Showing papers in "International Journal of Computer Mathematics in 1978"

Deletion in one-sided height balanced search trees

Abstract: We give an deletion algorithm for one-sided height balanced search trees of n nodes. Thus deletion appears to be simpler than insertion, since Hirschberg's insertion algorithm is .

A self correcting conjugate gradient algorithm

Abstract: In this paper we develop a new procedure for constructing a conjugate gradient direction equation The new equation is a linear combination of two orthogonal vectors one of which is the negative gradient This procedure is reduced to the method of Polak-Ribiere whenever line search is perfectly accurate Otherwise, as reflected by our computational results, the method is more effective than any other conjugate gradient algorithm we have tested

Fast methods for the iterative solution of linear elliptic and parabolic partial differential equations involving 2 space dimensions

Abstract: Within the last decade, attention has been devoted to the introduction of several fast computational methods for solving the linear difference equations which are derived from the finite difference discretisation of many standard partial differential equations of Mathematical Physics. In this paper, the authors develop and extend an exact factorisation technique previously applied to parabolic equations in one space dimension to the implicit difference equations which are derived from the application of alternating direction implicit methods when applied to elliptic and parabolic partial differential equations in 2 space dimensions under a variety of boundary conditions.

An algorithm for the linear complementarity problem

Abstract: A practical algorithm for solving the linear complementarity problem is presented. This algorithm is based on the n-cycle algorithm, which is known to converge if M is a nondegenerate Q-matrix. A brief survey of other available algorithms is also given. Some typical test results are included.

String representations of graphs

Abstract: In this paper we associate with each graph g of a very large class of graphs a set of formal languages. Each such language can serve as description of g in the sense that g can be reconstructed from it in a unique manner. Our approach permits the convenient representation of finite and infinite graphs both for human communication and for storing or processing graphs in a computer. Since the model only requires that nodes of the graph are labeled “locally unique”, the relabeling of nodes necessary for performing certain operations on graphs using their descriptions is kept to a minimum. The model further allows a nice classification of graphs based on results on formal languages and provides means to define sequences of graphs as requiredin applications such as developmental biology.

Arithmetic congruences and semantic composition

Abstract: Integer arithmetic as performed in fixed‐length representations sometimes leads to errors, the result obtained being congruent to the desired arithmetic result where the modulus of the congruence is dependent on the representation used. In some cases it is possible to perform a sequence of computations in such a way that the final result is correct even though the execution of each constituent is “incorrect”. (Indeed, it may be that the particular computation in question cannot be performed without causing an intermediate “error”). Put another way, there exist functions f and g such that y = f(x) and z = g(y) are erroneous for some x and yet their composition z=g ∘ f(x)=g(f(x)) is correct. We show how elementary theorems from Number Theory may be applied to verify some such computations.

On a matrix factorization algorithm

Abstract: A matrix factorization technique called the NF (Nonsingular Factorization) algorithm is introduced. It has the following properties with respect to computing the matrix-vector product MX: (i) if M is an (n × n) Kronecker matrix, then the total number of multiplications followed by additions, (denoted by ∑), is such that : (ii) if M is an (m × n) non-Kronecker matrix. The case implies that no savings in arithmetic operation are realized. The NF algorithm provides a means of readily obtaining factorizations associated with Kronecker matrices. In addition, it provides a unified approach for seeking factorizations of a class of non-Kronecker matrices (i.e., transformations) that are used in signal processing applications. In the past, the factorizations associated with such matrices have been sought using ad hoc techniques.