Accurate Multiple-Precision Gauss-Legendre Quadrature

Transcription

1 Accurate Multiple-Precision Gauss-Legendre Quadrature Laurent Fousse 1,2 1 Université Henri-Poincaré Nancy 1 2 INRIA Rocquencourt 18th IEEE International Symposium on Computer Arithmetic

2 Outline Motivation Numerical Quadrature Certified arithmetics Our Results Gauss-Legendre Quadrature Complete example Conclusion

3 Quadrature is ubiquitous In physics (crystallography [Jézéquel Chesneaux 2004]): g(a) = + 0 [(exp(x) + exp( x)) a exp(ax) exp( ax)] dx where 0 < a < 2 and g(a) is the crystal energy. probabilities [Trefethen, Kern 2002] 2 = 1 π π π dφ 3 4 cos φ + cos 2 φ + 16ɛ 2 (parameterized random walk in Z 2 ).

4 Quadrature is hard At least you can fail spectacularly. One example with Pari/GP 2.3.0: 10 0 x 2 sin(x 3 )dx?default(realprecision,115);intnum(x=0,10,x^2*sin(x^3)) %2 = [...]

5 Quadrature is hard At least you can fail spectacularly. One example with Pari/GP 2.3.0: 10 0 x 2 sin(x 3 )dx?default(realprecision,115);intnum(x=0,10,x^2*sin(x^3)) %2 = [...]?default(realprecision,116);intnum(x=0,10,x^2*sin(x^3)) %2 = [...] Which result do you trust?

6 Quadrature is hard At least you can fail spectacularly. One example with Pari/GP 2.3.0: 10 0 x 2 sin(x 3 )dx?default(realprecision,115);intnum(x=0,10,x^2*sin(x^3)) %2 = [...]?default(realprecision,116);intnum(x=0,10,x^2*sin(x^3)) %2 = [...] Which result do you trust? 10 0 x 2 sin(x 3 )dx = 1 (1 cos(1000)) 3 0,

7 Quadrature is hard A famous computer algebra software: \^/ Maple 10 (IBM INTEL LINUX)._ \ / _. Copyright (c) Maplesoft, a division of Waterloo Maple Inc \ MAPLE / All rights reserved. Maple is a trademark of < > Waterloo Maple Inc. Type? for help. > evalf(int(exp(-x^2)*ln(x), x=17..42));

8 Quadrature is hard A famous computer algebra software: \^/ Maple 10 (IBM INTEL LINUX)._ \ / _. Copyright (c) Maplesoft, a division of Waterloo Maple Inc \ MAPLE / All rights reserved. Maple is a trademark of < > Waterloo Maple Inc. Type? for help. > evalf(int(exp(-x^2)*ln(x), x=17..42)); > Digits:=20: evalf(int(exp(-x^2)*ln(x), x=17..42));

9 Quadrature is hard A famous computer algebra software: \^/ Maple 10 (IBM INTEL LINUX)._ \ / _. Copyright (c) Maplesoft, a division of Waterloo Maple Inc \ MAPLE / All rights reserved. Maple is a trademark of < > Waterloo Maple Inc. Type? for help. > evalf(int(exp(-x^2)*ln(x), x=17..42)); > Digits:=20: evalf(int(exp(-x^2)*ln(x), x=17..42)); > Digits:=50: evalf(int(exp(-x^2)*ln(x), x=17..42));

10 Quadrature is hard A famous computer algebra software: \^/ Maple 10 (IBM INTEL LINUX)._ \ / _. Copyright (c) Maplesoft, a division of Waterloo Maple Inc \ MAPLE / All rights reserved. Maple is a trademark of < > Waterloo Maple Inc. Type? for help. > evalf(int(exp(-x^2)*ln(x), x=17..42)); > Digits:=20: evalf(int(exp(-x^2)*ln(x), x=17..42)); > Digits:=50: evalf(int(exp(-x^2)*ln(x), x=17..42)); > Digits:=100: evalf(int(exp(-x^2)*ln(x), x=17..42)); [...] 10

11 Quadrature is hard exp( x 2 ) log xdx 0, Precision Maple answer Correct digits 10 0, , , , , , , , 7

12 Miscomputations are bad Ariane 5

13 Miscomputations are bad Ariane 5 Patriot anti-missile

14 Miscomputations are bad Ariane 5 Patriot anti-missile Vancouver Stock Exchange

15 Miscomputations are bad Every time you miscompute a kitten gets killed. Please, think of the kitten! = mimic the IEEE 754 standard as closely as possible for numerical integration.

16 Gauss-Legendre Quadrature in short rule of interpolatory type: b a f (x)dx b n 1 a x xj n 1 f (x i ) = w i f (x i ) xi x j i=0 the x i s are the roots of the n-th Legendre polynomial P n the weights w i are easily computed w i = mathematical error bounded by 2 (1 x 2 i )P 2 n (x i ). (b a) 2n+1 (n!) 4 (2n + 1)[(2n)!] 3 f (2n). i=0

17 Gauss-Legendre Quadrature in CRQ Correctly Rounded Quadrature Compute P n via (n + 1)P n+1 (X) = (2n + 1)XP n (X) np n 1 (X) Root isolation (Uspensky) and refinement (interval Newton) Forward error analysis to derive bounds and their proof

18 Hypothesis y f(x) b a f(x)dx a b x f is a C black-box function in arbitrary precision. bounds M k on f (k) are known.

19 Quadrature Algorithm INPUTS: â, b a, (ŵ i ), f, ( v i ), n where v i = ( 1+x i 2 ). OUTPUT: Î b a f (x)dx 1: for i 0 to n 1 do 2: t (( b a) v i ) 3: x i (t + â) 4: fi (f ( x i )) 5: ŷ i ( f i ŵ i ) 6: end for 7: Ŝ sum(ŷ i, i = 0... n 1) 8: D ( b a)/2 9: return ( DŜ) = Î

20 Theorem Theorem Let δ byi = 11 4 ulp(ŷ i ) + 6M 1 ŵ i ulp( x i ), p 2 the working precision. With n integration points the error Î I is bounded by: B total = 21 4 ulp(î) + 5n 4 D max(δ byi ) + (b a)2n+1 (n!) 4 (2n + 1)[(2n)!] 3 M 2n.

21 A glimpse of the proof Let v i = 1+x i 2 Then v i v i 1 2 ulp( v i ) x i = ( ( v i ( b a)) + â). ( v i b a) v i (b a) 5 2 ulp( ( v i b a))

22 A glimpse of the proof Let v i = 1+x i 2 Then v i v i 1 2 ulp( v i ) x i = ( ( v i ( b a)) + â). x i x i 1 2 ulp( x i ) ulp( ( v i b a)) ulp(â) 17 4 ulp( x i ).

23 Let v i = 1+x i 2 Then A glimpse of the proof v i v i 1 2 ulp( v i ) x i = ( ( v i ( b a)) + â). x i x i 1 2 ulp( x i ) ulp( ( v i b a)) ulp(â) 17 4 ulp( x i ). (f ( ˆx i )) f (x i ) (f ( ˆx i )) f ( ˆx i ) + f ( ˆx i ) f (x i )

24 A glimpse of the proof Let v i = 1+x i 2 x i x i 1 2 ulp( x i ) ulp( ( v i b a)) ulp(â) 17 4 ulp( x i ). (f ( ˆx i )) f (x i ) (f ( ˆx i )) f ( ˆx i ) + f ( ˆx i ) f (x i ) ulp( (f ( ˆx i ))) + M 1 ˆx i x i

25 Let v i = 1+x i 2 A glimpse of the proof x i x i 1 2 ulp( x i ) ulp( ( v i b a)) ulp(â) 17 4 ulp( x i ). (f ( ˆx i )) f (x i ) (f ( ˆx i )) f ( ˆx i ) + f ( ˆx i ) f (x i ) ulp( (f ( ˆx i ))) + M 1 ˆx i x i ulp( (f ( ˆx i ))) M 1ulp( ˆx i )

26 Our favorite example I = g(x) = e x 2 i=0 e x 2 log xdx h(x) = log x n ( ) n d f (n) i dn i (x) = g(x) h(x) i dx i dx n i After some computations: f (n) n n!e 172 ( (n + 1)42 n log 42 + (n 1)42 n 2).

27 B math vs. B rounding (p = 1000) math rounding log(error) n

28 Measured error vs. computed error bound (p = 1000) -200 measured total log(error) n

29 Quadrature coefficients computation time 3500 Computation time Time (ms) n

30 CRQ vs. Rest of the World I = 1 0 max(sin(x), cos(x))dx = 2 cos 1 Precision CRQ Time 26ms 110ms 1s 11s 108s 3138s Error 1 2.6e-2 9.2e Maple Time 6s 38s 371s 2618s Error 2.1e-1 1.7e-1 1.7e-1 2.9e-1 PARI/GP Time 20ms 80ms 480ms 3s 22s 301s Error 3.1e25 2.0e54 2.7e e e e1496

31 Summary Multiple-precision quadrature can be fast...

32 Summary Multiple-precision quadrature can be fast... and certified at the same time.

33 Summary Multiple-precision quadrature can be fast... and certified at the same time. Code available at

34 Summary Multiple-precision quadrature can be fast... and certified at the same time. Code available at TODOs: Speed-up Gauss-Legendre coefficients computation (parallel Newton iteration?), Look at other methods in the Gauss family, Automatic computation of bounds for specific functions, Optimal composition strategy.