Showing papers in "International Journal of Computer Mathematics in 1981"
TL;DR: In this paper, a new one-parameter family of methods for finding simple zeros of non-linear functions is developed, each member of the family requires four evaluations of the given function and only one evaluation of the derivative per step.
Abstract: A new one-parameter family of methods for finding simple zeros of non-linear functions is developed. Each member of the family requires four evaluations of the given function and only one evaluation of the derivative per step. The order of the method is 16.
79 citations
TL;DR: In this article, the fixed language of an endomorphism of a free monoid is shown to be free and a concide description of the basis of a fixed language is given.
Abstract: The fixed language of an endomorphism of a free monoid is shown to be method that provides a concide description of the basis of the fixed language. A supplementary disscussion shown that the adult language of an OL scheme is free and provides a simple listing procedure for the basis element.
32 citations
TL;DR: The notion of “pure grammar” is introduced as a semi-Thue system with no non-terminals and “definite Turing machines” that accept pure languages, pure parallel languages,pure relations and pure Post canonical systems are introduced.
Abstract: The notion of “pure grammar” is introduced as a semi-Thue system with no non-terminals. The basic properties of pure grammars and the “pure languages” that they generate are investigated. Other topics introduced are “definite Turing machines” that accept pure languages, pure parallel languages, pure relations and pure Post canonical systems.
22 citations
TL;DR: In this paper, it was shown that for each inside-out (or IO-) macro language $L, there is a λ-free grammar with the following property: for each $x$ in $L, there is derivation of length at most linear in the length of $x.
Abstract: Starting from Fischer's IO Standard Form Theorem we show that for each inside-out (or IO-) macro language $L$, there is a $\lambda$-free IO-macro grammar with the following property: for each $x$ in $L$, there is a derivation of $x$ of length at most linear in the length of $x$. Then we construct a nondeterministic log-space bounded auxiliary pushdown automaton which accepts $L$ in polynomial time. Therefore the IO-macro languages are (many-one) log-space reducible to the context-free languages. Consequently, the membership problem for IO-macro languages can be solved deterministically in polynomial time and in space $(\log n)^2$.
20 citations
TL;DR: There is no monotonicity principle for alphabetic code trees, contrary to what is claimed in [5], and the basic running time in all cases is O(N 3) for binary search trees, but this principle does not extend to optimal multiway search trees in general.
Abstract: Given Nweighted keysN+1 missing-key weights and a branching factor t the application of dynamic programming yields algorithms for constructing optimal binary search trees (t = 2), optimal multi-way search trees (t>2), and optimal leaf search trees (or alphabetic code trees) with leaf weights only. The basic running time in all cases is O(N 3)(in terms of the number of keys), but it can be reduced to O(N 2) for binary search trees by a “monotonicity” principle which restricts the number of candidates for the root at each step in the construction. This principle can also be applied for multiway search trees when the missing-key weights are zero. However it does not extend to optimal multiway search trees in general, as we demonstrate; in particular, there is no monotonicity principle for alphabetic code trees, contrary to what is claimed in [5].
20 citations
TL;DR: In this article, the applicability of the quadrant interlocking factorisation method to the solution of the banded systems of linear equations which occur in the finite difference and finite element discretisation of engineering problems is discussed.
Abstract: In this paper, the authors extend the applicability of the new quadrant interlocking factorisation method [7] to the solution of the banded systems of linear equations which occur in the finite difference and finite element discretisation of engineering problems.
19 citations
TL;DR: In this paper, the error estimate can be obtained more simply, and two third-order methods along similar lines are derived. But both methods are single-step and linearly implicit, and they are designed to avoid frequent evaluations of the Jacobian and inversion of the associated matrix.
Abstract: A recent paper by Steihaug and Wolfbrandt [6] gives a second-order method for stiff differential equations, which includes an error estimate. We show how, in most practical circumstances, the error estimate can be obtained more simply, and derive two third-order methods along similar lines. All the methods described are single-step and linearly implicit, and they are designed to avoid frequent evaluations of the Jacobian and inversion of the associated matrix.
18 citations
12 citations
TL;DR: In this paper, a method of numerical solution of singular integral equations of the first kind with logarithmic singularities in their kernels along the integration interval is proposed, which is based on the reduction of these equations to equivalent SIEs with Cauchy-type singularities.
Abstract: A method of numerical solution of singular integral equations of the first kind with logarithmic singularities in their kernels along the integration interval is proposed. This method is based on the reduction of these equations to equivalent singular integral equations with Cauchy-type singularities in their kernels and the application to the latter of the methods of numerical solution, based on the use of an appropriate numerical integration rule for the reduction to a system of linear algebraic equations. The aforementioned method is presented in two forms giving slightly different numerical results. Furthermore, numerical applications of the proposed methods are made. Some further possibilities are finally investigated
9 citations
TL;DR: It turns out that loop programs can be characterized as the class of all recursive functions that are computable with a polynomial number of steps by modified programs.
Abstract: By slightly modifying the original definition of loop-programs by Meyer and Ritchie, a modified hierarchy of loop(n) programs , is obtained, with the following characteristics. Let be the class of functions defined by programs in , be the Grzegorzcyk hierarchy. Then , where in particular has a natural counterpart in loop programs. It also turns out that can be characterized as the class of all recursive functions that are computable with a polynomial number of steps by modified programs.
8 citations
TL;DR: It is shown that the B-fuzzy pushdown automata can accept context sensitive languages by setting a threshold, while the ( fuzzy) pushdown automation can accept only context-free languages.
Abstract: FUZZY phushdown automata and B-fuzzy pushdown automata are defined as an extension of pushdown automata. It is shown that the B-fuzzy pushdown automata can accept context sensitive languages by setting a threshold, while the (fuzzy) pushdown automata can accept only context-free languages.
TL;DR: High order methods for the numerical solution of nonlinear scalar equations are proposed which are more efficient than known procedures, and a unified approach to various methods suggested in literature is given.
Abstract: High order methods for the numerical solution of nonlinear scalar equations are proposed which are more efficient than known procedures, and a unified approach to various methods suggested in literature is given.
TL;DR: Efficient, exact (infinite precision) algorithms, along with their computing time analysis, are presented for the implementation of Cauchy's little known rule for computing a lower (or upper) bound on the values of the positive roots of a polynomial equation.
Abstract: Cauchy's little known rule for computing a lower (or upper) bound on the values of the positive roots of a polynomial equation has proven to be of great importance; namely it constitutes an indispensable and crucial part of the fastest method existing for the isolation of the real roots of an equation, a method which was recently developed by the author of this article. In this paper efficient, exact (infinite precision) algorithms, along with their computing time analysis, are presented for the implementation of this important rule.
TL;DR: In this paper, it has been shown that some three-step methods exist which, as well as Neta's methods, require one derivative and three function evaluations per iteration, but have an asymptotic convergence rate 7 which is better than the 6 of Neta.
Abstract: It has been shown that some three-step methods exist which, as well as Neta's methods, require one derivative and three function evaluations per iteration, but have an asymptotic convergence rate 7 which is better than the 6 of Neta.
TL;DR: An algorithm, based on a 3-term recursion relation, is effective for computing the analytic continuation of the function in any of its parameters.
Abstract: We present an algorithm for computing the generalized hypergeometric function with unit argument 3 F 2(1). The algorithm, based on a 3-term recursion relation, is effective for computing the analytic continuation of the function in any of its parameters. As an application, we give a new algorithm for computing the Beta function.
TL;DR: It is shown that these Runge-Kutta methods lend themselves easily to the development of error estimators similar to those of Fehlberg or England.
Abstract: The computation of non-stiff systems of ordinary differential equations can be accomplished with explicit Runge-Kutta methods. A class of explicit Generalized Runge-Kutta is described which requires an accurate evaluation of a Jacobian at every step. Second and fourth order processes are also described. In addition a second class of explicit Generalized Runge-Kutta is introduced which requires that the Jacobian be evaluated less than once every step. Finally a third order process is described. It is shown that these methods lend themselves easily to the development of error estimators similar to those of Fehlberg or England.
TL;DR: The development of a dynamic algorithm for improving the estimates of the involved parameters is presented and it is seen that the attained rate of convergence is approximately O(h 1/2) and is better than the algorithm using estimated parameters in certain cases.
Abstract: The Preconditioned Simultaneous Displacement (PSD) iterative method is considered for the solution of symmetric, sparse matrix problems, The development of a dynamic algorithm for improving the estimates of the involved parameters is presented, These estimates are then used to accelerate the PSD method by employing semi-iterative techniques, The algorithm determines adaptively a sequence of parameters while the iteration is in progress without requiring preliminary eigenvalue estimates (only trivial input parameters are required), The performance of the algorithm is tested on a number of generalised Dirichlet problems, It is seen that the attained rate of convergence is approximately O(h 1/2) and is better than the algorithm using estimated parameters in certain cases.
TL;DR: The difference between the methods is shown to depend on whether a conditioning matrix R consists of components derived from a splitting or factorization of A, and some theoretical results for the iterative schemes are given.
Abstract: This paper generalises the preconditioning techniques, introduced by Evans [2], and defines sparse and compact preconditioned iterative methods for the numerical solution of the linear system Au = b.The difference between the methods is shown to depend on whether a conditioning matrix R consists of components derived from a splitting or factorization of A. Some theoretical results for the iterative schemes are given when A has particular properties such as consistent ordering, irreducibility, diagonal dominance, positive definiteness, etc., when derived from the finite difference discretisation of a 2nd order self-adjoint elliptic partial differential equation. Finally, the application of both forms of preconditioning to the Conjugate Gradient method is presented and computational results compared.
TL;DR: In this article, an implicit iterative method for improving the accuracy of the inverse matrix is presented and shown to possess superior convergence properties over the well known quadratically convergent Hotelling method.
Abstract: An implicit iterative method for improving the accuracy of the inverse matrix is presented and shown to possess superior convergence properties over the well known quadratically convergent Hotelling method. Numerical examples are included to illustrate the new method
TL;DR: A backtracking algorithm which generates all the kernels of a directed, graph in lexicographic.order is developed which compares very favourably with existing algorithms.
Abstract: Determining whether or not a directed graph has a kernel belongs to a class of hard combinatorial problems, known as NP-complete. In this paper we develop a backtracking algorithm which generates all the kernels of a directed, graph in lexicographic.order. Extensive computational experience on randomly generated graphs ranging from 10 to 100 nodes and from 30% to 90% densities has shown that this algorithm compares very favourably with existing algorithms.
TL;DR: The root α of the Riemann Zeta Function can be determined by using the iterative formula provided that x 0 is a good starting approximation to α and it is noted here that there are advantages in using the formula and, where λ is suitably chosen, this formula is equivalent to Aitken's; δ2extrapolation formula as discussed by the authors.
Abstract: The root α of the equation can be determined by using the iterative formula provided that x 0is a good starting approximation to α and It is noted here that there are advantages in using the formula and, where λ is suitably chosen, this formula is equivalent to Aitken's; δ2extrapolation formula. The iterative technique is found to be successful in speeding up the convergence of alternating series and has also been applied to finding zeros of the Riemann Zeta Function.
TL;DR: This work studies the influence of the number of considered brothers, the splitting of an overflowing node with respect to the storage utilization and the numberof input/output operations per insertion in the dense m-ary-tree insertion scheme.
Abstract: Insertion schemes for various classes of multiway search trees have been implemented in PASCAL and experimentally studied. While the original B-tree insertion scheme does not consider brothers the dense m-ary-tree insertion scheme considers all brothers before splitting an overflowing node. There are many possible schemes in between these two extremes. We study the influence of the number of considered brothers, the splitting of an overflowing node with respect to the storage utilization and the number of input/output operations per insertion.
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TL;DR: In this paper, a new insertion algorithm is given for son-trees used as node search trees, obtained by random insertions starting with the empty tree, and the results are compared to known results about another insertion algorithm.
Abstract: Son-trees can be used as node search trees. A new insertion algorithm is given for son-trees used as node search trees. Random son-trees, obtained by random insertions starting with the empty tree, are studied, when the new insertion algorithm is used. The results are compared to known results about another insertion algorithm. The old insertion algorithm will give trees with better storage utilization, but the new insertion algorithm will need less restructurings.
TL;DR: In this article, the authors pointed out the inadequacies of the standard mathematical approach to n- tuples, and developed an axiomatic base for linear dyadic structures (list structures).
Abstract: This paper (1) points out the inadequacies of the standard mathematical approach to n- tuples, (2) develops an axiomatic base for linear dyadic structures (list structures), (3) uses the axiomatic base to (a) prove a theorem which shows that the system requires an infinite model (Foundedness Theorem), (b) define n-tuples and (c) prove a theorem used in fast pattern matching (Fundamental N-tuple Theorem). Finally, (4) the system is shown to be consistent.
TL;DR: There exist two distinct infinite hierarchies of AFG which exhaust the derivation bounded AFG and are shown to be strongly incomparable to the other; that is, the first member of each generates some language not generated by a fixed but arbitrary member of the other.
Abstract: An abstract family of grammars (AFG) may be defined as a class of grammars for which the corresponding class of languages forms an abstract family of languages (AFL) as defined by Ginsburg and Greibach. The derivation bounded grammars of Ginsburg and Spanier is an example of an AFG which is properly included in the class of all context-free grammars (also AFG). The main result is that there exist two distinct infinite hierarchies of AFG which exhaust the derivation bounded AFG such that the AFL associated with the kth member of one of these AFG hierarchies is properly included in the AFL associated with the k-lst member of that same hierarchy. Each hierarchy is shown to be strongly incomparable to the other; that is, the first member of each generates some language not generated by a fixed but arbitrary member of the other. We designate these hierarchies as the hierarchies of left and right dominant grammars (languages)
TL;DR: It is shown that livelocks are not preserved by reduction, implying that reduction cannot be used directly in proving the absence of livelocks.
Abstract: This paper is the second of a two-part series exploring the subtle correctness criterion of the absence of livelocks in parallel programs. In this paper we are concerned with the issue of proving this correctness criterion. It is shown that livelocks are not preserved by reduction, implying that reduction cannot be used directly in proving the absence of livelocks. Two applicable proof techniques are also presented. One is based on the notion of establishing sufficient conditions for livelock-freedom; the other is an extension of the well-founded set method for proving termination in sequential programs.
TL;DR: In this article, the concept of new Gauss-Seidel-like iterative methods was extended so as to obtain a class of convergent GSE-like block iterative method to solve linear matrix equations Ax=b with an M-Matrix A.
Abstract: The concept of new Gauss–Seidel like iterative methods, which was introduced in [3], will be extended so as to obtain a class of convergent Gauss–Seidel like block iterative methods to solve linear matrix equations Ax=b with an M-Matrix A. New block iterative methods will be applied to finite difference approximations of the Laplace's equation on a square (“model problem” [8]) which surpass even the block successive overrelaxation iterative method with optimum relaxation factor in this example.
TL;DR: It is shown that if F generates only CF languages then all languages generated by F are indeed linear, and that for arbitrary OL forms F it is decidable whether or not F generatesonly CF languages.
Abstract: Let F be an OL form or a so called clean EOL form. We show: if Fgenerates only CF languages then all languages generated by F are indeed linear. From the proof technique employed it further follows that for arbitrary OL forms F it is decidable whether or not F generates only CF languages.
TL;DR: In this article, a method for solving single nonlinear equations of the form X = F(X) is given in which an approximation by a parabola whose axis is parallel to the line y = x is used.
Abstract: In this paper a method for solving single nonlinear equations of the form X=F(X) is given in which an approximation by a parabola whose axis is parallel to the line y = x is used. It is shown that proposed method is faster than Picard, Steffensen or Wegstein methods.
TL;DR: It is shown that data structures of cardinality > n can be specified by means of a bounded number of equations, not depending on n, which are each of length O(n) only.
Abstract: Having concerned ourselves with boundedness properties of algebraic data structure specifications in the sense of the ADJ Group (see [4, 6, 7]) before now—see [1], which is in a sense a follow-up of [3] of Bergstra and Tucker—we show in this paper that data structures of cardinality > n can be specified by means of a bounded number of equations, not depending on n, which are each of length O(n) only