Validating an A Priori Enclosure Using High-Order Taylor Series George F Corliss and Robert Rihm Abstract We use Taylor series plus an enclosure of the remainder term to validate the existence of a unique solution for initial value problems in ordinary di erential equations and to compute a coarse enclosure of that solution The signi cance of this result is its application to Lohner's AWA algorithm for validated solutions, not to the theory of ordinary di erential equations By using high-order Taylor series in Lohner's Algorithm I, we are able to validate the solution over much longer time steps than is done in the current AWA code For Lohner's enclosure by polynomials, the enclosures are expensive to compute, but it is easy to check for enclosure For our enclosure by Taylor series, the enclosures are free because they are already being computed, but checking for enclosure requires n polynomial root ndings Work is continuing on an implementation that will allow direct computational comparisons of the e ectiveness of the two methods Keywords: ordinary di erential equations, Lohner's algorithm, validated computation, Taylor series, enclosure methods Introduction The AWA (Anfangswertaufgabe) program by Rudolf Lohner 4, 5, 6, 7, 8] computes an enclosure of the solution of an initial value problem (IVP) in ordinary di erential equations (ODE) u0 = f (u); u(t0 ) = u0; (1) where we only know an interval enclosure u0] of the vector u0 in general Without loss of generality, Lohner assumes the system is autonomous only to simplify the proofs We assume that f is at least p ? times continuously di erentiable in a domain D with u0] D IR , p Then there is a unique at least p times continuously di erentiable solution u(t) in a neighborhood of t0 AWA is a single-step method At each integration time step, AWA applies two algorithms: n Algorithm I: (Existence and enclosure) Find a step size h and a coarse enclosure interval u ] D such that for t t ] := t ; t + h], the solution u(t) exists and satis es u(t) u ] 0 0 George F Corliss and Robert Rihm Algorithm II: (Tightening) Compute a tight enclosure for u(t) at t = t := t + h Throughout this paper, we use the term \validation" to mean \prove existence of u(t) for some t and nd a coarse enclosure u0]" The e ciency of AWA has been limited by the Euler step size enforced by Algorithm I as used in the current program We show that Taylor series plus an enclosure of the remainder term can be used not only in Algorithm II to tighten the enclosure, but also in Algorithm I for the validation step The consequence of this result for AWA is that Algorithm I can validate the existence of the unique solution over time steps appropriate for high-order methods employed for tightening during Algorithm II The basic idea of AWA is to enclose the Taylor coe cients and the remainder term of the solution It goes back to Moore who presented his method in 1965 (see 10, 11]) The Taylor coe cients are computed using recurrence relations derived from the ODE: f (0) (u) := f (u); (u)0 := u; ( ?1) ( ) (2) f (u) := @f@u f (u) ; i = 1; 2; : : :; p ? ; (u) := f ( ?1) (u); i = 1; 2; : : :; p : i The recurrences given by Equation (2) are not as intimidating as they might appear; this is just an expression of the computation of the Taylor coe cients for the solution They can be evaluated by using automatic di erentiation (see e.g 12]) If we begin the recurrences given by Equation (2) with the initial value u0 and compute in exact arithmetic, we get the Taylor coe cients (u0 ) for the solution expanded at t = t0 If we begin the recurrences given by Equation (2) with an interval vector u0] and compute in rounded interval arithmetic, then we get enclosures ( u0]) of the Taylor coe cients for all solutions starting from u0 u0] We can also begin with a coarse interval u0] and compute the interval vectors ( u0]) Then (b ? t0 ) ( u0]) contains the remainder of the p ? 1st degree Taylor polynomial t of u(b), if u(t) u0](t) for all t t0 ; b] t t We emphasize that the computation of these values does not pre-suppose the existence of a solution We are just computing sequences of numbers If we cannot compute any of these values, then we are not able to proceed with validation However, if we know an a priori enclosure u0] of the solution on t0], then i i i i i i i p u(t) u(t) p ?1 X p i=0 p (t ? t0) (u0 ) + (t ? t0 ) ( u0]) =: T (t; u0; u0]) or ? X i=0 i p i p p (t ? t0) ( u0]) + (t ? t0 ) ( u ]) = T (t; u0]; u ]), resp i i p p p (3) Validating an A Priori Enclosure for t t0], if only the interval polynomial T stays in u0] Moore as well as Lohner use (modi cations of) this formula to tighten an a priori enclosure u0] (see e.g 13]) p Current AWA Approach Validation and the computation of a tight enclosure are separate, though related, issues This paper addresses only the issue of validation, the role of Lohner's Algorithm I The techniques of his Algorithm II for tightening the enclosure are discussed in 4, 5, 6, 7, 8] Algorithm I uses the following theorem for validation Theorem ( 6]) Let u ] D be an interval vector satisfying ? u1] := u0 + 0; h] f u0] Then for t t0 ] = t0; t0 + h] u0 ] : u(t) u1] (and hence u(t) u0]) It follows directly from Theorem Let u](t) and v](t) D be interval vector valued functions, and let each p times continuously di erentiable function v(t) v](t) satisfy Zt (v(t)) := u0 + f (v( )) d u](t) v](t) (4) t0 for t t0 ] Then the solution u of Equation (1) satis es u(t) u](t) for t t0 ] Proof: If (4) holds and we start a Picard-Lindelof iteration u (t) = (u ?1)(t)) = u0 + (k ) Zt (k f (u( ?1) ( )) d ; k = 1; 2; : : : k t0 with a p times di erentiable function u(0) (t) v](t), then every successor is again p times di erentiable and lies in u](t) However, the sequence of Picard iterates should leave this interval function if the solution did so 4 George F Corliss and Robert Rihm It seems strange to let v(t) be p times di erentiable However, this assumption is used in the following section AWA nds a step size h and an interval u0] such that ? ? u0] u1] := u0] + 0; h]f u0] u] ; (5) for all t t0 ] := t0; t0 + h], and hence validates that the solution of Equation (1) exists on t0] and is contained in u0] Figure shows the a priori bound u0] = 0:2; 0:5] and ( u0 ]) for the logistic equation u0 = u(1 ? u), u(0) = 0:3, on the interval t 0; 0:6] In this gure, ( u0]) u0] in the interval t 0; 0:5] If we tighten u0] = 0:29; 0:5], then ( u0]) u0] in the larger interval t 0; 0:563] 0.6 0.5 0.4 u 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 t Figure A priori bounds with u0] = 0:2; 0:5] Lohner's Enclosure of u(p) The allowable step size h for which validation can be done using Theorem as described in Section is limited to a step appropriate for Euler's method, no matter how high the order of the method used during Algorithm II to tighten the enclosure Stetter showed in 14] that for the problem u0 = Au, A IR , the step size h obtained by applying Theorem may be multiplied by the degree p of the Taylor n n Validating an A Priori Enclosure polynomial used in Algorithm II Algorithm II yields a validated tight enclosure for t = t0 + ph because p!( u0]) = A u0] contains the p-th derivative u( ) (t) for all t t0; t0 + ph], even though u0] does not enclose the solution on this enlarged interval, in general Lohner 9] suggests using coarse enclosures for higher derivatives also in the general case Actually, he uses Theorem instead of Theorem and a special kind of interval polynomials for v](t) instead of a constant coarse enclosure as in the current AWA program Lohner guesses an enclosure v0 ] of u( ) (t)=p! satisfying p p p p v](t) := ?1 X p (t ? t0) (u0 ) + (t ? t0) v0 ] D: i p i i=0 The Picard-Lindelof operator reproduces the Taylor coe cients of the exact solution There is an interval vector v1 ] such that each p times di erentiable function v(t) v](t) satis es (v(t)) u](t) := p?1 X (t ? t0 ) (u0) + (t ? t0 ) v1 ]: i i p i=0 If v1] v0 ] holds, then we also have u](t) v](t), and according to Theorem 2, u(t) u](t) To obtain v1 ], one has to compute an enclosure of the remainder coe cient of the p ? 2nd degree Taylor polynomial of f (v(t)) for all p times di erentiable functions v(t) v](t) It also contains the remainder coe cient of the p ? 1st degree Taylor polynomial of (v(t)) For computing or enclosing these coe cients on a computer, one has to apply polynomial interval machine arithmetic given by Eiermann 3], which provides the enclosure of a xed number of Taylor coe cients of standard operations and standard functions on interval polynomialsas well as the enclosure of the respective remainders The advantage of this idea of Lohner over the validation strategy discussed in Section is that validation is possible for step sizes appropriate for a method of local order p Example The solution of the initial value problem u0 = ?u; u(0) = satis es u(t) = e? 0; 1] for t However, the application of Theorem for t u ] = 0; 1] requires a step size since u0 + 0; h]f ( u0]) u0] , + 0; h](? 0; 1]) 0; 1] , h 1: George F Corliss and Robert Rihm However, Lohner's approach yields validation for much longer steps If we choose p = and v0 ] = 0; 1], we get t4 t5 t6 v](t) = ? t + t2 ? t6 + 24 ? 120 + 720 0; 1]; t4 t5 f ( v](t)) = ? v](t) ?1 + t ? t2 + t6 ? 24 + 120 ? h ; 1]; and t4 t5 t6 u](t) = ( v](t)) = ? t + t2 ? t6 + 24 ? 120 + 720 ? h ; 1] for t t0 ; t0 + h] We have ? h ; 1] v0 ] = 0; 1] , h : Of course, we could enclose the sixth Taylor coe cient of f instead of the fth one to obtain h = However, in general we only assume f to be p ? times di erentiable In this example, the coe cients of f (v(t)) can easily be calculated However, there is no implementation of the method for the general case yet, although Lohner has described 9] such an implementation using the Eiermann operators for interval polynomial arithmetic Validated Enclosure Using Taylor Series The use of Taylor series for validation, as well as for tight enclosure, is suggested by pictures such as those in Figure 2, which shows Taylor series enclosures for the solution of the Lorenz system 20 18 16 14 12 10 0.01 0.02 0.03 t 0.04 0.05 Validating an A Priori Enclosure 30 20 10 0 0.01 0.02 0.03 0.04 0.05 0.03 0.04 0.05 t -10 40 30 20 10 0 0.01 0.02 t Figure Taylor Series Enclosures for the Solution of the Lorenz System As already mentioned, AWA uses the recurrence relations (2) and the formula (3) for computing a tight enclosure in Algorithm II after having validated a coarse enclosure u0] for a step size h in Algorithm I It would cause no additional costs if the Taylor expansion (3) could also be used in Algorithm I The following theorem shows that this can actually be done Theorem Let u int( u ]) D Let 0 u](t) := ?1 X p (t ? t0) (u0 ) + (t ? t0 ) ( u0]) i p i p u 0] ; (6) i=0 for t t0 ] Then u(t) u](t) for t t0] : Proof: It follows from our assumptions that Equation (1) has a solution u in some neighborhood of the point (t ; u ), for every u u ] We must prove that that neighborhood includes all of t ] If (6) holds, then u(t) u](t) as long as u(t) u ] 0 0 0 George F Corliss and Robert Rihm Assume that u(t) leaves u](t) =: u(t); u(t)], and hence u0] at b int( t0]) Without t loss of generality, we assume u (b) = u (b) = sup u0] , for some j f1; : : :; ng From t t u( ) (t) p!( u0]) , it follows that j p j j p u0 (t) ?2 X p (t ? t0) (u0 ) + (t ? t0 ) ?1 p( u0]) ; i p i p i=0 t t t and hence u0 (t) u0 (t) on t0; b] Therefore, u (b) = u (b) implies u (t) = u (t) for all t t0 ; b] If u is constant, then we have u0 = u (b) = sup u0] , which contradicts t t the assumption u0 int( u0]) Otherwise, u (b) is an isolated maximum of the p-th t degree polynomial u The same holds for u (t), since this function is at least p times continuously di erentiable in t0] Hence, u(t) cannot leave u](t) at b t j j j j j j j j j j This approach avoids complicated calculations and nevertheless takes the order p into account With p = 6, we can validate the coarse enclosure 0; 1] in Example for h = 2:1 , compared with h = for a constant bound enclosure and h = using Lohner's polynomial enclosure Example We want to validate the solution u(t) = 1=t of the initial value problem u0 = ?u ; u(1) = in the interval u0] = 0; 2] For the constant bound enclosure, we have u1] := + 0; h](? 0; 2]2) = ? 4h; 1] u0] , h 1=4 as the maximal step size we can reach by applying Theorem However, the approach of this section for p = yields a longer step: u](t) = ? (t ? 1) + 0; 8](t ? 1)2 p + 33 0:42 : 16 0; 2] , h Implications for AWA Using the results of the previous section, AWA Algorithm I is Task 1: Approximate solution Compute Taylor coe cients of an approximate solution u(t) := b ?1 X p (t ? t0) (u0 ) i i i=0 from recurrence relation (2) In practice, we must compute (u0) using rounded interval arithmetic to capture any round-o errors The work of Task is already being done in AWA Algorithm II i Validating an A Priori Enclosure Task 2: Guess a step h Use heuristics based on the radius of convergence 1] of the truncated series for u or on previous steps sizes b Task 3: Guess a constant enclosure u0] Bound the range of u( t0 ; t0 + h]), and in ate b it a little Task 4: Compute the remainder terms ( u0]) , for i = 0; : : :; p from the recurrence relation (2) The work of Task is already being done in AWA Algorithm II Task 5: Compute stepsize for validation The stepsize is the point nearest to t0 (and in t0]) at which i u](t) := p?1 X (t ? t0 ) (u0) + (t ? t0) ( u0]) i p i p i=0 leaves u0] This requires n polynomial root ndings For each polynomial root nding problem, it is su cient to compute only relatively coarse lower bounds for the smallest real root in the interval (t0 ; t0 + h], so a few iterations of a simpli ed interval Newton method should su ce Remark Variable order We must compute ( u ]) , : : :, ( u ]) ? in order to compute 0 p ( u ]) Hence, it is relatively inexpensive to compute the step size in Task for each order p The order used for validation in Algorithm I can be di erent from the order used for tightening in Algorithm II The information is readily available to make both Algorithms I and II fully variable order Remark Iteration If Task nds that u](t) u0] on t0], then we can repeat Tasks and with a larger h or else with u0] = u]( t0]) Remark Optimization Given guesses for h from Task and for u0] from Task 3, we compute in Task the largest step for which we can validate existence and containment Numerical experiments have shown 2] that a carefully chosen u0] allows step sizes 10 times as long as step sizes corresponding to u0] given by reasonable heuristics That suggests viewing Algorithm I as an optimization problem: p Maximize step size h by varying guessed h and u0] subject to u](t) := ?1 X p (t ? t0 ) (u0) + (t ? t0) ( u0]) i i p p u0 ] i=0 Remark E ectiveness Either Lohner's enclosure by polynomials or our enclosure by Taylor series is signi cantly more expensive than Picard-Lindelof iteration of constant bounds currently used by AWA However, AWA Algorithm II involves the solution of an ODE system of dimension n n, so almost any e ort we expend in Algorithm I that allows AWA to take longer steps improves the overall e ciency of AWA 10 George F Corliss and Robert Rihm Remark Which is better? For Lohner's enclosure by polynomials, the enclosures are expensive to compute, but it is easy to check for enclosure For our enclosure by Taylor series, the enclosures are free because they are already being computed, but checking for enclosure requires n polynomial root ndings Work is continuing on an implementation that will allow direct computational comparisons of the e ectiveness of the two methods Acknowledgments We thank the anonymous referee for helpful suggestions The work of Corliss was supported in part by National Science Foundation Grant No DMS{9413525 and in part by SUN Academic Equipment Grant No EDUD{US{940208 References 1] Y F Chang and G Corliss, Solving Ordinary Di erential Equations Using Taylor Series, ACM Trans Math Software, 8(1982), 114{144 2] G Corliss, Validating an A Priori Enclosure Using High-Order Taylor Series, presented at SCAN '95, Wuppertal, 1995 3] M Ch Eiermann, Adaptive Berechnung von Integraltransformationen mit Fehlerschranken, PhD thesis, Universitat Freiburg, 1989 4] R J Lohner, Anfangswertaufgaben im IR mit kompakten Mengen fur Anfangswerte und Parameter, Diplomarbeit, Inst f Angew Math., Universitat Karlsruhe, 1978 n 5] R J Lohner, Enclosing the Solutions of Ordinary Initial and Boundary Value Problems, in Computer Arithmetic: Scienti c Computation and Programming Languages, E W Kaucher, U Kulisch, and C Ullrich, eds., Wiley-Teubner Series in Computer Science, Stuttgart, 1987, pp 255{286 6] R J Lohner, Einschlie ung der Losung gewohnlicher Anfangs{ und Randwertaufgaben und Anwendungen, PhD thesis, Universitat Karlsruhe, 1988 7] R J Lohner, Interval Arithmetic in Staggered Correction Format, in Scienti c Computing with Automatic Result Veri cation, E Adams and U Kulisch, eds., Academic Press, San Diego, 1993 8] R J Lohner, On Step Size and Order Control in the Veri ed Solution of Ordinary Initial Value Problems, presented at SCAN '93, September 1993, Vienna Validating an A Priori Enclosure 11 9] R J Lohner, Step Size and Order Control in the Veri ed Solution of IVP with ODE's, presented at the SciCADE '95 International Conference on Scienti c Computation and Di erential Equations, Stanford, Calif., March 28 { April 1, 1995 10] R E Moore, The Automatic Analysis and Control of Error in Digital Computation Based on the Use of Interval Numbers, in Error in Digital Computation, Vol I, L B Rall, ed., John Wiley and Sons, New York, 1965, pp 61{130 11] R E Moore, Interval Analysis, Prentice-Hall, Englewood Cli s, N.J., 1966 12] L B Rall, Automatic Di erentiation: Techniques and Applications, vol 120 of Lecture Notes in Computer Science, Springer Verlag, Berlin, 1981 13] R Rihm, Interval Methods for Initial Value Problems in ODEs, in Topics in Validated Computations, J Herzberger, ed., Elsevier Science B.V., Amsterdam, 1994, pp 173{207 14] H J Stetter, Validated Solution of Initial Value Problems for ODE, in Computer Arithmetic and Self-Validating Numerical Methods Ch Ullrich, ed., Academic Press, San Diego, 1990, pp 171{186 Addresses: G.F Corliss, Marquette University, Department of Mathematics, Statistics, and Computer Science, P.O Box 1881 Milwaukee, Wisconsin 53201-1881, USA georgec@mscs.mu.edu R Rihm, Universitat Karlsruhe, Institut fur Angewandte Mathematik, 76128 Karlsruhe, Germany robert.rihm@math.uni-karlsruhe.de ... Equations Using Taylor Series, ACM Trans Math Software, 8(1982), 114{144 2] G Corliss, Validating an A Priori Enclosure Using High- Order Taylor Series, presented at SCAN ''95, Wuppertal, 1995 3]... Eiermann, Adaptive Berechnung von Integraltransformationen mit Fehlerschranken, PhD thesis, Universitat Freiburg, 1989 4] R J Lohner, Anfangswertaufgaben im IR mit kompakten Mengen fur Anfangswerte... relation (2) In practice, we must compute (u0) using rounded interval arithmetic to capture any round-o errors The work of Task is already being done in AWA Algorithm II i Validating an A Priori