12 bound for remainder interval in rigorous intergation josep galen 6 2008

Bound for Remainder Interval in Rigorous Integration Joseph Galante September 2008 1 The E stimate We will give a bound on the remainder formula for rigorous integration using Taylor Models which updates the ideas in (BM4). Suppose we are solving the ODE  ˙x = f(x) x(0) = x 0 (1) Using the algorithm in (BM4), we can generate a p olynomial invariant under A(x)(t) = x 0 +  t 0 f(x(τ ))dτ (the Picard Operator) by applying A (n + 1) times to the zero polynomial, and keeping the terms of degree ≤ n. Let P be the A invariant n th degree polynomial. To complete the application of Schauder’s Theorem we must find a Taylor Model which is invariant under A. We desire an interval I so that A(P +I) ⊂ P +I, ie A(P +I)−P ⊂ I We have A(P +I) = x 0 +  t 0 f(P +I)dτ. By FTTMA f(P + I) will be a Taylor Model. We can decompose as f (P +I) = Q +R + ˆ I where Q is all terms of degree (n − 1) or less, R is all degree n terms, and ˆ I is the remainder. Since A(P ) = n P and since deg(R) = n , then R will integrate to an (n + 1) order term, and we must have that P = x 0 +  t 0 Qdτ. Thus the other terms will contribute only to the remainder. ie A(P + I) − P ⊂  t 0 (R + ˆ I)dτ (2) 1 Edited by Foxit Reader Copyright(C) by Foxit Software Company,2005-2008 For Evaluation Only. We want to better understand this relation since ˆ I depends upon I. We actually have A(P + I) = x 0 +  t 0 f(P + I)dτ (3) = x 0 +  t 0 f(P ) + f  (P )I + f  (P ) 2 I 2 + dτ (4) = x 0 +  t 0 Q + R + f  (P )I + f  (P ) 2 I 2 + dτ (5) = P +  t 0 R + f  (P )I + f  (P ) 2 I 2 + dτ (6) (7) Suppose we are working in a class of functions which is real analytic (or at least analytic on a large enough domain, say range(P ) + 2). Then the ’ ’ will converge due to class of functions we are working in. Now suppose the interval I ⊂ [−d, d] w here d < 1. Then we would have A(P + I) − P ⊂  t 0 R + f  (P )I + f  (P ) 2 I 2 + dτ (8) ⊂  t 0 R + I · (f  (P ) + f  (P ) 2 + )dτ (9) (10) Notice that the Taylor expansion of f(P + 1) about the point P gives f(P + 1) = f(P ) + f  (P ) + f  (P ) 2 + (11) where the sum will converge due to the regularity class of the functions we are considering. Hence we have A(P + I) − P ⊂  t 0 R + I · (f (P + 1) − f (P ))dτ (12) ⊂ B(  t 0 Rdτ ) + IB(  t 0 (f(P + 1) − f(P ))dτ) ⊂ want I (13) 2 In the event that R = 0, ie f(P ) has no n th order terms, we can make an upper bound by replacing R = 0 with R = t n . We can now solve for I to get B(  t 0 Rdτ ) 1 − B(  t 0 (f(P + 1) − f(P ))dτ) ⊂ I. (14) Since we are choosing I, we choose it to the the left hand side, and we get our desired inclusion. However there is a catch. We have made the assumption that I ⊂ [−d, d] where d < 1. There is no reason why apriopi, this must be true. However we have one more variable which we can control. Now we can control t. Suppose t ∈ [−h, h]. Clearly for h = 0, then left hand side evaluates to zero, which is to say that to model the initial condition will have no error. Also the expression is continuous as a function of h, except at the point where the denominator is zero, ie the point where the bounds become (−∞, ∞). By the intermediate value theorem, there is some h so that we can have d < 1. And we are done. Notice that the remainder interval I scales with h. This makes sense since it is easier to model the flow for a short period of time, than for a longer period. What is hidden is exactly how well it scales. The term R which is the n th degree pieces of f (P + I) will behave like O(h n+1 ) so decreasing h (the time we are modeling the ODE for), or increasing n will result in a dramatic increase in accuracy. Remark: The above results carry through without difficulty into multidi- mensional system case. The only detail to note is the the remainder interval must be the same in each dimension. (This can probably be avoided with some more lengthy expressions for the remainder interval error, however we do not persue this). Remark: We did not necessarily need to replace R = 0 with R = t n . It simply allows a single formula which works in all cases. If instead we left R = 0, then it would boil down to requiring t to be so that B(  t 0 (f(P + 1) − f(P ))dτ) ⊂ [−α, α] for some 0 < α < 1. Remark: We can modify this formula to incorporate x 0 as a Taylor Model. If x 0 = G + J where G is a degree n polynomial, then we create the invari- ant polynomial P under ˜ A(x)(t) = G +  t 0 f(x(τ ))dτ . But then A(P )(t) = 3 G + J +  t 0 f(x(τ ))dτ = ˜ A(P )(t) + J = n P + J. We can then add this extra J into the above estimates to get the remainder b ound to get B(  h 0 Rdτ ) + J 1 − B(  h 0 (f(P + 1) − f(P ))dτ) ⊂ I. (15) With appropiate Shrink Wrapping, B(J) will be approximately zero, and this extra term won’t greatly ruin our degree n scaling for the remainder term. 2 Example Consider the simple linear ODE  ˙x = λx x(0) = 1 (16) We will consider a degree n = 3 Taylor Model and model up to time t = h. Iteration of the operator A gives the polynomial P (t) = x 0 · (1 + λt + (λt) 2 2 + (λt) 3 6 ) (17) And the remainder interval is now B(  h 0 λ(λτ ) 3 6 dτ) 1 − B(  h 0 (λ · (P + 1) − λ(P ))dτ) (18) Which gives d = (λh) 4 24(1 − λh) (19) Note that we require |λh| < 1 in order to avoid a useless bound. This constraints our choice of h. We are further constrained in that we need |λh| so small that d < 1. (For this particular value of n, a choice of h = min(0.99, 0.72 |λ| ) works.) For λh < 1, then we can power series expand d to get (λh) 4 24(1 − λh) = 1 24 (λh) 4 (1 + λh + (λh) 2 + (λh) 3 ) (20) 4 Comparing this to the power series of the actual solution at time t = h, Exp(λh) = (1 + λh + (λh) 2 2 + (λh) 3 6 + (λh) 4 24 + (λh) 5 120 + ) (21) We see that P (h) + I has larger coefficents for terms of order 4 and higher. Hence we have a valid enclosure. 3 References (DC) David Kincaid and Ward Cheney. Numerical Analysis: Mathematics of Scientific Computation. Third Edition. (IEEE) IEEE-ANSI. Standard for Binary Floating-Point Arithmetic. 1985. (KM) Ulrich W. Kulisch and Willard L. Mikranker. Computer Arithmetic in Theory and Practice. Academic Press, New York, 1981. (BM1) Martin Berz and Kyoto Makino. Taylor Models and Other Validated Functional Inclusion Methods. International Journal of Pure and Applied Mathematics. Vol 4. No. 4 2003, 379-456. (BM2) Martin Berz and Kyoto Makino. COSY Infinity 9.0 Programmers Manual. MSU Report MSUHEP 060803. August 2006. (BM3) Martin Berz and Kyoto Makino. Verified Global Optimization with Taylor Model-based Range Bounders. Transactions on Computers. Issue 11, Vol 4, November 2005. (BM4) Martin Berz and Kyoto Makino. Verified Integration of ODEs and Flows using Differential Algebraic Methods on High-Order Taylor Models. Reliable Computing 4 (1998) 361-369 (BM5) M. Berz and K. Makino. Suppression of the Wrapping Effect by Taylor Model-Based Verified Integrators: Long-Term Stabilization by Shrink Wrapping. International Journal of Differential Equations and Applications 10(4) (2005) 385-403 5 (BM6) M. Berz and K. Makino. Suppression of the Wrapping Effect by Tay- lor Model-Based Verified Integrators: The Single Step. International Journal of Pure and Applied Mathematics 36(2) (2006) 175-197 (BM7) M. Berz and K. Makino. Suppression of the Wrapping Effect by Taylor Model-Based Verified Integrators: Long-Term Stabilization by Pre- conditioning. 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(N1) Markus Neher. From interval analysis to Taylor models - An overview. Proc. IMACS 2005, Paris, France (2005), T2-I-102-0658. (N2) Markus Neher. Improved validated bounds for Taylor coefficients and for Taylor remainder series. J. Comput. Appl. Math. 152 (2003), 393-404. (RMB1) N. Revol, K. Makino, M. Berz. Taylor Models and Floating-Point Arithmetic: Proof that Arithmetic Operations are Validated in COSY. Jour- 6 nal of Logic and Algebraic Programming 64 (2005) 135-154 University of Lyon LIP Report RR 2003-11, MSU HEP report 30212 (Z1) Piotr Zgliczynski. Lecture notes for Conference on Computer Assisted Proofs in Dynamical Systems - 2007. http://www.ii.uj.edu.pl/ zgliczyn/cap07/cap07.htm 7