Showing papers in "International Journal of Computer Mathematics in 1987"
TL;DR: In this article, a family of multipoint iterative functions for finding multiple roots of equations was derived and a comparison of computational results with other well-known methods was made with other methods.
Abstract: In this paper, we have derived a family of multipoint iterative functions for finding multiple roots of equations. In addition, comparison of computational results are made with other well-known methods.
88 citations
TL;DR: In this article, a family of second order methods is developed for the numerical solution of variable coefficient fourth order parabolic partial differential equations in one space variable, which arise from a three-point recurrence relation for numerical solutions of systems of second-order ordinary differential equations.
Abstract: A family of second order methods is developed for the numerical solution of variable coefficient fourth order parabolic partial differential equations in one space variable. The methods arise from a three point recurrence relation for the numerical solution of systems of second order ordinary differential equations. The methods are analysed and tested on two problems which have appeared in the literature.
37 citations
TL;DR: This coding is used in order to generate all the trees with n nodes and m leaves as a list, by embedding them in a set of regular m-ary trees.
Abstract: A coding of binary trees, introduced in a previous paper [4], is generalized to regular k-ary trees. This coding is used in order to generate all the trees with n nodes and m leaves as a list, by embedding them in a set of regular m-ary trees.
31 citations
TL;DR: In this paper, the authors derived the following simple formula for t(Pn ) the number of spanning trees in the prism Pn, defined as the graph obtained by adding to the disjoint cycles all edges of the form ViWi The prism is sometimes denoted by K 2×Cn.
Abstract: Let the vertices of two disjoint, equal length cycles be labelled in one cycle and in the other. The prism Pn is defined as the graph obtained by adding to the disjoint cycles all edges of the form ViWi The prism is sometimes denoted by K 2×Cn . In this work we derive the following simple formula for t(Pn ) the number of spanning trees in .
28 citations
21 citations
TL;DR: In this paper, the point A.G.E. method is extended to obtain the solution of block tridiagonal linear systems derived from the discretisation of multidimensional elliptic boundary value problems.
Abstract: In this paper, the point A.G.E. method is extended to obtain the solution of block tridiagonal linear systems derived from the discretisation of multidimensional elliptic boundary value problems. The numerical results obtained agree with the theoretical results presented earlier.
20 citations
TL;DR: In this paper, a new class of non-linear methods based on Euler's integration formula for the numerical solution of ordinary differential equations is presented, considering the accuracy and stability of the proposed method and its applicability to stiff problems with supporting numerical evidence.
Abstract: A new class of non-linear methods based on Euler's integration formula for the numerical solution of ordinary differential equations is presented. Considerations are given to the accuracy and stability of the proposed method and its applicability to stiff problems with supporting numerical evidence.
17 citations
TL;DR: The convergence rate of the Schwarz Alternating Procedure (SAP) for solving Mathematical Physics problems was studied in this article, where the optimal overrelaxation factor, the maximum convergence factor and the convergence range of SAP for various values of ρ were given.
Abstract: This is the fourth of a series of papers about the convergence rate of the Schwarz Alternating Procedure (SAP) for solving Mathematical Physics problems. In this paper, some typical values of the optimal overrelaxation factor, the maximum convergence factor and the convergence range of SAP for various values of ρ (see Evans et al. [1]) are given.
17 citations
TL;DR: A new polygon class taking linear-time and space for triangulation, called an if-polygon, is defined and it is shown that some triangulating-linear classes previously known have the same property, called the if-property, as the newly defined class.
Abstract: A new polygon class taking linear-time and space for triangulation, called an if-polygon, is defined. After describing an algorithm for triangulating this class, we show that some triangulation-linear classes previously known, such as a convex polygon, a spiral polygon, an edge-visible polygon and a chain-visible polygon have the same property, called the if-property, as the newly defined class. Consequently, a monotone-separable polygon and a star-shaped polygon can be considered as a union of two if-polygons, respectively. Also, we present a modified algorithm for triangulating a star-shaped polygon without decomposition. As a result, the algorithm is simpler to implement and easier to understand and its correctness can be easily verified.
11 citations
TL;DR: In this paper, the convergence rate of the Neumann problem is analyzed for the Schwarz Alternating Procedure (SAP) for solving Mathematical Physics problems and it is shown that it converges at a rate of 1.
Abstract: This is the third of a series of papers about the convergence rate of SAP (the Schwarz Alternating Procedure) for solving Mathematical Physics problems. In this paper, the convergence rate for the Neumann problem is obtained.
10 citations
TL;DR: In this article, the optimal order of embedded pairs of Diagonally Implicit Runge-Kutta (DIRK) methods is examined, and an analysis of A and L-stability properties of q-stage order q DIRK methods with unequal diagonal elements is presented.
Abstract: In this paper, the optimal order of embedded pairs of Diagonally Implicit Runge-Kutta (DIRK) methods is examined. It is shown that a q-stage DIRK method of order p embedded in a q + 1 stage DIRK method of order p + 1 cannot have p = q + 1. Thus adopting embedding techniques to estimate the local truncation error results in giving up an order of accuracy for q<6. Embedded pairs of orders two and three for the basic method are derived with the additional stage being either explicit or implicit. Numerical results indicate that significant savings are realized when the extra stage is explicit. An analysis of A and L-stability properties of q-stage order q DIRK methods with unequal diagonal elements is presented. Necessary and sufficient conditions for A and L-stability are derived. To assess the potential of such methods, a number of embedded DIRK formulas are implemented. Numerical results for selected test problems are presented.
TL;DR: In this paper, it is shown that it is decidable whether a permutation-free morphism is an L code and the degree of L-ambiguity with respect to a set of words can be computed effectively.
Abstract: We show that it is decidable whether or not a permutation-free morphism is an L code. We also show that the degree of L-ambiguity with respect to a set of words can be computed effectively.
TL;DR: In this article, some mathematical properties of the shortest-length addition chain are found for certain integers whose binary patterns meet some special forms; and the correctness of these properties is proved.
Abstract: In this paper, some mathematical properties of the shortest-length addition chain are found for certain integers whose binary patterns meet some special forms; and the correctness of these properties is proved.
TL;DR: A synchronised parallel algorithm for the strong connectivity augmentation problem is presented and its depth is 0(log n) using 0(n 3) processors on a concurrent read, concurrent write parallel random access machine.
Abstract: A synchronised parallel algorithm for the strong connectivity augmentation problem is presented. Its depth is 0(log n) using 0(n 3) processors on a concurrent read, concurrent write parallel random access machine.
TL;DR: It is shown that the Rank-1and RANK-2 annihilation schemes compete favourably with existing systolic schemes for arbitrary matrix inversion, by trading basic inner product cells for simple delay registers which consume less area.
Abstract: The sysiolic principle is applied to ihe inversion of matrices by the methods of rank annihilation The systolic arrays presented are particularly effective for computing ihe inverse of a matrix which differs only partially from a matrix with a known inverse It is shown that the RANK-1and RANK-2 annihilation schemes compete favourably with existing systolic schemes for arbitrary matrix inversion, by trading basic inner product cells for simple delay registers which consume less area For Rank-1 annihilation we show that the computation time is , and for Rank-2 where n is ihe order of the matrix and r the number of applications of the annihilation formula The technique generalises to arbitrary matrix inversion resulting in 0(n 2) computational schemes but unfortunately they do not compete with the systolic Gaussian Eliminalion methods which are of 0(n) However they provide a more general architecture applicable to non-linear problems where partial changes in matrices can easily be constructed with a rel
TL;DR: An explicit annihilation procedure, based on an iterative scheme of Rudisill and Chu, is proposed for computing derivatives of eigenvalues and eigenvectors of parameter-dependent matrices and it is shown that it produces the exact solution with exact arithmetic.
Abstract: An explicit annihilation procedure, based on an iterative scheme of Rudisill and Chu, is proposed for computing derivatives of eigenvalues and eigenvectors of parameter-dependent matrices. It is shown that, with exact arithmetic, the method produces the exact solution. Numerical results establish the viability of the method in the presence of roundoff, even for subdominant eigenvalues. The method is compared with some alternative methods.
TL;DR: In this article, an algorithm for enumerating P(n,r) in the lexicographic order is presented, where the mapping from P n, r to the set of their inversions and the inverse mapping are established by a pair of coding and decoding algorithms.
Abstract: Let P(n,r) = {P n , r |P n , r = p 1 p 2…p r and p 1,p 2,…,p r ∊Z and p 1≠p j if i≠j}, where Z = {1,2,…,n}. An θ(r) algorithm for enumerating P(n,r) in the lexicographic order is presented. The mapping from P(n,r) to the set of their inversions and the inverse mapping are established by a pair of coding and decoding algorithms. Furthermore, ranking and unranking algorithms are also provided for establishing the mappings between P(n,r) and Z = {1,2,…,|P(n,r)|}. Indeed, the enumeration, ranking and unranking algorithms are more general than known algorithms.
TL;DR: The scope of languages defined by DTTs is investigated, and their dynamic nature enables production rules to be created and destroyed according to the context seen.
Abstract: A new device called a Dynamic Template Translator (DTT) is defined. DTTs are extensions of context free translation schemes in two directions: (1) their dynamic nature enables production rules to be created and destroyed according to the context seen, and (2) their template structure enables information contained within symbols to be manipulated. The scope of languages defined by DTTs is also investigated in this paper.
TL;DR: In this paper some systolic designs are presented for the implementation of the Graeffe root-squaring method for polynomial root solving with simulated soft-systolically in an OCCAM program.
Abstract: In this paper some systolic designs are presented for the implementation of the Graeffe root-squaring method for polynomial root solving. From a semi-systolic array, “retiming” transformations are applied to yield a purely systolic array that performs the squaring of the coefficients of an equation. The systolic array is then simulated soft-systolically in an OCCAM program listed in the Appendix. The overall design of a systolic system for the solution of equations based on the Graeffe method is also discussed.
TL;DR: In this article, the authors compare the finite element solution of some illustrative time-dependent problems involving parabolic and hyperbolic partial differential equations with alternative solution methods, and show that finite element solutions are not the best solution methods for these problems.
Abstract: In this paper the authors compare the finite element solution of some illustrative time-dependent problems involving parabolic and hyperbolic partial differential equations with alternative solution methods.
TL;DR: It is shown that the conversion from a tree permutation to the conventional representation of a binary tree using records (nodes) and pointers can be accomplished in 0(n) units of time.
Abstract: An efficient algorithm for enumerating all tree permutations of n integers in the natural order is presented. The enumeration problem is solved by considering that a tree permutation hLR is simply a linearized representation of the corresponding binary tree, such that h is the node value and L and R are its left and right subtrees, respectively. The best-case, average-case and worst-case time-complexities of the enumeration algorithm are 0(1)0 (3) and 0(n) respectively, whereas its space-complexity is 0(n). Furthermore, it is shown that the conversion from a tree permutation to the conventional representation of a binary tree using records (nodes) and pointers can be accomplished in 0(n) units of time.
TL;DR: In this article, the inconsistency in the criteria for adjudicating a Towers of Hanoi contest proposed by Wood is pointed out and a set of better criteria for better criteria is proposed.
Abstract: This paper points out the inconsistency in the criteria for adjudicating a Towers of Hanoi contest proposed by Wood. Counter examples to some of Wood's claims are given. In particular, the shortest path for transforming a configuration of n discs to another configuration may not be unique. Finally, a set of better criteria for adjudicating a Towers of Hanoi contest is proposed.
TL;DR: In this paper, the convergence properties of the EGS1 and EGS2 methods were investigated and the choice of their optimum factors on the basis of Generally Consistently Ordered (q r) matrices as defined by Verner and Bernal was shown.
Abstract: In this paper we investigate the convergence properties of the Extrapolated Gauss-Seidel 1 and 2 (EGS1 and EGS2) methods and the choice of their optimum factors on the basis of Generally Consistently Ordered (q r) matrices as defined by Verner and Bernal. It is shown how our results extend and correct some results in the literature.
TL;DR: It is shown that the second degree EAGS method is superior to the Accelerated Overtaxation (AOR) under the assumption that the matrix coefficient of the system is consistently ordered and positive definite and zero is not an eigenvalue of the corresponding Jacobi matrix.
Abstract: First and second degree Extrapolated Accelerated Gauss-Seidel (EAGS) methods for solving a system of linear algebraic equations are studied and it is shown that the second degree EAGS method is superior to the Accelerated Overtaxation (AOR) under the assumption that the matrix coefficient of the system is consistently ordered and positive definite and zero is not an eigenvalue of the corresponding Jacobi matrix.
TL;DR: In this paper, it was shown that labeling the productions of two regular grammars G 1 and G 2 in such a way that the Szilard language of G 1 is included in G 2 is NP-complete.
Abstract: It is shown that the problem of labeling the productions of two regular grammars G 1 and G 2 in such a way that the Szilard language of G 1 is included in the Szilard language of G 2 is NP-complete.
TL;DR: For a small number of processors the parallel Huard method is faster than the parallel Jordan method and slower otherwise, and the separation is obtained for p = 0.44n.
Abstract: We study the parallel implementation of two diagonalization methods for solving dense linear systems: the well known Gauss-Jordan method and a new one introduced by Huard. The number of arithmetic operations performed by the Huard method is the same as for Gaussian elimination, namely 2n 3/3, less than for the Jordan method, namely n 3. We introduce parallel versions of these methods, compare their performances and study their complexity. We assume a shared memory computer with a number of processors p of the order of n, the size of the problem to be solved, We show that the best parallel version for Jordan's method is by rows whereas the best one for Huard's method is by columns. Our main result states that for a small number of processors the parallel Huard method is faster than the parallel Jordan method and slower otherwise. The separation is obtained for p = 0.44n.
TL;DR: In this article, the ideas and theory of multistep multiderivative methods for solving ordinary differential equations are extended to Volterra integro-differential equations and a concept of zero-stability is given which, together with consistency, permits the derivation of an order-of-convergence result.
Abstract: The ideas and theory of multistep multiderivative methods for solving ordinary differential equations are extended to Volterra integro-differential equations. A concept of zero-stability is given which, together with consistency, permits the derivation of an order-of-convergence result. It is noted that lower order quadrature formulae may be used in the evaluation of the integrals involved in the derivatives of the function F without any deterioration in the global order of convergence. Numerical results are presented.
TL;DR: In this paper an attributed translation grammar (abbreviated as ATG) for the PL/0 is defined and the translation achieved is the generation of the P-codes, used in the original interpreter by Wirth.
Abstract: In this paper an attributed translation grammar (abbreviated as ATG) for the PL/0 is defined. The translation achieved by the ATG is the generation of the P-codes, used in the original interpreter by Wirth [7]. Due (o backtracking which may be followed by trimming of subtrees after code generation, it is never necessary to have the complete derivation tree. This is true even with recursion.
TL;DR: The proposed Restricted Pushdown Array of Counters-Finite Matrix Automaton (RPAC-FMA) is proposed and it is shown that a lanauge is RPAC-FM Language if and only if it is a EOL-RM Language.
Abstract: In formal language theory, a traditional topic is to characterize classes of languages by machine models. Motivated by the idea of extending the machine characterization of L systems to two dimensions and at the same time to generate interesting picture classes we propose in this paper a new model called EOL-Regular Matrix Systems (EOL-RMS). EOL-RM languages are obtained by substituting regular sets vertically into EOL languages. First a horizontal line of intermediates is generated by an EOL system. Then regular sets are substituted vertically for each intermediate eel! resulting in a rectangular array. We propose in this paper Restricted Pushdown Array of Counters-Finite Matrix Automaton (RPAC-FMA) and show that a lanauge is RPAC-FM Language if and only if it is a EOL-RM Language.