Round-off Error Analysis of Explicit One-Step Numerical Integration - PowerPoint PPT Presentation

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, ´ ENSTA ParisTech ⋆ we thank the IEEE for registration bursary

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives 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 2 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Ordinary differential equations (ODEs) y ′ ( t ) = f ( y,t ) . Exact resolution is hard ⇒ numerical methods. 3 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Numerical integration 4 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Numerical integration 5 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Numerical integration 6 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Numerical integration 7 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Numerical integration 8 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Numerical integration 9 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Numerical integration 10 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Numerical integration 11 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Euler method k 1 = hf ( t n ,y n ) y n + 1 = y n + k 1 + O ( h 2 ) 12 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives 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 ) 13 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Working assumptions • FP arithmetic; • neither underflow nor overflow, u = 2 − 53 • radix 2 double precision, 14 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Working assumptions • FP arithmetic; • neither underflow nor overflow, u = 2 − 53 • radix 2 double precision, • ODEs; • first-order, y ′ = λy • linear, • y ∶ R → R , 14 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Working assumptions • FP arithmetic; • neither underflow nor overflow, u = 2 − 53 • radix 2 double precision, • ODEs; • first-order, y ′ = λy • linear, • y ∶ R → R , • Methods. • explicit, • one step, • constant step. 14 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives RK methods on linear problems: linear stability { y 0 ∈ R y n + 1 = R ( h,λ ) y n ( R polynomial in hλ ) 15 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives 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 : 15 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives RK methods on linear problems: FP implementation { y 0 ∈ R { ̃ y 0 ≃ y 0 R (̃ h, ̃ y n + 1 = ̃ y n + 1 = R ( h,λ ) y n ̃ λ, ̃ y n ) 16 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives RK methods on linear problems: FP implementation { y 0 ∈ R { ̃ y 0 ≃ y 0 R (̃ h, ̃ y n + 1 = ̃ y n + 1 = R ( h,λ ) y n ̃ λ, ̃ y n ) Euler: • R ( h,λ ) = 1 + hλ ; • ̃ R (̃ h, ̃ y n ⊕ ̃ h ⊗ ̃ λ, ̃ y n ) = ̃ λ ⊗ ̃ y n . 16 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives RK methods on linear problems: FP implementation { y 0 ∈ R { ̃ y 0 ≃ y 0 R (̃ h, ̃ y n + 1 = ̃ y n + 1 = R ( h,λ ) y n ̃ λ, ̃ y n ) Euler: • R ( h,λ ) = 1 + hλ ; • ̃ R (̃ h, ̃ y n ⊕ ̃ h ⊗ ̃ λ, ̃ y n ) = ̃ λ ⊗ ̃ y n . RK4: 2 ( hλ ) 2 + 1 6 ( hλ ) 3 + 1 • R ( h,λ ) = 1 + hλ + 1 24 ( hλ ) 4 ; • ̃ R (̃ h, ̃ λ, ̃ y n ) = y n ⊕̃ h ⊘ 6 ⊗ ̃ y n ⊕̃ h ⊘ 3 ⊗ ̃ y n ⊕̃ h ⊗̃ h ⊘ 6 ⊗ ̃ λ ⊗ ̃ y n ⊕̃ h ⊘ 3 ⊗ ̃ λ ⊗ ̃ λ ⊗ ̃ λ ⊗ ̃ λ ⊗ ̃ y n ⊗ ̃ ̃ h ⊗ ̃ h ⊘ 6 ⊗ ̃ λ ⊗ ̃ y n ⊕ ̃ h ⊗ ̃ h ⊗ ̃ h ⊘ 12 ⊗ ̃ λ ⊗ ̃ λ ⊗ ̃ y n ⊕ ̃ h ⊘ 6 ⊗ ̃ λ ⊗ ̃ λ ⊗ ̃ λ ⊗ ̃ y n ⊕ ̃ h ⊗ ̃ h ⊘ 6 ⊗ ̃ λ ⊗ ̃ y n ⊕ ̃ h ⊗ ̃ h ⊗ ̃ h ⊘ 12 ⊗ ̃ λ ⊗ ̃ λ ⊗ ̃ λ ⊗ ̃ λ ⊗ ̃ y n ⊕ ̃ h ⊗ ̃ h ⊗ ̃ h ⊗ ̃ h ⊘ 24 ⊗ ̃ λ ⊗ ̃ λ ⊗ ̃ λ ⊗ ̃ λ ⊗ ̃ y n . ( > 60 flops!) 16 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives 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]. 17 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives 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

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives 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 18 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Method error vs roundoff error 19 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Method error vs roundoff error 20 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Method error vs roundoff error 21 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Local roundoff error vs global roundoff error Local error: ε 0 = ∣̃ y 0 − y 0 ∣ ∀ n ∈ N ∗ , ε n = ∣ ̃ R (̃ h, ̃ λ, ̃ y n − 1 ) − R ( h,λ ) ̃ y n − 1 ∣ . Global error: ∀ n ∈ N , E n = ̃ y n − y n . 22 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives From local to global roundoff error Local error: ε 0 = ∣̃ y 0 − y 0 ∣ ∀ n ∈ N ∗ , ε n = ∣ ̃ R (̃ h, ̃ λ, ̃ y n − 1 ) − R ( h,λ ) ̃ y n − 1 ∣ . Global error: ∀ n ∈ N , E n = ̃ y n − y n . Theorem 1: Global absolute error of RK methods Let C ∈ R ∗ + . Suppose ∀ n ∈ N ∗ ,ε n ⩽ C ∣ ̃ y n − 1 ∣ . Then, ∀ n ∈ N , C ∣ y 0 ∣ ∣ E n ∣ ⩽ ( C + ∣ R ( h,λ )∣) n ( ε 0 + n C + ∣ R ( h,λ )∣) . 23 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Relative roundoff errors Relative error: ∣ ̃ y n − y n ∣ ⩽ ( C + ∣ R ( h,λ )∣ C ∣ y 0 ∣ n ) ( ε 0 + n C + ∣ R ( h,λ )∣) . ∣ R ( h,λ )∣ y n 24 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Relative roundoff errors Relative error: ∣ ̃ y n − y n ∣ ⩽ ( C + ∣ R ( h,λ )∣ C ∣ y 0 ∣ n ) ( ε 0 + n C + ∣ R ( h,λ )∣) . ∣ R ( h,λ )∣ y n If C ≪ ∣ R ( h,λ )∣ , then: ∣ ̃ y n − y n C ∣ y 0 ∣ ∣ ≲ ε 0 + n ∣ R ( h,λ )∣ . y n In practice (Euler, RK2, RK4): C ≤ 200 u and 200 u ≪ ∣ R ( h,λ )∣ . 24 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives 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 / 34

Motivations and numerical methods Roundoff errors of RK methods Conclusion and perspectives Technical lemma for local roundoff errors Local error of stable Euler’s method ( − 2 ≤ hλ < 0 ): y n ⊕ (̃ h ⊗ ̃ ε n + 1 = ∣ ̃ λ ⊗ ̃ y n ) − ( 1 + hλ ) y n ∣ . 26 / 34