mathematics of computation volume 36.number 153 JANUARY 1981 A New Method for Chebyshev Approximation of Complex-Valued Functions By K. Glashoff and K. Roleff Abstract. In this paper we are concerned with a formulation of the Chebyshev approxima- tion problem in the complex plane as a problem of linear optimization in the presence of infinitely many constraints. It is shown that there exist stable and fast algorithms for the solution of optimization problems of this type. Some numerical examples are presented. 1. Introduction. We are going to consider the following approximation problem: Let C(7) be the normed space of complex-valued continuous functions / on a compact subset 7 of the complex plane C, equipped with the uniform norm ||/||00 = max|/(z)|. Let/(z) and w,(z), . . ., wn(z) be fixed given functions out of C(7). For any set of n complex parameters x = {x,, . . . , xn), let n E(x, Z) = 2 XrWr(Z)- r=l The problem is to determine an optimal set of parameters x such that (1) ||/ - L(x, z)|L < ||/ - L(x, z)\\x for all x. Problems of this type appear in many connections and have been treated by various authors; cf. [2], [12], [13]. Considering the computation of best rational Chebyshev approximations to complex-valued functions by descent algorithms, Ellacott and Williams [6] report that the main portion of computer time is spent in the solution of the linear subproblems. They have applied the very slowly conver- gent Lawson algorithm to these subproblems, and they remark that a fast linear algorithm would bring about a significant improvement in efficiency. In our paper we describe a new algorithm which seems to be the first one which works for the continuous case (where 7 is not a finite set), too. In our method we transform the complex approximation problem into a (real) linear optimization problem with infinitely many constraints, a so-called semi-infinite program (SIP); cf. [5], [8], [10]. This is solved by a two-step procedure: in the first step we apply the stabilized Simplex algorithm to a discrete approximation of the SIP and in the second we make use of the Newton-Raphson method in order to obtain very rapidly the solution of the SIP and thus of our original approximation problem. Received March 23, 1979; revised March 27, 1980. 1980 Mathematics Subject Classification. Primary 65E05; Secondary 30C30. © 1981 American Mathematical Society 0025-5718/81/0000-0018/$02.7 5 233 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use