Chebyshev approximation by exponential-polynomial sums (*) Charles B. Dunham (**) ABSTRACT It is shown that best Chebyshev approximations by exponential-polynomial sums are character- ized by (a variable number of) alternations of their error curve and are unique. Computation of best approximations via the Remez algorithm and Barrodale approach is considered. Let [a, 3] be a closed finite interval and C[a, 3] be are distinct and nonzero (in the case m = 0 we will the space of continuous functions on [a, 3]. For permit one of a n + 1 ..... a2n to be zero). It is easily g ~ C [a, 3] define seen that any parameter has an equivalent standard parameter. The degeneracy d (A) of a standard parameter Itgll=max (Ig(x)l:a<x<3). A is the number of zero coefficients in a I ..... a n. The Let w be a positive element of C[a, 3], an ordinary degree of F at standard parameter A is defined to be multiplicative weight function. For given fLxed n > 0, 2n + m - d (A). m >/ 0 define n m k-1 Haar subspaces F (A, x) = k~=l a k exp (an + k x) + k Z=I a2n + kX Definition The Chebyshev problem is given f~ C[a, 3] to find a A linear subspace of C [a, 3] of dimension ~ is a Haar parameter A* to minimize IIw(f-F(A, .))II. Such a subspace on [a, 3] if only the zero element has ~ zeros parameter A* is called best and F(A*, .) is called a on [a, 31. best approximation to f. It might perhaps be thought that the case m > 0 is not Lemma I of practical interest. However, Spath has developed The linear space generated by algorithms for discrete L 2 approximation for the case (exp (aix), xexp (aix) ..... x m(i) exp (aix)), n = 1, m = 1 and 2 in Spath [26] and Spath [27] respectively. Further, the model n = 1, m = 2 has been i = 1 ..... n, a 1 < ... < a n , is a Haar subspace of dimen- found to apply to a biochemical system (Heitkamp n et al. [19] equation (17)) and Spath [27] states that sion Z (m(i)+l). it is often needed in chemistry and radiation chemistry. i=1 The author has been informed by Dr. Alan Miller of The lemma is given in Meinardus and Schwedt [25, p. CSIRO that models with m = 1, n = 2 were of interest. 313] and also proven in Meinardus [24, p. 177): Such a model was used by Cantraine and Jortay [6]. Models with m = 1 and n larger are given by Cook and Property Z Taylor [10] and by Lemaitre and Malenge [22]. Models which can be converted by change of variable into the Lemma 2 case n = 1, m = 1 or 2, are discussed by Shah and Let F be of degree £ at standard parameter A, then Khatri [29]. F (A, .) - F (B, .) has at most ~ -1 zeros or vanishes The same model with m -- 1 is used by Gregory [30]. identically. The case where m = 0, w = 1, has been studied by Meinardus and Schwedt [25, p. 312-313] and also Proof appears in Meinardus [24, p. 177-178]. We analyze F (A, .) - F (B, .) is a linear combination of (1 ..... x m -1 } our problem with a theory also due to Meinardus and and at most 2n-d (A) non-constant exponentials. Schwedt. By Lemma 1 it has at most 2n + m -1 - d (A) zeros or A parameter A will be called standard if an+ 1""' a2n vanishes identically. (*) This work has been assisted by a grant from the National Research Council of Canada. (**) Ch. B. Dunham, The University of Western Ontario, Dept. of Computer Science, London Ontario N6A 5B 9, Canada. Journal of Computational and Applied Mathematics, volume 5, no 1, 1979. 53 The partial derivatives (i) set j = 0 and choose {x0 ..... X2n + m ) ' Define the partial derivatives where a < x 0 < ... < X2n+m ~ 3, F k (A, x) = 8 F (A, x), (ii) solve the (non-linear) system of 2n + m + 1 levelling equations then they are seen to be continuous in A, x. (*) f(xi) - F(A, xi) - (-1)1X/w(xi) = 0, The tangent space i =.0 ..... .2n + m, for 2n + m + 1 unknowns AJ and XJ , Define (iii) find the alternating extrema of the error curve 2n+m D(A,B,x) = b k ~ F (A, x) k=l w(f - F (Aj , .)) and call them {x0 ..... X2n + m } . n n This process of simultaneous exchanges is described = ~ bkex p (an+ k x) + k~=ibn+kakxexp (an+k x) k=l in Meinardus [24, p. 107] (iv) add 1 to j and go to (ii). m k-1 + Y~ k x k =1 b2n + This process is an infinite process. In an actual program it is terminated when the extrema of the error curve Len'/ma 3 found in (iii) are sufficiently equal in absolute value. Let F be of degree J~ at standard parameter A, then Finding the alternating extrema can be done exactly (D (A .... ) } is a Haar subspace of dimension £. as in variants of the Remez algorithm for other approxi- This follows immediately from Lemma 1. mation problems, for example Cody, Fraser, and Hart [9] for rational approximation. The major difficulty is Characterization of best approximations in solving the levelling equations (*). The most straight- forward way to solve them is to use Newton's method Fundamental to much of the characterization theory directly. In the special case n = 1, it is also possible to of Chebyshev approximation, both linear and non- re-arrange the system (*) to make the unknowns enter linear, is the alternation (equioscillation, equal ripple) less nonlinearly and then apply Newton's method. property. The case n= m=l, w= 1, is considered in Dtinham [11] Definition and it is not difficult to see from this analysis how to cover the general case of n = 1. Both approaches appear g ~ C[a, ~] alternates £ times on [ct, 13] if there exists to work reasonably well in the case n = 1 in the author's (x 0 ..... x£}, where a < x 0 < ... < xj~ < ~, such that experiments. A problem with both approaches is that Newton's method may converge to a "pseudo-solu- Ig(xi)l = IIg[I g(x i) = (-1)ig(x0) i= 0 ..... 9~. tion". Pseudo-solutions for the case n = m = 1 are given in Dunham [11, p. 211] for the method with re- Such a set (x 0 ..... x£} is called an alternant of g. arrangement. Whether the levelling equations (*) can easily be solved in the case where n > 2 and optimal Theorem 1 {an+k } are not well separated is still an open question. Let F have degree £ at standard parameter A. A neces- It should be noted that the levelling equations may sary and sufficient condition that F (A,.) be best to f not even have a solution. is that w (f - F (A, .)) alternate J~ times on [ct,/~]. Best approximations are unique. Example The theorem follows from Lemma 2, continuity of Let n=l, m=0, w=l and f(x0)=-l, f(xl)= f(x2)=l. the partial derivatives, Lemma 3, and theory of Meinardus and Schwedt [25, theorems 13 and 14], Suppose a solution A, X to (*) exists. The first possi- als0 given in Meinardus [24, p. 144-147] in the case bility is that a I <~ 0, in which case F (A,.) ~ 0, and we w = 1. The extension to a general w follows from cannot have f (Xl) f(x2) > 0. The second possibility theorem 1 of Dunham [12]. is that a I > 0, in which case F (A, .) > 0 and it can be seen by drawing a diagram that f (x0), f (x 1) cannot Computation of best approximations take their values. In case the best approximation is of maximum degree The variant of Meinardus and Schwedt [25, p. 320 ff], (2n + m), its error curve alternates at least 2n + m also given in Meinardus [24, p. 150-151], uses one times. This suggests that best approximations can be iteration of Newton's method to approximately solve determined by a variant of the Remez algorithm, the levelling equations. Barrar and Loeb [1] have general versions of which are given by Barrar and Loeb proven that if the best approximation is of maximum ([1], p. 389) and Dunham ([12], p. 228). degree, and the starting point (estimate of the parameters The Remez algorithm for our problem proceeds rough- and deviation of the best approximation and an alter- ly as follows : nant) is sufficiently close, the Remez algorithm will Journal of Computational and Applied Mathematics, volume 5, no 1, 1979. 54 converge (neglecting analytical and numerical diffi- (X,) of finite sets fdling out [ct, ~]. This should work culties in solving the levelling equations). Dunham if i~e best approximation is of maximum degree [ 13]. [12] proves that under favorable conditions, the con- vergence of the Remez algorithm and the variant of Discrete approximation Meinardus and Schwedt is quadratic. The author has Consider instead approximation on a finite set programmed a version of the Remez algorithm for the case n = 1 which appears to work reasonably well, X=(x 1 ..... x j), j>2n+m, x l<...<xj.Define provided the parameters of the best approximation [[glIx= max {Ig(x)l : x~X). can be closely estimated. Another approach is to reduce part of the problem We wish to minimize Hw (f - F (A, .))[IX" to a linear problem. Associate with each set an + 1 ....