Showing papers in "International Journal of Computer Mathematics in 1989"
TL;DR: A new kind of static analysis is defined: congruence analysis, which is conceived to discover the properties of the following type: “the integer valued variable X is congruent to c modulo m”, where c and m are automatically determined integers.
Abstract: In this paper, a new kind of static (or semantic) analysis is defined: congruence analysis, which is conceived to discover the properties of the following type: “the integer valued variable X is congruent to c modulo m”, where c and m are automatically determined integers. This analysis is then related to an algebraic framework and wholly characterized. Moreover, we show an example how it can be useful for automatic vectorization. Finally, we present some extensions of it, namely its combination with the analysis of bounds, and also some analyses defined when the modulus of congruences is given a priori.
166 citations
TL;DR: A direct and constructive proof is given that every regular language is the homomorphic image of a splicing language as defined by Head.
Abstract: A direct and constructive proof is given that every regular language is the homomorphic image of a splicing language as defined by Head. The class of splicing languages involved have a particularly simple presentation and are regular. Also presented are some examples which demonstrate the relation of the class of splicing languages to that of regular languages and an example of a regular language which is not a splicing language.
58 citations
TL;DR: The main result is that there exist sequential splicing systems with recursively unsolvable membership problem and the technique of the proof is to embed Turing machine computations in the languages.
Abstract: The notion of splicing system has been used to abstract the process of DNA digestion by restriction enzymes and subsequent religation. A splicing system language is the formal language of all DNA strings producible by such a process. The membership problem is to devise an algorithm (if possible) to answer the question of whether or not a given DNA string belongs to a splicing system language given by initial strings and enzymes. In this paper the concept of a sequential splicing system is introduced. A sequential splicing system differs from a splicing system in that the latter allows arbitrarily many copies of any string in the initial set whereas the sequential splicing system may restrict the initial number of copies of some strings. The main result is that there exist sequential splicing systems with recursively unsolvable membership problem. The technique of the proof is to embed Turing machine computations in the languages.
54 citations
TL;DR: In this article, the authors analyzed the minimum number of moves required to change an arbitrary initial configuration into an arbitrary final configuration, and showed that the number of required moves is approximately 0.52655 · 2 n and 0.25025 · 2n respectively for n disks.
Abstract: The Towers of Hanoi problem is analyzed with the aid of Hanoi graphs depicting legal configurations (i. e. arrangements of the disks on the pegs) and the transitions among them. Specifically, the mean and the standard deviation of the minimum numbers of moves required to change an arbitrary initial configuration into an arbitrary final configuration are computed, and are shown to be approximately 0.52655 · 2 n and 0.25025 · 2 n respectively for n disks.
32 citations
TL;DR: In this paper, the authors give a short proof of a formula, conjectured by Kirkman and first proved by Cayley, that counts the number of ways of placing m non-intersecting diagonals in a convex polygon of k sides.
Abstract: Schutzenberger's philosophy of getting algebraic generating functions out of context-free languages is practiced to obtain short proofs of results of Kreweras, Kreweras-Poupard and Kreweras-Moszkowski, concerning the refined enumeration of legal bracketings according to various parameters. We also mention some ramifications to the enumeration of ordered trees. As an “encore”, we give a short proof of a formula, conjectured by Kirkman and first proved by Cayley, that counts the number of ways of placing m non-intersecting diagonals in a convex polygon of k sides. 1980 Mathematics Subject Classification: 68R05, 68Q45, 05A15.
28 citations
TL;DR: This work discusses the seemingly formidable task of finding the promised polynomial-time decision algorithms, and briefly reviews and illustrates the use of powerful new nonconstructive tools that can be applied to guarantee membership in P.
Abstract: Although polynomial-time complexity theory has been formulated in terms of decision problems, polynomial-time decision algorithms generally operate by attempting to construct a solution to an optimization version of the problem at hand. Thus it is that self-reducibility, the process by which a decision algorithm may be used to devise a constructive algorithm, has until now been widely considered a topic of only theoretical interest. Recent fundamental advances in graph theory, however, have made available powerful new nonconstructive tools that can be applied to guarantee membership in P. These tools are nonconstructive at two distinct levels: they neither produce the decision algorithm, establishing only the finiteness of an obstruction set, nor do they reveal whether such a decision algorithm can be of any aid in the construction of a solution. We briefly review and illustrate the use of these tools, and discuss the seemingly formidable task of finding the promised polynomial-time decision algorithms wh...
23 citations
TL;DR: It is shown how this consistency relation may be used in an algorithm for computing quartic spline approximations to the solution and its higher derivatives for a linear two-point boundary value problem.
Abstract: This brief paper derives via quartic spline function a new consistency recurrence relation connecting the quartic spline function values at equidistant knots and the corresponding values of the second derivatives. It is shown how this consistency relation may be used in an algorithm for computing quartic spline approximations to the solution and its higher derivatives for a linear two-point boundary value problem. Some numerical evidence is also included to demonstrate the practical usefulness of the algorithm.
23 citations
TL;DR: In this paper, the authors propose a new algorithm to unify the optimising transformations of hoisting, loop optimisation, and strength reduction, which uses a refined framework for code placement that provides guarantees of improvement in execution efficiency.
Abstract: Unifications of optimising transformations are motivated by the desire to reduce the cost of optimisation. Two unifications using code placement techniques based on the suppression of partial redundancies have been reported in the literature [5, 8]. A common drawback of these unifications has been their failure to provide sufficient guarantees regarding improvement in execution efficiency of programs. This has tended to limit their effectiveness from a practical viewpoint. This paper proposes a new algorithm to unify the optimising transformations of hoisting, loop optimisation and strength reduction. The algorithm uses a refined framework for code placement that provides guarantees of improvement in execution efficiency of programs. The properties of this algorithm are formally proved.
22 citations
TL;DR: In this article, three point difference methods for the singular two-point boundary value problem were derived, and the O(h 4 ) and O (h 6 ) convergence of the methods was shown with the help of four examples.
Abstract: Three point difference methods, of orders four and six, for the singular two-point boundary-value problem: are derived . The O(h 4) and O(h 6)convergence of the methods is illustrated with the help of four examples
19 citations
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TL;DR: A new class of grammars with derivation restrictions (similar to the matrix, conditional ones, etc.) is presented, where if a rule of a subset peculiar to P is used in a derivation then it is compulsory to use only productions of that particular subset as long as it is possible.
Abstract: In this article a new class of grammars with derivation restrictions (similar to the matrix, conditional ones, etc.) is presented. In this situation, if a rule of a subset peculiar to P is used in a derivation then it is compulsory to use only productions of that particular subset as long as it is possible. According to the Chomsky hierarchy, naturally attributed to the grammars the generative power of the modular grammars is studied (Section 2), a series of results being obtained as well as a normal form similar to Chomsky's normal one. The following two paragraphs deal with the closure properties—only for ℳ 2 and (proving that ℳ 2 is AFL) and also with decision problems connected with the modular grammars, using algorithmical proofs in general.
19 citations
TL;DR: In this article, a class of one-step finite difference formulae for numerical solution of first-order differential equations is considered and the accuracy and stability properties of these methods are investigated.
Abstract: A class of one-step finite difference formulae for the numerical solution of first-order differential equations is considered The accuracy and stability properties of these methods are investigated By judicious choice of the coefficients in these formulae a method is derived which is both A-stable and third-order convergent Moreover the new method is shown to be L-stable and so is appropriate for the solution of certain stiff equations Numerical results are presented for several test problems
TL;DR: The combinatorial structure of n-prefix-suffix languages, called n-ps-codes in this paper, is investigated.
Abstract: The combinatorial structure of n-prefix-suffix languages, called n-ps-codes in this paper, is investigated. An n-ps-code is a language, each of whose subsets of cardinality at most n is a prefix code or a suffix code.
TL;DR: In this paper, nonlinear parabolic partial differential equations in one space dimension are solved numerically by using the Alternating Group Explicit (AGE) method (Evans, 1985) using alternating group explicit (AGE).
Abstract: In this paper, nonlinear parabolic partial differential equations in one space dimension are solved numerically by using the Alternating Group Explicit (AGE) method (Evans, 1985).
TL;DR: In this paper, a method for solving systems of two simultaneous nonlinear and/or transcendental equations in, which is based on reduction to simpler one-dimensional nonlinear equations is presented.
Abstract: A new method for solving systems of two simultaneous nonlinear and/or transcendental equations in , which is based on reduction to simpler one-dimensional non-linear equations is presented. This method to approximate a component of the solution does not require any information about the other component in each iteration. It generates a sequence of points in which converges quadratically to one component of the solution and afterwards it evaluates the other component using one simple computation. Moreover, it does not require a good initial guess of the solution for both components and it does not directly need function evaluations. A proof of convergence is given.
TL;DR: In this paper, a scaled runge-kutta-nystrom (SRKN) algorithm is proposed to determine an approximation of the solution and its derivative of the second order differential equation at intermediate points of a given integration step, as well as at the end of each step.
Abstract: New Runge-Kutta-Nystrom algorithms are presented which determine an approximation of the solution and its derivative of the second order differential equation y = f(x,y) at intermediate points of a given integration step, as well as at the end of each step. These new algorithms, called scaled Runge-Kutta-Nystrom (SRKN) methods, are designed to be used with existing Runge-Kutta-Nystrom (RKN) formulas, using the function evaluations of these methods as the core of the new system. Thus, for a slight increase of the cost, the solution may be generated within a successful step, improving so the efficiency of the existing RKN methods.
TL;DR: It is desirable to consider placement of 1- unit computations strictly in those nodes which contain the 1-unit definitions, even if such placement is unsafe, when node placement would be unsafe.
Abstract: NCHSA performs strictly safe hoisting. This necessitates edge placement of 1-unit computations when node placement would be unsafe. In Figure 3, a 1-unit computation would be inserted in the synthetic edge block 5-2 because the candidate expression is not anticipated along all paths through node 5. A tacit assumption here is that a 1-unit definition occurs along all paths reaching the exit of node 5. This may not aways be true, viz. if the 1-unit definition of node 3 is not present. Another difficulty with this placement approach is with the occurrence of many different 1-unit definitions in a program, viz. i≔ i+l in node 3 and i≔ i + 5 in node 4. It is not possible to have a single 1-unit computation in synthetic block 5-2 to cover the effect of these 1-unit definitions. Due to these reasons it is desirable to consider placement of 1-unit computations strictly in those nodes which contain the 1-unit definitions, even if such placement is unsafe. (This is not inconsistent with the placement philosophy of ...
TL;DR: A method for giving an interpretation for predicates other than the predicate q 0 indirectly is developed, which is based on the idea of using the oracle and a one to one mapping from a set of predicates to aSet of strings.
Abstract: This paper is concerned with an algorithm for identifying an unknown regular language from examples of its members and non-members. The algorithm is based on the model inference algorithm given by Shapiro. In our setting, however, a given first order language for describing a target logic program has countably many unary predicate symbols: q 0,q 1,q 2…. On the other hand, the oracle which gives information about the unknown regular language to the inference algorithm has no interpretation for predicates other than the predicate q 0. In such a setting,we cannot directly take advantage of the contradiction backtracing algorithm which is one of the most important parts for the efficiency of the model inference algorithm. In order to overcome this disadvantage, we develop a method for giving an interpretation for predicates other than the predicate q 0 indirectly, which is based on the idea of using the oracle and a one to one mapping from a set of predicates to a set of strings. Furthermore, we propose a mod...
TL;DR: In this paper, an ordered sequential algorithm for the global minimization of univariate functions over an interval is proposed, in which step-sizes are such that the function can be approximated locally by a polynomial with a precision of e.g.
Abstract: Ordered sequential algorithms for the global minimization of univariate functions over an interval proceed by evaluating this function at successive points corresponding to increasing values of the variable. Extending a recent idea of Wingo, we propose a new ordered sequential algorithm in which step-sizes are such that the function can be approximated locally by a polynomial with a precision of e. This algorithm explores efficiently the vicinity of the optimum, while other sequential algorithms do not. Computational experiments with several variants of the proposed method and with the algorithm of Vasil'ev and Ganshin are reported on.
TL;DR: In this article, a method using quintic splines which provides 0(h 6) uniformly convergent approximations for the solution of fourth order two-point boundary value problems was presented.
Abstract: It has been known for a long time that quintic spline collocation for fourth order two-point boundary value problems: y (4)+p(x)y=q(x), a
TL;DR: This is the second part of a paper investigating k-context-free languages and it is seen that for k≧3 the family of k-languages is incomparable with EOL languages.
Abstract: This is the second part of a paper investigating k-context-free languages. Here, using results obtained in Part 1 (Intern. J. Computer Math. 26, pp. 69-90) we show that for every k the family of k-languages is properly included in the family of (k + 1)-languages. Also, it it is seen that for k≧3 the family of k-languages is incomparable with EOL languages.
TL;DR: The authors' example give a justification to the oscillatory movement of the discrete solution of a well known problem by conditioning the discrete Galerkin system for a bounded elliptic linear problem by the Choleski factors of the Gramian.
Abstract: Preconditioning the discrete Galerkin system for a bounded elliptic linear problem, by the Choleski factors of the Gramian, we obtain a mesh independent condition number. Moreover, the rate of convergence of the multigrid method is also mesh independent. Our example give a justification to the oscillatory movement of the discrete solution of a well known problem.
TL;DR: In this article, generalized approximate inverse matrix (GAIM) techniques based on the concept of LU-type sparse factorization procedures are introduced for calculating explicitly approximate inverses of large sparse unsymmetric matrices of regular structure without inverting the factors L and U.
Abstract: Generalized Approximate Inverse Matrix (GAIM) techniques based on the concept of LU-type sparse factorization procedures are introduced for calculating explicitly approximate inverses of large sparse unsymmetric matrices of regular structure without inverting the factors L and U. Explicit first and second-order iterative methods in conjunction with modified forms of the GAIM techniques are presented for solving numerically three-dimensional initial/boundary-value problems on multiprocessor systems. Applications of the new methods on a 3D boundary-value problem is discussed and numerical results are given.
TL;DR: Bounds on sup norm for the derivative of a certain function f involved in the equation with different boundary values, are computed and provide the rate of convergence of the iterative sequence of approximate solutions obtained by Picard method to the exact solution.
Abstract: Implicit solution to a certain class of non-linear fourth order differential equation is presented. Bounds on sup norm for the derivative of a certain function f involved in the equation with different boundary values, are computed. These bounds provide the rate of convergence of the iterative sequence of approximate solutions obtained by Picard method, to the exact solution.
TL;DR: The row and column sweep parallel algorithms for the solution of the triangular linear system are discussed and the algorithm was implemented on the Loughborough NEPTUNE MIMD system for 1, 2, 3 and 4 processors and the performance results analysed.
Abstract: In this paper, the parallel solution of linear Volterra integral equations of the second kind are presented After a suitable quadrature technique, the row and column sweep parallel algorithms for the solution of the triangular linear system are discussed. The algorithm was implemented on the Loughborough NEPTUNE MIMD system for 1, 2, 3 and 4 processors and the performance results analysed
TL;DR: In this article, the average length of an Euclidean minimum spanning tree is estimated by using the ordered statistics, where the vertices of an EMST are uniformly distributed on a unit disk in R 2.
Abstract: In this paper we estimate the average length of an Euclidean minimum spanning tree. If the vertices of an EMST are uniformly distributed on a unit disk in R 2 we can derive the behavior of the edge lengths of the EMST constructed from these vertices by using the ordered statistics, and obtain an upper bound for the expected length of an EMST to be where n is the number of vertices of the EMST. This estimation is much closer to the actual values than the previous results. Experiment values are also given.
TL;DR: This work investigates context-free grammars the rules of which can be used in a productive and in a reductive fashion, while the application of these rules is controlled by a regular language.
Abstract: We investigate context-free grammars the rules of which can be used in a productive and in a reductive fashion, while the application of these rules is controlled by a regular language. We distinguish several modes of derivation for this kind of grammar. The resulting language families (properly) extend the family of context-free languages. We establish some closure properties of these language families and some grammatical transformations which yield a few normal forms for this type of grammar. Finally, we consider some special cases (viz. the context-free grammar is linear or left-linear), and generalizations, in particular, the use of arbitrary rather than regular control languages.
TL;DR: One object is to investigate whether transformations which yield high order methods with the trapezium rule have advantages over conventional methods.
Abstract: Some less common transformation methods for general and singular quadratures are considered. Efficient methods for computing the transformations themselves are given as these are no longer standard functions and extensive tests are carried out. One object is to investigate whether transformations which yield high order methods with the trapezium rule have advantages over conventional methods
TL;DR: This article propose a series of generalizations of Langford strings considered in Combinatorics and discuss the place of corresponding languages in Chomsky hierarchy, as well as their contextual ambiguity in the sense of algebraic linguistics.
Abstract: We propose a series of generalizations of Langford strings considered in Combinatorics and discuss the place of the corresponding languages in Chomsky hierarchy, as well as their contextual ambiguity in the sense of algebraic linguistics.
TL;DR: Here an interpolant with 0(h 6) local truncation error for the fifth order solution used in RKF4(5) method is presented, with a cost of “about” one extra function evaluation per integration step.
Abstract: In this paper a general interpolant for the Explicit Runge-Kutta methods is proposed. These interpolants are based on second derivatives on mesh-points of the integration interval, and first derivatives on interior points of each step. These first derivatives can be produced using lower order interpolants. Here an interpolant with 0(h 6) local truncation error for the fifth order solution used in RKF4(5) method is presented, with a cost of “about” one extra function evaluation per integration step.
TL;DR: In this paper, a numerical method was presented for the generalized exponential integral function E v (x), (x>0, v∊R), of interest for applications in astrophysics and nuclear physics in a general treatment context of this function.
Abstract: In a previous paper a numerical method was presented for the generalized exponential integral function E v (x), (x>0, v∊R), of interest for applications in astrophysics and nuclear physics in a general treatment context of this function. In the present work, we are mainly concerned with numerical features, above all as regards the relevant computational procedure and execution. Comparisons performed by means of a different algorithm ensure efficiency of the evaluation technique and the corresponding implementation, ERA, whose accuracy is confirmed by the results of the related error analysis.