APPROXIMATE DETERMINATION OF THE RATIONAL FUNCTION OF THE BEST RESTRICTED APPROXIMATION IN HAUSDORFF METRIC

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2 SERDICA Bulgaricae mathematicae publicationes 19 (1993) APPROXIMATE DETERMINATION OF THE RATIONAL FUNCTION OF THE BEST RESTRICTED APPROXIMATION IN HAUSDORFF METRIC P. MARINOV, A. ANDREEV, PL. YALAMOV 1. Definitions and notations. The problem of numerical determination of the polynomial of the best restricted uniform approximation for a given continuous function f on a finite interval Ω = [a,b] is considered in [1]. More precisely, let functions l,f,u satisfy the inequalities l(x) < f(x) < u(x), x Ω. Let K = K(H n ;l,u) = {p : p(x) = ni=0 a i x i ; l(x) p(x) u(x), x Ω} and E(K;f) = inf p K max x Ω f(x) p(x) = inf p K f p. The element q K which satisfies E(K;f) = f q is called polynomial of best restricted approximation ( note that q does not exist always). In practice (digital filters synthesis, [2]) it is important to approximate discontinuous functions by polynomials or rational functions under the restrictions that the approximate element lies between two given functions. For this purpose we use a Hausdorff metric [3]. Using the notations from [4],[5] let R = {x : < x < }, S M (R) = {[a,b] : a,b R, M a b M, M > 0}, A M Ω = {f : Ω SM (R), Ω = [a,b]} be the set of all segment-valued bounded functions in the interval Ω. If f A M Ω, then F(f) AM Ω, F(f;x) = [I(f;x),S(f;x)], where: I(f;x) = lim δ 0 I(δ,f;x) = lim δ 0 inf{y f(t) : t [x δ,x + δ] Ω}, S(f;x) = lim δ 0 S(δ,f;x) = lim δ 0 sup{y f(t) : t [x δ,x + δ] Ω} is called completed graph of f. Let f A M Ω. The functions l and u satisfy: 1. l(x) < +, u(x) >, for x Ω; 2. the sets Ω = {x Ω : l(x) = } and Ω + = {x Ω : u(x) = + } are open; This work is partially supported by the Bulgarian Ministry of Science, Education and Culture, Contract MM 208/92.

3 60 P. Marinov, A. Andreev, Pl. Yalamov 3. l A M Ω, u AM Ω+, where Ω = Ω\Ω, Ω+ = Ω\Ω + ; 4. l(x) < f(x) < u(x), for x Ω. If the functions f,g A M Ω the one-sided Hausdorff distance in Ω, with parameters α(x) > 0, β(x) > 0 from the function g to the function f is defined as in [5]: { } x ξ h(d,f) = h(ω,α,β;g,f) = max min max y η, (x,y) F(g) (ξ,η) F(f) α(x) β(x) and the Hausdorff distance in Ω, with parameters α(x) > 0, β(x) > 0 between g and f is the number H(f,g) = H(g,f) = H(Ω,α,β;f,g) = max{h(ω,α,β;f,g),h(ω,α,β;g,f)}. For a given function f A M Ω we denote E H (f) = E H (K,Ω,α,β;f,l,u) = inf{h(ω,α,β;q,f) : q K(H n ;l,u)} E h (f) = E h (K,Ω,α,β;f,l,u) = inf{h(ω,α,β;q,f) : q K(H n ;l,u)} The elements p,q K are called elements of the best h and H approximation of f if they satisfy E h (f) = h(p,f) and E H (f) = H(q,f) respectively. 2.The algorithm. Let d h (Ω,α,β;p,f;x) = min (ξ,η) F(f) max { x ξ α(x) } p(x) η,, β(x) D h (p,f;x) = D h (Ω,α,β;p,f;x) = sgn[f(x) p(x)] d h (Ω,α,β;p,f;x), sgn[(f p)(x)] = sgn[f(x) p(x)] = +1, p(x) < min{z : z f(x)} 1, p(x) > max{z : z f(x)} 0, p(x) f(x), h(p,f) = h(ω,α,β;p,f) = max{d h (Ω,α,β;p,f;x) : x Ω}, { max{dh (p,f;x), D D r (p;f,l,u;x) = h (p,l;x) + h(p,f)}, sgn[(f p)(x)] = +1, min{d h (p,f;x), D h (p,u;x) h(p,f)}, sgn[(f p)(x)] = 1, D r (p;f,l,u;x) = sgn[(f p)(x)] d r (p;f,l,u;x), r(p;f,l,u) = r(ω,α,β;p;f,l,u) = max{d r (p;f,l,u;x) : x Ω}, Note, that if p K(H n ;l,u), then h(p,f) = r(p;f,l,u). Modifying the algorithm in [1] we suggest the following numerical method of determining the polynomial p K(H n ;l,u), for which E h (K,Ω,α,β;f,l,u) = h(ω,α,β;p,f).

4 Rational Function of the Best Restricted Approximation in Hausdorff Step 0 Set s = 1, X s = {x s i }n+1 i=0, xs i < xs i+1, xs i Ω; X s f = Xs, X s l =, X s u =, X s = X s f + Xs l + X s u, f 0 = f, p 0 0. Step 1 Find p s K - element of the best h-approximation on discrete set X s for ˆf with restrictions ˆl,û, where: for x X s, ˆf(x) = f(x), ˆl(x) = l(x), û(x) = u(x). Step 1.1 Put t = 1, p(x) = p s (x), X f = Xf s, X l = Xl s, X u = Xu, s X = X s. Step 1.2 Define the discrete functions f,l,u on the discrete set X l(x) = ˆl(x) p(x) for x X l ; l(x) = for x X\X l ; u(x) = û(x) p(x) for x X u ; u(x) = + for x X\X u ; f(x) = D h (Ω,α,β;p,f;x) for x X f. Step 1.3 Solve the linear system nj=0 c j (x i ) j + ( 1) i c n+1 = f(x i ), x i X f nj=0 c j (x i ) j = l(x i ), x i X l nj=0 c j (x i ) j = u(x i ), x i X u put p s (x) = p(x) + q(x), e s = c n+1 and define D h (p s,f;x), e max = max x Xf {d h (p s,f;x)}, e min = min x Xf {d h (p s,f;x)} If {e max e min }/e max < ε or t > T max, then go to Step 2 otherwise put p(x) = p s (x), t = t + 1 and go to Step 1.2 Step 2 Find set of points X s+1 f, X s+1 l, Xu s+1 and X s+1 = X s+1 f + X s+1 l + Xu s+1 = {x s+1 i } n+1 i=0 such that D r (p s ;f,l,u;x s+1 i ) = ( 1) i σd r (p s ;f,l,u;x s+1 i ) where σ = sgn[d r (p s ;f,l,u;x s+1 0 )] = ±1, d s i = d r(p s ;f,l,u;x s+1 i ) there exists µ, 0 µ n for which d s µ = r(ps ;f,l,u) and sgnd r (p s ;f,l,u;x s+1 i ) = ( 1) i sgnd r (p s ;f,l,u;x s+1 0 ). Step 3 Define sets of points X s+1 f, X s+1 l, Xu s+1 as follows: Xf s+1 = {x s+1 i : D r (p s ;f,l,u;x s+1 i ) = D h (p s,f;x s+1 i )} Xl s+1 = {x s+1 i : D r (p s ;f,l,u;x s+1 i ) = D h (p s,l;x s+1 i ) + h(p s,f)} Xu s+1 = {x s+1 i : D r (p s ;f,l,u;x s+1 i ) = D h (p s,u;x s+1 i ) h(p s,f)} Step 4 Find the numbers d s max = max{d s i }n i=0 ; ds min = min{ds i }n i=0. If d s max ds min < ε or s > S max, then go to Step 5 else put s = s + 1 and go to Step 1. Step 5 If s S max, then we accept that p s is the required element of the best approximation; stop the computation.

5 62 P. Marinov, A. Andreev, Pl. Yalamov A M Ω+ 3.Convergence of the algorithm. Let Ω, Ω, Ω+, f A M Ω, l AM Ω, u, be given and let us define for δ > 0 the following functions: l(δ;ω,α,β;f,l;x) = max{l(x), I(δα,Ω,f;x) δ β(x)}, u(δ;ω,α,β;f,u;x) = min{u(x), S(δα,Ω,f;x) + δ β(x)}, where I(δα,Ω,f;x) and S(δα,Ω,f;x) are given by I(δα,Ω,f;x) = inf{y f(t) : t [x δα(x),x + δα(x)] Ω}, S(δα,Ω,f;x) = sup{y f(t) : t [x δα(x),x + δα(x)] Ω}. Let for δ > 0 the functions α, β, f, l, u satisfy H(l(δ;Ω,α,β;f,l), f) = H(u(δ;Ω,α,β;f,u), f) = δ. Let the best H-approximation satisfy E H (K,Ω,α,β;f,l,u) = λ δ, then: 1) H(l(δ;Ω,α,β;f,l), f) = H(u(δ;Ω,α,β;f,l), f) = λ. 2) There exists only one element of best H-approximation p K(H n ;l,u) such that E h (K,Ω,α,β;f,l,u) = h(ω,α,β;p,f) = λ = H(Ω,α,β;p,f) = E H (K,Ω,α,β;f,l,u). 3) The sequence of polynomials from H m, which is generated by the algorithm, converges uniformly to the polynomial of the best restricted approximation, i.e. lim = p, p K(H n ;l,u), {e s } increases and s lim = s e = E h (K,Ω,α,β;f,l,u) = h(ω,α,β;p,f). 4. Numerical Experiments. The suggested algorithm was used for calculating polynomials and the rational functions of the best restricted Hausdorff approximation of the following two functions: f 1 (x) = 1, 0 x < 0.25 [0,1], x = , 0.25 < x x 1, 0.50 < x , 0.75 < x , 0 x < 0.5 ; f 2 (x) = [0,1], x = 0.5 0, 0.5 < x 1.0 The restrictions l and u are given by the lines connecting the points: l 1 i x y ε 0 ε l 2 i x y ε 0 ε

6 Rational Function of the Best Restricted Approximation in Hausdorff u 1 i x y 1 + ε 1 + ε u 2 i x y 1 + ε 1 + ε where ε = Parameters of h distance are α(x) = 1, β(x) = 1, Ω = [0,1]. A modification of the above algorithm works in the case of rational functions R m,k = {f : f = p/q, p H m, q H k, q(x) > 0,x Ω} (see [4]). The next table shows the order of the h approximation (denoted by E h ) of the functions f 1 and f 2 for different m and k. The non-restricted approximation Eh o is given for comparison. for f 1, l 1, u 1 for f 2, l 2, u 2 m,k E h Eh o m,k E h Eh o 17,0 NONE ,0 NONE , , , , , , , , , , , , , , , , , , , , , , , , Fig.1 gives the functions f 1, l 1, u 1 and its elements of the best restricted 18 Hausdorff approximation q R 18,0 and p R 5,4. If we denote q(x) = c i x i and { 5 { p(x) = a i x }/ i 4 b i x }, i then: i=0 i=0 i=0 i a i i b i

7 64 P. Marinov, A. Andreev, Pl. Yalamov i c i i c i Fig.2 gives the functions f 2, l 2, u 2 and its elements of the best restricted Hausdorff approximation q R 14,0 and p R 3,3. If we denote q(x) = c i x i 14 and { 3 { 3 p(x) = a i x }/ i b i x }, i then: i=0 i=0 i a i i b i i c i i c i i=0

8 Rational Function of the Best Restricted Approximation in Hausdorff Fig.1 Fig.2 R EFERENCES [1] D. R. Gimlin, R. K. Cavin and M. C. Budge, A multiply exchange algorithm for calculation of best restricted approximations, SIAM, J.Numer.Anal. 11 (1974), [2] Справочник по теоретическим основам радио-электроники, том. 2 (под ред. Б. Х. Кривицкого), Энергия, Москва, [3] Бл.Сендов, Хаусдорфовые приближения, Изд. БАН, София, [4] P.G.Marinov and A.S.Andreev, A modified Remez algorithm for approximate determination of the rational function of the best approximation in Hausdorff metric, Comptes rend. de l Acad. Bulgare de Sciences 40 (1987), [5] P. G. Marinov, Approximate determination of the generalized polynomial of the best one-sided Hausdorff approximation. Matematica Balkanica, New Series 3 (1989), Institute of Mathematics Bugarian Academy of Sciences P.O.Box Sofia BULGARIA Received

Supplementary Notes for W. Rudin: Principles of Mathematical Analysis

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