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Several starting approximations for square root calculation by Newton's method are presented in a form to facilitate their use in IBM System/360 square root routines. These approximations include several for the range [~¢, 1], which is the interval of primary interest on IBM System/360.

It has been suggested by Householder [1] and by Householder and Bauer [2] that orthogonal similarity transformations of matrices are particularly stable with respect to the practical computation of proper values. It is the purpose of this note to examine this question and to demonstrate in terms of a “condition number” to be defined below a sense in which this conjecture is true. Broadly speaking, any problem may be termed “ill-conditioned” for which the solution is acutely sensitive to slight variations in the parameters of the problem. Examples of ill-conditioning occur in many contexts. Computers are familiar with the phenomenon as it manifests itself, for example, in the study of matrix inversion. Since our purpose is to study the conditioning of matrices specifically for the proper value problem, it is convenient to have a nomenclature and notation which will avoid confusion with conditioning as it is used in other senses.

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The Editors of the Algorithms and Numerical Analysis departments of the CACM wish to encourage the submission of algorithms with supporting numerical analysis papers and numerical analysis papers with related algorithnis.Hence contributions are solicited consisting of a paper plus an algorithm. If accepted, the algorithm will appear in the Algorithms department and the paper will appear in the Numerical Analysis department.The ground rules are as follows: 1. The "paper-algorithm" should be submitted in duplicate to the editor of either one of the two departments. The covering letter should state that this is a joint contribution.2. Referees will be chosen for the algorithm and for the numerical analysis paper by the respective responsible editors. The algorithm and the paper will be refereed according to the standards of the appropriate department.3. After both the algorithm and the paper have been accepted by the respective editors, both will appear in the same issue of the CACJM, each with reference to the other.^J . 1' ". THAUB AND J. G. HEERIOT

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Abstract. Various writers have dealt with the subject of optimal starting approximations for square-root calculation by Newton's method. Three optimality criteria that have been used can be shown to lead to closely related approximations. This fact makes it surprisingly easy to choose a starting approximation of some prescribed form so that the maximum relative error after any number of Newton iterations is as small as possible. | 1. Introduction. The choice of polynomial and rational starting approximations for square-root calculation by Newton's method has been the subject of various investigations (e.g., [l]-[5]). These approach the problem from several different points of view. In this paper we will show how approximations obtained from these different viewpoints are related and how some of them can be derived from others.