Showing papers in "Numerische Mathematik in 1959"

A note on two problems in connexion with graphs

Abstract: We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the branches definitely assignec~ to the tree under construction (they will form a subtree) ; II. the branches from which the next branch to be added to set I, will be selected ; III. the remaining branches (rejected or not yet considered). The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes), We start the construction by choosing an arbitrary node as the only member of set A, and by placing all branches that end in this node in set II. To start with, set I is empty. From then onwards we perform the following two steps repeatedly. Step 1. The shortest branch of set II is removed from this set and added to

Newton's method for convex programming and Tchebycheff approximation

Abstract: w 1. Introduction. The rationale of Newton's method is exploited here in order to develop effective algorithms for solving the following general problem: given a convex continuous function F defined on a closed convex subset K of E,~, obtain (if such exists) a point x of K such that F(x)_<_F(y) for all y in K. The manifestation of Newton's method occurs when, in the course of computation, convex hypersurIaces are replaced by their support planes. The problems of infinite systems of linear inequalities and of infinite linear programming are subsumed by the above problem, as are certain Tchebycheff approximation problems for continuous functions on a metric compactum. In regard to the latter, special attention is devoted in w167 27--30 to the feasibility of replacing a continuum by a finite subset in such a way that a discrete approximation becomes an accurate substitute for the continuous approximation.

The radius of univalence of the error function

Abstract: (x/2)'» = 1.25 • • , [Rogozin, 2], the largest positive root R, of x —arctan x — w, where x — (£R —1); i? = 1.51 • • • , [Reade, 3] . These bounds were obtained by different, rather general methods. Our methods are based on special properties of erf 0, and were suggested by a detailed study of actual numerical values of erf z, which were computed on the IBM 704 at the National Bureau of Standards by E. Brauer and J. C. Gager.

Numerical study of the representation of a totally positive quadratic integer as the sum of quadratic integral squares

Abstract: Altogether a total of almost 200,000 totally positive numbers from different fields were decomposed into squares; in no case were more than five squares required, although in many cases no number of squares sufficed. For some quadratic fields, from our evidence it would seem safe to completely characterize the couples for whichQ=5 or 0, particularly whenm?13; and furthermore, form=17 or 33 it seems possible to characterize all cases whereQ=4, 5, or 0. The whole calculation seems to be pointed toward the result that three squares are sufficient except for "special" cases. Incidentally, the analytic methods ofSiegel andMaass run parallel to the calculation in that these methods involve the third, fourth, and fifth power of a theta-function. The numberical evidence would therefore suggest that their methods point to an analytic (or even a purely algebraic) proof of the futility of using more than five squares in any case. The work was supported in part by the U. S. National Science Foundation Grant G-4222 and the computer services were contributed by the Argonne National Laboratory of the U. S. Atomic Energy Commission during the summer of 1958. The coding was performed by Mr.Alan V. Lemmon with remarkable economy of length of program and running time. The deepest debt of gratitude is owed to the lateDonald A. (Moll) Flanders whose contributions to the logical design ofGeorge had made the rapid execution of the program possible and whose personal interest made possible the availability of the computer for this work.