on the approximate location for every root together with its multiplicity on the basis of the argument principle.
Abstract: A graphical root-finding procedure is proposed for nonlinear algebraic equations. It employs two graphs of the zero-curves of the real and imaginary parts of the polynomial. It gives information on the approximate location for every root together with its multiplicity on the basis of the argument principle. Locally convergent iterative method for roots may enjoy this information for their implementation. Some numerical examples are shown with the graphs.

Journal ArticleDOI
TL;DR: In this article, an explicit expression for the inverse of an invertible, real tridiagonal matrix is obtained, and its principal structural properties are determined using an efficient and stable algorithm.
Abstract: An explicit expression for the inverse of an invertible, real tridiagonal matrix is obtained, and its principal structural properties are determined. An efficient and stable algorithm is developed by utilising these properties.

Journal ArticleDOI
TL;DR: This paper discusses the implementation of an algorithm due to Sreedharan for the minimization, subject to linear constraints, of an objective function composed of the sum of a piecewise-affine, convex function with a smooth, strictly conveX function.
Abstract: This paper discusses the implementation of an algorithm due to Sreedharan [8] for the minimization, subject to linear constraints, of an objective function composed of the sum of a piecewise-affine, convex function with a smooth, strictly convex function. Successful techniques for two subproblems arising in the algorithm, a projection problem and a line search problem, are described in detail. Computational experience with the algorithm on several test problems is presented.