Round-off Error Analysis of Explicit One-Step Numerical Integration Methods

Transcription

1 Round-off Error Analysis of Explicit One-Step Numerical Integration Methods 24th IEEE Symposium on Computer Arithmetic Sylvie Boldo 1 Florian Faissole 1 Alexandre Chapoutot 2 1 Inria - LRI, Univ. Paris-Sud et CNRS - Univ. Paris-Saclay 2 U2IS, ÉNSTA ParisTech we thank the IEEE for registration bursary

2 2 / 34 Table of contents 1 Motivations and numerical methods 2 Roundoff errors of RK methods Local roundoff errors Global roundoff errors of classical methods 3 Conclusion and perspectives

3 Ordinary differential equations (ODEs) y (t) = f(y, t). Exact resolution is hard numerical methods. 3 / 34

4 4 / 34 Numerical integration

5 5 / 34 Numerical integration

6 6 / 34 Numerical integration

7 7 / 34 Numerical integration

8 8 / 34 Numerical integration

9 9 / 34 Numerical integration

10 10 / 34 Numerical integration

11 11 / 34 Numerical integration

12 12 / 34 Euler method k 1 = hf(t n, y n ) y n+1 = y n + k 1 + O(h 2 )

13 13 / 34 RK2 method k 1 = h f(t n, y n ) k 2 = h f(t n + h 2, y n + k 1 2 ) y n+1 = y n + k 2 + O(h 3 )

14 14 / 34 Working assumptions FP arithmetic; neither underflow nor overflow, radix 2 double precision, u = 2 53

15 14 / 34 Working assumptions FP arithmetic; neither underflow nor overflow, radix 2 double precision, ODEs; first-order, linear, y R R, u = 2 53 y = λy

16 14 / 34 Working assumptions FP arithmetic; neither underflow nor overflow, radix 2 double precision, ODEs; first-order, linear, y R R, Methods. explicit, one step, constant step. u = 2 53 y = λy

17 15 / 34 RK methods on linear problems: linear stability { y 0 R y n+1 = R(h, λ)y n (R polynomial in hλ)

18 15 / 34 RK methods on linear problems: linear stability { y 0 R y n+1 = R(h, λ)y n (R polynomial in hλ) Stable R(h, λ) < 1:

19 16 / 34 RK methods on linear problems: FP implementation { y 0 R y n+1 = R(h, λ)y n { ỹ0 y 0 ỹ n+1 = R( h, λ, ỹ n )

20 16 / 34 RK methods on linear problems: FP implementation { y 0 R y n+1 = R(h, λ)y n { ỹ0 y 0 ỹ n+1 = R( h, λ, ỹ n ) Euler: R(h, λ) = 1 + hλ; R( h, λ, ỹ n ) = ỹ n h λ ỹ n.

21 RK methods on linear problems: FP implementation { y 0 R y n+1 = R(h, λ)y n { ỹ0 y 0 ỹ n+1 = R( h, λ, ỹ n ) Euler: R(h, λ) = 1 + hλ; R( h, λ, ỹ n ) = ỹ n h λ ỹ n. RK4: R(h, λ) = 1 + hλ (hλ) (hλ) (hλ)4 ; R( h, λ, ỹ n ) = ỹ n h 6 λ ỹ n h 3 λ ỹ n h h 6 λ λ ỹ n h 3 λ ỹ n h h 6 λ λ ỹn h h h 12 λ λ λ ỹ n h 6 λ ỹ n h h 6 λ λ ỹ n h h h 12 λ λ λ ỹ n h h h h 24 λ λ λ λ ỹ n. (> 60 flops!) 16 / 34

22 17 / 34 State-of-the-art Roundoff errors in numerical methods (N = nb of iterations): probabilistic result: error in N [Henrici,1963]; in practice (implicit RK): error in N [Hairer&al, 2008]; interval analysis [Bouissou-Martel, 2006]; numerical integration (fine-grained): Newton-Cotes, Gauss-Legendre,... [Fousse, 2006].

23 State-of-the-art Roundoff errors in numerical methods (N = nb of iterations): probabilistic result: error in N [Henrici,1963]; in practice (implicit RK): error in N [Hairer&al, 2008]; interval analysis [Bouissou-Martel, 2006]; numerical integration (fine-grained): Newton-Cotes, Gauss-Legendre,... [Fousse, 2006]. Our approach: fined-grained analysis; use of mathematical properties of the methods (stability). 17 / 34

24 18 / 34 Table of contents 1 Motivations and numerical methods 2 Roundoff errors of RK methods Local roundoff errors Global roundoff errors of classical methods 3 Conclusion and perspectives

25 19 / 34 Method error vs roundoff error

26 20 / 34 Method error vs roundoff error

27 21 / 34 Method error vs roundoff error

28 Local roundoff error vs global roundoff error Local error: ε 0 = ỹ 0 y 0 n N, ε n = R( h, λ, ỹ n 1 ) R(h, λ)ỹ n 1. Global error: n N, E n = ỹ n y n. 22 / 34

29 23 / 34 From local to global roundoff error Local error: ε 0 = ỹ 0 y 0 n N, ε n = R( h, λ, ỹ n 1 ) R(h, λ)ỹ n 1. Global error: n N, E n = ỹ n y n. Theorem 1: Global absolute error of RK methods Let C R +. Suppose n N, ε n C ỹ n 1. Then, n N, E n (C + R(h, λ) ) n C y 0 (ε 0 + n C + R(h, λ) ).

30 24 / 34 Relative roundoff errors Relative error: n y n C + R(h, λ) C y 0 ỹn ( ) (ε 0 + n y n R(h, λ) C + R(h, λ) ).

31 24 / 34 Relative roundoff errors Relative error: n y n C + R(h, λ) C y 0 ỹn ( ) (ε 0 + n y n R(h, λ) C + R(h, λ) ). If C R(h, λ), then: ỹn y n y n ε 0 + n C y 0 R(h, λ). In practice (Euler, RK2, RK4): C 200u and 200u R(h, λ).

32 25 / 34 Table of contents 1 Motivations and numerical methods 2 Roundoff errors of RK methods Local roundoff errors Global roundoff errors of classical methods 3 Conclusion and perspectives

33 26 / 34 Technical lemma for local roundoff errors Local error of stable Euler s method ( 2 hλ < 0): ε n+1 = ỹ n ( h λ ỹ n ) (1 + hλ)y n.

34 27 / 34 Technical lemma for local roundoff errors Local error of stable Euler s method ( 2 hλ < 0): ε n+1 = ỹ n X1 ( h λ X2 ỹ n y ) ( 1 α1 + hλ α2 ) ỹ n y. Lemma 2 Let y R. Let C 1, C 2 R +. Let α 1, α 2 R. Let X 1, X 2 F s.t. X 1 α 1 y C 1 y and X 2 α 2 C 2. Then: X 1 (X 2 y) (α 1 + α 2 )y y (C 1 + C 2 + u ( α α 2 + C 1 + 2C 2 ) +u 2 (C 2 + α 2 )).

35 27 / 34 Technical lemma for local roundoff errors Local error of stable Euler s method ( 2 hλ < 0): ε n+1 = ỹ n X1 ( h λ X2 ỹ n y ) ( 1 α1 + hλ α2 ) ỹ n y. Lemma 2 Let y R. Let C 1, C 2 R +. Let α 1, α 2 R. Let X 1, X 2 F s.t. X 1 α 1 y C 1 y and X 2 α 2 C 2. Then: X 1 (X 2 y) (α 1 + α 2 )y y (C 1 + C 2 + u ( α α 2 + C 1 + 2C 2 ) +u 2 (C 2 + α 2 )). ỹ n X1 1 α1 ỹ n y 0 C1 ỹ n, h λ hλ 6u y X2 α2 C2 (Gappa [Melquiond]).

36 28 / 34 Local roundoff errors of higher-order methods Stable RK4 method: C ỹ n 0 [ h 6 λ] 1 2u (Gappa) [ỹ n + h 1 6 λỹ n ] = ỹ n h 6 λ ỹ n 4u (Lemma 2) [ h 1 3 λ] 4u (Gappa) [ h h 1 6 λ λ] 12u (Gappa) [ h h h 1 12 λ λ λ] 28u (Gappa) [ h h h h 1 24 λ λ λ λ] 53u (Gappa) (7 Lemma 2) [ỹ n + + h h h h 1 24 λ λ λ λỹ n ] 194u (Lemma 2)

37 29 / 34 Local roundoff errors of classical methods Lemma 3: Local error of Euler method Suppose 2 hλ and 2 60 h 1. Then: n N, ε Euler n u ỹ n. Lemma 4: Local error of RK2 method Suppose 2 hλ and 2 60 h 1. Then: n N, ε RK2 n u ỹ n. Lemma 5: Local error of RK4 method Suppose 3 hλ and 2 60 h 1. Then: n N, ε RK4 n+1 194u ỹ n.

38 30 / 34 Table of contents 1 Motivations and numerical methods 2 Roundoff errors of RK methods Local roundoff errors Global roundoff errors of classical methods 3 Conclusion and perspectives

39 31 / 34 Global roundoff errors of classical methods Bounds on global errors: Euler: C = 11.01u E n (11.01u + R(h, λ) ) n 11.01u y 0 (ε 0 + n 11.01u + R(h, λ) ) ; RK2: C = 28.01u E n (28.01u + R(h, λ) ) n 28.01u y 0 (ε 0 + n 28.01u + R(h, λ) ) ; RK4: C = 194u E n (194u + R(h, λ) ) n 194u y 0 (ε 0 + n 194u + R(h, λ) ). no compensation but reasonable bounds.

40 32 / 34 Table of contents 1 Motivations and numerical methods 2 Roundoff errors of RK methods Local roundoff errors Global roundoff errors of classical methods 3 Conclusion and perspectives

41 33 / 34 General methodology y n+1 = M i=0 α i y n ỹ n+1 = M i=0 α i ỹ n (α i = hλ, hλ 3, hλ 6, h4 λ )

42 33 / 34 General methodology y n+1 = M i=0 α i y n ỹ n+1 = M i=0 α i ỹ n (α i = hλ, hλ 3, hλ 6, h4 λ ) Methodology to bound roundoff errors:

43 33 / 34 General methodology y n+1 = M i=0 α i y n ỹ n+1 = M i=0 α i ỹ n (α i = hλ, hλ 3, hλ 6, h4 λ ) Methodology to bound roundoff errors: 1) bound ε 0 = ỹ 0 y 0 ;

44 33 / 34 General methodology y n+1 = M i=0 α i y n ỹ n+1 = M i=0 α i ỹ n (α i = hλ, hλ 3, hλ 6, h4 λ ) Methodology to bound roundoff errors: 1) bound ε 0 = ỹ 0 y 0 ; 2) bound the error on each term α i (Gappa + stability);

45 33 / 34 General methodology y n+1 = M i=0 α i y n ỹ n+1 = M i=0 α i ỹ n (α i = hλ, hλ 3, hλ 6, h4 λ ) Methodology to bound roundoff errors: 1) bound ε 0 = ỹ 0 y 0 ; 2) bound the error on each term α i (Gappa + stability); 3) bound local errors by M applications of Lemma 2;

46 33 / 34 General methodology y n+1 = M i=0 α i y n ỹ n+1 = M i=0 α i ỹ n (α i = hλ, hλ 3, hλ 6, h4 λ ) Methodology to bound roundoff errors: 1) bound ε 0 = ỹ 0 y 0 ; 2) bound the error on each term α i (Gappa + stability); 3) bound local errors by M applications of Lemma 2; 4) bound the global error by instantiating Theorem 1.

47 34 / 34 Conclusion and perspectives Conclusion: fine and mechanical analysis of roundoff errors; results on useful and classical methods (Euler, RK2, RK4); linear growth of roundoff errors; no compensation (for explicit one-step methods). Perspectives: overflows and underflows; formalization in Coq (proof assistant): based on the Flocq library [Boldo-Melquiond], based on the gappa and interval tactics [Melquiond], more general ODEs: complex, matricial, non-linear; more general methods: multi-step, variable step, implicit.