Linear adaptive information for approximating a zero of f is studied where f belongs to the class of polynomials of unbounded degree. A theorem on constrained approximation of smooth functions by polynomials is established

Sikorski, Krzysztof A.

English

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study of Linear Information for Classes~of Polynomial Equations CUCS-94-84 by K. Sikorski Department of Computer Science Columbia University New York January 1984 This research was supported in part by the National Science Foundation under Grant MCS-7823676. AD.tract Linear adaptive information for approximating a zero of f is studied where f belongs to the class of poly-nomials of unbounded degree. A theorem on constrained approximation of smooth functions by polynomials is established. Subject Classifications: AMS(MOS): 6SH10, CR: 5.15. 1. summary. For a given positive £ we seek a point x* such that Ix*-a I < c, where a is a zero of a real polynomial p p p in the interval [a,b]. We assume that p belongs to the class Fl of polynomials having a root in [a,b] or to the class F2 of polynomials which are nonpositive at a, nonnegative at b and have exactly one simple zero in [a,b]. The information on p consists of n values of arbitrary linear functionals which are computed adaptively. The point x* is constructed by means of an algorithm which is an arbitrary mapping depending on the information on p. We show that if Ie ~ (b-a)/2 then there exists no information and no algorithm for finding x* for every p from F l , no matter how large the value of n. This is a stronger result than that obtained for smooth functions in [7]. For the class F2 we can find a pOint x* for arbitrary p and c. An optimal algorithm, i.e., an algorithm with the smallest error, is the bisection of the smallest known interval containing a root of p. We also exhibit optimal information operators, i.e., the linear functionals for which the error of an optimal algorithm that uses them is 1 2 minimal. It turns out that in the class of nonadaptive information, i.e., ~hen functionals are given simultaneously, optimal information consists of the evaluations of a polynomial at n-equidistant points in [a,b]. This is a stronger result than that obtained for continuous functions in [9, p. 166]. In the class of adaptive continuous information, i.e., ~hen the next continuous functional depends on the values of all previously computed functionals, optimal information consists of evaluations of a polynomial at n points generated by the bisection method. This is a ~ stronger result than that obtained for C functions in [6]. To prove this result we establish a theorem on constrained approximation of smooth functions by polynomials. More precisely we prove that a smooth function can be arbitrarily well uniformly approximated by a polynomial which satisfies constraints given by n arbitrary continuous linear functionals. OUr results indicate that the problem of finding an c-approximation to a real zero of a real polynomial is essentially of the same difficulty as the problem of finding an €-approximation to a zero of infinitely differentiable function, see [6,7]. This makes the results of [6] and [7] stronger. We stress that we did not assume the knowledge of the degree of a polynomial. The problem of finding an c-approximation to a zero of a polynomial of known degree has been studied in many recent papers, e.g., [1,2,3,4,8] . 3 4 2. B .. ic definition. and results. Let P - P[a,b] be the set of all real polynomials on the interval [a,b] in a, let S(p) be the set of all zeros co co of p in [a,b] for pEP, and let C - C [a,b] be the space of infinitely differentiable functions in [a,b]. Define two subclasses Fl and F2 of P by (2. I) FI = (p E P: S(p) ~ J', lip!! ~ I}, co where 11.11 is an arbitrary seminorm. in C and PEP: pea) ~ 0, pCb) 2 0, (2 .. 2) S(p) is a singleton and f' (S(p» 1= 0 For a given £, £ > 0, define the set (2.3) S (p, c) = (x e [ a, b]: di s t (x, S (p» < f!}, Vp e p. The set S(P,e) is of course not empty for p E Fl U F2 e The problem is to find an t-approximation to a zero of a polynomial p from Fl or F2 , i.e., a point x* such that (2 .4) To find x* satisfying (2.4) we use an information operator N and an algorithm ~ using N. These are defined n n as in [gr- _ 00 Let-~ € C and (2.5) where y. = L. (f~Yl"" ,Yo 1) ~ ~ ~-and (2. 6) df 00 L. f{') = L'{·~Yl'···'Y' 1): C ~ It ~, ~ ~-is a linear functional, i = 1,2, ... ,n. If L. f{') = L. (.), ~, ~ ~i, i.e., L. f does not depend on the previously computed ~, values Yl""'Y i - l the information operator ~ called nonadaptive~ otherwise it is called adaptive. The total 5 number of functional evaluations n is called the cardinality of N . n Knowing N (p) we approximate x* by an algorithm ~. n ~ which is a mapping (2.7) CD': N (F.) ~ [a,b], ~ n ~ i = 1,2. The error of the algorithm~. in the class F. is defined by ~ ~ (2 • 8) e (~;) = sup dis t (s (p) ,~. (N (p»). ... ~ n pEF. ~ Thus x* = CD' (N (p» satisfies_(2.4) for every ~ n iff e(~.) <~. Note that (2.8) can be restated as ~ in F. ~ (2. 9) where the local error e(~i'p) is given by (2.10) Define the radius of the information-operator N (briefly n radius of information) by (2" 11) r(N ,F.) = sup n J. €F P . J. r.(N ,p), J. n where the local radius r. (N ,p) is given by J. n (2 • 12) --= N (p)}. n Let I. = I. (N ) be the class of all algorithms of the form ~ 1 n (2.7) using the information operator N. It is obvious that n (2 . 13 ) and (2 • 14) inf ('D.el, 1 l. e«('D"p) 1 = r.(N ,p), l. n e«('D') = r(N ,F.). l. n l. 'TIp e F. ~ We are interested in algorithms for which the error e(~.) 1. is minimal. (2 • 15) o An algorithm ('D. is optimal iff l. o e(~.) = r(N ,F.). l. n 1 7 The radius of information measures the strength of an information operator. We can solve the problem (2.4) iff r(N ,F.) < c. For a given n we want to find the functionals n l. in (2.5) such that the radius of information is minimized. More precisely, let ~n be a class of information operators with cardinality at most n. Then the information operator o 0 . . 1 ;ff N , N e ~ l.S optl.ma • n n n (2. 16) o r(N ,F.) = n l. inf r (N, F. ) . l. Ne"n In this paper, we solve the following problems: (2.17) (2 • 18 ) In Section 3 we prove that if e < (b-av2 then there exist no information and no algorithm for finding x* for every p from F l , no matter how large the number n of functional evaluations·. This is a stronger result than that obtained for the class of infinitely differentiable functions in [6]. In Section 4 we prove that the optimal nonadaptive information for solving (2.4) in the class F2 consists of evaluations of a polynomial at n equidistant points. in [a,b]. This is a stronger result than that obtained in [9, p. 166] for (2.19) the class of continuous functions changing a sign at the endpoints of [a,b]. In Section 5 we first prove Theorem Sol which is of intrinsic interest. Namely we assume that N n of the form (2.5) is continuous, i.e., that i = 1,2, ... ,n, and show that for an arbitrary 00 function fEe and arbitrary N there exists a n 8 polynomial P having the same information as f, Nn(P) = Nn(f), such that IIp- f lloo and IIp'-f'IIco are arbitrarily small. Using Theorem 5.1 we prove that the optimal adaptive continuous information for solving (2.4) in the class F2 is the evaluation of a polynomial at n points generated by the bisection method (Theorem 5.2). This is a stronger result than that obtained in [6], assuming continuity of information. We also stress that using the same proof technique as in the proof of Theorem 5.2 one obtains Theorem 4.1 of [10] for the case of real polynomials and continuous information~ 3. Class F 1-In thi. section we show that there exists no infor-mation and no algorithm to solve (2.4) in the class Fl with c < (b-a)/2. A similar result was established in [7] 9 for the class of infinitely differentiable functions. Here we present a sketch of the proof, since the idea is similar to that presented in [7]. Namely we prove Theorem 3.1: (3 • 1) for arbitrary n of the form (2.5). and arbitrary adaptive information N n proof: Setting ~(N(p)) = (a+b)/2 we get e(~) ~ (b-a)/2. o Thus r(Nn,F l ) ~ (b-a)/2 due to (2.14). To prove the reverse inequality we construct for every y, 0 < y < (b-a)/2, .. IIW two polynomials p and p from Fl such that N (p) = N (p) n n -- .. and dist(S(p),S(p» 2 b-a-2y. Then (3.1) will follow from (2.11) with y ~ o. Construction of the polynomials p and is ~ similar to the construction of functions f and f from [7, section 2]. as in [7], i. e. , Define the functions h., i = 1,2, ... ,n+l ~ 4 2 2 exp(l6«n+l)/y) )eJCp(-l/«x-Xi _ l ) (x-xi») if x E [xi_1,X i ] o otherwise, where x. - a + iy/(n+l), i - O,l, ... ,n+l, and y is an 1. arbitrary number, 0 < y < (b-a)/2. Let p. be the polynomials such that 1. -2 max , Pi (x) - h i (x) , ~ 10 I ( n+ 1) · xe[a,b] Let d = max(IIIll, max lip· II), and take a positive 6 such I~i~n+l 1. that {II (4 (n+l) d) ! < +00 if d > 0, if d = O. Applying N to the constant polynomial ~(x) = ~ we get n the information operator N , see (2.6) n,~ N (p) = [L1 (p), ... ,L (p)]. n,! ,6 n,~ Let c = [c1' ... ,cn+ l ] be a nonzero solution of the homo-geneous system n+l L:. 1 c.L. (p.) = 0, 1.= ~ J, ~ ~ j = 1,2, ... ,n. max Ie. t· l~i~n+l ~ Define the polynomial p* by 10 -2 -2 Then for c € ((1+10 /(n+l))/(l-lO /(n+l)),3] let Pc = [: + cp* (J - cp* if if If d =" 0 then IIp /I = O. If d > 0 then c Observe that -2 -2 P (x.) 2 ~ - 3(n+l)10 /(n+l)~ = 6(1-3·10 ) > 0 c ~ and 11 -2 -2 Pc((~-l +~)/2) ~ ~ - c& (1-10 I (n+l)) ~ 6 (1- (1+10 / (n+l)) < O. Thus p has a zero in [a,b]. The definition of p implies c c that S(p ) c [a,a+yJ. The polynomial p is defined as c p = p for some c as above. c .... To construct p we proceed as above with x. replaced by ~ x~ = b - iy/(n+l), i = O,l, ... ,n, compare [7]. ~ o Theorem 3.1 states that the error of any algorithm is at least (b-a)/2. Thus if e ~ (b-a)/2 then there exists no algorithm using linear information to solve the problem (2.4). 4. Cla.s F2 - Op.timal Nonadaptive Information. In thi •• ection we prove that the optimal nonadaptive information for solving (2.4) in the class F2 consists of evaluations of a polynomial at n equidistant points in [a,b]. This is a stronger result than that established in [9, p. 166] for the class of continuous functions. Let ~~on be the class of all nonadaptive information operators of the form (2.5) with cardinality at most n. Let N (p) = [p(Xl), ... ,p(x )], n n where x. = a + i(b-a)/(n+l), i = 1,2, ... ,no Let p be 1. an arbitrary polynomial from F2 and j = j(Nn(p» be the index such that P(X j ) ~ 0 and P(Xj + l ) 2 0 where Xo = a, x 1 = b. Then it is clear that a zero of p lies in n+ [X. , x. 1] ·and zeros of all polynomials p having the same J J+ information as p lie in [x., x. 1]. J J+ Thus (2.11) and (2.12) imply that (4. l) Then we prove Theorem 4.1: non "" , i.e., I;n o The information N is optimal in the class n 12 13 b-a 2 (n+l)· o Proof: For arbitrarily small ~ > 0 and information non N (.) = [LI(·), ... ,L (. )], N € ~ , we construct two n n n n polynomials PI and P2 from F2 such that Nn(P I ) = Nn (P2 ) and dist(S(P l ),S(P2 » 2 (b-a)/(n+l) -!. Then Theorem 4.1 will follow from (2.11), (2.12) and (4.1) with I ~ o. Let a = [aO, ... ,an] be a·non-zero solution of the homogeneous system of n linear equations with n + 1 unknowns: n i i:. ° a.L.(x) = 0, ~= ~ J Define the polynomial p(x) n i = b. 0 a.x ~= ~ j = 1,2, ... ,n. Since p is of degree not larger than n there exists a subinterval [c,d] of the interval [a,b], a < c, d < b, such that d - c 2 (b-a)/(n+l) - & and p is of a constant sign in [c,d]. without loss of generality suppose that p is positive in [c,d], see Fig. 4.1. 14 1 t{x)- IT (g(x») m- m 2 / Fig~ 4.1 -rn+l Tben take a normed. Chebyschev polynomial t(x} = 2 T (g(x)}, rn see Fig. 4~l, where 9 is a linear transformation of [c,d] onto [-1,1], i.e., g(x) 2 d+c = --x - --, m d-c d-c is sufficiently large odd integer and ~ is sufficiently small positive number, such that the following inequalities hold: -rn+l 2 < min p(x} xe[c,d] t(x} < -lp(x) 1 x e [a, c- iI) (4.2) t(x) > 1 p (x) I x e (d+,.."b] t I (x) > Jp' (x) I x € [c-.,." c] U [d,d+~] c-" > a and d+" < b. The numbers,,, and m exist due to well known properties of Chabyschev polynomials. Define Pl(x) = t(x) + p(x), (4.3) . P2 (x) = t (x) - p (x) . Then N (PI> = N (t) = N (P2) and p. (a) < 0, p. (b) > 0, n n n ~ ~ i = 1,2. Moreover each of PI and P2 has a single and simple zero. S(Pl ) C [c-~,c], S(p2 ) C [d,d+~]. Thus Pi e F 2 ' ¥ i. Since the proof is completed. D IS 16 5. Class F2- Optimal Continuous Adaptive Information~ In this aection we prove that the bisection information Nbis , defined a8 in [6], is optimal in the class of all n adaptive continuous information ~. This is a stronger c result than that obtained in [6], assuming the class of continuous information operators. We first prove co Theorem 5.1: For every function f E C [a,b], information N E ~ and ~ > 0, Y > 0, there exists a polynomial n c W € P[a,b] such that IIw I - f I ILJO ~ y, and (5.2) N (w) = N (f). n n Proof: Recall that N f(e) = [L l f(e), ... ,L f(·)]' see n, , n, (2.6). consider the functionals Li, ... ,Lk which form n c the maximal set of linearly independent functionals among Ll,f, ... ,Ln,f. Since Li, ... ,Lk are linearly independent n 00 and continuous on C [a,b], then they are linearly independent on P[a,b]. Therefore there exist polynomials p~, i = l, ... ,k , ~ n p~ € P[a,b], such that ~ L ~ (p~) = S. ., Vi, j . J ~ ~, J 00 Consider a sequence of polynomials (wm}m=l such that (5.3) as m ~ 00. Since L~ are continuous, then J (5.4) L~(f-w ) ~ 0 as m ~ 00, j = l, ... ,k , J m n and also L. f(f-w ) ~ 0 as m ~ 00, j = 1, ... ,n. For each J, m w define a polynomial p by m m (5.5) k n = ~. 1 L~(f-w ).p~. J= J m J Then L~(p ) = L~(f-w ), Yj, J m J m k k 17 and IIPmlloo ~ i: j ~llLj (f-wm) I' I!Pj !Ioo ~ l~;;n lip; 1l00i:j~11 Lj (f-wm) 1 Conditions (5.3) and (5.4) imply that there exists an index mO such that for every rn 2 rno the following inequal-ities hold: 18 k IIPjllco I:j~l~Lj(f-Wm)' ~~, k max l~l~k n II Pj' 1100 ~~l \Lj (f-wm) , ~ ~, Define the polynomial w* by (5.6) Then L~(w*) = L~(f), j = l, ... ,k and also L. f(w*) = L. f(f), J J n J, J, j = l, ... ,n, which means that N (w*) = N (f). Moreover n n and which means that w* satisfies (5.1) and (5.2). o In [6] the class of infinitely differentiable £unct-ions with simple zeros is studied. We use here the same notation as in [6] and assume that the reader is familiar with the proof technique presented there. Now we are ready to prove Theorem 5.2: bis The bisection information N is optimal in n the claaa ~ , i.e., c (5.7) Proof: For every n c, 0 < e < (b-a)/(2 n), and every information N E ~ we construct two polynomials wI and n c o Then the proof of Theorem 5.2 will follow from (2.11) and (2.12) with ~ tending to zero. Consider the function f constructed by induction in n Lemma 2.2 of [6] with fl in the proof replaced by -2 [ a, a-+1] , - exp (- (x - a - e:l2 ) ) x E 19 (5.8) fl(x) = 0 x € [ a -+1 ' x 1 -~] , -2 [x 1-f, b] . exp{-{x-x l +€!2) ) x E Then as in the proof of Optimality Theorem of [6], construc_t 00 f* and f**, f*,f** E C [a,b], such that Nn(f*) = Nn(f**), each of f*, f** has exactly one, . simple zero a* = S (f* ) , a** = n S(f**) and a** - a* 2 (b-a)/2 - n€. The choice {S.B} of f1 guarantees that f (a) < 0 and f (b) > 0, which yields n n that f*(a) < 0, f*(b) > 0 and f**{a) < 0, f**{b) > O. Let y - min ( f* • (a*) , f* '* • (a* *) } . The number -y is positive, aince f* and f** are strictly increaaing in neighborhoods 1* - (~ - c/2 ,~) and 1'** = (~* ,x~* + cl2) of their zeros. Define U* = (u1 ,u2 ) and U** =(Vl ,v2 ), u* c 1*, U** c 1** to be neighborhoods of a* and a** such that f*'(x) )':.(/2 for X E U* and f** • (x) ) :;/2 for X E U**~ Let y = :;/4 and 20 1 ~ = 2' mine min \ f* (x) " min , f** (x) \ } " xE[a,u11u[u2,b] xE[a, v1 lu[v2 ,b] The definition of f*,f**,U* and u** implies that ! is positive. Applying Theorem 5.1 with the above ~ and y to the functions f* and f** we obtain two polynomials: WI and w2 ' each of them having exactly one simple zero, n distance between these zeros not less than (b-a)/2 - n~ This completes the proof of Theorem 5.2. o 21 Acknowledgements I am indebted to J.F. Traub and H. wozniakowski for their valuable comments on the manuscript and to T. Boult for the remarks on the proof of theorem 5.2. References. [1] Kuhn, H.W., Finding Roots of Polynomials by Pivoting, in Fixed Points: Algorithms and Applications, S. Karamardian (ed.), Academic press, 1977. [2] Kuhn, H.W., Wang, Zeke, Xu, Senlin, On the cost of Computing Roots of polynomials, submitted for publication, 1983. [3] Murota, K., Global convergence of a Modified Newton Iteration for Algebraic Equations, SIAM J. Nurner. Anal. 19, 1982, 793-799. [4] Renegar, J., On the cost of Approximating all Roots of a Complex polynomial, submitted for publica-tion, 1983. [5] Shub, M., Smale, S., On the Average cost of Solving polynomial Equations, to appear in the proceed-ings of the 1982 Rio Conference on Dynamical Systems. [6] Sikorski, K., Bisection is Optimal, Nurn. Math., 40, 1982, 111-117. [7] Sikorski, K., wo~niakowski, H., For Which Error criteria Can We Solve Nonlinear Equations, Dept. Compo Science, columbia University Report, 1983, submitted for publication. [8] Smale, S., The Fundamental Theorem of Algebra and Complexity Theory, Bulletin of the American Mathematical Society 4, 1981, 1-36. [9] Traub, J.F., WoZniakowski, H., A General Theory of Optimal Algorithms, Academic Press, 1980. [10] Waailkowski, G., Any Iteration for Polynomial Equations Using Linear Informatic1n has Infini te Complexity, Theoretical Computex' Science, 22, 1983, 195-208. 22 

Study of Linear Information for Classes of Polynomial Equations

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Study of Linear Information for Classes of Polynomial Equations

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