Correctly Rounded Newton-Cotes Quadrature
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Correctly Rounded Newton-Cotes Quadrature Laurent Fousse. N° 5605 Juin 2005. ISSN 0249-6399. ISRN INRIA/RR--5605--FR+ENG. Thème SYM. apport de recherche.
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(34) !"!$#!J%>&MO]icvj;KNT_]h< MO9;> > HLb[]h>7KhKNVsT_jTv ]hT_<KN>.dMhVWT0j J w i. wi. = = = = = =. Rb. 1 l (x)dx a i Rh n−1 li (a + xh)dx R0n−1 Q (a+xh−xj ). (xi −xj ) dx (a+xh−(a+jh)) dx R0n−1 Qj6=i (x−j)h(i−j) j6=i (i−j) dx 0 (−1)n−1−i R n−1 Q j6=i (x − j)dx. i!(n−1−i)! 0. R0n−1 Qj6=i. 8 9;>Ec_]OVYcvR[Us>di9;cvj;w_> x 7→ n − 1 − x KO9[TQKMh9;cM w = w Q >aM cvj;g L Mh9[> cvjCMOVYgL>=]hVscvMOVs_> T_ l KNZ+di9 MO9;cvM (x − j) l (x) = ( ) u = 8 9[>7j>daT0<?=b[(−1) ZLMO>&Mh9[>>=Vsw_9CMhK c_K n−1−i. ? i. i. i. j6=i. (−1)n−1−i i!(n−1−i)!. n−1−i. i. ? i. Li (0) = 0. a> M. n−1 i. n!. ,I. wi = ui Li (n − 1). ;]OT0< MO9;>T0]O<SZ[UYcT0j[>ud=c_jj[T_MOVYda>MO9;>u>7VWw090MiK?cv]h>]icMhVWT0j;cvU8 9;>=Id7cvjMO9JZ;K?Rf> daT0<Pb;ZLMO>.g > H[c_dMhUWIc_K w = RJI UWw0T_]hVMh9[< R+>7UWT x #
(35) % ¥ &} | ' >7 MOT_j[`T_MO>7K d=TC> daVs>=jCMhK bi δ. i. () δ ← lcm(2, 3, . . . , n) *+) l? ← (x − 1)(x − 2) . . . (x − n − 1)δ ,+).- 0 i ← 0 to n − 1 R n−1 ? / ) Li ← 0 li (x)dx ¥) x−i ? li+1 ← x−(i+1) li? ¦) bi ← n−1 Li i 0 ) 1 2+). ¥}. ¦¥. VsKQc_jVsjCMO>7w_>=] Z. ;bfg[cvMO.>VYLKgLT_j;>VWjb[UYc_da> i. ¡ ¦ ¥} } ¡ | } (b , b , . . . , b , δ · n!) 8 Vs<?VWj[wCK>=]h>)gLT0j[>QT_jc\%>7j0MhVWZ;<S d=T_<?b[ZLMh>=] VWMO9Du\ %e 430 _>=]iKOVWT0j?t[ c_j;gcv]h>)KO9[Tj T0jz+w_Z[]h>tL58 9[>&MOVs<?>daT0<Pb;UW > HJVWM`I VYK O(n log n) T0]>7c0di9UsTJT_bMO9JZ;K O(n log n) VsjMhTvMicvU 0. 1. bn/2c. 2. ¡ £¢"¤!¥¦§¥. 3. 2.
(36) n. (" ! 9$$ . 6 Computation time. 5. time (ms). 4. 3. 2. 1. 0 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100. n. Vsw_Z[]h>tJ TJ> daVs>=jCMiKdaT_<?b[Z[MhcMhVWT0jMOVs<P>&T0]QKN<c_UWU n A! 9;>=jb+>7]NT0]O<?Vsj[wcjJZ[<?>=]hVsd7cvUVWjCMh>=w_]icMhVWT0jpRJIu<?>7c_j;KETvc >7 MOT0jL`T_MO>.K)<?>aMh9[TLg2Mh9[>=]h>?cv]h> M`TKOT_Z[]ida>.K
(37) T_>=]h]OT0]hK
(38) MOTSdaT_j+KNVYgL>=0] JMh9[>Q<cMh9[>=<cMhVsd7cvU[>7]O]hT_]%Mh9;cMdaT_<?>.K]OT0<Mh9[><?>aMO9;TJg?VWMhKO>=UW` c_j;g Mh9[>]OT0Z[j;gLT_>7]O]hT_]VsjMh9[>udaT0<Pb;ZLMhcvMOVsT_j9[VYdi9g[>=bf>=j;g[KPT0j MO9[> cI>zVs<?b[Us>=<?>=jCM?MO9[> c_UWw0T_]hVMh9[< >&;]iK`MQw0VW0> RfT_Z[j+g[KT_jMO9;><?cvMO9[>7<cMOVYd=c_U>=]h]OT0] VMh9>7UW>7<?>=jCMhc_]OIb[]hTJTv K7 .
(39) !"$#%'&(& )& *.9- ' ()¡J ¥ } ¡ ,=+
(40) "
(41) (
(42) $n 0) &) ¡ #""! " $ POP )$ (".-9 ()" " ! "E
(43) " [a, ("
(44) $b] )P )n ! PO $ $ ≤n n ≤ n−1. !"!$#!J [T0]&c_jJI MO9[>?<?>aMh9[TLgpVYK& > H[c_daM T0]&b+T0UWIJj[T0<PVYcvUYK Tvg[>=w_]h>=>SZ[bMOT Mh9;cM)d7c_KO>)Mh9[> c_w_]incvj[w0>)VsjCMO>=]hbfT_UYcMOVsj[w?bfT_UsICj;T_<?Vsc_UVsK f > H[c_daMOUsI_. n−1. R+>.d=cvZ+KN>PVWj. a !.
(45)
(46) !" # $&%'!()(" 3. >aM%j[T n Rf>TLg[g28 9[> di9[T0Vsd=> Tv;MO9[>>=cvUsZ;cvMOVsT_jbfT_VsjCMhK
(47) T_]MO9;><?>aMh9[TLgw0VW0>7K x +x >aM g(x) = (x − x )(x − x ) . . . (x − x ) a+b . i. 0. g. . 1. a+b −x 2. . n−1−i. =. n−1. n−1 Y. =. i=0. a+b − x − xi 2. n−1 Y. . a+b − x − (a + b − xn−1−i ) 2 i=0 n−1 Y a+b = − − x + xn−1−i 2 i=0 n−1 Y a + b n = (−1) + x − xn−1−i 2 i=0 a+b +x = −g 2 =. . P c_j;g Mh9[>=j R KOVWj+da> 8 9[> >= MhT_jLNTvMh>7K<?>=MO9[TLgVYK > H[c_dM?T0]b+T0UWg(x)dx IJj[T0<PVYcvUY=KPTv0=gL>=w0]O>7> nw−g(x c1 )M<?T0KNM7 g(xcvj+g) =T0] 0g 9[Vsdi9 9;c0K?gL>=w0]O>7> n 9[>=j+da>RJI UsVsj[>7c_]OVWM`IVsKQ > H[c_daMT0]QcvUsUb+T0UWIJj[T0<?Vsc_UsK T_gL>7w_]h>=>cM<?T0KNM n b a. . . . n−1 i=0. i. i. i. '$ ' )& . j MO9[VYK KN>.dMOVsT_jz> >7KNMhcvR;UWVYKN9 MO9[> > HLb[]h>7KhKNVsT_jT_2MO9[> <cMO9;>=<cMhVsd7cvUf>=]h]OT0]w_Vs_>7j VWj@J>7daMOVsT_j [T_] Mh9[VYK MO9[>&T0]O<c_UWVYKN< T_
(48) Mh9[>\%>7c_j[TL_>=]hj[>7UfT_%c_jVsjCMO>7w_]icMOVsT_j<?>aMO9;TJgVYKQcPb+T>=]OZ[U MhTCT0U R ;T_]&czXCZ;c_g[]hcvMOZ[]h><?>aMh9[TLg MO9;>S>7]O]hT_] d7cvj Rf>KO>=>=jc_KcPUsVWj;>7cv] Z;j;dMhVWT0j CI : C([a, b])([a,→b])R 5→R>9;c0>)Mh9[>&T_EUsUWT:fVsj[7→w?]h>7KOZ[fU(x)dx M0J − I(f ) * ' ¡J¥ } ¡ + b a. ν+1. ν+1. . . (. (x − . . E[p] = 0. Kν. 1 E[x 7→ (x − t)ν+ ] ν!. Kν (t) =. "("
(49) P
(50) =
(51) " (
(52) $. $ (
(53) P
(54) . ¡ £¢"¤!¥¦§¥. \%>7c_j[T&L_>=]hj[>7U. p. t)ν+. =. . (x − t)ν 0. )) ( ^$. E[f ] =. " . Z ν. b a. . if x > t otherwise. ν. ". f (ν+1) (t)Kn (t)dt. E. . f ∈ C ν+1 ([a, b]). .
(55) (" ! 9$$ . %]hVMhVWj;wSMh9[>8cICUsT_]KO>=]hVW>.K c_KhKNTLdaVYcMh>7gzVMh9 f cMT0]OVsw_Vsj. !"!$#!J. f (x). = pν (x) + = pν (x) + "Z b. Z. Z. a. J. x. 1 (x − t)ν f (ν+1) (t)dt ν!. b. 1 (x − t)ν+ f (ν+1) (t)dt, ν! #. a. a. 1 (x − t)ν+ f (ν+1) (t)dt ν! a Z b 1 E (x − t)ν+ f (ν+1) (t)dt. = ν! a. E[f ] = E. 8 9[VYK MO9[>7T_]h>=< UsVW=j LLK MO9;>S<cMh9[>=<cvMOVYd=cvU>=]h]hT_])VWMO9pMO9[>P<"c HLVs<?c_Ug[>=w_]h>=>STvMh9[>Pb+T0UWIJj[T0<PVYcvU Mh9[><?>aMh9[TLgSVsjCMO>7w_]icMO>.K> H[c_daMOUsIm,VWMhK5<cHJVs<cvULT0]hgL>7] Ia j8 9[>7T_]h>=< * > b[]hT_>.gMh9;cM5Mh9[>T_]igL>=] T_ c_j nbfT_VsjCMhK >= MOT0jLNTvMO>.K&<?>aMO9;TJg VYKcvMUW>.c_KNM n − 1 T_] n >7_>=j c_j;g n T0] n TLg[g > c0gL<?VMMO9+cMMO9[VYK VsKT0bLMOVs<cvU cvj;g MO9;cvMMh9[>&\>.cvj[T L_>7]Oj[>7U+T_2MO9;> >= MhT_jLNTvMh>7K<P>=MO9[TLggLTJ>7Kj[TvM di9+cvj[w0>EKOVWw0jT0j JMO9[>7j>&9;c0>EcP<?>aMh9[TLgzMhTdaT_<?b[Z[MO>EMO9[>d=TJ > daVs>=jCM c w_Vs_>7jzVsjMicvR[Us> VWMO9Mh9[><?>7cvj[a,cvUsb]Z[>&MO9;>=T_]h>=<#MO9[>7]O>& > HLVYK`MiK ζ ∈]a, b[ KOZ;di9zMh9;c0M J Z ,)t I E[f ] = f (ζ) K (t)dt c_j;g>&T_R[MhcvVsj ( R j K (t)dt if is even, R c := j K (t)dt if is odd. ;T_] > H[c_<Pb;UW>ET_] Mh9[> bfT_Vsj0MiK <?>aMO9;TJg LJj[Tjc_ K !$ $
(56) L>w_>=M 3 n. b. (ν+1). ν. a. b 1 n hn+1 a b 1 n+1 hn+2 a. n. E[f ] =. Z. b a. a+b b−a (f (a) + 4f ( f (x)dx − ) + f (b)) , 6 2 K(t) =. 1 Ex ((x − t)3+ ), 6. a !.
(57) .
(58) !" # $&%'!()(" . 6K(t) = = = Z. b. K(t)dt = a. =. c_j;g>E;j;gMh9[>cvUsZ[>. a+b b−a (a − t)3+ + 4( − t)3+ + (b − t)3+ 6 2 a Z b b−a a+b 3 3 3 (x − t) dx − − t)+ + (b − t) 4( 6 2 (t a+b (b−t)4 3 3 − b−a if t < a+b 4 6 4 4( 2 − t) + (b − t) 2 , (b−t) b−a 3 − 6 (b − t) if t ≥ a+b 4 2 Z a+b Z b 2 b−a (b − a) a + b (b − t)4 3 − (b − t) dt − ( − t)3 dt 24 36 9 2 a a 5 (b − a)5 (b − a)5 (b − a)5 1 b−a − − =− 120 144 576 90 2 Z. b. (x − t)3+ dx −. 1 c3 = − 90. w0VW0>=jVWjMO9;> Mh9[Vs]hg]hT Tv%8
(59) c_R[Us> . '&
(60) )& . !" ! . jT_]igL>7]MhTR+>c_R[UW>EMhTw_Vs_>&cvjcvR;KOT_UsZLMO>&RfT_Z[j+gT0jzMh9[><cMh9[>=<cMhVsd7cvU >=]h]hT_] >&j[>7>7gzMOTRfT_Z[j+g Mh9[> c d=TJ> daVs>=jCMhK7>AgL>aMicvVsU
(61) 9;>=]h>MO9[>AR+T0Z[j;g c_j;gMO9[>Sb[]hTCT_%T_] n >7_>=j +cvj+gw_Vs_>Mh9[>]h>7KOZ[UM T0j[UsI&T_] n TLg[g2e>.d=c_UWUJMh9;cM%T_] n >=_>7j2Mh9[> >7 MOT_j[`T_MO>7K<?>aMh9[TLgAVYK%> H[c0dM%T_]5cvUsULbfT_UsICj;T_<?Vsc_UsK Z;bMhTgL>=w0]O>7> n − 1 8c L0> T0] f cP<?T_j[VYdgL>7w_]h>=>a n bfT_UsICj;T_<?Vsc_U p VWj>7XCZ;cvMOVsT_Uj , t IMhTw_>=EM J cn. n. En [p] = p(n) (ζ). Z. b. Kn−1 (t)dt. c_j;gzMO9[>7j c = KOVsj;da> p (ζ) = n! jb;c_]NMhVsd=Z[Usc_]T_] ;>&9;c0> p(x) = (x − x )(x − x ) . . . (x − x ) n. En [p] n!hn+1. a. (n). 0. En [p] =. Z. 1. b. p(x)dx − a. n−1. n−1 X. wi p(xi ) =. Z. b. p(x)dx a. OK T VMVsK>7j[T_Z;w_9MOT RfT_Z[j;g
(62)
(63)
(64) R p(x)dx
(65)
(66)
(67) Vsj T_]igL>=]?MhT R+T0Z[j;g c >uVsUWU)Z;KN>u]h>=bf>7cvMO>.gLUWI MO9[> T0UWUsTVsj[wPKOVs<Pb;UW>Us>=<?<c=J . ¡ ~ + + ¡" (u, x, v) ∈ R $! (" u ≤ x ≤ v |x − u||x − v| ≤ } ¥ ¥ { | ¥ +
(68) i=0. b a. n. (v−u)2 4. 3. !"!$#!J ]xi0 , xi0 +1. ∀x ∈ [a, b], |p(x)| ≤. >=M KNZ;di9 MO9;cvM [ ;cvj+xg ∈Mh9CZ+[a,K b]. p(x) 6= 0. hn (n − 1)! . 4 i0 ∈ [0, n − 2]. 8 9[>=jMO9[>7]O>VYK. |x − xi0 ||x − xi0 +1 | ≤. ¡ £¢"¤!¥¦§¥. h2 . 4. . >7<?<?c * * . KOZ;di9Mh9;cM. x ∈.
(69) . o. (" ! 9$$ . 8 9;>=j |p(x)|. n−1 Y. ≤. |x − xi |. i=0 2. h 4. ≤. ". Q. Y. (i − i0 )h. i>i0 +1. #". Y. ". Y. (i − i0 )h. i>i0 +1. (i0 + 1 − i)h. i<i0. #. #". (i0 + 1 − i)h ,. i<i0. ≤ (n − 1 − i0 )!(i0 − 1)!hn−2. En [p] = cn hn+1 n!. L9[VYdi9ICVs>=UYg[K. >Ew0>aM. 1 n−1 ≤ . 4n 4. ;T_] TJg;g>)Mh)c L_> bfT_VsjCMhKc_nj;gMO9[><?VYg[gLUs>p(x) Tv Mh9[=>Vs(xj0Mh−>=]hxcvU2)(xc_K'−7>=x]hTJ)...(x >7KTv p− x @JVs<?VWUYcv]Qd=T_<?b[ZLMicMhVWT0j;K w_Vs_> J 0. c_j;gVMh9. #. hn (n − 1)! . 4 b − a = (n − 1)h
(70) Z
(71) Z
(72) b
(73) b hn+1 (n − 1)!(n − 1)
(74)
(75) |p(x)|dx ≤ . |En [p]| =
(76) p(x)dx
(77) ≤
(78) a
(79) 4 a |cn | ≤. Mh9[>=j. Y. ≤ (n − 1)!hn−2 ,. |p(x)| ≤. jMO9;>TvMO9;>=]9;c_j;gz>&9;c_>. VWMO9. |x − xi |. i6={i0 ,i0 +1}. h2 4. ≤. Y. |p(x)| ≤. 1. n−1 )(x −. a+b 2 ). > 9;c_>Mh9[>E>=cvUsZ;cMhVWT0j. hn+1 (n − 1)!(n − 1) 8.
(80) Z
(81)
(82) b
(83) hn+2 (n − 1)!(n − 1)2
(84)
(85) p(x)dx
(86) ≤
(87)
(88) a
(89) 8. En = cn hn+2 p(n+1). >&w_>=M. |cn | ≤. 1 (n − 1)2 ≤ . 8n(n + 1) 8. a !.
(90) .
(91) !" # $&%'!()(" . . # !"$#!
(92) #
(93) . . ;T_]Mh9[> >7]O]hT_]c_j;cvUsILKNVYK&Tv Usw_T0]OVWMO9;< t[
(94) >j[>=>.g c>= Z;KO>aZ[UUs>=<?<c_Kd=T_j;d=>=]hj[VWj;wMO9[>.Z;UWb d7cvUYdaZ[UsZ;K.i0c_K%>=UsU;c_KKOT_<?>gL>=;j[VWMOVsT_j;K758 9[>';T0cvMOVsj[wv bfT_Vsj0M5jJZ[<SR+>7]hK5c_]O> ]h>=b[]h>7KO>=jCMO>.gAVWMO9]ic_g[V H ,Mh9[VYKd=T_Z[UYgRf> w_>7j[>=]icvUsV7>7gET0]%c_jJIE]ic_g[V HR[Z[M]ic_gLVH VYK%KNVs<?b[UW>7]5cvj;gAVsKj;cvMOZ[]icvUCT_jPdaT_<?b[Z[MO>=]iK I 2;T_]EMO9;VsKKO>7daMOVsT_j2 VYK&Mh9[>?T0] LJVsj[wb;]O>.daVYKNVsT_j22c_j;g2>c_KhKOZ[<?>?c_UWU +T0cMhVWj;wv b+T0VWjCM&jCZ;<ARf>=]iKcv]h> p j;T_]h<?c_UWV =>.gJ9[VYdi9<?>.cvj;K VsjT_Z[] j[T_MhcMhVWT0j;KMO9+cMMO9;> > HLb+T0j[>=jCM]icvj[w0>EVYK Z[jJRfT_Z[j;g[>7g ¡ " +| ¥ + ¥
(95) ¡ | ¡" ( ! 0 ("
(96) 9! x E(x) := 1+blog |x|c $! (. 2 ≤ |x| < 2 ¡ " +| ¥ + # ¡"(m! 0 ("
(97) nO x &) ulp(x) := 2 ;T_])c?]h>7c_U c_j;guc?T_] LCVsj[w?b[]h>7d=VsKOVWT0j >Ac_UW cILK 9;c_> Vx YKc ;TCcMhxVWj[6=w_0bfT_VsjCMjCZ;<ARf>=].2MO9;>=j ulp(x)pVYK&MO9[>>=Vsw_9CMT_Mh29[>UWulp(x) >.c_KNMAKNVsw_≤j;V|x|+d7cv<jCM2R[VWulp(x) MF =>7]OT T0]&j[TvMEF VsjMh9[> p R[VWM&<c_j0MhVsKhKhczT_ x [T0]cvUsU5]O>.cvU x ulp(x) VsKc_UW cILK)w0]O>.cMO>7])Mh9;cvj =>7]OTRJI g[>a;j[VWMOVsT_j . ¡ ~ + c 6= 0 (" x 6= 0 c · ulp(x) < 2 · ulp(cx) !"!$#!J%VW VWMQVYK _T_VYg IgL>=;j[VWMOVsT_jTv ulp(x) >&9;c0> T0]QcvUsU c > 0 J c<0 . . . . E(x)−1. . . . . .
(98) . . . 2. E(x). . . . . E(x)−p. p−1. p. . 2p−1 ulp(x) ≤ |x|. c_j;g KOT. ¡ ~ + . |cx| < 2p ulp(cx) c · 2p−1 ulp(x) ≤ |cx| < 2p ulp(cx).. $ $ $ P " "!# 9$ #$0 &%mE $ Pt(
(99) P
(100) P E (0 ("
(101) ( ^(' $ P1 P ^$ ◦(x) x 0) P ("Nx) " ulp(x) !" O R≤ulp(◦(x)) (12 !"!$#!J 2E(x)−1 ≤ |x| < 2E(x) ulp(x) = 2E(x)−p H E(x)−1 E(x) E(x) E(x)−1. p>u9;c0>. -. 2 ≤ | ◦ (x)| ≤ 2 ulp(◦(x)) ≥ 2E(x)−p ≥ ulp(x). ¡ ~ +. . KOVsj;da>. 2. cvj;g c_j;g 2. MO>=]]OT0Z[j;gLVsj[w >uw_>aM cv]h> > [c_daMOUsI O] >7b[]O>.KN>7jCMhcvR;UW>0 MO9[>7]O>=T_]h>. P $P1 ("$ " P $ * ( "O0 ("
(102) (" ◦(x) p |x| ≤ x. (1 + 2−p )| ◦ (x)|. %RJIzgL>=;j[VWMOVsT_jTv%]hT_Z[j;g[VWj[wSMOTj;>7cv]h>7KNM >&9;c0>. !"!$#!J. |x − ◦(x)| ≤. ¢. ¢. +
(103) , (- ./- 0.132547698.;:34769<
(104) 2. ¡ £¢"¤!¥¦§¥. 1 1 ulp(◦(x)) ≤ 21−p | ◦ (x)|, 2 2. |x| ≤ | ◦ (x)| + 2−p | ◦ (x)|..
(105) . t. (" ! 9$$ . ¡ ~ + . $ . p. ( q N!0 '0(" P1) ! P! O $ $ ( $ ) (. P U("a
(106) P
(107) n"bP1 0) $ ulp(a) + ulp(b) ≤. !"!$#!JK. 3 ulp(◦(a + b)). 2. M5KOZ da>7K2MOTEdaT_j+KNVYgL>=] MO9[> d=c0KN> 9[>=]h> a cvj;g b cv]h>bfT0KOVMhVW0>_
(108) 8 9;>gL>=;j[VWMOVsT_jT_;Z[Usbw_Vs_>.K J 2p−1 ulp(a) ≤ a < 2p ulp(a),. Mh9JZ;K . 2p−1 ulp(b) ≤ b < 2p ulp(b). ulp(a) = ulp(b) >Ew0>aM. 2p−1 [ulp(a) + ulp(b)] ≤ a + b < 2p [ulp(a) + ulp(b)].. 2p ulp(a) ≤ a + b < 2p+1 ulp(a). _c j;gMO9;>=]h>aT_]h> ulp(◦(a + b)) ≥ ulp(a + b) ≥ 2ulp(a) = ulp(a) + ulp(b) , >7<?<?c * Ic_j;gMO9[> Us>=<?<cP9[T_UYg[K7 Q)Mh9[>=]hVsKO>> d=c_jSc_KhKOZ[<?>VWMO9[T0ZLM5UWTCKOKT_;w_>7j[>=]icvUsVM`IEMh9;cM MO9;cvM5VsK ulp(a) ≥ ulp(a) > ulp(b) 5 > L g . > L g ; Z = d )J > 2 · ulp(b) 3 ulp(a), 2 , ulp(◦(a + b)) ≥ ulp(a + b) ≥ ulp(a) ulp(a) + ulp(b) ≤. c_j;gMOT0w_>aMh9[>=]VWMO9 b;]OTJTv`
(109) KJ >aM c_j;gdi9;T0KO>]hT_Z;j;gLVsj[wMOT j[>7c_]O>.K`M0J p = 4 j;TvMhcvMOVsT_j2 a + b = 1.1101 ◦(a + b) = 1.110 ulp(a) + ulp(b) = 2 + 2 = 2 = ulp(◦(a + b)). . . −3. −4.
(110) && )&(. 3 −3 2. >=<?<c * ISMh9[VsKzdaT_j+daUsZ;gL>7KMO9[> . a = 1.010 b = 0.1001. Vsj R[Vsj;cv]hI. 3 2. j T0]hg[>=]?MOT b[]OTJVYgL> cvj >7]O]hT_]RfT_Z;j;g T_j Mh9[> jJZ[<?>=]hVsd7cvUQ]h>7KOZ[UWMzw_Vs_>7j RCIMO9;> >7 MOT_j[`T_MO>7K <?>=MO9[TLg[>&j[>=>.gMOT?9;c_>c?KNMO>=b[RJIC KNMO>7bzUsTJT)L VsjCMOT Usw_T0]OVWMO9;< tL 8 9[VYK KNMO>7b?VsK5T_MO>7jj[>=w0UW>.dMO>.gP9[>=jgLT0VWj[wjJZ[<?>7]OVYd=c_U[VWjCMh>=w_]icMhVWT0j29[>7]O>>7]O]hT_]c_j;cvUsILKNVYK%K`MhT_b;K ]hVsw_9CMcMO>7]K`MicMhVWj[w&Mh9[>>=Us=U LJj[TjSRfT_Z;j;gST_]5Mh9[><cMO9;>=<cMhVsd7cvUL>=]h]hT_]. jP c_daM7vMh9[>> HLbf>=]hVW<?>7j0M VsUsUWZ;KNMO]icMh>7gSVWj VWw0Z[]h> * KO9[TQK
(111) MO9;cvM<AZ;di9P]O>7<cvVsj;K
(112) MOT&R+>gLT0j[>MOTdaT0jCMO]hT_UCMO9[>>7]O]hT_]T_jSMO9;>]h>7KOZ[UWM7 ;T_]Mh9[VsKKN>.dMhVWT0j>pgL>7j[TvMh>uRJI Mh9[> cvUsZ[> c0dMOZ+cvUsUWIdaT_<?b[Z[MO>7g , V >_ VWMO9 cvUsU)]hT_Z;j;gLTv >7]O]hT_]i K I2T0]cw0VW0>=j N > H[c_da.M c_UWZ;> x Lc_xbK T0Z[UYgSRf> d=T_<?b[ZLMh>7gVWMO9 c_jVWj[;j[VWMO>)b;]O>.daVYKNVsT_j?]hT_< MO9[> Rf>=w0VWj;j[VWj;wPT_
(113) Mh9[>cvUsw_T0]OVWMO9;< j?c_g[g[VMhVWT0jAMhT)Mh9[> b;cv]icv<?>aMh>=]iK T_fc_UWw0T_]hVMh9[<kt)>j[>=>.gSc_jAZ[b[bf>=]5RfT_Z;j;g Tv T0j VW n VsK >=_>7j2_T0] cvj?Z[b;b+>7] RfT_Z[j;g?T_ |f | V n VYK5TLg[g p VsKMO9[>QT_] LJVWj;wEb;]O>.MdaVYKNVsT_j?|f > HLb[|]h>7KhKO>7[a,gPVWb]j . (n). (n+1). a !.
(114) *.
(115) !" # $&%'!()(" . x#
(116) % ¥ }& |'. 7> MOT_j[`T_MO>7KVsjCMO>=w0]hcvMOVsT_j aJ b aJ , b, f, n ab Ib ().- ¥ } i ← 0 to n − 1 ¦ ¥ . . *+) ,+). / ). ¥) ¦). t ← ◦(b a ∗ (n − i − 1)) u ← ◦(bb ∗ i) x ← ◦(t + u) x ← ◦(x/(n − 1)) yi ← ◦(f (x)) 1 yi- ← ◦(yi ∗ ni ). 0) 2+) +) S ← sum(yi , i = 0 . . . n − 1) ( § ) U ← ◦(S/d(n − 1)) (() D ← ◦(b − a) (*+) ◦(DU ). ¡ ¦ ¥}. .. VMh9 ? => <?<?>=U
(117) cvj;g QVYg[ccvUsw_T0]OVWMO9;< t . }¡| }. 200. 0.
(118)
(119) !
(120)
(121)
(122) . log(error). -200. -400. -600. -800. -1000. -1200 0. 20. 40. 60. 80. 100. 120. 140. 160. 180. 200. n. Vsw_Z[]h> * J j > H[cv<?b[Us>T_>=]h]OT0]<P>.c_KOZ[]h>=<?>=jCMQT0]QMO9;> >7 MOT0jL`T_MO>.K<?>aMO9;TJguT_] f : x 7→ x PT_<?b[Z[MhcMhVWT0j;KE>7]O>?gLT0j[>SVWMO9Mh9[>gL>a c_Z[UM "
(123) b[]h>7d=VsKOVsT_j T_ R[VWMhK7 Mh9[> [a, jJZ[<Sb]R+=>7]Q[0,Tv59]bfT_VsjCMhKVYKQgLVYKNb;UscI0>7gT_jMO9;>AcvR;KhdaVYKhKOc;[cvj;gVsjT_]igLVWj+cMO>.K MO9;>R;c_KO>a 2 UWT0w053c_]OVWMO9[< Tvn%MO9[> >7]O]hT_]. . 2. . ¡ £¢"¤!¥¦§¥.
(124) . (" ! 9$$ . Mh9[>EjCZ;<ARf>=]Tv R[VWMhKTv MO9[>E<cvjCMOVYKOKhc[ a cvj;g b ]OT0Z[j;gL>.g?MOTPj[>.cv]h>7KNM p R[VWMR;TCcMhVWj[w_bfT_VsjCMjJZ[<SR+>7]hK w0VWJVsj[w ba c_j;g bb m c_juZ[b[bf>=] R+T0Z[j;guT_ |f | T_j [a, b] ' jMO9[>S]O>.K`MQT_Mh9[VYK KO>7daMOVsT_ju>VWUsU
(125) b;]OT0> T0Z[]<cvVsjzMh9[>=T0]O>7i< J * ' ¡J¥ } ¡ +
(126) i P1 O ("
(127) !()(" f 9$P1
(128) P & (
(129) 0. "N $
(130) $$ " Etot. . '. 45 b + 1 |U b | ulp(bb) + ulp(b + 21 · 2−p )ulp(I) a) 2 2 n+2 1 b−a M b (1 + 2−p )n · D 8 n−1 n+1 +3 max(δybi ) + d(n − 1) 1 b−a M 4 n−1. = (. . n. $0). ,. $ + . 8 9[>c_UWw0T_]hVMh9[<#d=c_jRf>cvj+cvUsI27>7gVWjuKO>=0>=]icvUK`Mh>=b;KEJ MO9[>d=T_<?b[ZLMicMOVsT_jTv MO9[>?>=Vsw_9CMhK TvMh9[>?<?>aMO9;TJg2 ;T_] >7 MOT_j[`T_MO>7K7 MO9[TCKN>>7VWw09CMhKc_]O> ]icMhVWT0j;cvUcvj;gd=wT_<?b[iZL∈Mh>7[0,g > nH[c_−dMh1]UWIJ w = 9;>=]h> n , d ∈ Z KOT j[T ]OT0Z[j;gLVsj[w?>=]h]hT_] TLd=daZ;]hK cMMh9[VYKQK`Mh>=b2 tLMO9[>d=T_<?b[ZLMicMhVWT0jT_ x 8 9[VYKVYKgLT_j;>&cvMQUWVsj[>AmSTv Usw_T0]OVWMO9[<tJ i. ni d. i. i. i. . x bi = ◦. ◦(◦((n − 1 − i) · b a) + ◦(i · bb)) n−1. !. .. j T0]hg[>=]MOT?KNVs<?b[UWVWI?MO9;>Ej;TvMhcvMOVsT_j;K> ]hVMh> t = (n − i − 1)a u = i · b c_j;gMh9[>=Vs] VWj;> H[c_daM daT0Z[jCMO>=]hb;c_]NMiK bt = ◦((n − i − 1)ba) ub = ◦(i bb) b = 0 T0] i = 0 MO9[>S>=]h]OT0]QT_j ub VsK=>7]OT+ Q)MO9[>7]OVYKO> Mh9[>>=]h]hT_] >7KNMOVs<cMOVsT_jIJVW>7Usg;K J. >=<?<c_K * 3 cvj+g * @JVs<PVsUYcv]hUWIV a = 0 T0] n − i − 1 = 0 Mh9[>p>7]O]hT_]T0j bt VYKm7>=]hT;cvj+g TvMO9;>=]hVsKO>u>pw_>aM |b t − t| ≤ ulp(b t) >=MZ;Kj[Tc0KOKOZ[<?>PMO9;cvM cvj+g 9;c_>PMh9[> KOc_<?>?KOVsw_j ,9[VYdi9d7cvjR+> 7>=]h2T I Mh9[VsKUW>.c_g[K >7c0KNVsUWI MOT?MO9;>& c_dMMh9;cM bt cvj+ag ub 9+cb_>EMO9[>Khcv<?>KOVWw0jU,MO9[VYKQc0KOKOZ[<?bLMOVsT_jVYKj[>7d=>7KhKOc_]OI?MhTR+> cvR[Us>QMhTZ;KO> >7<?<?c * o2I Mh9[VYK VYK j;TvMMh9[> d7c_KO>Q> d=cvj KOb[UsVMMO9[> VWjCMh>=w_]icMhVWT0j?VWjCMO>7]Oc_U+cM DuT0]O>7T_>7]c0KOKOZ[<?> VWMO9;T_ZLMQUsT0KhK Tv%w_>7j[>=]icvUsVM`IPMh9;cM 0 ≤ a < b L9[VYdi9w_Vs_>.K 0 ≤ ba ≤ bb 0 | ◦ (i · bb) − ib| ≤ ≤. 1 b 2 ulp(◦(ib)) 3 b 2 ulp(◦(ib)). + 2i ulp(bb) = 23 ulp(b u).. . 3 2. | ◦ (b t+u b) − (t + u)| ≤ ≤. >=<?<c +* ulp(b o u)). 1 b b)) + 3 (ulp(b t) 2 ulp(◦(t + u 2 11 b ulp(◦( t + u b )). 4. . a !.
(131) m .
(132) !" # $&%'!()(" . 8
(133) c)LJVWj[w?VsjCMOTc_d7daT_Z;j0M Mh9[>>=]h]OT0]d=T_<?Vsj[wP]OT0<#MO9;>gLVWJVYKNVsT_jRJI δxbi = |xi − x bi | ≤ 21 ulp(b xi ) + 1 xi ) + ≤ 2 ulp(b ≤ 6 · ulp(b xi ).. >w0>aM0J b ulp(◦(b c >7+<Pd)) <c0K * 3 c_j;g * " ulp(b x). 11 4(n−1) 11 2. n−1. i. MO9[> d=T_<?b[ZLMicMhVWT0jpTv f (x ) P>c_KhKNZ;<P>?>S9+c_>?cvjVW<?b[Us>=<?>=jCMicMOVsT_jTv f VWMO9 daT_]h]h>7dM ]OT0Z[j;gLVsj[w+fc_j;gu>Sd=c_UWU MO9[>SZ[j;daMOVsT_j ]h>7XCZ[>.K`MhVWj;wMO9[>S]OT0Z[j;gLVsj[wMhTzj[>.cv]h>7KNM)Tv5Mh9[>A> H[c_dM cvUsZ[>VMh9b[]h>7daVYKOVWT0j p %@JZ+di9?d=T_]h]O>.dMhUWfI]hT_Z[j+gL>7gPVs<Pb;UW>7<P>7jCMhcMhVWT0j;K%Tv <cMh9[>=<cMhVsd7cvULZ;j;d MOVsT_j;KVWMO9cv]hR[VWMO]icv]hISb[]h>7d=VsKOVsT_jzT0j Mh9[> ]h>7KOZ[UWM d7cvjR+>ET_Z;j;gT0] > H[cv<?b[Us> VsjDu\ e 43E T_] j[T_j[ Mh]OVsJVsc_U Z[j;dMhVWT0j;K UsV L0> exp sin arctan cvj;gjJZ[<?>=]hT_Z;KT_MO9[>7]hK7 VMh9MO9[>c_UW]h>7c0gLI>7KNMOVs<cMh>7g>=]h]OT0]T0j xb >E9+c_>)J *. i. i. 0. |f (b xi ) − f (xi )| = |f (θi )(b xi − xi )|, θi ∈ [min(xi , x bi ), max(xi , x bi )]. cvj;gVWMO9cvjzZ;b[b+>7] R+T0Z[j;gT_j >&d=cvjR+T0Z[j;gzMO9[VYK >=]h]hT_]c_R;KOT_UsZLMO>7UWI0 >aM R+>&Mh9[> ;TCcMOVsj[w_bfT_VsjCM jJZ[<SR+>7]QfdaT0<Pb;ZLMO>.g MMO9[VYKQKNMO>7b>j[T 9+c_>)J 0. daT0<Pb;ZLMhcvMOVsT_jTv
(134) MO9[>. |fbi − f (xi )|. yi = f (xi ) · ni. fbi = ◦(f (b xi )). ≤ |f 0 (θi )(b xi − xi )| + 21 ulp(fbi ) ≤ 6m · ulp(b xi ) + 12 ulp(fbi ).. 8 9;>&c0d=d=Z[<AZ[UYcMh>7g>=]h]OT0]KNTP cv]0J. yi ) δybi = |b yi − yi | ≤ |ni | · |fbi − fi | + 21 ulp(b 1 ≤ 6|ni | · m · ulp(b xi ) + |n2i | ulp(fbi ) + 2 ulp(byi )* 3 3 yi ). ≤ 6|ni | · m · ulp(b xi ) + 2 ulp(b. => <?<c_K cvj;g * " e>=<c_]LJ 9[>7j R+T0Z[j;gLVsj[wMO9[>>=]h]OT0]zT0j c_K>7UWU&c_K QMO9;>pMh>=]h< VWMO9 x ) cvj[VYKN9;>7KPV MO9[>>7]O]hT_]PT_j xb VYK7>=]hT;Q j;>xbd7cvj fb >.c_KOVWUsI KN9;T ybMO9;cvMVMh9T0Z[]c_KhKNZ[<?ulp(b bLMhVWT0j MO9;cvMj[TZ;j;gL>=];T TLd=d=Z[]iK=5VW xb = 0 Mh9[>=j MO9[>>7]O]hT_]ST_j xb VsK&7>=]hwT ,V >_ x = 0IScvj;g MO9[>VsUsU gL>a+j[>7g XCZ;cvjCMhVM`I ulp(bx ) cvj[VYKO9[>7K7 [T0]&MO9[>>7]O]hT_]&RfT_Z[j;g> L_>=>7bpMh]hc0 d L TvT_j[UsI max(δ ) mLKNZ;<P<cvMOVsT_j T_MO9[> y KEJ MO9[VYKEVsK&gLT0j[>AVWMO9 ? >7<?<P>7U5cvj;g )Vsg;czKOZ[<?<?cvMOVsT_jcvUsw_T_]hVWMO9[< t 9[VYdi9 w_Z;c_]hc_jCMO>=>.Kcvju>7]O]hT_]T_ cM)<?T0KNM 1.5 ulp T0jMO9[>A;j;cvU
(135) ]O>.KNZ;UM.Q8 9[VsK cvUsw_T0]OVWMO9[< Z;KO>7K cPUsc_]Ow0>=] T0]LJVsj[wPb[]O>.daVYKNVsT_j p ≈ p + log (n) >aM S = P y i. i. i. i. i. i. i. i. i. y bi. i. 0. 2. |Sb − S| ≤. n−1 i=0. 3 b + n · max(δyb ). ulp(S) i 2. i. n[gLVsCVYKOVWT0jTv S RJI d(n − 1) J U = p8 9[>daT0<Pb;ZLMhcvMOVsT_j Tv VWjCMh>=w_>7]Qcv]hVMh9[<?>aMhVsd&cvj+gVYK > H[c0dM758 9[>>7]O]hT_]cvMMh9[VYKQK`Mh>=bVYKMO9JZ;KEJ S d(n−1). b − U| ≤ |U ≤. ¡ £¢"¤!¥¦§¥. 1 b 2 ulp(U ) 7 b 2 ulp(U ). + +. d(n − 1). >=<?<c_K * 3 c_j;g * . 3 n b bi ) 2d(n−1) ulp(S) + d(n−1) max(δy n max(δ ). y bi d(n−1). . VYKAgLT0j[> VWMO9.
(136) . n 3. (" ! 9$$ . <AZ[UWMOVsb[UsVsd7cMhVWT0jRJI. 5>j;TvMO>. cvj+g. J b = ◦(bb − b D a) b − a I = (b − a)U D =b−a h i b − D| ≤ 1 ulp(D) b + ulp(b |D a) + ulp(bb) . 2. . >9+c_>&RJI 9JICbfTvMh9[>7KOVYK 9[>=]h>. ulp(0) = 0. bb ≥ b − 1 ulp(bb), 2 b a ≤ a + 21 ulp(b a). RJIzdaT0jC0>=jCMOVsT_jcvj;gzMO9;>=]h>aT_]h>. ulp(b a) + ulp(bb) ≥ b − a − ulp(bb).. bb − b a ≥ b−a−. 1 2. pj Mh9[>?TvMh9[>=]&9;c_j;g >LJj[T bb > ba KOT>P9;c_> 9[VYdi9w0VW0>7K ulp(bb) ≤ 2(bb − ba) 8 9[>7j Q. bb − b a ≥ max( 12 ulp(bb), b − a − ulp(bb)). 7> <P<c * " , * I >Sb[ZLM&cvUsU2MO9;>A]h>7KOZ[UMiK)c_j;gR+T0Z[j;g[K w0cvMO9[>7]O>.gKNT c_]7f>Ad=c_ju]O>.c_di9MO9;>T_UsUsTVWj[w?+j;cvU >=]h]OT0] T_j Ib = ◦(Db U)b J b. D ≤ bb − b a + ulp(bb) ≤ 3(bb − b a) ≤ 3(1 + 2−p )D. |Ib − I| ≤ ≤ ≤ ≤. 1 b 2 ulp(I) + 1 b 2 ulp(I) + 1 b 2 ulp(I) + 1 b 2 ulp(I) + −p. 3(1 + 2. . bU b − D · U| |D b b − D| + |D| · |U b − U| | U | · |D b b b · |U b − U| |U | · |D − D| + 3(1 + 2−p )|D| b | · |D b |U − D|+ 7 b ) + n max(δyb ) b ulp(U )|D| 2. d(n−1). . j[>.X0Z+cvUsVM`IN, * I . i. >7<?<?c0K * 3 cvj;g * ". −p b + |U b | · |D b − D| ≤ ( 43 )ulp(I) 2 + 21 · 2. . b. )n·D +3 (1+2 bi ) d(n−1) max(δy 45 −p b + 1 |U b | ulp(bb) + ulp(b a ) ≤ ( 2 + 21 · 2 )ulp(I) 2 −p. 7> <?<?c0K * 3 cvj;g * " 8 9[VYKRfT_Z[j+gzT0] MO9[>>7]O]hT_] VYKKOcvMOVYK` c0dMhT_]hI?T_]Z;KOVWj;wPVWMQVsjMh9[>cvUsw_T0]OVWMO9;<LRf>7d7cvZ;KO>&VMQVYK <c_gL> T_XCZ;c_j0MhVMhVW>.K MO9+cMQ>&d7cvjdaT0<Pb;ZLMO>Rf>aT0]O>EMO9;>cvUsw_T_]hVWMO9[<#VWMhKO>=UW , p nIa[T_]9;Vsdi9cv]h>&j;cMhZ[]hc_UWUsI d=T_<?b[ZLMh>7gVsjzMh9[> ;T T_
(137) Mh9[>cvUsw_T0]OVWMO9;< , Ib Ub bb ba Db d δ I ;T_] MO9;>&;j;cvU >=]h]OT0]RfT_Z[j+g>&j[>=>.gMOTc_g[gcPRfT_Z[j;gT0jzMh9[><cMO9;>=<cMhVsd7cvU>=]h]hT_0] J VW n VsK TLg[g , 1 b−a M , I 8 n−1 E ≤ TvMh9[>=]hVYKN> 1 b−a M b. )n·D +3 (1+2 bi ). d(n−1) max(δy −p. . ybi. n+2. n+1. 4. n−1. a !.
(138) &3.
(139) !" # $&%'!()(" . 9;Vsdi9 VYK>.c_KOVWUsId=T_<?b[ZLMh>7g c0K>=UsU v j Mh9[>zd=T_Z[]iKN>T_MO9[>zb[]OT0w_]icv< >zc_]O> c_R[Us>MOTpdi9;T0KO>MO9[> ]hT_Z;j;gLVsj[w)<?TLgL>T_]>=0>=]hI&d=T_<?b[ZLMicMhVWT0j28 9;VsK%cvUsUsTQK Z;K
(140) MhT Z;KO>b+>.KOKOVW<?VYK`MhVsd]hT_Z[j+gLVWj;w)<?TLgL>7K%cvj+g Mh9JZ;Kc_T0Vsg MO9[>b[]hT_R;UW>7<#Tv%]OT0Z[j;gLT_ >=]h]OT0]hKVsjMh9[>daT0<?b[ZLMhcvMOVsT_jTvMO9[>>7]O]hT_]RfT_Z[j;gVWMhKO>=UW` S
(141) ! UWw0T_]hVMh9[<tP c_KQVW<?b[Us>=<?>=jCMh>7gZ;KOVWj;w?MO9[>AD \ %e UWVsR[]icv]hI 430 R j c0g[gLVWMOVsT_jMOTMh9[>]h>7KOZ[UWMQTv%MO9[> VsjCMO>7w_]icMOVsT_j 0Mh9[>b[]hT_w0]hc_< w0VW0>7Kc_j>7]O]hT_] RfT_Z[j;gT0jzMh9[>daT0<Pb;ZLMO>.gz]h>7KOZ[UWM)KNb[UsVWMQVWjT_Z[]QMO>=]h< K J MO9[><cvMO9[>7<?cvMOVYd=c_U>=]h]OT0]7J9[TCKN>& > HLb[]h>7KhKOVWT0jzVYK w_Vs_>=jVsj>.XCZ;cMhVWT0j tLMO9[> OKNMhcMhVsd E>7]O]hT_] E = ( + 21 · 2 )ulp(I)b * MO9[ > OgLVWf>7]O>7j;da > E>7]O]hT_] E = |U|b ulp(bb) + ulp(ba) + ulp(D)b MO9[>>7cvUsZ;cMhVWT0j>7]O]hT_] E
(142) = 3 max(δ ) ;T_]T_Z[] > HLbf>=]hVW<?>=jCMiK >di9;T0KO> cZ[j+dMOVsT_j?cvj+gcvjVsjCMO>7w_]icMOVsT_jSgLT_<cvVsj9[>7]O> MO9[> > H[c0dM
(143) c_UWZ[> VYRK LCj;Tj2CKOTMh9;cM > d=c_j <?>7c0KNZ[]h>Qb[]h>7d=VsKO>=UsISMO9[>&c_daMOZ;c_U+>7]O]hT_]T_2MO9;>Ed=T_<?b[ZLMicMOVsT_tj , gL>7j[TvMh>7gRJI R Ia VWw0Z[]O> KO9[TQK MO9;>SgLVWf>7]O>7jCM >=]h]hT_]iK 9[>=jpd=T_<?b[ZLMhVWj;w MO9;>AVsj0Mh>=w0]hc_U VWMO9 E R[ VMiKTvT0] LJVsj[wPb[]O>.daVYKNVsT_j JMO9[>jJZ[<SR+>7]Tv
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(171) Unité de recherche INRIA Lorraine LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France).
(172). . ISSN 0249-6399.
(173)
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