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Direct Chebyshev approximation Henderson, John Robert 1963

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DIRECT CHEBYSHEV APPROXIMATION by JOHN ROBERT HENDERSON B . A . , The University of British Columbia, I960 A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T OF . T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF ARTS in the Department of MATHEMATICS We accept this thesis as conforming to the . required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1963 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y shall'make' i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department- or by h i s r e p r e s e n t a t i v e s . . I t i s understood t h a t copying, or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of '\6-Th-e. I ^ w ^ l v' c S The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date Abstract The Approximation P r o b l e m and s p e c i f i c a l l y , " d i r e c t " rational Chebyshev approximation i s discussed. A brief summary i s made of " d i r e c t " Chebyshev approximation. The remainder of the thesis i s devoted to various aspects of a "Remes-type" A l g o r i t h m for rational Chebyshev approximation, as proposed by F r a s e r and Hart. It i s fi n a l l y concluded that the inherent di f f i c u l t i e s of the method would generally outweigh the advantages of the rational approximation which it obtains. We accept this abstract as conforming to the required standard - iv -Ac kno wl e dg me nt . I wish to thank D r . T . E . Hull and D r . D. Derry for thei valuable criticisms of the presentation of this thesis, and the Mathematics Department, University of British Columbia, for a Summer Grant (1962), and a Teaching Assistantship over the pa two years (1961-1963). i i Table of Contents Page I Notation 1 II Introduction 2 III Basic Theorems 8 IV Methods of Direct Chebyshev Approximation 10 A) A Method Using Zeros of the E r r o r Curve 10 B) Iterative Methods Using the Absolute Extreme of the E r r o r Curve 11 C) Programming Methods 13 D) Other Methods 15 V The Re mes Rational Algorithm and Associated Problems 17 A) Difficulties of the Iterative Procedure 18 B) Difficulties of the Overall Procedure 24 C) Other Problems - Timing 2 6 - Stability 28 VI Numerical Results 31 VII Conclusion 33 VIII Bibliography 34 i i i T a b l e s , D i a g r a m s , G r a p h s : E r r o r B o u n d s f o r A p p r o x i m a t i o n o f e x ( T a b l e I) F o l l o w i n g p a g e 6 T h e R e m e s R a t i o n a l A l g o r i t h m ( F l o w D i a g r a m I) " " 17 ( N o t a t i o n f o r F l o w D i a g r a m II ( M o v e m e n t o f C r i t i c a l P o i n t s ( F l o w D i a g r a m II) " " 18 C o m p a r i s o n o f k , n c ( T a b l e II) M M 2 8 S t a b i l i t y a n d W o r d l e n g t h f o r e x ( G r a p h I) " " 3G ( S u m m a r y o f M o v e m e n t o f C r i t i c a l P o i n t s S i n ( 3 , 3) a p p r o x i m a t i o n o f e x ( G r a p h s o f s a m e ( G r a p h s I I - V ) " " 32 E * f o r v a r i o u s f u n c t i o n s ( G r a p h s V I - V I I I ) " " 32 - 1 -N o t a t i o n T h e f o l l o w i n g a r e the m a i n c o n v e n t i o n s and n o t a t i o n u s e d t h r o u g h -out the t e x t : - P ( x ) / P ( x ) , Q(x) denote p o l y n o m i a l s ; R(x ) z / Q ( x ) d e n o t e s a r a t i o n a l f u n c t i o n ; f(x) i s r e t a i n e d f o r the f u n c t i o n b e i n g a p p r o x i m a t e d . T h e a p p r o x i m a t i o n i n t e r v a l i s i n g e n e r a l Ca,b3 o r m o r e s p e c i f i c a l l y C- 1, 13 . - A n a s t e r i s k " * " d e n o t e s " C h e b y s h e v " o r " b e s t u n i f o r m " . - A s u b s c r i p t d e n o t e s a m a x i m u m d e g r e e e.g. dev^es a C h e b y s h e v (best u n i f o r m ) p o l y n o m i a l of m a x i m u m d e g r e e n . (k) - S u p e r s c r i p t s e g . V '(x) denote e i t h e r a d e r i v a t i v e o r d e r o r the n u m b e r of a s tep of a n i t e r a t i v e p r o c e d u r e . In any p a r t i c u l a r c a s e , the c h o i c e b e t w e e n t h e s e a l t e r n a t i v e s w i l l be c l e a r . ttc. = m-»-m + 2. i s two m o r e than the s u m of the m a x i m u m d e g r e e s of a r a t i o n a l f u n c t i o n ^ n ^ . 5 Q m ( x ) - En ,m"f o r s i m p l y E * n , m o r E*f i f t h e r e i s no a m b i g u i t y , Pn*(x ) denote the C h e b y s h e v e r r o r i n a p p r o x i m a t i n g f by Q m » ( x ) • " ( n , m ) a p p r o x i m a t i o n " d e n o t e s a n a p p r o x i m a t i o n of the f o r m *n{f\. U m ( x ) - 2 -Introduction In numerical work i t i s commonly desirable to replace a "complicated" continuous function by a " s i m p l e r " one; one which represents i t " s u f f i c i e n t l y w e l l " for some p a r t i c u l a r purpose. In this way the "complicated" continuous function i s "condensed" into a more tractable form. "Complicated" i m p l i e s there are d i f f i c u l t i e s concerning the length of time required for function evaluation and/or, with computer space l i m i t a t i o n s , for the commands required. F o r example, the solution of a p a r t i c u l a r l y complicated f i r s t order di f f e r e n t i a l equation can be represented to some extent by tabulated values for equal increments i n x ( i . e . by interpolation), but possibly considerably more accurately by a low degree polynomial approximation determined by its zeros XJ at specially chosen points. The sense of the approximation, i . e . recognition of what details of the function we are w i l l i n g to give up for " s i m p l i c i t y " , i s i m p l i c i t i n the phrase "s u f f i c i e n t l y good approximation". This art of choosing a suitable norm has too many ramifications to even begin to consider here, but it i s for example brought out by the difference between least squares and Chebyshev approximations. The main advantage of least squares approximations i s the s i m p l i c i t y by which they can be obtained - - a feature which i s p a r t i c u -l a r l y evident i n the hist o r y of approximation theory. Laplace posed one of the f i r s t Chebyshev approximation problems (solution of an inconsistent set of l i n e a r equations) i n 1799. His procedure for solving the problem as i t stood was completely i m p r a c t i c a l and - 3 -remained so until 1804 when Legendre substituted the L 2 (least square) norm for the Chebyshev norm. With the success of his procedure, the use of the least squares norm has never slackened. On the other hand, Chebyshev approximations provide a known upper bound (the least possible for the p a r t i c u l a r interpolation class used) on the e r r o r expected when using them. This i s probably the usual goal, but unfortunately,with present algorithms the work required to generate such approximations of any p r a c t i c a l u t i l i t y i s so great that it i s prohibitive without the aid of d i g i t a l computers. With this inaccessible character, it is natural that interest i n algorithms for Chebyshev approximation has been revived only in the last few y e a r s . As far as.Chebyshev approximation algorithms themselves are concerned, we w i l l distinguish two fundamental types. The distinction i s i n terms of where the e r r o r of the approximation genera-ting algorithm (as distinct f r om the e r r o r of the approximation itself) i s generated. The two types are: 1) " i n d i r e c t " algorithms with an i n i t i a l e r r o r introduced by the approximation generating algorithm, and 2) " d i r e c t " algorithms with e r r o r of the procedure being i n t r o -duced during a final i terative procedure. Although this terminology i s somewhat ambiguous, it i s retained i n honour of Maehly [ l 5 l , a modern pioneer in the approximation algorithm f i e l d . D iagramatically, this difference can be represented as follows: - 4 -D i r e c t Algorithms (usually, non-terminating iterative procedures) e r r o r of in d i r e c t methods A ty p i c a l pair of such methods i s "Economizing" (an indirect algorithm) as proposed by Lanczos [ l 3 l and the direct procedure known as the Remes A l g o r i t h m for polynomials [2 l] which w i l l be discussed l a t e r . In the remainder of this thesis only direct algorithms w i l l be discussed. The class of " s i m p l e r " functions, i . e . the subspace of approxi-mations, has commonly been taken to be polynomials of specified maxi-mum degree although 'other subspaces of trigammetric polynomials, classes of exponential functions,and rational functions have been con-sidered. Disregarding piecewise polynomial functions, since only the basic operations of (+,x) are available, rational functions are essent i a l l y the most general function form for digital computer com-putations. For this reason, this thesis w i l l be r e s t r i c t e d to rational function approximation and the special case of polynomial approximation. Two further points should be noted. There i s considerable value i n making transformations before applying approximation Indirect Algorithms (commonly having a finite number of steps) i n i t i a l app roximation - 5 -algorithms. Unfortunately, however, there i s no general rule available which gives the ideal transformation required. Some indication i s given by the T a y l o r series remainder t e r m in the case of polynomial approxi-mation. The choice of the appropriate transformation is another " a r t " , w e l l exemplified by Hastings C9!l . r Secondly, there i s a problem of motivation. What i s the value of considering rational function approximation algorithms when i n comparison (as w i l l be seen later) analogous polynomial algorithms t ha-ve none of their troubles? The answer of course i s the degree of approximation. Hopefully, the extra freedom allowed i n the choice of f o r m w i l l significantly reduce the maximum e r r o r (E*)- In practice, varying degrees of "reduction" are observed (see graphs in Numerical Section). A more analytical insight into this problem is given by the following considerations: Let R(x) = Pn(x) be such that Qm(x) Pn(xj) ,. .. . ,\ = f(xj) at n c - l points x; ...... ...I) take an a r b r i t r a r y y such that -I i x, <y < X*,-i - ' and consider: g(x) = f(x) Qm(x) - Pn(x) - S^MQM'W] . (Y-KYX-*IY • M*,-^  then g(x,.) = g( X a.) = .... = g{*nc_i) = g(y) = 0 i . e . g(x) vanishes n c - i times on £-1, l3 , so that by successive (n - 1) applications of Rolle's Theorem, g v c" ' ( x) vanishes at least once i n (-1, 1) i . e . there exists a 9 6 (-1,1) such that g ^ n c _ 1 ^ ( 9 ) = 0 r, ^ P n (x ) now let f(x), Q m ( x ) coincide in particular at the ttc-1 zeros of Tvv-i Lt>> - % ( V ^ e ' cos Ov-i^ e 5 x - cos © '-2.) from which, taking absolute values, f i v. V - ' ...3> Pn* where denotes the irreducible Chebyshev approximation to f on [.-1, l] , and Qm is determined by 2), | ) . Equation 3) gives some idea of the relative value of polynomial and rational function Chebyshev approximations. To proceed further with the general case, one needs an expression for min |Qm*(x)| and a means of eliminating Qm(x) from the right-hand side of 3). The solution of these problems does not seem to have been attempted. For two special cases (m = 0 - the polynomial case - and m = 1) we can readily get a somewhat deeper insight. For polynomials (m = 0) the two above-mentioned difficulties disappear and we obtain from 3), This result has been given by Shohat [25"] who has also produced a lower bound: />H>) assuming f ( ^ l ' (Q) > 0, by a consideration of 3) in the polynomial case, without introducing absolute value. A sample application of these bounds for e x , x 6 [-1, 1] is given in Table I, the "true value" being obtained numerically from the Remes polynomial algorithm. In the case m = 1 we have Qm(x) = Q,(x) = 1 + Qzx- ^ w e Pn(x) assume that our choice 2) produces r^j(x) with no pole in the interval (see Theorem introducing Section V), then | < | so that Table I E r r o r Bounds on En(f) for e x — According to Shohat xet-1, 11 Polynomial True Value: Degree Upper Bound ^ En* (e x) £ Lower Bound 0 .272 X 10' . 154 X 10' .368 X 10° 1 .680 X 10° .279 X 10° .920 X IO" 1 2 .113 X 10° .450 X IO" 1 . 153 X IO" 1 3 . 142 X 10" 1 .553 X 10"2 . 192 X 10"2 4 . 142 X I O - 2 .547 X IO" 3 . 192 X IO" 3 5 .118 X ID" 3 .452 X 10-4 . 160 X IO" 4 6 .843 X I Q ' 5 .321 X IO" 5 . 141 X IO" 5 7 .527 X 10" 6 .200 X 10" 6 .713 X IO" 7 8 .329 X i o - 7 .111 X IO" 7 .445 X IO" 8 9 . 183 X 10" 8 .552 X 10-9 .248 X 10-9 10 .915 X I O " 1 0 .250 X I O " 1 0 . 124 X I O " 1 0 _ 7 -In p a r t i c u l a r , for e x, x 6 C-l, l l we have &y - pji!k>l ^ i r ^ i L ± _ - - — — ...7) where Q 1 * ( x ) = 1 + 0.2*x Thus, we have E E * , which i s not an extremely attractive result, but i t i s thought that p a r t i c u l a r functions f may be much more rewarding. The following work has two purposes: 1 ) to describe b r i e f l y the main direct polynomial and rational function algorithms for Chebyshev approximation on finite i n t e r v a l s , and 2 ) to discuss in some detail c ertain aspects (stability, timing, and problems unique to rational approximation) of a method of rational Chebyshev approxi mation as proposed by F r a s e r and Hart £7] , with the object of evaluating its p r a c t i c a l implementation. 8) I - 8 -Basic Theorems Before turning to the various algorithms, we w i l l divert our attention slightly to some of the basic theorems upon which the algorithms are constructed. In the f i r s t place, existence and uniqueness are guaranteed for rational function approximations: PS(x) Theorem: There exists a unique Chebyshev approximation Q ^ ^ J to a continuous function f(x) on Ca>b3 . Proof: The reader i s r e f e r r e d to C. de l a Vallee Poussin &0] and Chebyshev [3] for the m = o (polynomial) case, and to Chebyshev [3] , Rice t23] in the general case. C h a r a c t e r i -zations are given by the following theorems: Pfi(x) , Theorem: The Chebyshev rational function approximation Q m ( x ) * 0 reduced to lowest terms, to a continuous function f(x) on Pn(x) \a., b"] i s characterized by f(x) - Q ^ ^ J assuming with alternating sign the value I to - I at not less than n + m+2 - min ( ) points i n [a, b3where h-jl ^*n-V are the highest powers of x o c c u r r i n g i n Pn(x), Qm(x) respectively whose coefficients^? 0. Proof: See Achieser [l\ and Rice [23, p. 3-60J . As a special case, for polynomials, we have m = o (and H- 0 ) so: Theorem: The Chebyshev polynomial approximation PK(x) to a continuous function f(x) on [a,b] i s characterized by f(x) - Pn(x) assuming - 9 -with alternating sign the value: at not less than n + 2 points in ja, b] The latter two theorems determine certain "critical points which in turn, determine the best approximation - - s o defining an approximation algorithm. The following theorem is of a different nature. It is stated here not only because it is the basis of an approxi-mation algorithm but also because of the insight it gives into the nature of Chebyshev approximation. polynomials of degree n) define L 'P ' f as the least pth degree approxi-mation to f on [a, b) (in a manner analogous to the least squares (p = %) approximation). Then: Proof: Theorem: There is a subsequence of p = 1 ,2 , . . . which converges to the Chebyshev approximation to f on [a, b] . In the polynomial case, the stronger result of L^P^f itself -Chebyshev approximation has been shown by Jackson [l0\ . The general case is given by Polya [191. - 1 0 -Methods of D i r e c t Chebyshev Approximation A) Methods Using Iteration on the Zeros of the E r r o r Curve These methods were proposed by Maehly |[l53 for both polynomial and rational function approximation. Let ( ^ i \ > • - • n c - i with ^11 Xi < X^<U < +1 be n c - 1 otherwise a r b i t r a r y i n i t i a l points. The successive application of the following two steps defines an i t e r a t i o n procedure: ,x ^ (k) P n ( k ) ( x ) 1) Pn , or more generally, ~ (k), . i s determined on the Qm v '(x) kth step by: >0O 2 ) Modify \ ^ \ (each method giving a d i s t i n c t i t e r a t i o n procedure), so obtaining \ ^ so that e v n " M k ; w i l l tend to be of equal value on the next (k + l)th step. In the polynomial case, i t i s to be noted that i s a lin'ear system having associated with it an (n+l)th order Vandermonde determinant i n the ix^\ • Thus, if the are kept distinct and i f f(xj) ^ 0 simultaneously for a l l j = l , 2 , . . . n c - l , then the l i n e a r system i s always non-singular and the ite r a t i o n can i n p r i n c i p l e be continued indefinitely. Under certain severe assumptions Maehly has shown where k i s some constant and xj'" denotes the position of XJ for the best Pn*(x) Chebyshev approximation ^ r - . This result forms the basis of a rule Qm (x) for step 2) above, which reduces (increases) | > ^ — "X ||M \ ^'JV'VI ^ i s too large (too small) i n comparison with other values (j = 1, 2 , . ..n c-2). E m p i r i c a l l y he noted that the use of this rule gave quadratic convergence for both polynomial and rational function approximations. B) Methods Using Iteration on Absolute E x t r e m a of E r r o r Curve .As an example of such a method let us f i r s t consider what is commonly known as the "Exchange Method" e.g. Stiefel [26, p.220]. For nth degree polynomial approximation the procedure i s as follows: 1) Select any n c = n + 2 points • _\ « yj'ic ... < +1 2) Solve: thus obtaining the best approximation Pn'°'(x) on the finite set \ X J ( o ) ^ - [l2, p.223] 3) Determine an x* such that: 4) Let: W) ~ \\*?\ ~*?\ "\*'\ where x ^ 0 ) i s so chosen that Pn'°' ( x ^ 1 ) ) - f ( x j ^ ^ ) takes alternating sign for j = 1,2,3, ...nc= n + 2. This i s always possible by the "Exchange Theorem" ^ S t i e f e l [26, p.220], Novodvorskr and P i n s k e r [ l 8 ] . Step 2) i s repeated with superscript increased by one; etc. , the procedure being repeated indefinitely. - 12 -A c c o r d i n g to V e i d i n g e r [27, p. 10o], the g e n e r a l p r o o f of convergence of Re mes [2l] c a n be us e d to show that the above-de s c r i b e d p r o c e d u r e i s c o n v e r g e n t . It a l s o f o l l o w s f r o m [18] i f E^°^0. A n o t h e r m a i n v a r i a t i o n on t h i s theme i s R e m e s Second A l g o r i t h m f o r p o l y n o m i a l a p p r o x i m a t i o n , i n w h i c h the f o l l o w i n g change i s made i n step. 3): 3) L e t \yt \ 4w\iXi.-Mtr\ w i t h -\•• >,<yv < . . . < y ^ , = +1 be the n + 1 z e r o s of P n ( o ) ( x ) - f( x ) , along w i t h f c l} . R a t h e r than m o d i f y $ X J ^ ° ^ by a s i n g l e p o i n t x*, n c points xjj^ = 1, 2 , . . . n c) a r e cho s e n s u c h that yj •£ yg4, \-\$ y^  i X £ y<4l Step 4) then becomes : ^« l> l j •' • ^ « 4') ^ x j ( D ? ~~Uf\ A s s u m i n g that N o v o d v o r s k i and P i n s k e r ' s p a p e r [18*] p r o v e s the con v e r g e n c e of t h i s a l g o r i t h m , V e i d i n g e r [27]has shown the co n v e r g e n c e to be q u a d r a t i c i f 'f 6 C . The d e t e r m i n a n t a s s o c i a t e d w i t h the l i n e a r s y s t e m ob t a i n e d u s i n g t h i s a l g o r i t h m ^ 0 - a fea t u r e w h i c h w i l l be evident i n l a t e r d i s c u s s i o n . S e v e r a l other m o d i f i c a t i o n s of Remes Second A l g o r i t h m have a p p e a r e d i n the l i t e r a t u r e : B a r t h [2], H a s t i n g s [9], M u r n a h a n and W r e n c h [ l 7 ] , Selfridge[24"] . F o r the m a i n p a r t these c o n s i s t m e r e l y of t a k i n g ^XJ*"^ at the p o s i t i o n s of l o c a l r a t h e r than ab s o l u t e e x t r e m a of the e r r o r c u r v e . P a t h o l o g i c a l c a s e s c a n a l w a y s be c o n s t r u c t e d to defy such a l g o r i t h m s - i . e . a c o m p u t e r p r o g r a m can a l w a y s be beaten! A s t r a i g h t f o r w a r d g e n e r a l i z a t i o n of the R e m e s Second A l g o r i t h m to r a t i o n a l C h e b y s h e v a p p r o x i m a t i o n has been p r o p o s e d by F r a s e r and 13 H a r t . R a t h e r t h a n s o l v e t h e l i n e a r s y s t e m P n t x j ) - ( - 1 ) J E = f ( x j ) j = 1 ,2 , . . . n c = n + 2 t h e n o n - l i n e a r s y s t e m - ( - I ) ' E = f ( x j ) j = 1 ,2 , . . . n c = n + m + 2 Q m ( x j ) w h e r e 1 - Q m ° ( x ) = Q m ( x ) h a s c o n s t a n t t e r m = 1 o r r a t h e r , P n ( x j ) - [ fc-O* E - Kx^") <L(*0 H ^ e : ^ i ) i » » > * , - ~ * c i s s o l v e d b y i t e r a t i o n o n E . T h i s a l g o r i t h m w i l l be c o n s i d e r e d i n s o m e d e t a i l i n S e c t i o n V . C ) P r o g r a m m i n g M e t h o d s T h e r e l a t i o n s h i p b e t w e e n p r o g r a m m i n g m e t h o d s a n d a p p r o x i -m a t i o n a l g o r i t h m s i s r e a d i l y s e e n b y c o n s i d e r a t i o n o f the f o l l o w i n g l i n e a r p r o g r a m m i n g a p p r o x i m a t i o n p r o b l e m : P r o b l e m : F i n d t h e n t h d e g r e e C h e b y s h e v a p p r o x i m a t i o n to a c o n t i n u o u s f u n c t i o n f ( x ) , x 6 £- 1, l l . I n o t h e r w o r d s , w e w i s h to c h o o s e j = 0 , 1 , 2 , . . . n s u c h t h a t m a x 1 HA "X 3 - i x 1 - ^ f o r s m a l l e s t p o s s i b l e . 3 A g a i n e q u i v a l e n t l y , w e w a n t to m i n i m i z e ^ u n d e r t he r e s t r i c t i o n s o f t h e 2 ( n - 2 ) i n e q u a l i t i e s : s o o b t a i n i n g a b e s t a p p r o x i m a t i o n o n t he f i n i t e s e t ^ x j ^ , j = l , 2 . . . n c . A s f o r p r e v i o u s l y m e n t i o n e d m e t h o d s , t he ^ x j ^ a r e m o d i f i e d t o i n c l u d e p o i n t s o f m a x i m u m d e v i a t i o n , a n d t h e p r o c e d u r e i s r e p e a t e d , e t c . A l t h o u g h o f s o m e c o n s i d e r a b l e i n t e r e s t , s u c h p r o g r a m m i n g m e t h o d s h a v e o n e o b v i o u s , y e t s e r i o u s d i s a d v a n t a g e : t he e x c e s s i v e - 14 -amount of work required. In p a r t i c u l a r , consider the solution of the above-mentioned problem using the simplex algorithm. As the number of operations (multiplications or divisions) required per vertex using this method or any of its variants i s approximately n 2 , the total maximum number of operations per ite r a t i o n is approximately: as compared to the constant number of approximately: per iteration by solving the corresponding line a r system. An advantage of the above-mentioned types of approximation procedures i s their generality. They can be used i n any type of approximation where a basis: ^ • e.*V \ l,» y . > v V ' ^ o r n t h d e 8 r e e polynomial approximation, can be selected for the n-dimensional subspace of approximations . However, they cannot be applied to approximation by rational functions. F o r rational approximation, one must turn to Concave, Convex and/or Dynamic Programming rather than L i n e a r Programming. This i s an exceptionally recent approach to the approximation problem (the e a r l i e s t papers appearing i n 1958). Consequently, only isolated algorithms have been proposed, so far, without the publication of any numerical r e s u l t s , and moreover, except for [43 (see also below) without proofs of convergence. A ty p i c a l sample of the l i t e r a t u r e i s Loeb [l4l, for approximation using the quotient of l i n e a r forms. Let us now consider the rational approximation algorithm of Cheney and Loeb [53, [43. For (n,m) approximation on a finite set 15 -we make the following definitions: for any r e a l valued vector c = (cj), i = 1,2,..n c =n4m+2 we define: The algorithm i t s e l f i s then defined by iteration on step 2) after the i n i t i a l step 1) i n the following scheme: 1) Choose any c(°) = (c.(°)) with |Co|<| , 6 = > > X> . .. v\c and corresponding Q(c(°),x) > O for Xs£-I>i] 2) Knowing c(^), minimize as a function of c: by means of convex programming, so obtaining a vector c = c ( k + 1 ) . F o r other than a finite set, the minimization of 9) presents formidable d i f f i c u l t i e s . The authors of this method have proven it s convergence [ 4 l . D) Other Methods One class of other methods (known as de l a Vallee Poussin Algorithms) are those which obtain the best approximation as the li m i t i n g case, as the numbers of points increases, of best approxima-tion on a finite set. That i s , suppose (in the nth degree polynomial case) that P n^(x) i s the best (Chebyshev) approximation to f(x) on a finite subset of points s£-»>r] . We want to take: o The methods of finding P n l ( x ) are numerous, see for example Stiefel [l2, pp.2 17-232); such methods being based on "Exchange Theorems" s i m i l a r to the one previously mentioned. Convergence of the de l a Valine Poussin A l g o r i t h m has been shown under quite general condi-tions on f(x) and the approximation subspace [22*]. Rate of convergence - 16 -for optimal selection of the q points has also been considered t.2Z]. De l a Vallee Poussin [20], [22] also proposed an equivalent algorithm; finding at each step the Chebyshev approximation to an equivalent sys-tem of inconsistent equations. The problem on a finite point set has been considered by several authors: Zuhovickii [28], Cheney and Goldstein [8*], [6], Rice (see above), and StiefelC261. The latter has an excellent summary of available methods . As a last method to be considered f rom the class of more or les s r e a l i s t i c proposals, there i s the Polya algorithm, i . e . an algorithm based on the last theorem quoted i n the previous section. - 17 -The R e m e s R a t i o n a l A l g o r i t h m and A s s o c i a t e d P r o b l e m s M e n t i o n has been made p r e v i o u s l y of t h i s n a t u r a l e x t e n s i o n of the R e m e s P o l y n o m i a l A l g o r i t h m to an a l g o r i t h m f o r r a t i o n a l f u n c t i o n a p p r o x i m a t i o n . In m o r e d e t a i l , the method i s as f o l l o w s f o r the Pn(x) a p p r o x i m a t i o n of f by r a t i o n a l f u n c t i o n s ^ V '. . Qm(x) 1) S e l e c t a r b i t r a r y \ xj ^  j = 1, 2 n c = n + m+2 and i n i t i a l E-(°) w i t h — \ * V, < v 4 < . . . <, X ^ < | 2) Set up the n o n - l i n e a r s y s t e m : n n { ? i \ + (-I)' E = f(x.) j = l , 2 , . . . n c Qm(xj) J 3) I t e r a t e o n E ^ , 4= 1,2, ... ( i . e . p e r i t e r a t i o n s o l v e the c o r r e s p o n d i n g n c non-homogeneous l i n e a r equations i n n c v a r i a b l e s ) i n the scheme: P*(x^ E * - teal ^  K->* Eu1, f(x^ . • • \ where f o r u n i q u e n e s s 1 + Qm°(x) = Qm(x) i s t a k e n to have constant t e r m = 1 (for a " b e s t " i r r e d u c i b l e a p p r o x i -m a t i o n the constant t e r m ^ o , o t h e r w i s e t h e r e i s a pole at x=o and .'. we c a n take i t =1 without l o s s of g e n e r a l i t y ) . (0) H o p e f u l l y L i m E v e x i s t s . 4) Move the XJ i n a manner analogous to the R emes Exchange f o r the P o l y n o m i a l A l g o r i t h m ; i n p a r t i c u l a r the o r d e r i n g 10) i s p r e s e r v e d . 5) Repeat step 2) e t c . T h i s p r o c e d u r e i s o u t l i n e d i n F l o w D i a g r a m I, and has been coded i n a v e r s i o n of F o r t r a n II f o r U.B.C.'s I.B.M. 1620. F l o w £ > a ^ & * r x L . N O Sol vs. C o r r y ^ ^ ; « < Y6S Wave. +Wc No - 18 -As far as the programming i s concerned, there are two distinct parts: 1) solution of a system of non-homogeneous equations (this was done by Gaussian elimination with row interchanges), and 2) the di f f i c u l t one - movement of the c r i t i c a l points x j to extrema points of the e r r o r curve. This was done by a stepping procedure - where there i s a set upper bound to the number of steps allowed between any two successive c r i t i c a l points on [-1, l ] . The second flow diagram (II) i l l u s t r a t e s the procedure (for s i m p l i c i t y , with fixed endpoints). Pn(x) Besides the p o s s i b i l i t y of the approximation of f by Q ^ X ) having exactly n c c r i t i c a l points (see the characterization theorem), two basic problems are evident, a satisfactory answer to the f i r s t one being i t s e l f prerequisite to a solution of the second. 1) Under what conditions on f does the iterative scheme 12) converge for a p a r t i c u l a r choice of n, m, x^ , E^°^? 2) Does there exist a convergence proof for this algorithm along the lines of the Novodvorski and P i n s k e r proo or of Rice's proof [23]? Anticipating our discussion of the second question, i t s answer i s , i n brief, no, and in fact, various types of simple counterexamples can be constructed. That we s t i l l might have convergence for some highly r e s t r i c t e d class of functions f, i s s t i l l an open question. A) D i f f i c u l t i e s of the Iterative Procedure Our minimum hope, and as i t turns out, our basic l i m i t a t i o n , i s expressed in the following theorem: Theorem: The non-linear system N o t a t i o n f o r F l o w D i a g r a m II The f o l l o w i n g n o t a t i o n i s u s e d i n the next d i a g r a m : - h denotes the step s i z e i n the stepping p r o c e d u r e to f i n d the poi n t s of m a x i m u m d e v i a t i o n of the c u r r e n t e r r o r c u r v e . Thus i n g e n e r a l , f o r each k, fl^hr \ X ^ - Y ^ \ where n j i s the m a x i m u m number of steps a l l o w e d and xk has i t s u s u a l meaning (the k t h a p p r o x i m a t e c r i t i c a l point on C - l , 1]). - n c = n + m + 2 a l s o has i t s u s u a l m e aning. - k i s c o n s i s t e n t l y u s e d i n denoting x^; j i s u s e d i n denoting steps of magnitude h beyond x^, i . e . x^ 4 jb . - e k j = e (xk 4 jh) = val u e of the e r r o r c u r v e at xk 4 j h . - X L B denotes a " l o w e r bound" on x^ f o r a p a r t i c u l a r k, and i s u s e d to p r e s e r v e the o r d e r i n g p r o p e r t y of the [x^\. It s h o u l d be noted that a l l d e t a i l s of output, t e s t f o r c o n v e r g e n c e , e s t i m a t i o n of e r r o r , e t c . have been o m i t t e d , f o r the sake of c l a r i t y , i n F l o w D i a g r a m I I . P l o w D*i d<^r±w\ NO YES wait 44 A 4 p»r»Wo\.'e. _ NO To < ... . NO - 19 -has at most one solution Q m ( ^ ) with no pole in [-1, l ] (and this solution is then, necessarily irreducible). Proof: Suppose not; i.e. suppose we have two solutions C Pw' (x> , ^  l*> > e' where <#»!to , Qy^(x) both ( P^'to •> M ^ E " have constant term = 1 then since £ ) J l u ^ 0 , #J! (x^ * 0 for any KK « f^Cx^ « fl»> Cxp _ jJ \ upon subtraction so that, p ; ^ Assume E E1 , and without loss of generality, that (E"-B,V) fl^M $ V ! > > 0 throughout [-1, l ] and consider the polynomials: [ Kto <^"w - Kto ^ ' w we note that the former alternately agrees, and agrees except for sign with the latter at at least n c = n-m-2 points in [-1, l ] . Hence P^M - P*' Cx) Q Cx"1) must have at least n c - l = n+m-1 zeros in £- 1, l] which is impossible for E"^ E* > since it has maximum degree m+-n "S 0 which implies E ' E i . '. E is uniquely determined. We appeal to the next Pn(x) theorem (page 2 0) to show that the corresponding ^ ^ ^  is also unique. That the corresponding solution Q ^ ^ ) p' i x \ is irreducible follows immediately since if —• is a $w>-K (.*) solution of the non-linear system, then so also is P V K ^ ^ V * ^ and we can apply our uniqueness result. Hopefully then, our iterative procedure generates this solution, whenever it exists. - 20 -There are then, three problems associated with the iterative procedure: i) the possibility of L i m not existing; ii) the possibility of producing Q^ |^ ) o n some <& th step in the iterative procedure 12) such that Qirf^ Hx ) - 0 for some Xs^-iiQ ( i . e . an "approximation" with a pole in the approximating i n t e r v a l ! ) , this is a s p e c i a l case of the non-linear system 12) having only solutions with poles i n Ca, b3 and wil l be included in that discussion; i i i) the possibil ity of singular l inear systems 12) arising and the associated p r o b l e m of 11) having no solutions at a l l . Before turning to these p r o b l e m s , we first note that a solution Pn(x), Qm(x), E of the non-l inear system 11), if it exists, is generally uniquely determined by the value E . T h e o r e m : If E satisfies the non-l inear system 11) and f (xj )Jp (-l)-'E for at least one j (j = 1, 2, . . . n c ) then E determines Pn(x), Qm(x) uniquely. Proof: E satisfies the non-linear system and therefore there is at least one solution ^ P ^ W ^ ^ l x ) ^ ^ ^ ^ of: Ph *[W&- q w °(x^ *ir*k + n*C> ... 13) which is a l inear system of n c non-homogeneous equations in n c - l variables, and therefore has at most one such solution. M o r e o v e r , any solution of 12) obviously satisfies 13). A s a consequence of this result, we wil l commonly speak of E as being "the" solution of a particular non-linear system 12). The following simple result is also of interest: Pn(x) T h e o r e m : Pn(x), Q m ( x ) , E , (J m (x) irreducible is a solution of 12) iffit is a solution of: P^Cx^+ t-o'e * M * ^ * fCxj) •••'^ - 21 -Proof: We note that 14) is simply a rearrangement of: PwCxO + ( - ^ E < L ( X O * ^ ( x ^ . . . ^ Now if jPn(x), Qm(x), E ^ is a solution of 12) then it clearly is a solution of 13). On the other hand, if ^Pn(x), Qm(x),E^ is a solution of 14) or equivalently of 15) then it will be a solution of 12) if Qm(xj) 0 for all j (j = 1, 2, . . .n c). But if Qm(xi|) = 0 for some j in 15) imply-Pn(x) ing Pn(xj) = 0 so that Qm(x) would be reducible . Turning to the first problem, we see that the instances of LiwsE not existing are numerous. Not only might we get a singular system for some Ji , and therefore in practical application not be able to take the limit,! but even if only non-singular systems do arise, the limit need not exist since its existence is in no way dependent on how near \x.$\ are to the $xj*| . Noticing that: Ph(*£ A- \ t - ^ E ^ *(x0\ <C(*0 + * H*b ^>V-if non-singular jean be considered of the form E^»p(k^ ^\ where F is a rational function of degrees (m,m), the following two theorems shed some light on the situation. For proofs see Montel [163. Theorem: Let ^ 0.^= "j 0*""! define an iterativeprocedure in which we assume there is no round-off, and suppose in the neighbourhood of a double point ft, (a point such that c^ -. ) exists and moreover, |<j*(,S.^| <• I i.e. is "attractive", then for |fl.-4ol sufficiently small, the procedure con-verges to Ot . Theorem: Under the same assumption as the above theorem but with l^ '^ l >> rather than | ^ (K))* | , i.e. 5. is "repulsive", the initial iterates OLo^ , will diverge from 0t . - 22 -Note however that the latter theorem does not say we have ultimate divergence. F o r example, Montel gives a simple counterexample: let g(x) 1 x(x-2)jthen 0 i s a repulsive double point and we have both: , &^  = .^. = 3 and J7 + \ , o.| ' X ? ^-s . * 0 Applying these ideas to (n,m) approximation, our iterative procedure i s defined by: where ^ \ * ^ ^ F r o m which (dropping superscript): I Vt Xi • • • • +1 10 That f can readily be suitably chosen so that \ p '^g^ ^. ) i s more readily seen by considering the special case of (p)\) ', here the iterative scheme 16) i s defined by: ^ • • u r ^ + M ^ ' - V i . e " } . L i , . , . Solving 17) for E ^ \ thus obtaining the form E ^ = F ( E ^ " ' *), and imposing the r e s t r i c t i o n 1 Pl(e)l < I (where F -Fd^ ) ), we obtain the following condition for convergence after some lengthy algebra: •7) - 2 3 ~ E v e n i n t h e c a s e ^ x j ^ = ^xj*"^ , E - E** i t i s e v i d e n t t h a t s u i t a b l y c h o s e n f n e e d n o t s a t i s f y t h i s c o n d i t i o n . W e n o w t u r n to t h e l a s t - m e n t i o n e d p r o b l e m . T h e i t e r a t i v e s c h e m e 13) g i v e s r i s e o n a n y p a r t i c u l a r s t e p ( the s u p e r s c r i p t h a s b e e n d r o p p e d ) t o a d e t e r m i n a n t o f c o e f f i c i e n t s o f t h e f o l l o w i n g f o r m : I X x X? • • • X^ Uth i x^ '1> It i s w e l l k n o w n t h a t t h e d e t e r m i n a n t 19) ^ O i n t h e p o l y n o m i a l c a s e ( m = o ) . T h e d e t e r m i n a n t t a k e s t h e f o r m : I Xj y * Y? " I . 10) 1 Xvwl ' ' w h e r e - I 4 X, < Xt< X-\< . - . < X^ 4 4 E x p a n d i n g t h i s d e t e r m i n a n t b y t h e l a s t c o l u m n , w e n o t e t h a t i t i s t h e s u m o f n + 2> V a n d e r m o n d e d e t e r m i n a n t s , a l l h a v i n g t h e s a m e s i g n ( (-0* ) d u e t o t h e o r d e r i n g o f t h e ^"X^ a n d h e n c e t h e d e t e r m i n a n t 2 0 ) *f 0 . B e f o r e c o n s i d e r i n g t h e m o r e g e n e r a l f o r m 1 9 ) , l e t u s c o n s i d e r w h e r e ?M s S W f +, X f > ^ s Z ^ t l X 1 w - 24 -By an i m p l i c i t function theorem e.g. Kantorovitch QtQ we know that 21) does not have a solution ^ t>|» ^ ^ E ^ i f the Jacobian of the system vanishes at ^ Is^a^E^ . Now since, Jacobian = det where C-^£-^i*. l-o'e ^ ( x i ) we note the vanishing of the above determinant,22) i s (assuming JQm(x)| •>£ ^  XeC-'iB ) simply a generalization on the problem presented by 19). However, the property exhibited by 20), for m=o, does not generalize to V*> | i n either of these cases, as can eas i l y be seen by taking a Laplace expansion of the last m-1 columns. That i s to say, not only might an inverse f a i l to exist on a pa r t i c u l a r iteration, but also no ultimate solution need exist. B) D i f f i c u l t i e s of the O v e r a l l Procedure In order to discuss why we do not have convergence for the Remes Rational Algorithm, apart f r o m those reasons associated with the non-linear system, we w i l l consider the problem i n the light of the convergence proof of Novodvorski and P i n s k e r Cl8] and Rice [23] for the Second A l g o r i t h m of Remes. Definition: The set \ ^ C x u . i s said to form a Chebyshev set in C- 1, l ] i f the difference U ^ x i - U ^ (U^rf 1M\ having more than n zeros i n [-1, l ] i m p l i e s flj = 4.'c, or equivalently, i f ( C c £ ) -(<Pc i s non-singular for every set of distinct X J . - 25 -The above-mentioned proofs hold for sets of unisolvent functions (Rice 23, p.3-55), which are slight g e n e r a l i -zations on Chebyshev sets (in p a r t i c u l a r LO^c,*^ need not be li n e a r i n the ^(x -) ). S p e c i f i c a l l y , the following two properties are required: 1) If A,tt)-^ (£> ha s more than n zeros (counting m u l t i p l i c i t i e s ) then A ,W »_ 2) Any n+1 points S (fc J „ „ A ^ a r b i t r a r y r e a l numbers uniquely determines a A(fc)eJl* which consequently (Rice 23, p. 3-56 ) depends continuously on the points i n the following sense: let correspond to Cfcij&ii) '>*•>•"*+\ then for a l l 6>o there exists an -v\>o such that: |Ai-A^ | * ^ ) for a l l tat-»»0 In order to generalize 1) to a two parameter interpolation class of rational functions -XL^ -*^  jjp-j^ w e rnust r e s t r i c t ourselves to i r r e d u c i b l e rational functions of degrees (n,m) which have no poles i n the approxi-mating i n t e r v a l (the hope that a convergence proof might go through even when "pole positions" are chosen as c r i t i c a l points has not been sub-stantiated even by any numerical attempts). The major dif f i c u l t y a r i s e s when we attempt to have a property analogous to 2) for our interpolation c l a s s . The class of i r r e d u c i b l e rational functions, as a whole, fa i l s not only because a given set may not correspond to a rational function Qm(x)' also, because i f a solution does exist, it may have poles i n the i n t e r v a l , e.g. for interpolation in Slo^\ on C-1, l ] , the unique rational function (with constant t e r m i n denominator = 1) corresponding to:a) (-1,1), (1,1) is non-existent, b) (0, -1), (1, 1) i s ~ — which has a pole at x = 1/2. 1 - 2x i r t i c u l a r c h o i c e of \ e.g. =. z e r o s of TAJX) , i t i s not known T h e r e a r e two o b s t a c l e s c o n c e r n i n g the p o l e s i n £-1, l ] : 1) f o r any \ > fc $ a s u i t a b l y "bad" f can be c o n s t r u c t e d s u c h that the i n i t i a l n o n - l i n e a r s y s t e m has only s o l u t i o n s w i t h p o l e s i n the i n t e r v a l ; 2) i f a g i v e n continuous f u n c t i o n f does i n fact have a best Pn(x) Qm(x) a P P r o x ^ m a t i Q n w i t h n c c r i t i c a l p o i n t s , then t r i v i a l l y , by s e l e c t i o n of the c o r r e c t s o l u t i o n of the n o n - l i n e a r s y s t e m , o u r o v e r a l l p r o c e d u r e c a n p r o d u c e t h i s a p p r o x i m a t i o n f r o m the i n i t i a l s e l e c t i o n : \.*^ " \*i>*\ , E^ 0) = E * . H o w e v e r , c o n v e r s e l y , s t a r t i n g w i t h a pa] f o r what continuous f we w i l l a l w a y s have one p o l e - f r e e s o l u t i o n throughout o u r i t e r a t i v e p r o c e d u r e . In c o n c l u s i o n , a p a r t f r o m the r e a s o n s a s s o c i a t e d w i t h the ( n o n - l i n e a r s y s t e m , f a i l u r e of the R e m e s R a t i o n a l A l g o r i t h m to con-v e r g e i s evident f o r o t h e r than a h i g h l y r e s t r i c t e d , and so f a r unknown i n t e r p o l a t i o n c l a s s . We w i l l now t u r n o u r d i s c u s s i o n to two o t h e r p r o b l e m s ( T i m i n g and S t a b i l i t y ) whose r e l e v a n c e should be c o n s i d e r e d i n the l i g h t of m u c h m o r e e n c o u r a g i n g r e s u l t s i n the a l g o r i t h m ' s p r a c t i c a l i m p l e m e n t a t i o n (see next s e c t i o n ) than i s o t h e r w i s e evident f r o m t h i s s e c t i o n . C) O t h e r P r o b l e m s - T i m i n g It i s the p u r p o s e of t h i s d i s c u s s i o n to make a c o m p a r i s o n of the w o r k r e q u i r e d to o b t a i n r a t i o n a l a p p r o x i m a t i o n s , as c o m p a r e d to p o l y n o m i a l a p p r o x i m a t i o n s . A s the d e t a i l e d e x p e n d i t u r e of t i m e (work) to o b t a i n an a p p r o x i -m a t i o n i s d i r e c t l y dependent upon the s p e c i f i c type of d i g i t a l c o m p u t e r use d , t h i s p r o b l e m ca n m o r e p r o f i t a b l y be p h r a s e d i n t e r m s of - 27 -machine operations (multiplications and divisions), and as a separate category, function evaluations. The number of machine operations is determined as follows: Pn*(x) if in the course of determining a best approximation — to f, Qm v(x) nm moves of the critical points are required, each taking an average of iterations on the non-linear system, and the maximum number of steps being allowed in the moving procedure is n g (see page 18), then: 1 -1) Approximately "n^ncpij^ machine operations are required for the solution of the non-linear systems . 2) The approximate minimum/maximum number of machine operations for moving the critical points are respectively 3nc nm and n s n c ^ n m ; we will denote average by h&^t^v^ 3) The number of function evaluations required for setting up the non-linear systems is approximately n c n(jn m . 4) The approximate minimum/maximum number of function evaluations needed in moving the critical points are respectively: 3 n c n m and n s n c n m (average denoted ^s^c^ ). It should be understood that the above figures are realistically crude, as an actual individual element (n c , n m , n )^ may for various reasons differ by 1 or 2, thus making the above values more nearly correct for larger n, m values, where more useful approximations are obtained. Suppose further that a function evaluation can be considered as requiring k (constant) machine operations. The total number of machine operations required is then given approximately by: NOP = ^ ^ [ v ^ U ^ d 4 K}M+SB\] ..-23) - 28 -The following conclusions are arrived at from 23): 1) Except for extremely low degree approximations, So that k needs to be at least comparable to n c (where n c is typical of an approximation having practical utility) for the function evaluations to have any significant lien on the total computing time (see Table II). 2) As a consequence of 1), if k is small, there is considerable advantage in keeping n<j small (and, if necessary, increasing ^ s ) . This advantage is lost if k is large. 3) is a function only of n+- m and not of n or m individually. If n m were independent of m (the degree of Qm(x)), approxi-mations lying along lines of constant n + m (see Numerical Section) could be obtained with equal work. However as n m increases somewhat with m, approximations with small m are always most easily obtained. Other Problems - Stability We will speak of the Rational Algorithm as being stable (Maehly T l 5 ] ) for particular n, m, wordlength L and particular move k, if up to this move, the mean absolute error at the kth approximate critical points has strictly increased. Moreover, in a practical sense, we will speak of it as being convergent (for a particular n s value) if on some move, no critical points x^  can be moved with this value of n s . If for particular f, m, n, n s the algorithm converges, a basic and difficult unsolved problem is to determine a pr ior i tke. mini-mum wordlength required in order that it is stable for all moves up Table. H Co vmpAr! sov\ of K i h t •for e>* •> (see Hxf p^t 19) r'U&WiAe. oper*"V;o/\s fcre, el&Hrr*;h.eJ * s s * « i i * ^ c&/\vV*»>H Series arc sWe^. (= V-z) bo A n d *V\tAi«*W>«« 0 3 3 . 1 &7AW>° f H >T\(o X|o"' 4-5" . 8?3xtox & " f . xio1* 5* 7 5 0 ^ZWxiS4 7 «* 7 • W x l o * 8 10 * 8 M "* . 2 J o * l o 7 7 1* \o 10 II n e x c l u s i o n k fit ^c. for tAse-U ^PproXmyHoviS °F «^ (=ws (»*!^ ) ( s"«^ • M- 7 • 155" to* 7 ip 9 .135" lo1 0 «Z eve*. We-fter ^pproxi^^ohS K < ^ c - 29 -to and i n c l u d i n g the one on w h i c h i t does c o n v e r g e . C o n s i d e r a p a r t i c u l a r l i m i t i n g c r i t i c a l point x* and the e r r o r c u r v e e(x) i n the neighbourhood of t h i s p o i n t . If: ~ ~ !>'> a —T- i s the a p p r o x i m a t e m i n i m u m step s i z e , and b i s the c o m p u t e r base, then i n s t a b i l i t y can be e x p e c t e d . T h i s c o n d i t i o n i s of c o u r s e t r i v i a l i f h i s s u f f i c i e n t l y s m a l l and E''^ < b , but i t can r e a d i l y o c c u r f o r much l a r g e r E * f . We have e(x) = f(x) - • so that, a s s u m i n g C and t a k i n g : L H > X*X* (which i s not n e c e s s a r i l y t r u e f o r x5,i an endpoint of [-1, we o b t a i n the f o l l o w i n g a p p r o x i m a t e c o n d i t i o n f o r s t a b i l i t y : w here i t i s to be u n d e r s t o o d that P,Q a r e r e s p e c t i v e l y P n * ( x ) , Qm*(x), and the f u n c t i o n s a r e e v a l u a t e d at x* . L i t t l e c an be d e t e r m i n e d f r o m 24) although i t w i l l be noted that S v a r i e s d i r e c t l y w i t h n s ( e v e r y t h i n g e l s e r e m a i n i n g the same) - the l a r g e r n s i s the g r e a t e r the p o s s i b i l i t y of i n s t a b i l i t y (though the g r e a t e r the p o t e n t i a l a c c u r a c y of the a p p r o x i -m a t i o n ) . The p r o b l e m of the dependence of S on n c , L i s m u c h m o r e c o m p l i c a t e d . The d i f f i c u l t y of c o m p l e t e l y r e s o l v i n g t h i s p r o b l e m i s evident even i n the r e l a t i v e l y s i m p l e m= o ( p o l y n o m i a l ) case . C o n d i -t i o n 24) i n the case m = o i s : I f'V> - ?*\+>\ > b" L U n f o r t u n a t e l y , to t h i s author's knowledge, the nature of the second d e r i v a t i v e of a C h e b y s h e v p o l y n o m i a l a p p r o x i m a t i o n has not been i n v e s t i g a t e d . - 30 -The p a r t i c u l a r case of e x, x € [-1, l] , n§ = 60 i s i l l u s t r a t e d i n Graph.I, the isopleth L bounding those points (n,m) which are stable for wordlength L,, from those which are unstable. i i -3 o - 31 -N u m e r i c a l R e s u l t s w i t h R e mes R a t i o n a l A l g o r i t h m H a v i n g d i s c u s s e d the a l g o r i t h m i n a " t h e o r e t i c a l " s e nse, we w i l l now t u r n to some p r a c t i c a l r e s u l t s . In p r a c t i c e , we note i n i t i a l l y , the a l g o r i t h m i s not n e a r l y as bad as might be thought. M o r e o v e r , i f i t w e r e a l w a y s p o s s i b l e f o r g i v e n f,n,m, to r e a d i l y s e l e c t ^ , c. s u c h that no p o l e s w e re g e n e r a t e d — see p r e v i o u s d i s c u s s i o n — the a l g o r i t h m c o u l d be h i g h l y r e c o m m e n d e d . The next 4 g r a p h s ( G r a p h s II-V) o u t l i n e the stages i n a t y p i c a l a p p l i c a t i o n of the R e m e s R a t i o n a l A l g o r i t h m . The case i n point i s that P ? * ( x ) of f i n d i n g the Qg*(x) a p p r o x i m a t i o n to e x on [-1, l ] c a r r y i n g w o r d l e n g t h 12. A s u m m a r y of v a r i o u s a s p e c t s of t h i s s a mple r u n p r e c e d e s the g r a p h s . In p a r t i c u l a r , i t w i l l be noted, the a l g o r i t h m i s stable u n t i l , i t " c o n v e r g e s " . The f o l l o w i n g g r a p h s ( G r a p h s V I - V I I I ) show r e s u l t s of the a l g o r i t h m ' s use i n a m o r e s u c c i n c t f o r m ; i . e . f o r v a r i o u s f u n c t i o n s g r a p h s of E * vs (n,m) have been c o n s t r u c t e d . A l t h o u g h E*n,m i s a d i s c r e t e - v a l u e d f u n c t i o n , i s o p l e t h s of equal error''" have been d r a w n as i f continuous f o r c l a r i t y . The f o l l o w i n g s i g n i f i c a n t f e a t u r e s of these g r a p h s shoul d be noted: 1) V a r i a t i o n of e r r o r * along a l i n e of constant n + m ( i n t h i s way we a r e c o m p a r i n g the e r r o r t e r m s of a p p r o x i m a t i o n s w h i c h can be e v a l u a t e d , f o r a p a r t i c u l a r a rgument, w i t h a p p r o x i m a t e l y equal w o r k ) . 2) Instances of d i f f i c u l t y w i t h the a l g o r i t h m : a) S i n g u l a r s y s t e m s (denoted by S) b) P o l e s p r o d u c e d i n the i n t e r v a l (denoted by*) - 32 -In the case of the last-mentioned d i f f i c u l t y , the following simple s u f f i c i e n c y condition, together with D e s c a r t e s Rule of Signs, was sufficient to demonstrate i n almost a l l cases that a p a r t i c u l a r " a p p r o x i m a t i o n " had a pole i n C-1, l ] , without actually c a l c u l a t i n g the roots of Qm(x). T h e o r e m : Qm(x) = I + ft4X + 0.3X* + , . . + <W,X has no z e r o s i n [-1, l ] if <L I M < I but ) O^X + <X% *x * .. .+fli**iX*l i\<<$ M«V»U-*|v*|f < I O i l * J? a 5* * ~ ^ • £ is J~ ^  w ~ 2 "2 2 5- ^- - 2 x < X X * * 3 ,9 o -o o _ — ff- ft J— i-O a S~ f^- to cc C> CT" 0— « -^ C1 tO " *. X. * X rt or oo « • _ *- |-» K r* > eo O-vS ~9 ^ j , <r- O O A t, fcj V> In p 0° •o r ~» o tl ui -»• + X c x — * J o 4-O a -si * •4 I k z < > O al 3 erf, a— o 3 to o d »/. «< 0O cr *-o «0 o -a o 3" va o r t i l 1 L v •> ^ 5 0_ <± 4 4. CO cr o o c r In fl" I £ - 33 -C o n c l u s i o n s H a v i n g d i s c u s s e d the Remes R a t i o n a l A l g o r i t h m i n some d e t a i l , and h a v ing found i t c o n s i d e r a b l y l a c k i n g , i t i s now evident that t h e r e i s need f o r s i m i l a r s t u d i e s of other r a t i o n a l C h e b y s h e v a l g o r i t h m s . In th i s way, r e a l i s t i c , c o m p a r a t i v e c o n c l u s i o n s c o u l d be f o r m e d . A s f o r the Remes R a t i o n a l A l g o r i t h m i t s e l f , the f o l l o w i n g m a j o r p r o b l e m s r e m a i n : 1) U s e f u l e r r o r bounds f o r r a t i o n a l a p p r o x i m a t i o n (e.g. i f p o s s i b l e i n t e r m s of h i g h o r d e r d e r i v a t i v e s of f ) . 2) D e t e r m i n a t i o n of whether o r not a r e s t r i c t e d c l a s s of fun c t i o n s e x i s t s f o r w h i c h we have o v e r a l l c o n v e r g e n c e . 3) T i m i n g : to study the dependence of n m on m, the degree o f Q m ( x ) . 4) S t a b i l i t y : the b a s i c p r o b l e m of that s e c t i o n . In p r a c t i c e , f o r those continuous f u n c t i o n s (see g r a p h s ) f o r w h i c h the z e r o s of T n c ( x ) s e r v e d as "good" i n i t i a l s ets > a d i s t i n c t c o n c a v i t y was o b s e r v e d i n the e r r o r c u r v e s — j u s t i f y i n g to some extent the i n t e r e s t i n r a t i o n a l a p p r o x i m a t i o n . It i s o ur f i n a l c o n c l u s i o n that the i n h e r e n t d i f f i c u l t i e s i n the use of the R emes R a t i o n a l A l g o r i t h m a r e e x c e s s i v e f o r i t s s t a n d a r d ( i . e . s u b r o u t i n e ) use — to such an extent that t h e r e w o u l d be few t i m e s i n p r a c t i c e when the s l i g h t g a i n i n the degree of a p p r o x i m a t i o n c o u l d w a r r a n t the use of t h i s a l g o r i t h m , r a t h e r than the R emes P o l y n o m i a l A l g o r i t h m . - 34 -Bibliography N.I. A c h i e s e r , Theory of Approximation (English Translation) Ungar, New York, 1956. W. Barth, E i n Iterationsverfahren zur Approximation durch Polynome, Z. angrew. Math, und Mech. 38 (1958) pp. 258-260. P.L. Chebyshev, Collected Papers (in Russian) , French Translation, Chelsea, New York, 1962. E.W. Cheney and H.L. Loeb, On Rational Chebyshev Approxi- mation Num. Math, 4 (1962) pp. 124-127. E.W. Cheney and H.L. Loeb, Two Algorithms for Rational  Approximation, Num. Math, 3 (1961) pp. 72-75. W. Cheney and A. A. Goldstein, Note on a Paper by Zuhovickii Concerning the Tchebycheff P r o b l e m for L i n e a r Equations, S.I.A.M. Journal, 6 (1958) pp. 233-239. W.' F r a z e r and J . F. Hart, On the Computation of Rational Approximations to Continuous Functions, unpublished. A. A. Goldstein and W . Cheney, A Finite A l g o r i t h m for the Solution of Inconsistent L i n e a r Equations and Inequalities  and the Tchebycheff Approximation of Inconsistent L i n e a r  Equations, P a c i f i c J . Math, 8 (1958) pp. 415-427. C. Hastings, Approximations for D i g i t a l Computers, Princeton Un i v e r s i t y P r e s s , 1955. D. Jackson, On Functions of Closest Approximation, Trans. Amer. Math. S o c , 22 (1921) pp. 117-128. L. Kantorovitch, The Method of Successive Approximations for  Functional Equations, Acta Math., 71 (1939) pp. 63-97. R.E. Langer (Ed.), On Num e r i c a l Approximation, University of Wisconsin P r e s s , Madison, 1959. C. Lanczos, T r i g o m e t r i c Interpolation of E m p i r i c a l and Analytic  Functions, J . Math. Phys . , 16 ( 1138 ) pp. 123-199. H.L. Loeb, Algorithms for Chebyshev Approximation Using  the Ratio of L i n e a r Forms, S.I.A.M. Journal, 8 (1960) pp. 458-465. - 35 -15. H.J. Maehly, Rational Approximation for Trancendental Functions, Research Report RC-86 Jan. 16, 1959, I.B.M. Corporation, Yorktown Heights, New York. 16. P. Montel, Lecons sur les Recurrences et l e u r s Applications, P a r i s , G a u t h i e r - V i l l a r s , 1957. 17. F.D. Murnaghan and J . W . Wrench, J r . , The Determination of the Chebyshev Approximating Polynomial for a Differentiable  Function, Math. Tables, A i d . Comp., 13 (1959) pp. 185-193. 18. E.N. Novodvorskii and I. P i n s k e r , On a P r o c e s s of Equalization of Maxima (English Translation), New Y o r k U n i v e r s i t y . 19. G. Polya, Sur un Algorithme toujours Convergent pour Obtenir les Polynomes de M e i l l e u r e Approximation de Tchebychef  pour une Fonction Continue quelconque, C.R. Accd. S c i . P a r i s , 157 (1913) pp. 840-843. 20. C. de l a Vallee Poussin, Lecon sur 1'Approximation des Fonctions d'une Var i a b l e Reele, Gauthier V i l l a r s , P a r i s , 1952. 21. E . Re me s, Sur le ca l c u l E f f e c t i f des Polynomes d 1 Approximation de Tchebycheff, C.R. Acad. S c i . P a r i s , 199 (1934) pp. 337-340. 22. J.R. Rice, On the Convergence of an A l g o r i t h m for Best Tchebycheff Approximations, S.I.A.M. Journal, 7 (1957) pp. 133-142. 23. , The Approximation of Functions, Mathematics Group, General Motors Research L a b o r a t o r i e s . 24. R.C. Selfridge, Approximation with Least Maximum E r r o r , P a c i f i c J . Math., 3 (1953) pp. 247-255. 25. J . Shohat, The Best Polynomial Approximation of Functions Possessing D e r i v a t i v e s , Duke Math. J . , 8 (1941) pp. 376-385. 26. E. S t i e f e l , Note on Jordon E l i m i n a t i o n , L i n e a r Programming and Tchebycheff Approximation, Num. Math. , 2 (I960) pp . 1 - 17 . 27. L. Veidinger, On the N u m e r i c a l Determination of the Best Approxi-mations i n the Chebyshev Sense, Num. Math., 2 (I960)-pp. 99-105. 28. S.I. Z u h o v i c k i i , An A l g o r i t h m for the Solution of the Chebyshev Approximation P r o b l e m i n the Case of a Finite System of  Inconsistent L i n e a r Equations, P a c i f i c J . Math., 8 (1958) pp. 415-427. 

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