INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Gal’s Accurate Tables Method Revisited
Damien Stehlé, Paul Zimmermann
N° 5359
Octobre 2004
ISSN 0249-6399
ISRN INRIA/RR--5359--FR+ENG
THÈME 2
apport
de recherche
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Unité de recherche INRIA Lorraine
LORIA, Technopôle de Nancy-Brabois, Campus scientifique,
615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy (France)
Téléphone : +33 3 83 59 30 00 — Télécopie : +33 3 83 27 83 19
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‹Œ—Š ¸ ŒZ‡ ¸ ‰ '¸ Š ¼ ½
U~d7\‘’gEd_avaOfZpd7b+Pbd7kZ\[PvgENQPObjKM]1xF–•—±+}.J?d_xZx—p+Pvgg+Pvg¥bjKMPsg+Pvav]_XMxƒpd7XZ^…Pp+Pvx—fMaObjY[]_X]7œDbjKMPmufZY[a+¦
tZK;d_gP]7œ!bjKMPaed7\[aOfZ\¨d7bjY[]_Xˆ]7œ f (x) ¢_JKMPENQPObjKM]1xY[g)^…POXMPOpd7\‘±_kZfZb)KMPOp+PEŸPbd7¦…Pd_gP<¤Md7NƒtZ\[Pvg
d7XMx f (x) = sin x ¢ZJKMP~Y[xZPedƒ]7œbjKMPSg+Pvav]_XMxpd7XZ^…P~p+Pvx—fMaObjY[]_X¯Y[gb+]ƒŸpjYlb+P`­
f (x) = 2
x
2 x = 2 x0 · 2 h ,
Ÿ KMPOp+P x Y[g©YlX•d$bd7kZ\[P€]7œx—Y[gjbjYlXZ^`fZY[g+KMPvxšt?]_YlXnb+gd7XMxˆY[gEgjb+]_p+PvxˆŸYlbjK•d7Xd7tZtZp+]e¤1YlNšd7bjY[]_X¯]7œ
Ylb+gEav]_pjp+Pvgjt?]_XMx—YlXZ^ 2 ”5p+Pvg+t²¢ sin x d7XMx cos x ˜ h = x − x Y[gEgjNšd7\l\‘­…p+]_fZ^`KZ\lišg+tPed7¦1YlXZ^M±
Y[ggjNšd7\l\[POpEbjK;d7X x—Yl–uY[xZPvxˆknišbjKMPcXufZNSk?POp]7œhx—Y[g+bjYlXZ^`fZY[gjKMPvxšt?]_YlXnb+g¢ cœ5b+POpEbjKZY[gbd7kZ\[P<›
|h|
k;d_gPvx/pd7XZ^…Pp+Pvx—fMaObjY[]_X²± 2 ”5p+Pvgjt²¢ cos h d7XMx sin h˜Y[g¥av]_NƒtZfZb+Pvxˆd7tZtZp+]v¤—YlNšd7b+PO\liƒknifMgjYlXZ^
dˆ\[]Ÿ xZPO^`p+PvP/t]_\li1XM]_NƒY¨d7%\ ”5p+Pvgjt²¢bŸ]\[]eŸ xZPO^`p+PvP/t?]_\liuXM]_NƒY¨d7\[g ˜sŸKZY[aK aO\[]`g+PO\ližd7tZtZp+]e¤1Y ›
Nšd7b+Pvg 2 ”5p+Pvgjt²¢ sin h d7XMx cos h˜ŸKMPOX h Y[g¥aO\[]`g+Pb+] 0 ¢uJh]WµZ¤ƒbjKMPY[xZPed_ge±…ŸEPcaed7XˆgjfZtZt]`gP
bjK;d7bbjKMPvg+Pst]_\li1XM]_NƒY¨d7\[gd7p+PcbjKMP€Jdeiu\[]_pP<¤—t;d7XMgjY[]_XMg]7œhbjKMPcœ5fZXMaObjY[]_XMgd7b 0 ±ukZfZbYlbEY[gEt]`gj›
g+YlkZ\[Pb+]•xZ]®k?PObjb+PONp F .R J’¢ =‘œŸEPƒav]_XMgjY[xZPOp$bjKMP/xZ]_fZkZ\[PƒtZp+PvaOY[gjY[]_›X ”5Y‘¢’P`¢ 53 kZYlb+g~]7œNšd7XnbjY[g+g3d ˜<±
g+YlXMavPšŸPd7p+PˆYlX bjKMP®mufZY[a+¦“tZK;d_g+P`±bjKMPˆaed7\[aOfZ\¨d7bjY[]_XMgQg+KM]_fZ\[x“k?PšNšd_xZP®YlX³xZ]_fZkZ\[P®tZp+PvaOY ›
g+Y[]_X3d_g]7œ5b+POXžd_gt]`ggjYlkZ\[P`±Md7XMx•YlbY[gYlXub+POp+PvgjbjYlXZ^šb+]/K;de–…P~bd7kZ\[PvgŸYlbjKžd7tZtZp+]v¤—YlNšd7b+PO\li 2
x—Y[g+bjYlXZ^`fZY[gjKMPvx¯PO\[PONQPOXnb+g±Zœ¬]_paed_a+KMPS]_tZbjYlNƒIY ™d7bjY[]_XIp+Ped_g+]_XMag ”¡g+PvWP F {—^± HJ˜<¢
JKMPX;d7Yl–…PŸdvi³]7œ€aKM]u]`g+YlXZ^ bjKMP•x—Y[gjbjYlXZ^`fZY[gjKMPvxªPO\[PONQPOXnb+gšY[gb+]£bd7¦…PbjKMPON&Pvmnf;d7\l\li
g+t;d_avPvxD±nbjKufMgNƒYlXZYlNƒIY ™vYlXZ^ƒbjKMPcNšd¤—YlNSfZN –`d7\lfMPsbd7¦…POX®kni |h| ¢—U~d7\‘’gEY[xZPedY[gEb+]Sp+PO\¨d¤®–…POpji
g+\lYl^`Knbj\li bjKMPINšd¤1YlNfZN –_d7\lfMPIbd7¦…POX˜kni |h| YlX ]_p+xZPOp®b+] a+KM]1]`g+P3k?PObjb+POp•x—Y[gjbjYlXZ^`fZY[gjKMPvx
t?]_YlXnb+g­…KMPbd7¦…Pvgd7\lNQ]`gjbEp+PO^`fZ\¨d7pj\lišgjt;d_avPvxšx—Y[g+bjYlXZ^`fZY[gjKMPvxšt?]_YlXnb+g x gjfMa+KˆbjK;d7b©bjKMPcg+b+]_p+Pvx
d7tZtZp+]e¤1YlNšd7bjY[]_X°]7œ ”5p+Pvgjt²¢
d7XMx
˜SY[gfZXnfMgjf;d7\l\li aO\[]`g+P®b+]IYlb+gbjpjfMP®–`d7\lfMP`¢
]_p+PˆtZp+PvaOY[g+PO\li…±YlX 2xZ]_fZkZ\[P®tZp+PvsinaOY[gjxY[]_X²±œ¬]_p/cosd7Xui x 0 ≤ i < 2 ± x Y[g/d3Nšd_a+KZYlXMPXnfZNkPOp
aO\[]`gPvgjbb+] + gjfMaK¯bjK;d7be­
aONQ]ux 1 ≤ 2 œ¬]_p 2
2 · 2
aONQ]ux 1 ≤ 2 d7XMx 2 · cos x aONQ]1x 1 ≤ 2 œ¬]_p sin x,
2
· sin x
ŸKMPOp+P
Yœ
d7XMx ]_bjKMPOpjŸY[g+P`¢ avav]_p+x—YlXZ^Qb+]/bjKMPWpd7XMxZ]_NNQ]1xZPO\‘± g+KM]_fZ\[x
k?PƒP<¤—bjp+ePO=NQPO1\li aO|x\[]`g+|Pƒ≤b+] + 0 ±?YlNƒtZ\li1YlXZ^ždXMPO^`\lYl^`YlkZ\[PˆYlXMaOp+Ped_g+PQ]7œbjKMPš–_d7\lfMPƒxbjK;d7baed7X
k?Pbd7¦…POX kui |h| ¢ ]_p+Pv]e–…POpe±¥bjKZY[gˆg+PObš]7œ~x—Y[gjbjYlXZ^`fZY[gjKMPvx t]_YlXub+gQ^`Yl–…Pvg®d£NQ]_p+P3d_avaOfZpd7b+P
YlNƒtZ\[PONQPOXubd7bjY[]_X•]7œ?bjKMPœ5fZXMaObjY[]_X²­ugjYlXMavP |x − x | Y[g¥k?]_fZXMxZPvxQkni ≈ 2 ±`bjKMPµMX;d7\]_fZbjtZfZb
œ¬]_p f (x) aed7X¯k?PWNšd_xZP~d_avaOfZpd7b+P~ŸYlbjK ≈ 63 kZYlb+g]7œ)tZp+PvaOY[gjY[]_XId7\lbjKM]_fZ^`K¯bjKMP$aed7\[aOfZ\¨d7bjY[]_XMg
d7p+PINšd_xZP£ŸYlbjK˜xZ]_fZkZ\[Pvge¢ c^nd7YlX YlX˜bjKMP3pd7XMxZ]_N NQ]1xZPO\‘±bjKZY[g®YlNƒtZ\lY[PvgbjK;d7b®bjKMPI–`d7\lfMP
sin x = sin x0 · cos h + cos x0 · sin h,
0
x0
0
1
2
0
0
h
h
10
0
x0
1
2
0
i
0
i
0
xi0
−10
π
6
1
2
53
i
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x
−10
i
0
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211
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10
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−12
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•
]_fZbjtZfZbœ¬]_p (x) Y[g•av]_pjp+PvaObj\li p+]_fZXMxZPvx˜ŸYlbjK˜tZp+]_k;d7kZYl\lYlbi
d7p+]_fZXMx 1 − 2 ±ŸKZY[aKY[g
KZYl^`KMPOpcbjK;d7X f99.9%
¢
JKMPbd7kZ\[Pvgd7p+Pav]_XMgjbjpjfMaOb+PvxˆkniQP<¤—K;d7fMgjbjYl–…Pcg+Ped7p+aK²±ng+]~bjK;d7b©bjKMPOp+Psd7p+P€d7tZtZp+]v¤—YlNšd7b+PO\li
j
b
pjY¨d7\[gœ¬]_p€d7Xui¯x—Y[gjbjYlXZ^`fZY[gjKMPvxžPO\[PONQPOXnbW]7œ¥bjKMP ›‘bd7kZ\[P`±d7XMxžd7tZtZp+]v¤—YlNšd7b+PO\li 2 bjpjY¨d7\[g
2
œ¬]_pcd7Xni•x—Y[gjbjYlXZ^`fZY[g+KMPvx¯PO\[PONQPOXnb€]7œbjKMP sin x›‘bd7kZ\[2P`¢
¼ ¸ ®¸ h½ ¸ ‹7‰ ¼ ªŠ ¸ POb n kPšbjKMPˆtZp+PvaOY[g+Y[]_X³d7XMx f : [ , 1[→ [ , 1[ kPˆbjKMP®av]_XMgjY[xZPOp+Pvx9œ5fZXMaObjY[]_X²¢ m k3!
i^ x œ¡]_p f Y[gsd n›‘kZYlbsNšd_a+KZYlXMPXnfZNkPOpWgjfMaKIbjK;d7bWd7bs\[Ped_gjb n + m kZYlb+gWd7p+PXMPvPvxZPvx3b+]
p+]_fZXMx f (x) av]_pjp+PvaObj\li…¢ ]_p+PWtZp+PvaOY[g+PO\libjKMPvg+P~d7p+P$bjKMP x ’gYlX [ , 1[ gjfMaK¯bjK;d7be­
aONQ]1x 1| ≤ 2 œ¡]_pbjKMP x—Ylp+PvaOb+Pvxp+]_fZXMx—YlXZ^ NQ]1xZPvg ”5b+]Ÿd7p+xZg 0 ± +∞
|2]_p · f (x)
<
˜
±
−∞
|2 · f (x) − aONQ]1x 1| ≤ 2 œ¬]_pbjKMP$p+]_fZXMx—YlXZ^ˆb+]/XMPed7p+PvgjbNQ]1xZP`¢
€]_bjY[avP¯bjK;d7bˆYlX bjKZY[gˆxZP<µMXZYlbjY[]_X²± m aed7X°kPId7Xniªp+Ped7\cXufZNSk?POpe9¢ O€POp+P¯bjKMP¯œ5fZXMaObjY[]_XMgˆŸEP
av]_XMg+Y[xZPOp$K;dv–…Pƒ]_fZbjtZfZb~pd7XZ^…Pvg~x—Y DPOp+POXnbWœ¡p+]_N [ , 1[ ±²g+]®ŸEPQd_xZ]_tZb$bjKMPƒxZP<µMXZYlbjY[]_X b+]]_fZp
aed_gP`¢¶ Pgdei bjK;d7b x ∈ [ , 1[ Y[gƒd m›‘k;d_x aed_gPšœ¬]_p 2 Y œ |2 aONQ]ux 1| ≤ 2 ±d
›‘k;d_x aed_g+Pšœ¡]_p cos x Y œ |2 · cos x aONQ]ux 1| ≤ 2 ±d m›‘k;d_x9aed_g+Pšœ¬]_p sin x Y œ |2 ·
m
aONQ]ux 1| ≤ 2 ŸKMPOX x ∈] , 1[ d7XMx |2 · sin x aONQ]ux 1| ≤ 2 Y œ x ∈ [ , [ ¢
sin x
JKMP¦uXM]Ÿ\[Pvx—^…P©]7œZbjKMP d!ZX’i9;]ZX! ±OY‘¢’P`¢vbjKMP¥Nšd¤1YlNfZN gjfMaK~bjK;d7b d_x—NƒYlb+g
d m›‘k;d_x“aed_g+P`±Nšd7¦…PvgSYlbt]`ggjYlkZ\[PQb+]IYlNƒtZ\[PONQfPOXnb f P<™šaOY[POXubj\li…±k?Pvmaed7fMg+PˆbjKMPQNšfd¤—YlNSfZN
XufZNSk?POp¥]7œ!kZYlb+gŸKZY[a+Kšd7p+PXMPvPvxZPvx/Y[g)¦1XM]eŸX/YlXQd_x—–_d7XMavP`¢`JKMPx—pdeŸk;d_a+¦ƒ]7œ;bjKZY[g¥d7tZtZp+]…d_a+K
Y[g®bjK;d7bbjKZY[g®–_d7\lfMPIY[g®K;d7p+x b+] av]_NƒtZfZb+P`¢JKMPžP<¤—K;d7fMgjbjYl–…P£g+Ped7p+a+KY[g®kni œ¬d7p®YlX—œ¡Ped_g+YlkZ\[P
k?Pvaed7fMg+P]7œ!Ylb+g ≈ 2 av]_NƒtZ\[P<¤—Ylbi…¢ P<œ¬LO–1p+Pd7XMx fZ\l\[POptZp+]_t?]`g+Pvxƒd ≈ 2 d7\l^…]_pjYlbjKZN FIHe|—±
HOZ‚ Jš±ŸKZY[a+K
Ÿd_g3gjf—™šaOY[POXub3b+]³t?POp,œ¡]_pjN g]_NQP gji1g+b+PONšd7bjY[a ŸE]_pj¦˜YlX
xZ]_fZkZ\[P tZp+PvaOY[g+Y[]_X²¢
rub+POKZ\[V`± P<œ¬LO–1p+Pd7XMŽx œ!YlNƒNQPOpjNšd7XZX F R`.R J1^nde–…Pd^…POXMPOpd7\lIY ™d7bjY[]_XQ]7œMbjKZY[gd7\l^…]_pjYlbjKZN kni~fMg+YlXZ^
\¨d7bjbjY[avP<›‘k;d_gPvx ]_tZtPOp+gjNƒYlbjK²’gWŸ]_pj¦3b+]šµMXMx£gjNšd7\l\p+]1]_b+gW]7œ©NQ]1x—fZ\¨d7p€t?]_\li1XM]_NƒY¨d7\[Žg F |—±,Z‚ J¢
JKMPOi9]_kZbd7YlXMPvx d7X³d7\l^…]_pjYlbjKZN ŸYlbjK dIKMPOfZpjY[gjbjY[a
av]_NƒtZ\[P<¤—Ylbi…±)ŸKZY[a+K³Ÿd_g/\¨d7b+POp
YlNƒtZp+]–…Pvxšb+] ≈ 2 F R`.| J¢ s\lbjKM]_fZ^`K®bjKMPvg+Pav]_NƒtZ\[≈P<¤—2YlbiQk?]_fZXMxZgd7p+PckPObjb+POpEbjK;d7XšbjKMPs]_XMP
]7œ P<œ¬LO–up+P`’g~d7\l^…]_pjYlbjKZN¯±DYlb$g+PvPONQgWbjK;d7b$bjKMPtZpd_aObjY[aed7\©aOfZb,›] kPOb«ŸEPvPOX k?]_bjK NQPObjKM]1xZgWY[g
d7p+]_fZXMx¯bjKMP~xZ]_fZkZ\[P~P<¤—b+POXMxZPvx•tZp+PvaOY[gjY[]_X²±MY‘¢’P`¢ n = 64 ¢
−10
9
x
18
1
2
1
2
1
2
n
n
−m
1
2
n
−m
1
2
53
1
2
52+x
−m
53
−m
π
6
−m
x
53
54
−m
n
1 π
2 6
2n/3
3n/5
5n/7
m8_8_6A1 R /
L/ 7254
5: Z:U/ Y 4
s fZp©YlNƒtZp+]e–…PONQPOXnbY[gk;d_g+Pvxš]_XQbjKMPY[xZPedWbjK;d7b¥Ylb¥Y[g¥t]`g+g+YlkZ\[Pb+]Wp+PvmnfZYlp+PpjfZXMgE]7œNQ]_p+PbjK;d7X
av]_XMg+PvaOfZbjYl–…P~Y[xZPOXnbjY[aed7\²kZYlb+gcd_gcU~d7\²xZ]uPvge¢1¶ P€µMp+gjbxZPvgaOpjYlkPWŸK;d7bcd7p+PWbjKMPWk?Pvgjbg+POb+g]7œ
10
x—Y[g+bjYlXZ^`fZY[gjKMPvxšt?]_YlXnb+g©œ¬]_p 2 d7XMx sin x ±ud7XMxšbjKMPOX®g+KM]eŸ bjK;d7b¥œ¬]_p 2 ±—U$d7\‘’g©NQPObjKM]1xQk;d_gPvx
x
x
¶ }
"!$#%&(')'*!
]_X“bjKZY[gSg+PObSaed7X9kP/fMgPvx œ¬]_p$bjKMPQŸKM]_\[PšPO–`d7\lf;d7bjY[]_X9ga+KMPONQPm”5k?]_bjK9mufZY[a+¦ d7XMx“d_avaOfZpd7b+P
tZK;d_gPvg˜<¢
† ŒZ‡ ¸ ŒZ½ ¸ž¼ ŒZ½SŒZ‰ ¸ ‰ «¼ ‹šŠ ¸ A‹ ¸ Ž…Š ¸ ½ ˆ¼ ½ '¼ ½ ¸ ‰
sXMxZPOp)bjKMPpd7XMxZ]_N NQ]1xZPO\Zd_ggjfZNƒtZbjY[]_X²±`Ylb)Y[g)t]`ggjYlkZ\[Pb+]$gjbjp+POXZ^`bjKMPOXƒbjKMPp+PvmnfZYlp+PONQPOXub+g©]7œ
U~d7\n]_X~bjKMP¥\[POXZ^`bjKMg]7œZbjKMPpjfZXMgh]7œMav]_XMg+PvaOfZbjYl–…P©Y[xZPOXnbjY[aed7\1kZYlb+ge¢(KM]_pbjKMP 2 œ¡fZXMaObjY[]_X²±vŸEPEaed7X
bd7¦…P$d_gg+POb]7œx—Y[gjbjYlXZ^`fZY[g+KMPvxt]_YlXub+gd7\l\bjKMP 42›‘k;d_x•aed_g+Pvge¢—JKMPOp+P$d7p+P$d7tZtZp+]v¤—YlNšd7b+PO\li 2
]7œ¥bjKMPON d7XMxIbjKMPNšd¤1YlNfZN x—Y[g+bd7XMavPk?PObŸPvPOX bŸ]®av]_XMgPvaOfZbjYl–…P]_XMPvg€Y[g€k?PO\[]eŸ 2 ±
gPvPr1PvaObjY[]_X {—¢ EKM]1]`gjYlXZ^gjfMa+K dšgPObs]7œ©x—Y[gjbjYlXZ^`fZY[gjKMPvxž–`d7\lfMPvgsxZPvaOp+Ped_gPvgsbjKMPPOpjp+]_p€Nšd_xZP
]_X 2 YlX
kZfZbg+YlXMavP$bjKMPWNšd¤1YlNfZN x—Y[gjbd7XMavP~kPO2b«ŸEPv=POX32 b«ŸE·]ˆ2x—,Y[gjbjYlXZ^`fZY[gjKMPvx•–`d7\lfMPvgY[gXM]_bcxZPvaOp+Ped_gPvxD±
bjKMPˆPOpjp+]_pSNšd_xZPšŸKZYl\[PšPO–`d7\lf;d7bjYlXZ^IbjKMPšt]_\li1XM]_NƒY¨d7\d7tZtZp+]e¤1YlNšd7bjYlXZ^ 2 œ¬]_p h aO\[]`g+PQb+] 0
Y[gp+]_fZ^`KZ\li9bjKMP®g d7NQP`±Y‘¢’P`¢
­gjYlXMavP®bjKMPav]uP<™šaOY[POXub+gQ]7œcbjKMP®t]_\li1XM]_NƒY¨d7\d7p+PxZ]_f—›
kZ\[Pvg/d7XMx“g+YlXMavPQbjKZY[g~Y[g/d7\[g+]3≈bjKM2PQŸE]_pj¦1YlXZ^žtZp+PvaOY[gjY[]_X²±²bjKMP®xZPO^`p+PvP<› 1 av]1P<™šaOY[POXnbY[gav]_pjp+PvaOb
ŸYlbjK³POpjp+]_pk]_fZXMxZPvx9kni ≈ 2 ¢ €gd3av]_XMg+PvmufMPOXMavP`±bjKMPˆPOpjp+]_pNšd_xZPˆYlX bjKMPšmufZY[a+¦
tZK;d_gPSY[g€p+]_fZ^`KZ\li¯bjKMP/gd7NQP`¢ fZbWd_g€ŸEPŸYl\l\g+PvPYlXIbjKMPXMP<¤1bWgjfZkMg+PvaObjY[]_X²±bjKZY[gWa+KM]_Y[avP/]7œ
x—Y[g+bjYlXZ^`fZY[gjKMPvx•t?]_YlXnb+gNšd7¦…PvgcbjKMP~d_avaOfZpd7b+P~tZK;d_g+P~NQ]_p+P~P<™šaOY[POXnbe¢
KM]_psbjKMP
œ5fZXMaObjY[]_X²±?gjYlXMavPSbjKMPOp+Pd7p+PbŸ]ˆp+PO\¨d7b+Pvx3œ¡fZXMaObjY[]_XMgsYlXIbjKMPgPvav]_XMx3pd7XZ^…P
p+Pvx—fMaObjY[]_X ”5X;sind7NQxPO\li sin x d7XMx cos x˜<±DŸEPˆaed7XZXM]_bP<¤—tPvaOb$pjfZXMg]7œav]_XMg+PvaOfZbjYl–…PQY[xZPOXubjY[aed7\
kZYlb+gd_g)\[]_XZ^~d_gœ¡]_p 2 Z¢ =XMxZPvPvxD±7bjKMPk?PvgjbaKM]_Y[avP]7œ!x—Y[gjbjYlXZ^`fZY[gjKMPvx/–`d7\lfMPvg)Y[gNšd_xZP]7œd7\l\—bjKMP
’g)bjK;d7bEd7p+PgjYlNSfZ\lbd7XMPv]_fMg+\li 21›‘k;d_xšaed_g+Pvg)œ¬]_p sin x d7XMx cos x ¢`JKMPOp+Psd7p+Pd7tZtZp+]v¤—YlNšd7b+PO\li
x
]7œbjKMPON¯±Zd7XMxbjKMP$dv–…POpd7^…P$x—Y[g+bd7XMavPsk?PObŸPvPOX¯bŸ]ƒav]_XMg+PvaOfZbjYl–…P$]_XMPvgY[gkPO\[]Ÿ 2
2
d_gEgjKM]eŸXˆkni/bjKMPcP<¤—t?POpjYlNQPOXnb+g]7œhr1PvaObjY[]_X•{—¢…JKZY[g©NQPed7XMg©bjK;d7bEbjKMPcPOpjp+]_p+g¥Nšd_xZPs]_X sin x
d7XMx cos x YlX
d7p+P$xZPvaOp+Ped_gPvx®œ5p+]_N ≈sin2 x =b+] sin≈ x2 · cos¢—¶ hP$+d7pjcos
^`fMP$xbjK;·d7sinbbjKMhP$POpjp+(1)]_p+gNšd_xZPWYlX•bjKMPWt]_\li1XM]7›
NƒY¨d7\PO–_d7\lf;d7bjY[]_XMgWav]_pjp+Pvgjt?]_XMx—YlXZ^®b+] d7XMx aed7Xžd7\[g]Qk?P~Nšd_xZP~bjK;d7b€gjNšd7\l\‘±!YlXžd
–…POpjiP<™šaOY[POXnbsŸdei…¢M¶ P~xZP<µMXMP cos h sin h
d7XMx C(h) = 1 − 1 h + a h ,
S(h) = h − a h + a h
2
ŸKMPOp+P a Y[g©bjKMP€xZ]_fZkZ\[PsaO\[]`gPvgjb©b+] œ¬]_p i ∈ {3, 4, 5} ¢ €gjYlXZ^bjKMPœ‘d_aOb©bjK;d7b |h| ≤ 2
d7\[]_XZ^šŸYlbjK d7^`pd7XZ^…P`’gk?]_fZXMxD±ZŸP~]_kZbd7YlX3bjK;d7be­
x
12
−9.914
x0
x
x0
h
h
−10
−53−10
x
12
−9.977
0
0
0
−63
3
i
0
−74
3
5
5
1
i!
3
5
h
h
+ a3 −
| sin h − S(h)| ≤ sin h − h −
+
6
120 ¶0¶¸·¹Lº»º¼
2
4
4
−10.977
1 3 1 5
|h| + a5 −
|h |
6
120 He
†aVb';X‹Zj ')V$VWVŽXLX
≤ 2−76.839 · 2−12.299 + 2−56.584 · 2−32.931 + 2−62.906 · 2−54.885
≤ 2−88.314 ,
4 2
h
1
h
+ a4 − |h|4
+
| cos h − C(h)| ≤ cos h − 1 −
2
24
24 ≤ 2−65.862 · 2−9.491 + 2−58.584 · 2−43.908
ŸE]£Nšd_a+KZYlXMP
uX¶ fZP®NSXMk?]POŸp+gEd7P<¤—tZtZtZ\¨p+d7]eYl¤1X³YlNšKMd7]eŸ´
bjYlXZb+^ ]£cosPO–`d7x\lf;d7d7b+XMP®x Pvsinmuf;d7xbjY[]_¢ Xi/(1)av]_¢ XMgjbjPOpjb fMaOcbjY[]_d7XšXMŸx PsK;dvk?–…PšP |cbjKMP®−b«cos
x |, |s −¢
¢
O
P
b
7
d
M
X
x
n
¢
s
w
v
P
e
a
7
d
l
\
Z
\
j
b
;
K
7
d
b
sin
x
|
≤
2
C
(h)
=
C(h)
−
1
S
(h)
=
S(h)
−
h
h
≤
¶ tZp+P¥PvaOµMY[g+p+Y[gj]_bX²av­ ]_NƒtZfZb+Pd7tZtZp+]v¤—YlNšd7bjY[]_XMg]7œ S (h) d7XMx C (h) ŸYlbjKfO€]_pjXMPOpe’gNQPObjKM]uxYlX2 xZ]_fZkZ\[P
≤ 2−75.352 .
0
0
−74
0
0
0
∗
0
∗
∗
0
0
−10.977
∗
k ← o(h2 )
|k − h2 | ≤ 2−54−21 ≤ 2−75
0
3
k ← o(h · k)
|k − h | ≤ 2−10.977−75 + 2−54−32 ≤ 2−84.988
S ∗ ← o(a5 · k)
|S ∗ − a5 · h2 | ≤ 2−6.906−75 + 2−54−28 ≤ 2−80.952
S ∗ ← o(S ∗ − a3 )
|S ∗ − (−a3 + a5 · h2 )| ≤ 2−80.952 + 2−54−2 ≤ 2−55.999
∗
0
∗
∗
S ← o(k · S ) |S − S ∗ (h)| ≤ 2−84.988−2.584 + 2−55.999−32.930 + 2−54−35 ≤ 2−86.754
C ∗ ← o(a4 · k)
|C ∗ − a4 · h2 | ≤ 2−4.584−75 + 2−54−26 ≤ 2−78.777
|C ∗ − (− 12 + a4 · h2 )| ≤ 2−78.777 + 2−54−2 ≤ 2−55.999
C ∗ ← o(C ∗ − 12 )
C ∗ ← o(k · C ∗ )
|C ∗ − C ∗ (h)| ≤ 2−75−1.000 + 2−55.999−21.953 + 2−54−22 ≤ 2−74.824
0
JKMPSaed7\[aOfZ\¨d7bjY[]_XI]7œ sin x POXMxZgŸYlbjK¯bjKMPSgjfZN¯­
k ≤ 2−21.953
|k0 | ≤ 2−32.930
|S ∗ | ≤ 2−28.859
|S ∗ | ≤ 2−2.584
|S ∗ | ≤ 2−35.513
|C ∗ | ≤ 2−26.536
|C ∗ | ≤ 2−1.000
|C ∗ | ≤ 2−22.951 .
s = s0 + c0 · h + (s0 · C ∗ + c0 · S ∗ ),
bjKMP/b+POpjN c · h/k?POYlXZ^3PO–_d7\lf;d7b+Pvx YlX“d7X P<¤1b+POXMxZPvx tZp+PvaOY[g+Y[]_X²±?œ¡]_p~P<¤Md7NƒtZ\[PƒkuiIfMgjYlXZ^žd
muf;d_x—pjfZtZ\[P`±Md/xZ]_fZkZ\[P<›xZ]_fZkZ\[P`±;]_pkui®g+YlNSfZ\¨d7bjYlXZ^QbjKMP$P<¤—b+POXMxZPvxtZp+PvaOY[gjY[]_X¯ŸYlbjK•bjKMPWfMg+PW]7œ
dœ¡Nšd/]_t?POpd7bjY[]_X²¢YKM]_pcP<¤Md7NƒtZ\[P$ŸP~xZ]/bjKZY[gŸYlbjKIdœ¡Nšd—¢
h ← o (s +¢ c · h) ¢
t ← o(h − s ) ¢
l ← o (t − c · h)
ruYlXMavP
±ZŸP~K;de–…P h ≥ d7XMx•bjKMPSg+Pvav]_XMx3]_t?POpd7bjY[]_X3Y[gP<¤Md_aObe¢MJKMPOp+P<œ¬]_p+P~ŸP
K;de–…P`­ |c · h| ≤
d7XMx |(h + l) − (sin x + cos x · h)| ≤ 2 + 2
h +l = s +c ·h
≤2
.
av]_pjp+¶ PvaOPbj\liXM]bjŸKMP$avp+]_PvNƒg+fZtZ\lfZbb+NQPƒ]`g+]_gjbcNQ]7PSœ)\[bjPvKMgPWg€g+bjYlYlNQ^`XZP`Y µ;¢ aed7Xnb$kZYlb+ge±!ŸKZY[a+K YlX£t;d7pjbjY[aOfZ\¨d7p~gjf—™šavPb+]®p+]_fZXMx
0
0
f ma
0
0
0
0
f ma
s0
2
0
0
0
0
0
0
0
s0
2
0
0
−74
−74−10.977
−73.999
S ∗ ← o(c0 · S ∗ ) |S ∗ − cos x0 · S ∗ (h)| ≤ 2−74−35.513 + 2−0.188−86.754 + 2−54−35 ≤ 2−86.631 |S ∗ | ≤ 2−35.700
C ∗ ← o(s0 · C ∗ ) |C ∗ − sin x0 · C ∗ (h)| ≤ 2−74−22.951 + 2−0.249−74.667 + 2−54−23 ≤ 2−74.610 |C ∗ | ≤ 2−23.199
¶ "!$#%&(')'*!
HH
C ∗ ← o(S ∗ + C ∗ ) |C ∗ − (cos x0 · S ∗ (h) + sin x0 · C ∗ (h))| ≤ 2−86.631 + 2−74.610 ≤ 2−74.609
C ∗ ← o(l + C ∗ )
|(h0 + C ∗ ) − sin x| ≤ 2−74.609 + 2−73.999 + 2−1−106 ≤ 2−73.271 .
¶ KMPOX bjKMP¯x—Y[g+bd7XMavP¯k?PObŸEPvPOX 2 · sin x d7XMx Z Y[gQKZYl^`KMPOpšbjK;d7X 2 · 2 =
§IŸKMPOp+P e Y[g 1 YlX bjKMPšaed_g+Pƒ]7œcd®p+]_fZXMx—YlXZ^3b+]•bjKMPƒXMPed7p+Pvg+b~NQ]uxZP`±d7XMx 0 œ¡]_p
2
^`Yl–…Pvg®bjKMP£av]_pjp+PvaOb•]_fZbjtZfZbe¢JKZY[g
d9x—Ylp+PvaOb+Pvx p+]_fZXMx—YlXZ^³NQ]uxZP¯§³bjKMP d_xZx—YlbjY[]_X
YlNƒtZ\lY[PvgbjK;d7bkni®fMgjYlXZ^QgjfMa+K3d/ga+KMPONQP$bjKMPWp+PvgjfZh\lb+Y[gCav]_pjp+PvaObj\lip+]_fZXMxZPvx•ŸYlbjK•tZp+]_k;d7kZYl\lYlb«i
d7b©\[Ped_gjb 1 − 2 ±…ŸKZY[a+KšNšd7¦…PvgEbjKMP 99.9% Pvg+bjYlNšd7b+Pc]7œU$d7\;YlXMaOp+Ped_gPb+] ≈ 99.9998% ¢
† Ž`Š ¼ ZŒ hŒ—Š ¼ Ž ¸ ¸ ŒZ‰ ¸ ½ ¼ ®Œ l‰ ¸ Š ¼ ½
=«X bjKZY[ggjfZkMg+PvaObjY[]_X“ŸPšxZPvg+aOpjYlk?PQKM]Ÿ ]_XMPˆav]_fZ\[x“PO–`d7\lf;d7b+P
YlX xZ]_fZkZ\[P
tZp+PvaOY[g+Y[]_XšŸYlbjKˆav]_pjp+PvaObp+]_fZXMx—YlXZ^kniƒfMgjYlXZ^S]_XZ\liˆU~d7\‘’g¥NQPObjKM2]1xD±n: Yl[XQ,bjKM1[→
Psaed_[1,g+Pc]72[œhd$x—Ylp+PvaOb+Pvx
p+]_fZXMx—YlXZ^žNQ]1xZP`¢ €g~fMgjf;d7\‘±hbjKMPOp+P®d7p+PˆbŸ]3tZK;d_gPvge­²bjKMP®mnfZY[a¦ tZK;d_g+Pˆd7XMx9bjKMPˆd_avaOfZpd7b+P
tZK;d_gP`¢
¸ 5Ž ŒZ‰ ¸
¶ PSfMg+PQU$d7\‘’g€NQPObjKM]1x3ŸYlbjK£bjKMPbd7kZ\[PNšd_xZP/]7œ©bjKMP 42›‘k;d_x aed_g+Pvg¢
JKZY[gYlXMx—fMavPvgd7XPOpjp+]_pk?]_fZXMxZPvxˆkni œ¡]_pbjKMPsb+POpjN ±1d7XMxˆbjKMPstZp+]_kZ\[PON Y[gp+Pvx—fMavPvx
b+]žav]_NƒtZfZbjYlXZ^“d7tZtZp+]v¤—YlNšd7b+PO\li 2 Ÿ2KMPOp+P |h| Y[gg+Nšd7\l\[POp2 bjK;d7X 2
”¡g+PvP•r1PvaObjY[]_X 3{ ˜<¢
¶ P/xZ]®bjKZY[gWkni£PO–`d7\lf;d7bjYlXZ^•bjKMPt?]_\li1XM]_NƒY¨d7\ P (h) = 1 + a h + . . . + a h ±ŸKMPOp+P a Y[g
bjKMPsxZ]_fZkZ\[PcbjK;d7bY[g©aO\[]`gPvgjb©b+] j¢ =b¥Y[gEt?]`g+gjYlkZ\[Pb+]a+KMPva¦QbjK;d7b P (h) aed7XˆkPsPO–_d7\lf;d7b+Pvx
ŸYlbjK ]_XZ\li ]_t?POpd7bjY[]_XMg~]_X xZ]_fZkZ\[Pvg~b+]¯d7tZtZp+]e¤1YlNšd7b+P 2 ŸYlbjK“POpjp+]_p~k?]_fZXMxZPvx kni ≈ 2
ŸKMPOX |h| ≤ 2 ¢—JKZY[gYlNƒtZ\lY[PvgbjK;d7bY œ x Y[gXM]_bsd 10›‘k;d_x¯aed_g+P`±—bjKMPW–`d7\lfMP$aed7\[aOfZ\¨d7b+Pvx
g]œ¬d7p~œ¡]_p Y[g~av]_pjp+PvaObe¢ ]_p+Pv]e–…POpe±)d_gd¯gjY[xZPQP DPvaObe±²Y œ Y[gd ›‘k;d_x9aed_g+P`±hbjKMPƒp+PvgjfZ\lb
aed7X d7\[g+]ˆk2Pav]_pjp+PvaObj\li3p+]_fZXMxZPvxD­œ¬]_pWPed_a+K POXnbjpji (x , 2 )x]7œ©bjKMP42bd7kZ\[P`±ŸEPƒd_xZx£dškZYlbW]7œ
YlX—œ¬]_pjNšd7bjY[]_Xb+PO\l\lYlXZ^ƒŸKMPObjKMPOpbjKMP€g+b+]_p+Pvxˆ–`d7\lfMPcœ¡]_p 2 Y[ggj\lYl^`Kubj\liš\¨d7pj^…POp]_pgjNšd7\l\[POpbjK;d7X
Ylb+gp+Ped7\²–_d7\lfMP`¢
¸ ŒZŽ…Ž h‹7Œ—Š ¸ ŒZ‰ ¸ rufZtZt?]`g+PsXM]eŸ bjK;d7bEbjKMP€mnfZY[a¦QtZK;d_g+PsŸd_gXM]_bgjf—™šaOY[POXnbb+]^`f;d7p,›
d7Xub+PvP®dIav]_pjp+PvaObp+]_fZXMx—YlXZ^£]7œbjKMP®]_fZbjtZfZbe¢¶ Pš¦…PvPOt bjKMPšg d7NQPšt;d7Ylp
d7XMx ]_XZ\li
aK;d7XZ^…PcbjKMPct]_\li1XM]_NƒY¨d7\?PO–_d7\lf;d7bjY[]_XgjfZk—›‘tZK;d_g+P`¢u¶ PcPO–`d7\lf;d7b+PcYlX®mnf;d_x—pj(xfZtZ\[,Ps2tZp+)PvaOY[gjY[]_XˆbjKMP
t?]_\li1XM]_NƒY¨d7\ Q(h) = 1 + b h + . . . + b h ŸKMPOp+P b Y[gƒbjKMP•muf;d_x—pjfZtZ\[P•tZp+PvaOY[gjY[]_X Nšd›
aKZYlXMP$XnfZNkPOpsbjK;d7bY[gaO\[]`g+Pvgjbb+] _¢ =baed7X•k?P$a+KMPva¦…Pvx¯bjK;d7b Q(h) aed7X•k?P$PO–_d7\lf;d7b+Pvx
ŸYlbjK ]_tPOpd7bjY[]_XMg]_X9muf;d_x—pjfZtZ\[PvgSb+]Id7tZtZp+]v¤—YlNšd7b+P 2 ŸYlbjK9POpjp+]_pSk?]_fZXMxZPvx kui 2
ŸKMPOX |h| ≤ 2 ¢ POb Q̄(h) kP~bjKMP$–_d7\lfMPav]_NƒtZfZb+Pvx•œ¬]_p Q(h) ±d7XMx y k?P$bjKMP$–`d7\lfMP
œ¬]_p 2 bjK;d7bY[ggjb+]_p+Pvx¯YlX•bjKMP$bd7kZ\[P`¢Z¶ PWµMX;d7\l\li¯d7tZtZp+]v¤—YlNšd7b+P 2 kui y · Q̄(h) ±ZbjKMP~POpjp+]_p
k?POYlXZ^ƒk?]_fZXMxZPvx•kni?­
53+e
53+e
−73.271
−19.271+e
0
∗
−19.271
1
2
x
x0
−41
h
−10.914
1
(ln 2)i
i!
4
4
h
i
−63
−10.914
x
0
x0
x0
x0
0
1
(ln 2)i
i!
7
7
i
h
−106.821
−10.914
x0
x
0
0
2−106.821+1 + 2−41−53 · 1.000360 + 2−114+1 ≤ 2−93.999 .
¶0¶¸·¹Lº»º¼
HeR
†aVb';X‹Zj ')V$VWVŽXLX
JKZY[gcYlNƒtZ\lY[PvgcbjK;d7bsY œ Y[gXM]_bWd
›‘k;d_x aed_g+P`±MbjKMPOX3bjKMP$p+PvgjfZ\lbsY[gp+]_fZXMxZPvxIav]_pjp+PvaObj\li…¢
J KMPOp+P<œ¬]_p+P`±_ŸEPK;de–…Pd7xXƒYlNƒtZ\[PONQPO41.999
Xnbd7bjY[]_XQp+PObjfZpjXZYlXZ^~bjKMPav]_pjp+PvaObp+]_fZXMx—YlXZ^Wœ¬]_p¥d7XniSYlXZtZfZb x
P<¤ZavPOtZb©bjKM]`gPcbjK;d7bd7p+P
›‘k;d_x•aed_g+Pvg©kZfZbXM]_b ›‘k;d_xaed_g+Pvge¢n¶ PµZ¤QbjKZY[gEt]_YlXubEYlXˆbjKMP
œ¬]_\l\[]eŸYlXZ^$Ÿdvi…¢_¶ Pav]_XM41.999
gjY[xZPOp©dsbd7kZ\[PNšd_xZP]_XZ\li]7œ 4242.00072›‘k;d_xˆaed_g+PvgYlXMgjb+Ped_xƒ]7œ 42›‘k;d_x
aed_gPvge±²ŸKZY[aK9NQPed7XMgbjK;d7bŸPšp+PONQ]–…P 6543825232174115/2 œ¡p+]_N bjKMPˆ\[]1]_¦n›‘fZt9bd7kZ\[P
d7XMx9av]_XMgjY[xZPOpYlbd_gd3gjt?PvaOY¨d7\Eaed_g+P`¢JKZY[gSp+PONQ]–_d7\xZ]uPvgSXM]_baK;d7XZ^…PšbjKMPˆk]_fZXMx9]_X ¢
]_p+Pv]e–…POpe±—YlbY[gPed_gjiQb+]/g+PvP€bjK;d7bbjKMPWPOpjp+]_pd7X;d7\li1gjY[gq,fMgjbd7k]–…P€XM]eŸ^`Yl–…Pvgd7X®µMX;d7\DPOpj|h|p+]_p
k?]_fZXMxZPvx£kni 2 ¢?JKMPOp+P<œ¬]_p+PƒbjKMP/p+PvgjfZ\lb~Y[g$av]_pjp+PvaObj\li£p+]_fZXMxZPvx k?Pvaed7fMg+P/ŸEP/NfMgjb~k?P/YlX
]_XMPƒ]7œbjKMP/bjKZp+PvPœ¡]_\l\[]ŸYlXZ^3gjYlbjf;d7bjY[]_XMge­ Y[gWXM]_bSd ›‘k;d_x aed_gP`±?bjKMPPOpjp+]_pSk]_fZXMx£^`Yl–…Pvg
bjK;d7bSbjKMP/–_d7\lfMPƒp+PObjfZpjXMPvx œ¡]_p 2 Y[g~av]_pjp+PvxaObj\li p+]_fZXMxZ42Pvx x Y[gSd 42.000072›‘k;d_xªaed_gP`±?bjKZP
]_fZbjtZfZbcY[ggjb+]_p+Pvx•YlX•bjKMP$bd7kZ\[P ;]_bjKMPOpjŸY[g+P x Y[g 6543825232174115/2 ¢
53
−94
x
53
c2 96A/ /3:2<-s2/ r1]®œ‘d7pe±DŸPƒkZfZYl\lbS]_fZp$œ¡fZXMaObjY[]_X“PO–`d7\lf;d7bjY[]_X“g+a+KMPONQP®d_g$Y œŸEPƒ¦1XMPOŸyP<¤—tZ\lY[aOYlbj\ližbjKMP/bd7kZ\[Pvg
ŸP/xZPvg+aOpjYlkPvxD¢ €PO–…POpjbjKMPO\[Pvg+ge±bjKMPbd_gj¦3]7œav]_NƒtZfZbjYlXZ^•bjKMPbd7kZ\[PvgWY[g€kui¯œ‘d7psXM]_XubjpjYl–uY¨d7\‘­?Y œ
ŸPSŸPOp+PSfMg+YlXZ^®d7XIP<¤—K;d7fMgjbjYl–…Pg+Ped7p+a+K d_g$U$d7\xZ]1Pvge±MbjKMPav]`gjbs]7œ©av]_NƒtZfZbjYlXZ^bjKMPSbd7kZ\[P]7œ
POYlbjKMPOp ]_p Ÿ]_fZ\[x¯k?P aed7\l\[gcb+] ”5p+Pvgjt²¢ ˜EYlXIP<¤1b+POXMxZPvx3tZp+PvaOY[gjY[]_X²L¢ KM]_p
bjKMPWtZp+]_2kZ\[PON sinY[gPexd_gjYl\liˆg+]_\l–…Pvx•2]_XMavPWYlbY[gXM2]_b+PvxbjK;d7bcsinbjKMP x42›‘k;d_xIaed_g+Pvgaed7X•kPWaed7\[aOfZ\¨d7b+Pv2x
kui P<œ¬LO–up+P`’g$d7\l^…]_pjYlbjKZN FIHe|—^± HOZ‚ J]_psbjKMPS\¨d7bjbjY[avP<›‘k;d_gPvx£d7\l^…]_pjYlbjKZN ]7œrub+POKZ\[V`± P<œ¬LO–up+Pd7XMx
œ!YlNƒNQPOpjNšd7XZ
X F R`R—±+R`.| J’¢ =«X3bjKZY[gWg+PvaObjY[]_X²±Ddœ¡b+POps^`Yl–1YlXZ^®g]_NQPSk;d_a¦u^`p+]_fZXMx ]_Xž\¨d7bjbjY[avPvg$d7XMx
\¨d7bjbjY[avP~p+Pvx—fMaObjY[]_Xžd7\l^…]_pjYlbjKZNQge±;ŸEP~xZPvg+aOpjYlk?PSd/NQPObjKM]1x•b+]Qav]_XMg+bjpjfMaObcgjYlNSfZ\lbd7XMPv]_fMg+\li•k;d_x
aed_gPvg~œ¬]_pb«ŸE]Iœ¡fZXMaObjY[]_XMgen¢ KM]_p/d3^`Yl–…POX³tZp+PvaOY[gjY[]_X ±²bjKZY[gƒd7\l^…]_pjYlbjKZN d_x—NƒYlb+gƒd¯KMPOfZpjY[gjbjY[a
av]_NƒtZ\[P<¤—YlbiI]7œ ≈ 2 §YlXMgjb+Ped_xž]7œ ≈ 2 œ¬]_pcbjKMPnP<¤1K;d7fMgjbjYl–…Pg+Ped7p+aK²±d7XMx ≈ 2 ŸYlbjK
P<œ¡LO–1p+P`’gcd7\l^…]_pjYlbjKZN¯¢
† «¸…»®¸ ½ ¸ ‹‰ ¼ ©Œ—ŠvŠ AŽ ¸ ‰
=«X~bjKZY[ggjfZkMg+PvaObjY[]_XSŸP¥kZpjY[P DMi$^`Yl–…PEg+]_NQPXMPvavPvg+gd7pji$k;d_a¦u^`p+]_fZXMx]_X~\¨d7bjbjY[avPvg)d7XMxS]_X\¨d7bjbjY[avP
p+Pvx—fMaObjY[]_X³d7\l^…]_pjYlbjKZNQge¢²¶ PQp+P<œ¡POp~b+}] FIHH`^± H(—.JEœ¡]_pSNQ]_p+PQxZPObd7Yl\[gS]_X k?]_bjK bjKMPšd7\l^…]_pjYlbjKZNƒY[a
d7XMx¯Nšd7bjKMPONšd7bjY[aed7\d_gjt?PvaOb+ge¢
U*'U
i Y[gd x—Y[g+aOp+POb+P£gjfZkZ^`p+]_fZt˜]7œ ±©]_pPvmnfZYl–`d7\[POXnbj\li°bjKMP3g+POb]7œd7\l\€\lYlXMPed7p
YlXub+PO^`pd7\av]_NkZLYlX;d7bjY[]_XMgs]7œ ` ≤ n \lYlXMPed7pj\liYlXMxZPORt?POXMxZPOXnbc–…PvaOb+]_p+g ‡ ]–…POp R ±—bjK;d7bY[ge­
X̀ ‡
x
L={
| x ∈ Z}.
€]_bjY[avPbjK;d7bƒbjKMP•pd7XZ¦ ` ]7œWbjKMP\¨d7bjbjY[avP¯aed7XªkP•g+Nšd7\l\[POpƒbjK;d7XªbjKMP¯x—YlNQPOXMgjY[]_X n ]7œsbjKMP
PONkPvxZx—YlXZ^ g+t;d_avP`¢ €g/g+]1]_Xªd_g ` ≥ 2 ±dI^`Yl–…POX³\¨d7bjbjY[avP L d_x—NƒYlb+gšd7X³YlX—µMXZYlbi³]7œck;d_g+Pvg
x
52
n/2
x
x
n
2n/3
n
i
i i
i
i=1
¶ He|
"!$#%&(')'*!
p+PO\¨d7b+Pvx°b+] Ped_a+K˜]_bjKMPOpˆkui fZXZYlNQ]ux—fZ\¨d7pbjpd7XMg,œ¬]_pjNšd7bjY[]_XMge¢E¶ PIxZP<µMXMPIbjKMP !V$')XXY
]7œ/bjKMP \¨d7bjbjY[avP L d_g det(L) = Q ||‡ || ±ŸKMPOp+P ‡ , . . . , ‡ Y[g•bjKMP U€pd7N/› r1a+KZNƒY[x—b
]_pjbjKM]_^…]_X;d7\lYI™d7bjY[]_X']7œ/d9\¨d7bjbjY[avPžk;d_gjY[g ‡ , . . . , ‡ ¢EJKZY[g•muf;d7XnbjYlb«i xZ]1PvgXM]_b•xZPOtPOXMx˜]7œ
bjKMP/aKM]_Y[avPQ]7œ©bjKMP/k;d_gjY[ge¢ ]`gjb$]7œEbjKMPbjYlNQP`±D]_XZ\li3k;d_g+PvgWŸKZY[aK av]_XMgjY[gjb$]7œg+KM]_pjb€–…PvaOb+]_p+g
d7p+Ps]7œDYlXnb+POp+Pvgjbe¢1JKMP i kCWV$')XL'V$jV ]7œ?bjKMPc\¨d7bjbjY[avP L Y[g¥bjKMPsgjNšd7\l\[Pvgjb r g+fMa+KšbjK;d7bEbjKMPk;d7\l\
avPOXub+POp+Pvx•YlX ¯d7XMx•]7œhpd_x—YlfMg av]_Xnbd7YlXMgcd7b\[Ped_gjb \lYlXMPed7pj\li®YlXMxZPOt?POXMxZPOXnb\¨d7bjbjY[avPW–…PvaOb+]_p+ge¢
K;]_pšP<¤Zd7NƒtZ\[P¯bjKMPµMp+gjbQNƒYlXZYlNr fZN λ (L) Y[gƒbjKMP•i\[POXZ^`bjK ]7œSd£gjKM]_pjb+Pvg+bQXM]_X—;› ™ePOp+] \¨d7bjbjY[avP
–…PvaOb+]_pe¢
ruYlXMavPWYlbav]_pjp+Pvg+t]_XMxZgb+]/]_fZpgjYlbjf;d7bjY[]_X•YlXbjKMP€œ¡]_\l\[]ŸYlXZ^QgjfZkMg+PvaObjY[]_XŸEPWXM]Ÿ
gjfZtZt?]`g+P
bjK;d7b ` = 4 d7XMx L ⊂ Z j¢ =b©Y[gEaO\¨d_g+g+Y[aed7\ FIHHJbjK;d7bEYlXˆbjKZY[gEaed_g+PcbjKMPOp+P€d7\lŸdvi—gP<¤1Y[gjb+gd$k;d_g+Y[g
p+Ped_aKZYlXZ^³bjKMP¯œ¬]_fZpšµMp+g+bšNƒYlXZYlNšd—±gjfMa+K√d“k;d_gjY[gšk?POYlXZ^³aed7\l\[Pvx 'X_lgnil"'k!Z_
i! ±sd7XMx
bjK;d7bbjKMP$µMp+gjbNƒYlXZYlNSfZN ]7œ L Y[gk?PO\[]eŸ 2 · det(L) ¢ ]_p+Pv]–…POpe±MbjKMPWŸEPO\l\ ›‘¦1XM]eŸX d7\l^…]_pjYlbjKZN FIHe.{ J¥aed7X£kPfMg+PvxIb+]QµMXMx dš–…PvaOb+]_pWŸKZY[a+K Y[g€XM]_bsNfMa+K \[]_XZ^…POpWbjK;d7X£bjKMPµMp+gjb
NƒYlXZYlNfZN¯¢
d\WEIU*'U
i L ⊂ Z ‹' \ M = VN( || || ‹%€
¸u¼ ‹ ¸ “')&ZX 't
[ , . . . , ]
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Z‚z3Z'V rd&('*!ja'Xm*'VN1 O(log34 M ) P') [ 1, . . . , 4 ] *'))\[Z')Xz
√
|| 1 || ≤ 4λ1 (L) ≤ 4 2 det(L)1/4 .
5
i
i
J KZY[g)Y[g)XM]_b¥]_tZbjYlNšd7\MkPvaed7fMgPEbjKMP]_fZbjtZfZbk;d_g+Y[g)Y[g)XM]_bXMPvavPvg+gd7pji YlXZ¦…]eŸg+¦uY ›‘p+Pvx—fMavPvxD±
d7XMx~bjKMPOp+P<œ¡]_p+P©bjKMP¥–…PvaOb+]_p+g²Nšdei$XM]_b²k?P©d_ghgjKM]_pjbd_g²t?]`g+gjYlkZ\[P`¢eJKMP^`p+PvPvx—iSd7\l^…]_pjYlbjKZN ]7œ FIHe}.J
]_fZbjtZfZb+gWdƒk;d_g+Y[gp+Ped_a+KZYlXZ^bjKMP~µMp+gjbœ¬]_fZpcNƒYlXZYlNšd—±?d7XMx¯K;d_gsbjKMPd_xZx—YlbjY[]_X;d7\d_x—–_d7Xubd7^…P]7œ
d_x—NƒYlbjbjYlXZ^d O(log M ) av]_NƒtZ\[P<¤—Ylbik?]_fZXMxD¢
† † ¼ ‹ ¨Š «¼ ‹QŠ ¸ sin xº ŒZ‡ ¸
JKMPšg+Ped7p+a+K“]–…POp [ , 1[ Y[g~x—Yl–1Y[xZPvx YlXnb+] mufZY[a+¦ g+Ped7p+aKMPvgS]–…POp~YlXnb+POpj–`d7\[gS]7œ\[POXZ^`bjK ¢
JKMPvgP$mnfZY[a¦gPed7p+a+KMPvgsd7p+P~tPOp,œ¬]_pjNQPvx•kni®fMgjYlXZ^QbjKMP~d7\l^…]_pjYlbjKZN ŸEP~xZPvgaOpjYlkP$k?PO\[]eŸ~¢
U€Yl–…POX3b«ŸE]ˆœ5fZXMaObjY[]_XMg d7XMx ±!dƒtZp+PvaOY[g+Y[]_X ±;dQk;d_x1›aed_g+PSk?]_fZXMx d7XMxždšg+Ped7p+a+K
k?]_fZXMx T ±_bjKMPœ¡]_\l\[]ŸYlXZ^d7\lf^…]_pjYlbjKZNyf bjpjY[Pvg©b+]WµMXMxšd7\ln\MbjKMPNšd_a+KZYlXMPsXnfZNSk?POp+mg x ∈ [− , ]
g+fMa+K¯bjK;d7be­
aONQ]1x 1| ≤ 2 œ¬]_p i ∈ {1, 2}.
|2 · f (x)
=«XIœ¬d_aObWYlb$g+]_\l–…Pvg€bjKZY[gWtZp+]_kZ\[PON œ¡]_p$d7Xni N d7XMx M YlX£tZ\¨d_avPƒ]7œ 2 d7XMx 2 R¢ =«bWgjbd7pjb+g€kni
d7tZtZp+]e¤1YlNšd7bjYlXZ^Sk]_bjK d7XMx kuiSt?]_\liuXM]_NƒY¨d7\[g±_d7XMxbjKMPOX/bjpjY[Pvgb+]Wg+]_\l–…P©bjKMPtZp+]_kZ\[PON œ¬]_p
bjKMPvgPt]_\li1XM]_NƒY¨d7\[gEYlXMfgjb+Ped_xQ]7fœDbjKMPYlXZYlbjY¨d7\;œ5fZXMaObjY[]_XMge±…kniƒfMgjYlXZ^ ]_tZt?POp+gjNƒYlbjK²’gENQPObjKM]uxšb+]
µMXMxgjNšd7\l\…p+]1]_b+gh]7œZNSfZ\lbjYl–`d7pjY¨d7b+PNQ]ux—fZ\¨d7pt]_\li1XM]_NƒY¨d7\[|g F |—±,Z‚ J¢7sfZpd7\l^…]_pjYlbjKZN Y[g²KMPOfZpjY[g+bjY[a`­
d7\l\Ylb+g$]_fZbjtZfZb+gd7p+Pƒav]_pjp+PvaObe±DkZfZbWYlb$Nšdvi£PO–…POXnbjf;d7\l\ližœ¬d7Yl\‘¢€PO–…POpjbjKMPO\[Pvg+ge±?bjKMP/KMPOfZpjY[g+bjY[a/Y[g
2
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†aVb';X‹Zj ')V$VWVŽXLX
um fZYlb+Pp+Ped_g]_X;d7kZ\[Psd7XMxQbjKMP€d7\l^…]_pjYlbjKZN ŸE]_pj¦…Pvxˆ–…POpjiŸPO\l\!YlX®]_fZpEP<¤—tPOpjYlNQPOXub+gExZPvgaOpjYlkPvxˆYlX
r1PvaObjY[]_Xž{—­ZŸEPWœ¬]_\l\[]eŸ'bjKMP~gjbjpd7b+PO^`i]7œK;d7\l–uYlXZ^ T fZXubjYl\hbjKMPSd7\l^…]_pjYlbjKZN xZ]uPvgXM]_bœ¬d7Yl\‘±Md7XMx
YlbK;d7tZt?POXMgbjK;d7bcbjKMP~dv–…POpd7^…P T av]_pjp+Pvg+t]_XMxZgcb+]/ŸK;d7bbjKMP$bjKMPv]_pjitZp+Pvx—Y[aOb+ge¢
=X bjKMPQd7\l^…]_pjYlbjKZN ŸPƒfMg+PQd®p+]_fZbjYlXMP d7bjbjY[avPvwcPvx—fMavP`±²ŸKZY[aK“aed7X“POYlbjKMPOp$k?P/bjKMP d7\l^…]_pjYlbjKZN FIHe.{ J©]_p€bjKMP/^`p+PvPvx—i d7\l^…]_pjYlbjKZN ]7œ FIHe.} J¢DJKMP/YlXZtZfZbWœ5fZXMaObjY[]_XMgd7p+P/Nšd_xZP/YlXMxZP<›
t?POXMxZPOXnbs]7œ N ­ F (t) = N f (t/N ) ¢
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<ε
3
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0 ε 1/2 3 |Pi (t) −0 Fi·5(t)|
42Q P̃ (τ ) = bC · P (T
+. i ∈ {1, 2} τ )e
M = b 1/M +ε c 6 C = 3M 653
i
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º :G ?0
eg11 == 1C6 6 eg22==τC6 e·3τ =6 gτ3 =6 eP̃41=(τ υ) +6 e3υ
2)?V+/3φ
?0WE/?d·
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2? 4 × 5 –·
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3
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1C+ V ·54 Q (τ ) ? 1;1;/0 ?B4]9 ·2E 1 Z :VG ?0
M←[ LatticeReduce(L)
K
3
¼ ^+X
2?T1?I?02L1\
31 i ∈ {1, 2, 3} 1;-2/D7
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t /E4 1| <3 1 t+.0 IntegerRoots(q,
(t
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M
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¼ ‹7‹ ¸ Ž`Š ¸ ‰e‰ ¼ Š ¸ † ¼ ‹ AŠ JKMP$œ¡]_\l\[]ŸYlXZ^QbjKMPv]_p+PON ^`Yl–…PvgbjKMPSav]_pjp+PvaObjXMPvg+gc]7œ)bjKMPSd7\l^…]_pjYlbjKZN¯¢
¸u¼ ‹ ¸ xX
^€vZ‚z3Z'V !3"X *X x ‹ '*AZ&Z
iZ
^*~[‹5'Uw(wR'*
Œr€b3
^*Œ[EZŠ')XUz t ∈ [−T, T ] _
|F (t) VN(! 1| < \^Z i ∈ {1, 2} w
“d(\ Pvaed7fMg+P]7œ?bjKMPµMX;d7\;a+KMPva¦YlXrub+POt ±…ŸEP]_XZ\li/K;dv–…Pb+]$a+KMPva+¦ƒbjK;d7bXM]~ŸE]_p+gjb©aed_gP
Y[g²NƒY[g+g+PvxD¢rufZtZt?]`g+PbjKMPOp+PY[g t ∈ [−T, T ] Ÿ11YlbjK |F (t ) aONQ]1x 1| < œ¡]_p i ∈ {1, 2} ¢ KZp+]_N
bjKMPSxZP<µMXZYlbjY[]_Xž]7œ ±
aONQ]1x
¢!ruYlXMavP
œ¬]_p |τ | ≤ 1 ±—kni•a+KMP]1]`gjYl|PXZ^ (tτ )= ŸP$1|^…<POb |P̃ +(τε)≤aONQ]1x C| < 3|C¢ · P (T τ ) − P̃ (τ )| ≤
¶°KMPOXMavPbjKMP ’g€K;dv–…Pƒdˆav]_NƒNQ]_Xžp+]1]_b
±!NQ]1x—fZ\[] ¢?ruYlXMavP
bjKMP Q ’gd7p+Pš\lYlXMPegd7pYlXub+PO^…POpav]_NkZYlX;d7bjY[]_XMgƒ]7(tœbj/T,
KMP gυ ’,geφ±²bj)KMPO∈i“[−1,
d_x—Nƒ1]Ylb/d¯av]_NƒNQ]_X³Cp+]u]_bYlX
NQ]ux—fZ\[] C ±?d7XMx PO–…POX9]e–…POp~bjKMP/p+Ped7\[g~g+YlXMavP || || , || || , || || ≤ C ’¢ KYlX;d7\l\li
[−1, 1]
[
Y
c
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Xub+PO^…POpcp+]1]_b]7œ q(t) d7XMx•ŸYl\l\hkP€œ¬]_fZXMx3d7bsrub+POt 11 ¢
t
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Q *' ?<·
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o
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>=?1 ·k3 ?0>?0E?d·
M+ iZ[τ ] 3
C · Pi (T τ )
'*1
¶ He{
"!$#%&(')'*!
¼ ‹ ~‹ ¸ Ž 5‰ ¼ =«Xžrub+POt H`±MŸEPaed7X3fMg+P‰D;]…d7bjYlXZ^_›‘t]_YlXub$av]uP<™šaOY[POXub+gcYlXIbjKMPSJdeiu\[]_p
P<¤—t;d7XMgjY[]_X P (t) YlXMgjb+Ped_xž]7œ©gjiuNk]_\lY[aav]1P<™šaOY[POXnb+ge±Dd_gs\[]_XZ^d_gsYlbcYlXubjp+]ux—fMavPvgWXM]ˆPOpjp+]_p€YlX
rub+POt³|•ŸKZYl\[Pˆav]_NƒtZfZbjYlXZ^ P̃ (τ ) ±²œ¡]_p i ∈ {1, 2} ¢ POb a k?PƒbjKMP j ›‘bjK Jdvi1\[]_pav]1P<™šaOY[POXnb
]7œ f ’¢ =«Xž]_p+xZPOpWb+]®^…PObSdšav]_pjp+PvaOb P̃ (τ ) d7bSrub+POt |—±bjKMPPOpjp+]_p~]_X CN · a NSfMgjbWk?P
¢;ruYlXMavP
\[PvggbjK;d7X ±1bjKnfMgbjKMP$POpjp+]_p]_X a NfMgjbk?PW\[Pvg+gbjK;d7X
±1YlbbjKnfMg
N ≥T
g+f—™šavPvg©b+]av]_NƒtZfZb+P a ŸYlbjK log (2CN ) ≤ 2n kZYlb+gdœ5b+POpEbjKMPkZYlX;d7pjiƒt?]_YlXnbe’¢ ”5¶ PcŸYl\l\g+PvP
k?PO\[]eŸ bjK;d7b C = O(N ) ¢ ˜
¼ ¸ AŠ ¹ † hŒ ¹ ‰ 5‰ ¼ Š ¸ † ¼ ‹ AŠ €]eŸ˜bjK;d7bbjKMPWav]_pjp+PvaObjXMPvg+g]7œhbjKMP$d7\l^…]_pjYlbjKZN´Y[gtZp+]e–…PvxD±—ŸP€Pvg+bjYlNšd7b+PWYlb+gav]_NƒtZ\[P<¤—Ylbi…¢ POb
k?P€bjKMP~Jdeiu\[]_p€av]uP<™šaOY[POXub+gc]7œ f œ¡]_p i ∈ {1, 2} Y¢ O€POp+PWŸEPd_g+gjfZNQP$bjK;d7bœ¬]_pcd7Xni
a ,a ,...
dk 7XMx i ± |a | = O(1) ¢
Œ ¹ ¼ ‹ l‰ /¼ h½ ruYlXMavP®ŸEP®XMPO^`\[PvaObƒJdvi1\[]_pQav]1P<™šaOY[POXnb+gƒ]7œsxZPO^`p+PvP®bjKZp+PvP•d7XMx KZYl^`KMPOpe±
bjKMPSPOpjp+]_pcNšd_xZP~YlX3bjKMPd7tZtZp+]v¤—YlNšd7bjY[]_Xžb+]
kui Y[g
¢ruYlXMavP
ŸP d7p+P£\[]u]_¦1YlXZ^³œ¬]_pg+YlNSfZ\lbd7XMPv]_fMgj\li˜k;d_x
Naed_·g+Pvfg(ŸYl)bjK |PP(t)(t)aONQ]1≈x a1| · <T · N±ŸP£Ÿd7Xnb
¢
±—Y‘¢’P`¢
T ·N =O
T =O
¸ˆ¼ Š ¸ ¼D¸ Ž ¸ Še‰ ¼ Š ¸ P̃ l‰ JKMP€xZPO^`p+PvP<› 2 t]_\li1XM]_NƒY¨d7\[g P̃ (τ ) = p̃ +
¸
av]_NƒtZfZb+Pvx•d7brub+POt|S]7œ²bjKMP€d7\l^…]_pjYlbjKZN´gd7bjY[g,œ¡i p̃ = O(M · T · N ) ¢
p̃ τ + p̃ τ ∈ Z[τ ]
=«XMxZPvPvxD±`gjYlXMavP P (t) = p +p t+p t Y[g)bjKMPxZPO^`p+PvP<› 2 Jdvi1\[]_p©P<¤—t;d7XMgjY[]_XQ]7œ N ·f ±_ŸP
K;de–…P p = O N ¢ ]_p+Pv]–…POpe± P̃ Y[gxZP<µMXMPvxkni P̃ (τ ) = p̃ +p̃ τ +p̃ τ = bC ·P (T τ )e ¢
JKMP$œ¬d_aObbjK;d7b C = Θ(M ) av]_XMaO\lfMxZPvgsbjKMPWtZp+]1]7œ,¢
Š ¸ ¶ P€K;dv–…PWb+]/p+Pvx—fMavP€bjKMPW\¨d7bjbjY[avP$gjt;d7XZXMPvx®kniˆbjKMP
¸ Œ—Še‹ ¼ ²Š ¸ ½ªŒ—Š
p+]Ÿgc]7œ)bjKMP€œ¡]_\l\[]ŸYlXZ^šNšd7bjpjY ¤D¢
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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)
ISSN 0249-6399