Expected Behavior of Bisection Based Methods for Counting and. Computing the Roots of a Function D.J. KAVVADIAS, F.S. MAKRI, M.N.

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1 Expected Behavior of Bisection Based Methods for Counting and Coputing the Roots of a Function D.J. KAVVADIAS, F.S. MAKRI, M.N. VRAHATIS Departent of Matheatics, University of Patras, GR Patras, GREECE, and University of Patras Articial Intelligence Research Center{UPAIRC, University of Patras, GR Patras, GREECE. URL: Abstract: -We present a new bisection based ethod for counting and coputing roots of a function in a given interval. Our ethod is focused on very large probles, i.e. instances with nuber of roots of the order of the hundreds. The ethod draws its power fro the fact that the roots are expected to be any, in order to discover a large proportion of the very eciently. Its ain advantage, apart fro its eciency, is its siplicity which aes it a suitable preprocessing step to ore robust and expensive ethods in order to reduce the size of the proble. The algorith is accopanied by a probabilistic analysis of its behavior, which shows that a siple existence criterion lie the Bolzano's rule, can be a powerful tool in the root nding process. Keywords and phrases: Expected behavior, bisection based ethods, counting and coputing the roots of a function, very large probles. 1 Introduction In this paper we present a bisection based ethod for the proble of counting and coputing roots of a single equation: f(x) =0; (1) where f:[a; b] IR IR is continuous in the given interval [a; b]. The ethod is focused on very large instances of the proble (with roots of the order of the hundreds or ore). It is well nown that probles of such size eerge in several elds of atheatics and engineering. On the other hand, it is also well nown that solving such instances is a foridable tas and in soe sense the true testing eld for any root nding ethod. The proposed ethod taes advantage of the abundance of roots, in an attept to very cheaply (in ters of coputing resources) locate a large nuber of roots, thus reducing the size of the proble. The algorith alost blindly searches for roots by eploying only the Bolzano's existence criterion (see next section). It turns out however that even this siple rule is sucient to guide the algorith in discovering a large proportion of the total set of roots. Thus with very few function evaluations, the proble can be considerably reduced. But this is already ore than it could be expected fro such a siple ethod: while new roots are being discovered, the cost gets increasingly high to a point where it becoes unprotable to continue. Then the unsearched parts of the interval ust be searched by a dierent ethod. Thus the algorith ust be viewed as a preprocessing step of a ore robust and expensive ethod. Its ain advantage is its siplicity which follows precisely fro the expectation that with high probability even a siple search for a root will prove successful due to the abundance of roots. In this paper we also study analytically the expected behavior of the ethod and give theoretical justication of its good perforance. We would lie to point out here that recently, there has been a surge of interest concerning the expected behavior of nuerical algoriths [1,2,3,14] and [5, 6, 8, 19]. The traditional approach inevaluating a nuerical ethod, usually involves a nuber of experients on a nuber of inputs, either of individual interest or randoly constructed by altering certain paraeters of the proble. Very few nuerical ethods exist that are accopanied

2 by a robust analysis of their expected behavior. The analysis presented here follows the fraewor of [8]. A by-product of the algorith is an estiation of the nuber of roots in an interval. This proble is interesting on its own, either in order to a priori evaluate the size of a proble or as in this case, to establish a stopping criterion for the ethod. In the next section we give soe bacground aterial on the bisection ethod, as was odied in [15, 16]. In section 3 we briey discuss the ain steps of the algorith in order to present its theoretical analysis which follows in section 4. Then in section 5 we give a ore detailed description of the ethod which also refers to the theoretical analysis. We end in section 6 with soe conclusions and future wor. 2 Bacground aterial A siple oracle on the existence of a solution of f(x) = 0 in soe interval (a; b) where the function f is continuous in [a; b] is the following criterion: f(a) f(b) < 0; or sgn f(a) sgn f(b)=,1; where sgn is the well nown three valued sign function. This criterion is nown as Bolzano's existence criterion (for a generalization of this criterion to higher diensions see [17]). Note that this oracle ay introduce a one{sided error, i.e. a positive response eans that at least one root exists but a negative response ay correspond to the existence of an even nuber of siple roots. The siplicity of this criterion is what aes it attractive even though it has the above disadvantage. Thus it will be our ain tool in what follows. More elaborate relations can give ore inforation on the existence of roots. For exaple, interval analysis uses the range of the function to decide on the existence or not of a root (see e.g. [9, 10, 11]). An even ore coplicated oracle which gives the exact nuber of roots N r is based on topological degree theory using Kronecer's integral on a Picard's extension [4, 12]. This oracle was used in [8] as part of the rst phase of an algorith for the isolation of all siple roots of a function f(x) inan interval (a; b). The algorith of [8] uses in its second phase, bisection in order to copute the roots. Specically, it uses the following siplied version described in [15]: x i+1 = x i + c sgnf(x i )=2 i+1 ; i =0;1;:::; (2) where c = sgnf(a)(b,a): The sequence (2) converges to a root r 2 (a; b) if for soe x i ; sgnf(x 0 ) sgnf(x i )=,1; for i =1;2;:::: Furtherore, the nuber of iterations, which are required in obtaining an approxiate root r such that jr, r j"for soe " 2 (0; 1) is given by: = dlog(b, a) ",1 e; (3) where the logarith in the above relation and also in the rest of the paper is taen with base two. Instead of the iterative forula (2) we can also use the following one: x i+1 = x i, c sgnf(x i )=2 i+1 ; i =0;1;:::; (4) where c = sgnf(b)(b,a): The reason for choosing the bisection ethod is that it always converges within the given interval (a; b) and it is a globally convergence ethod. Moreover it has a great advantage since it is optial, i.e. it possesses asyptotically the best possible rate of convergence [13]. Also, using the relation (3) it is easy to have beforehand the nuber of iterations that are required for the attainent of an approxiate root to a predeterined accuracy. Finally, it requires only the algebraic signs of the function values to be coputed, as it is evident fro (2) or (4), thus it can be applied to probles with iprecise function values. As a consequence for probles where the function value follows as a result of an innite series (e.g. Bessel or Airy functions) it can be shown [18, 20] that the sign stabilizes after a relatively sall nuber of ters of the series and the calculations can be speed up considerably. 3 An inforal description Here is an inforal description of the algorith. The algorith begins by subdividing the interval into a nuber of equal subintervals (it will becoe clear that it is convenient for the nuber of subintervals to be a power of two and its specic value will be xed later). The ain body of the algorith consists of three steps: Step 1. We copute the sign of the function at the endpoints of the subintervals. Depending on the nuber of roots in each subinterval, its endpoints will have the sae signs (in a case of even roots) or opposite signs (in a case of odd roots). We therefore have certainty on the existence of at

3 least one root in an interval with opposite signs, so we proceed in discovering a root in those intervals using ordinary bisection. Step 2. Notice that at this point all intervals in our pattern of subdivisions have the sae sign at their endpoints. Depending now on a stopping criterion, we decide whether it is worthwhile continuing the algorith (in ters of the cost of discovering new roots). Step 3. If we decide that it is, we subdivide the class of subintervals of greatest length into two halves and go bac to step 1. Otherwise we output the discovered roots and halt. 4 Wor estiations Our stopping criterion is based on the expected, per root, nuber of function evaluations that the roots that will be discovered in the next iteration, will require. The probability space on which we ae our calculations and the whole fraewor can be found in [8]. We briey ention here that we view each root as a rando point in the interval (a; b). We also ae the assuption that the roots are uniforly distributed, i.e. intervals of equal length have equal probability of containing a root. Central to our decision turns out to be an estiation of the total nuber of roots in the original interval. This is of no surprise: if there is an abundance of roots in the interval and we have only discovered a few, then it is natural to expect that with high probability the next iteration will reveal a lot of roots. The estiation on the total nuber of roots is revised after each iteration, when new roots have been discovered. Hence at the beginning of the algorith there is only a rough estiation on the nuber of roots, but as we proceed the additional inforation allows us to be ore accurate in our prediction. For our estiation ethod we shall need the following two propositions: Proposition 1 Assue that the total nuber of roots in the interval is N. Then the probability p odd that a subinterval of length ` contains an odd nuber of roots is given by: Proof: See [7]. p odd = 1, (1, 2`)N : (5) 2 Throughout this paper we assue without loss of generality that the given interval is noralized, i.e. its length is 1. Consequently the length ` of a subinterval gives also the probability of a root to lie within this subinterval. Notice that for a nuber of roots N of the order of twenty orso,p odd is very close to 1/2 for sall enough `. Therefore, roughly half of the intervals provably have at least one root. This is natural, since when the nuber of roots is large, the intervals with odd and the intervals with even roots are about the sae. In such a case, after coputing the roots, it is worthwhile continuing by subdividing the largest intervals. When however we have discovered a nuber of roots close to the total, the nuber of intervals with zero roots begin to increase, and this is precisely what we tae advantage of to decide if it is tie to stop. In order however to quantify the above, we use a nown condence interval for the probability p of the binoial distribution. We ay view the faily of subintervals of interest as rando variables which follow, for the property of having or not odd roots, the binoial distribution with probability given by (5). Proposition 2 Assue that at a certain point of subdivision we have subintervals of largest size, out of which we have observed subintervals with opposite signs at their endpoints. Then with probability at least 1,, the probability of having odd roots p odd is between: where: and Proof: See [7]. p lower p odd p upper ; p lower =, z =2 p upper = + z =2 q (,) ; q (,) In the above forulas z =2 is a constant which can be deterined fro: Prob(,z =2 Z z =2 )=1,; where Z is a rando variable following the standard noral distribution. For exaple when = 0:05 then z =2 =1:96. We use the above condence interval to estiate p odd and then using (5) to estiate N. This is done by coputing two values for N, call the N lower and N upper. The rst is coputed by solving the equation (with respect to N): p upper = 1, (1, 2`)N ; (6) 2 :

4 and the second by solving the equation: we get the following: p lower = 1, (1, 2`)N : (7) 2 Notice that in soe cases N upper and even N lower can be innite. This will happen when insucient data are available for deterining exact values for the condence interval. But for our purposes this is iaterial: in either of these cases we continue subdividing the intervals. When after soe iterations, both ends of the condence interval are well dened, wechoose soe preferable value for N. A good choice sees to be (N lower +N upper )=2. Or, weay decide to be on the safe side and choose as N to be N lower. Whichever our choice ight be, we use it to decide whether it is tie to stop. Our criterion is based on the expected wor (in function evaluations) that the next root that will be discovered, will require. Assue that at a certain point our estiation for the total nuber of roots is N. Moreover, the length of the greatest subintervals is ` and their nuber is. (Notice that if this happens at the i-th iteration, ` =2,i ). Now, the wor that the algorith will do at the next iteration consists of (a) the function evaluations at the idpoints of the largest subintervals and (b) if sub-subintervals are discovered with odd nuber of roots, log (`,1 ) function evaluations for discovering a root in these subintervals. The total cost therefore of the next iteration is: + log ` : (8) Now, since the probability ofhaving sub-subintervals with odd nuber of roots is: p odd(1, p odd ), ; we get that the expected cost of the next iteration is: E cost = Since: and X =0 X =0 X =0 + log ` p odd(1, p odd ), : p odd(1, p odd ), =1; p odd(1, p odd ), = p odd ; E cost = + p odd log ` : Taing into consideration that the expected nuber of roots of the next iteration is p odd we get that the expected cost per root, E cost, of the next iteration is: E cost = 1 p odd + log ` : (9) The above expected cost constitutes our stopping criterion. The ost natural choice is to stop once this cost as given by (9) exceeds soe predened threshold c th. The intention is to stop coputing roots using the current ethod if the cost of discovering new roots becoes very high. But there are also other approaches to this end: for exaple we aychoose to end the algorith once a prede- ned \budget" of function evaluations has expired. Current wor involves experients for evaluating c th and the dierent ethods for stopping. 5 The algorith In this section we give a detailed description of the algorith based on the previous discussions. 1. Divide the interval in 32 equal subintervals; 2. Let A be the set of subintervals with opposite signs and B the set of subintervals with the sae signs at their endpoints; 3. Find one root in each interval in A using bisection; 4. Estiate the total nuber of roots using Relations (6) and (7); 5. If both N lower and N upper are nite copute the expected cost of new roots using Relation (9); 6. If the cost is less than the threshold or at least one of the N is innite then divide the subintervals in B and go to Step If the condition in Step 6 fails then output the roots and halt. The initial subdivision in 32 subintervals is forced by the theory supporting the condence interval in Proposition 2. In order for this interval to hold, has to be greater than 30. The subdivision into intervals which are fractions that are power of two is justied as follows:

5 Recall that after we identify an interval with opposite signs at its endpoints, we proceed in discovering a root using ordinary bisection (this is Step 3 above). But this has the side-eect that these intervals will be subdivided further in the process of discovering the root. Now if we choose the initial subintervals to be a power of two, all these subdivisions that eerge fro bisection, can be used in the future should the algorith proceeds to next iterations and thus soe function evaluations can be saved. 6 Conclusion We have addressed the proble of coputing the roots of a function in a case where the nuber of roots is very large. This is a foridable proble with any applications in various elds of science. It sees however possible to attac the proble by taing advantage precisely of its size. To this end we have presented an algorith that eectively discovers roots using an alost \blind" search up to a point where the original size of the proble has been severely reduced. Then, ore robust and expensive ethods can be used to copletely solve the proble. We have also given theoretical justi- cation of its good perforance, based on a probabilistic fraewor. The ain advantages of the proposed ethodology are its siplicity which results in fairly siple prograing and its eciency which increases with the proble size. Current research includes experiental evaluation of a nuber of paraeters in order to better tune the algorith. References [1] S. Graf, R.D. Mauldin, S.C. Willias, \Rando hoeoorphiss", Adv. Math., vol.60, 1986, pp.239{359. [2] S. Graf, E. Nova, \The average error of quadrature forulas for functions of bounded variation", Rocy Mountain J. Math. [3] S. Graf, E. Nova, A. Papageorgiou, \Bisection is not optial on the average", Nuer. Math., vol.55, 1989, pp.481{491. [4] B.J. Hoenders, C.H. Slup, \On the calculation of the exact nuber of zeros of a set of equations", Coputing vol.30, 1983, pp.137{ 147. [5] D.J. Kavvadias, F.S. Mari, M.N. Vrahatis, \Locating and coputing arbitrarily distributed zeros and extrea", 4th Hellenic European Conference on Coputer Matheatics and its Applications, E.A. Lipitais Ed., 1998, in press. [6] D.J. Kavvadias, F.S. Mari, M.N. Vrahatis, \Locating and coputing arbitrarily distributed zeros", SIAM J. Sci. Coput., in press. [7] D.J. Kavvadias, F.S. Mari, M.N. Vrahatis, \Expected behavior of bisection based ethods for counting and coputing the roots of a function", Division of Coputational Matheatics and Inforatics, Departent of Matheatics, University of Patras, Technical Report TR/1999-1, [8] D.J. Kavvadias, M.N. Vrahatis, \Locating and coputing all the siple roots and extrea of a function", SIAM J. Sci. Coput., vol.17, no.5, 1996, pp.1232{1248. [9] R.B. Kearfott, \Rigorous Global Search: Continuous Probles", Kluwer Acadeic Publishers, Dordrecht, The Netherlands, [10] R.B. Kearfott and M. Novoa III, \INTBIS: A portable interval Newton=bisection pacage", ACM Trans. Math. Software, vol.16, 1990, pp.152{157. [11] A. Neuaier, \Interval Methods for systes of equations", Cabridge University Press, Cabridge, [12] E. Picard, \Sur le nobre des racines counes a plusieurs equations siultanees, Journ. de Math. Pure et Appl. (4 e serie), vol.8, 1892, pp.5{24. [13] K. Siorsi, \Bisection is optial, Nuer. Math. vol.40, 1982, pp.111{117. [14] J.F. Traub, G.W. Wasilowsi, H. Wozniaowsi, \Inforation{based coplexity", Acadeic Press, New Yor, [15] M.N. Vrahatis, \Solving systes of nonlinear equations using the nonzero value of the topological degree", ACM Trans. Math. Softw., vol.14, 1988, pp.312{329. [16] M.N. Vrahatis, \CHABIS: A atheatical software pacage for locating and evaluating roots of systes of nonlinear equations", ACM Trans. Math. Software, vol.14, 1988, pp.330{ 336. [17] M.N. Vrahatis, \A short proof and a generalization of Miranda's existence theore", Proc. Aer. Math. Soc., vol.107, 1989, pp.701{703.

6 [18] M.N. Vrahatis, T.N. Grapsa, O. Ragos, F.A. Zaropoulos, \On the localization and coputation of zeros of Bessel functions", Z. angew. Math. Mech., vol.77, 1997, pp.467{475. [19] M.N. Vrahatis, D.J. Kavvadias, \Expected behavior of bisection based ethods for counting and coputing all the roots of a function", Proceedings of the Sixth International Colloquiu on Dierential Equations, D. Bainov, Ed., VSP International Science Publishers, Zeist, The Netherlands, 1996, pp.353{360. [20] M.N. Vrahatis, O. Ragos, F.A. Zaropoulos and T.N. Grapsa, \Locating and Coputing Zeros of Airy Functions", Z. angew. Math. Mech., vol.76, 1996, pp.419{422.