Gal's Accurate Tables Method Revisited

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(1)Gal’s Accurate Tables Method Revisited Damien Stehlé, Paul Zimmermann. To cite this version: Damien Stehlé, Paul Zimmermann. Gal’s Accurate Tables Method Revisited. [Research Report] RR-5359, INRIA. 2004, pp.23. �inria-00070644�. HAL Id: inria-00070644 https://hal.inria.fr/inria-00070644 Submitted on 19 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) 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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(39) {. .

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(42) ¼ ½ ¸ ¸ ¶ Pav]_XMgjY[xZPOpbjK;d7bd$5fZXMaObjY[]_X®k?POK;dv Pvgd_gd$pd7XMxZ]_N kZ\¨d_a+¦uk]e¤KMPOX®bjKMPµMp+gjbEkZYlb+g]7²Ylb+g ]_fZbjtZfZbd7p+PbjKZp+]eXdvdvi ¢ZrufMa+Kd7Xd_ggjfZNtZbjY[]_XY[g¥]7Dav]_fZp+g+Pc POpjigjbjp+]_XZ^d7XMxQav]_NtZ\[POb+PO\li ¶0¶¸·¹Lº»º¼.

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(44) j ')V$VWVXLX . KMPOfZpjY[g+bjY[a`¢ sPO POpjbjKMPO\[Pvg+gEP<¤1t?POpjYlNQPOXnb+g¥b+POXMx/b+]W_d7\lY[xMd7b+PbjKZY[g)Kuiut?]_bjKMPvgjY[g©d7XMx/YlXt;d7pjbjY[aOfZ\¨d7p bjKMP$P<¤t?POpjYlNQPOXnb+g]7¥r1PvaObjY[]_XI{/xZ]QXM]_bcav]_Xnbjpd_xY[aObsYlbe¢ ^`Yl POX3¡fZXMaObjY[]_X f : [ , 1[→ [a, b[ ¬]_pg+]_NQP a < b (wzw 2 : [ , 1[→ [1, 2[sY[gg+PvPOX d_g~d®pd7XMxZ]_N kZ\¨d_a+¦uk]e¤??¬]_pSd7Xni d7XMx \¨d7pj^ POp~bjK;d7X g+]_NQP/bjKZp+PvgjKM]_\[xZg±DY Y[g aKM]`g+POX®pd7XMxZ]_N\li®d7XMxfZXZY ¡]_pjN\liQYlnX [ , 1[k ± bj<KMPOkX®bjKMPckZYlb+gd7bt]`g+YlbjY[]_XMg k d7XMx k ]7²xbjKMP kZYlX;d7pji P<¤1t;d7XMg+Y[]_X ]7 f (x) d7p+PYlXMxZPOtPOXMxZPOXubQpd7XMxZ]_N `d7pjY¨d7kZ\[PvgfZXZY ¡]_pjN\li xY[gjbjpjYlkZfZb+Pvx YlX {0, 1} ¢!JKZY[gYlNtZ\lY[Pvgcg+]_NQP$fMgP<5fZ\DtZp+]_t?POpjbjY[Pvge g+J]_KMNQPsPWtZ^`p+]_Yl k;POd7X¯kZYlt?\lYl]`b«gjiYlbj]7Y[]_X]_kZ§bd7YlX¯YlXZbjYlXZKM^PWkZdYlX;pjfZd7X¯pji]7P< ¤pt;avd7]_XMXMgjY[g+]_PvX3aOfZ]7bj Yl P$Y[xZPOXuY[g bjY[aed7\²kZ¢ Ylb+g §gjbd7pjbjYlXZ^/¡p+]_N f (x) 2 ]7sbjKMPON&gjfMaK³bjK;d7bbjKMPkZYlX;d7pji P<=c¤1t;d7PXMgjavY[]_]_XMX9gjY[]7xZ POp 2 avK;]_d_XMgSg+Pvd®aOfZpjfZbjYlX9 P ]7x ge± avbj]_KMXMPOg+p+PvPaOfZY[bjg Yl O(1) PY[xZPOXnbjY[aed7\EkZYlb+g~gjbd7pjbjYlXZ^¡p+]_N g+]_NQP p ^`Yl POX¯t]`g+YlbjY[]_X²f¢ (x) kZJYlKMb+g¥P§g+POgjbbd7]7pjbjYlbjXZKM^P 5p+x]_N ∈ g+[]_NQ, 1[Ps^`Ylgj fMPOaXK9t?bj]`K;gjd7Ylbjb Y[]_fX(x)§/Y[gK;]7d_hgaedd7p+pjxfZYlX³X;d7]7\lYl b«ip d7avp+]_]_XMfZgXMPvx aOfZbjYl P±Y[xZd7POXMXuxbjY[bjaeKMd7P \ Nd¤1YlNfZN xY[gjbd7XMavPkPOb«EPvPOXb«E]~av]_XMg+PvaOfZbjYl PPO\[PONQPOXub+gY[g)\[Pvg+gbjK;d7X ≈2 1 + n − · ¡g+PvPWbjKMPd7tZtPOXMxY ¤¬]_pcdtZp+]u]7 <¢ 2 ¬]_pSg+]_NQP ¶ P^ POXMPOpd7\lIY ePbjKZY[gSNQ]uxZPO\©b+]b]¯¡fZXMaObjY[]_XMg "w zw sin x d7XMx cos x <¢nJKMPOi®d7p+Pcg+PvPOXd_g©pd7XMxZ]_N YlfXMxZ, POftPO:XMxZ[ PO,Xu1[→ bkZ\¨d_[a,a+¦ub[k]e¤Pvg µMp+gjbj\lia bj<KMPOi b k?]_bjKd7p+PkZ\¨d_a¦nk?]v¤ZPvge±nd7XMxg+Pvav]_XMx\li ±`¬]_p©d7Xni n d7XMx k , k \¨d7pj^ POpEbjK;d7Xg+]_NQPbjKZp+PvgjKM]_\[xZg± Y x Y[g$a+KM]`g+POXpd7XMxZ]_N\li£d7XMx fZXZY ¬]_pjN\liYlX [ , 1[ ±bjKMPOX bjKMPkZYlbSd7b$t?]`gjYlbjY[]_X k ]7bjKMP kZYlX;d7pjiQP<¤t;d7XMgjY[]_X®]7 f (x) d7XMxQbjKMPskZYlbd7bEt]`gjYlbjY[]_X k ]7²bjKMPkZYlX;d7pjiQP<¤t;d7XMgjY[]_X]7 f (x) d7p+PcYlXMxZPOtPOXMxZPOXubpd7XMxZ]_N `d7pjY¨d7kZ\[Pvge¢ KMp+]_N gjfMaK®d7XKni1t]_bjKMPvgjY[gEEPaed7X®d7\[g]SxZPOpjYl Pg+]_NQP tZp+]_t?POpjbjY[Pvge g+J]_KMNQPsPctZ^`p+Yl]_ k;POd7X®kZt?Yl\lYl]`b«gjiYlbj]7Y[]_X]_kZ§bYld7XYlXZbjYlKMXZPc^kZdYlX;pjd7fZpjX¯i]7P< ¤1pt;d7avXM]_gjXMY[g+]_PvXMaOgfZbj]7Ylh k?P$]_Y[bjxZK POXubjY[aed7\²kZd7YlXMb+x g §gjbd7pjbjYlY[XZg ^/¡p+]_N ¢ ]7bjKMPOfN (x)gjfMa+K9bjK;fd7bS(x)bjKMPQkZ2YlX;d7pji P<=¤1t;Ed7P®XMgjavY[]_]_XMXMgjg/Y[xZ]7POcp k?2]_bjK av]_XMg+PvaOfZd7bjXMYl x P x ge±DbjKMK;POdvp+ PPY[g dIO(1) pjfZX³]7 p av]_XMg+PvaOfZbjYl PY[xZPOXnbjY[aed7\kZYlb+g gjbd7pjbjYlXZ^¡p+]_Ng+]_NQP$^`fYl (x) POX¯t?]`gjYlbjY[f]_X²(x)¢ K;de PdpjfZX3]7 av]_XMg+PvaOfZbjYl P Y[JxZKMPOXnP~bjg+Y[aePObd7\¥]7kZ Ylbjb+KMg~P g+xbd7∈pjbjYl[XZ^, 1[5p+]_N gjfMag+K¯]_NQbjK;P/d7^`b Yl fPOX(x)t?]`d7gjXMYlbjx Y[]_fX (x) Y[g$]7aed7p+xYlX;d7\lYlbi d7pp+]_fZXMx 2 ± d7XMx bjKMPINd¤1YlNfZN xY[gjbd7XMavPIk?PObEPvPOXb«E] av]_XMg+PvaOfZbjYl P£PO\[PONQPOXnb+g®Y[g\[PvggQbjK;d7X ≈ ¢ (1 + n − p) · 2 s]_bjY[avPbjK;d7bYlbY[gPed_gjib+]Nd7¦ P$d7\l\bjKMPvg+PWgjbd7b+PONQPOXnb+gpjYl^ ]_p+]_fMgd7XMx®b+]^ POXMPOpd7\lIY ePWbjKMP NQ]1xZPO\Db+]NQ]_p+P$bjK;d7X3bE]Q5fZXMaObjY[]_XMg¢ 1 2. x. 1 2. 1. 1 2. 2. n. 1. 2. 1−p. p. 1 2. n. n−p. p 2. n−p. 1. 1 2. 1. 1 2. 2. 1. 2. 1. n. 2. 2. 1. 2p. 1. 1 2. 2. 2−2p. 2. n. 1. 2. n−2p. n−2p. ¶ .

(45) .

(46) "!$#%&(')'*! B. ® l/ ¸ Z ¸ '¸ ¼ ½ U~d7\gEd_avaOfZpd7b+Pbd7kZ\[PvgENQPObjKM]1xF±+}.J?d_xZxp+Pvgg+Pvg¥bjKMPsg+Pvav]_XMxpd7XZ^ Pp+PvxfMaObjY[]_X]7DbjKMPmufZY[a+¦ tZK;d_gP]7!bjKMPaed7\[aOfZ\¨d7bjY[]_X]7 f (x) ¢_JKMPENQPObjKM]1xY[g)^ POXMPOpd7\±_kZfZb)KMPOp+PEPbd7¦ Pd_gP<¤Md7NtZ\[Pvg d7XMx f (x) = sin x ¢ZJKMP~Y[xZPed]7bjKMPSg+Pvav]_XMxpd7XZ^ P~p+PvxfMaObjY[]_X¯Y[gb+]pjYlb+P` f (x) = 2 . . . x. 2 x = 2 x0 · 2 h ,. KMPOp+P x Y[g©YlXd$bd7kZ\[P]7 xY[gjbjYlXZ^`fZY[g+KMPvxt?]_YlXnb+gd7XMxY[gEgjb+]_p+PvxYlbjKd7Xd7tZtZp+]e¤1YlNd7bjY[]_X¯]7 Ylb+gEav]_pjp+Pvgjt?]_XMxYlXZ^ 2 5p+Pvg+t²¢ sin x d7XMx cos x h = x − x Y[gEgjNd7\l\ p+]_fZ^`KZ\lig+tPed7¦1YlXZ^M± Y[ggjNd7\l\[POpEbjK;d7X xYluY[xZPvxknibjKMPcXufZNSk?POp]7hxY[g+bjYlXZ^`fZY[gjKMPvxt?]_YlXnb+g¢ c5b+POpEbjKZY[gbd7kZ\[P< |h| k;d_gPvx/pd7XZ^ Pp+PvxfMaObjY[]_X²± 2 5p+Pvgjt²¢ cos h d7XMx sin h Y[g¥av]_NtZfZb+Pvxd7tZtZp+]v¤YlNd7b+PO\liknifMgjYlXZ^ d\[] xZPO^`p+PvP/t]_\li1XM]_NY¨d7%\ 5p+Pvgjt²¢b]\[]e xZPO^`p+PvP/t?]_\liuXM]_NY¨d7\[g sKZY[aK aO\[]`g+PO\lid7tZtZp+]e¤1Y Nd7b+Pvg 2 5p+Pvgjt²¢ sin h d7XMx cos hKMPOX h Y[g¥aO\[]`g+Pb+] 0 ¢uJh]WµZ¤bjKMPY[xZPed_ge± EPcaed7XgjfZtZt]`gP bjK;d7bbjKMPvg+Pst]_\li1XM]_NY¨d7\[gd7p+PcbjKMPJ deiu\[]_pP<¤t;d7XMgjY[]_XMg]7hbjKMPc5fZXMaObjY[]_XMgd7b 0 ±ukZfZbYlbEY[gEt]`gj g+YlkZ\[Pb+]xZ]®k?PObjb+PONp F .R J¢ =EPav]_XMgjY[xZPOp$bjKMP/xZ]_fZkZ\[PtZp+PvaOY[gjY[]_X 5Y¢P`¢ 53 kZYlb+g~]7Nd7XnbjY[g+g3d <± g+YlXMavPPd7p+PYlX bjKMP®mufZY[a+¦tZK;d_g+P`±bjKMPaed7\[aOfZ\¨d7bjY[]_XMgQg+KM]_fZ\[xk?PNd_xZP®YlX³xZ]_fZkZ\[P®tZp+PvaOY g+Y[]_X3d_g]75b+POXd_gt]`ggjYlkZ\[P`±Md7XMxYlbY[gYlXub+POp+PvgjbjYlXZ^b+]/K;de P~bd7kZ\[PvgYlbjKd7tZtZp+]v¤YlNd7b+PO\li 2 xY[g+bjYlXZ^`fZY[gjKMPvx¯PO\[PONQPOXnb+g±Z¬]_paed_a+KMPS]_tZbjYlNIY d7bjY[]_XIp+Ped_g+]_XMag ¡g+PvWP F {^± HJ<¢ JKMPX;d7Yl Pdvi³]7aKM]u]`g+YlXZ^ bjKMPxY[gjbjYlXZ^`fZY[gjKMPvxªPO\[PONQPOXnb+gY[gb+]£bd7¦ PbjKMPON&Pvmnf;d7\l\li g+t;d_avPvxD±nbjKufMgNYlXZYlNIY vYlXZ^bjKMPcNd¤YlNSfZN `d7\lfMPsbd7¦ POX®kni |h| ¢U~d7\gEY[xZPedY[gEb+]Sp+PO\¨d¤® POpji g+\lYl^`Knbj\li bjKMPINd¤1YlNfZN _d7\lfMPIbd7¦ POXkni |h| YlX ]_p+xZPOp®b+] a+KM]1]`g+P3k?PObjb+POpxY[gjbjYlXZ^`fZY[gjKMPvx t?]_YlXnb+g KMPbd7¦ Pvgd7\lNQ]`gjbEp+PO^`fZ\¨d7pj\ligjt;d_avPvxxY[g+bjYlXZ^`fZY[gjKMPvxt?]_YlXnb+g x gjfMa+KbjK;d7b©bjKMPcg+b+]_p+Pvx d7tZtZp+]e¤1YlNd7bjY[]_X°]7 5p+Pvgjt²¢ d7XMx SY[gfZXnfMgjf;d7\l\li aO\[]`g+P®b+]IYlb+gbjpjfMP®`d7\lfMP`¢ ]_p+PtZp+PvaOY[g+PO\li ±YlX 2xZ]_fZkZ\[P®tZp+PvsinaOY[gjxY[]_X²± ¬]_p/cosd7Xui x 0 ≤ i < 2 ± x Y[g/d3Nd_a+KZYlXMPXnfZNkPOp aO\[]`gPvgjbb+] + gjfMaK¯bjK;d7be

(47) aONQ]ux 1

(48)

(49)

(50) ≤ 2 ¬]_p 2

(51)

(52) 2 · 2

(53) aONQ]ux 1

(54)

(55) ≤ 2 d7XMx

(56)

(57) 2 · cos x aONQ]1x 1

(58)

(59) ≤ 2 ¬]_p sin x,

(60) 2 · sin x KMPOp+P Y d7XMx ]_bjKMPOpjY[g+P`¢ avav]_p+xYlXZ^Qb+]/bjKMPWpd7XMxZ]_NNQ]1xZPO\± g+KM]_fZ\[x k?PP<¤bjp+ePO=NQPO1\li aO|x\[]`g+|P≤b+] + 0 ±?YlNtZ\li1YlXZ^dXMPO^`\lYl^`YlkZ\[PYlXMaOp+Ped_g+PQ]7bjKMP_d7\lfMPxbjK;d7baed7X k?Pbd7¦ POX kui |h| ¢ ]_p+Pv]e POpe±¥bjKZY[gg+POb]7~xY[gjbjYlXZ^`fZY[gjKMPvx t]_YlXub+gQ^`Yl Pvg®d£NQ]_p+P3d_avaOfZpd7b+P YlNtZ\[PONQPOXubd7bjY[]_X]7?bjKMP5fZXMaObjY[]_X²ugjYlXMavP |x − x | Y[g¥k?]_fZXMxZPvxQkni ≈ 2 ±`bjKMPµMX;d7\]_fZbjtZfZb ¬]_p f (x) aed7X¯k?PWNd_xZP~d_avaOfZpd7b+P~YlbjK ≈ 63 kZYlb+g]7)tZp+PvaOY[gjY[]_XId7\lbjKM]_fZ^`K¯bjKMP$aed7\[aOfZ\¨d7bjY[]_XMg d7p+PINd_xZP£YlbjKxZ]_fZkZ\[Pvge¢ c^nd7YlX YlXbjKMP3pd7XMxZ]_N NQ]1xZPO\±bjKZY[g®YlNtZ\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 0. x. −10. i 0. −10. i 0. i. 211. 0. ¶0¶¸·¹Lº»º¼. 10. i 211. 52. 53+e. 0. −12.

(61) aVb';XZ

(62) j ')V$VWVXLX . ]_fZbjtZfZb¬]_p (x) Y[gav]_pjp+PvaObj\li p+]_fZXMxZPvxYlbjKtZp+]_k;d7kZYl\lYlbi d7p+]_fZXMx 1 − 2 ±KZY[aK

(63) Y[g KZYl^`KMPOpcbjK;d7X f99.9% ¢ JKMPbd7kZ\[Pvgd7p+Pav]_XMgjbjpjfMaOb+PvxkniQP<¤K;d7fMgjbjYl Pcg+Ped7p+aK²±ng+]~bjK;d7b©bjKMPOp+Psd7p+Pd7tZtZp+]v¤YlNd7b+PO\li j b pjY¨d7\[g¬]_pd7Xui¯xY[gjbjYlXZ^`fZY[gjKMPvxPO\[PONQPOXnbW]7¥bjKMP bd7kZ\[P`±d7XMxd7tZtZp+]v¤YlNd7b+PO\li 2 bjpjY¨d7\[g 2 ¬]_pcd7XnixY[gjbjYlXZ^`fZY[g+KMPvx¯PO\[PONQPOXnb]7bjKMP sin xbd7kZ\[2P`¢ ¼

(64) ¸ ®¸

(65) h½ ¸ 7 ¼ ª ¸ POb n kPbjKMPtZp+PvaOY[g+Y[]_X³d7XMx f : [ , 1[→ [ , 1[ kPbjKMP®av]_XMgjY[xZPOp+Pvx95fZXMaObjY[]_X²¢ m k3!. i^ x ¡]_p f Y[gsd nkZYlbsNd_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 ¡]_pbjKMP xYlp+PvaOb+Pvx

(66) p+]_fZXMxYlXZ^ NQ]1xZPvg 5b+]d7p+xZg 0 ± +∞ |2]_p · f (x) < ± −∞ |2 · f (x) − aONQ]1x 1| ≤ 2 ¬]_pbjKMP$p+]_fZXMxYlXZ^b+]/XMPed7p+PvgjbNQ]1xZP`¢ ]_bjY[avP¯bjK;d7bYlX bjKZY[gxZP<µMXZYlbjY[]_X²± m aed7X°kPId7Xniªp+Ped7\cXufZNSk?POpe9¢ OPOp+P¯bjKMP¯5fZXMaObjY[]_XMgEP av]_XMg+Y[xZPOp$K;dv P]_fZbjtZfZb~pd7XZ^ Pvg~xY DPOp+POXnbW¡p+]_N [ , 1[ ±²g+]®EPQd_xZ]_tZb$bjKMPxZP<µMXZYlbjY[]_X b+]]_fZp aed_gP`¢ ¶ Pgdei bjK;d7b x ∈ [ , 1[ Y[gd mk;d_x aed_gP¬]_p 2 Y |2 aONQ]ux 1| ≤ 2 ±d k;d_x aed_g+P¡]_p cos x Y |2 · cos x aONQ]ux 1| ≤ 2 ±d mk;d_x9aed_g+P¬]_p sin x Y |2 · m aONQ]ux 1| ≤ 2 KMPOX x ∈] , 1[ d7XMx |2 · sin x aONQ]ux 1| ≤ 2 Y x ∈ [ , [ ¢ sin x JKMP¦uXM]\[Pvx^ P©]7ZbjKMP d!ZXi9;]Z

(67) X! ±OY¢P`¢vbjKMP¥Nd¤1YlNfZN gjfMaK~bjK;d7b d_xNYlb+g d mk;d_xaed_g+P`± Nd7¦ PvgSYlbt]`ggjYlkZ\[PQb+]IYlNtZ\[PONQfPOXnb f P<aOY[POXubj\li ± k?Pvmaed7fMg+PbjKMPQNfd¤YlNSfZN XufZNSk?POp¥]7!kZYlb+gKZY[a+Kd7p+PXMPvPvxZPvx/Y[g)¦1XM]eX/YlXQd_x_d7XMavP`¢`JKMPxpdek;d_a+¦]7;bjKZY[g¥d7tZtZp+] d_a+K Y[g®bjK;d7bbjKZY[g®_d7\lfMPIY[g®K;d7p+x b+] av]_NtZfZb+P`¢JKMPP<¤K;d7fMgjbjYl P£g+Ped7p+a+K

(68) Y[g®kni ¬d7p®YlX¡Ped_g+YlkZ\[P k?Pvaed7fMg+P]7!Ylb+g ≈ 2 av]_NtZ\[P<¤Ylbi ¢ P<¬LO1p+Pd7XMx fZ\l\[POptZp+]_t?]`g+Pvxd ≈ 2 d7\l^ ]_pjYlbjKZN FIHe|± HOZ J±KZY[a+K d_g3gjfaOY[POXub3b+]³t?POp,¡]_pjN g]_NQP gji1g+b+PONd7bjY[a E]_pj¦YlX xZ]_fZkZ\[P tZp+PvaOY[g+Y[]_X²¢ rub+POKZ\[V`± P<¬LO1p+Pd7XMx !YlNNQPOpjNd7XZX F R`.R J1^nde Pd^ POXMPOpd7\lIY d7bjY[]_XQ]7MbjKZY[gd7\l^ ]_pjYlbjKZN kni~fMg+YlXZ^ \¨d7bjbjY[avP<k;d_gPvx ]_tZtPOp+gjNYlbjK²gW]_pj¦3b+]µMXMx£gjNd7\l\ p+]1]_b+gW]7©NQ]1xfZ\¨d7pt?]_\li1XM]_NY¨d7\[g F |±,Z J¢ JKMPOi9]_kZbd7YlXMPvx d7X³d7\l^ ]_pjYlbjKZN YlbjK dIKMPOfZpjY[gjbjY[a av]_NtZ\[P<¤Ylbi ±)KZY[a+K³d_g/\¨d7b+POp YlNtZp+] Pvxb+] ≈ 2 F R`.| J¢ s\lbjKM]_fZ^`K®bjKMPvg+Pav]_NtZ\[≈P<¤2YlbiQk?]_fZXMxZgd7p+PckPObjb+POpEbjK;d7XbjKMPs]_XMP ]7 P<¬LOup+P`g~d7\l^ ]_pjYlbjKZN¯±DYlb$g+PvPONQgWbjK;d7b$bjKMPtZpd_aObjY[aed7\©aOfZb,] kPOb«EPvPOX k?]_bjK NQPObjKM]1xZgWY[g d7p+]_fZXMx¯bjKMP~xZ]_fZkZ\[P~P<¤b+POXMxZPvxtZp+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

(69) /.

(70)

(71) .

(72) . L/ 7254 . . . 5: Z:U/ Y 4. . s fZp©YlNtZp+]e PONQPOXnbY[gk;d_g+Pvx]_XQbjKMPY[xZPedWbjK;d7b¥Ylb¥Y[g¥t]`g+g+YlkZ\[Pb+]Wp+PvmnfZYlp+PpjfZXMgE]7NQ]_p+PbjK;d7X av]_XMg+PvaOfZbjYl P~Y[xZPOXnbjY[aed7\²kZYlb+gcd_gcU~d7\²xZ]uPvge¢1¶ PµMp+gjbxZPvgaOpjYlkPWK;d7bcd7p+PWbjKMPWk?Pvgjbg+POb+g]7 10 xY[g+bjYlXZ^`fZY[gjKMPvxt?]_YlXnb+g©¬]_p 2 d7XMx sin x ±ud7XMxbjKMPOX®g+KM]e bjK;d7b¥¬]_p 2 ±U$d7\g©NQPObjKM]1xQk;d_gPvx x. x. ¶ .

(73) }. .

(74) "!$#%&(')'*!. ]_XbjKZY[gSg+PObSaed7X9kP/fMgPvx ¬]_p$bjKMPQKM]_\[PPO`d7\lf;d7bjY[]_X9ga+KMPONQPm5k?]_bjK9mufZY[a+¦ d7XMxd_avaOfZpd7b+P tZK;d_gPvg<¢ Z ¸ Z½ ¸¼ Z½SZ ¸ «¼ ¸ A ¸ ¸ ½ ¼ ½ '¼ ½ ¸ sXMxZPOp)bjKMPpd7XMxZ]_N NQ]1xZPO\Zd_ggjfZNtZbjY[]_X²±`Ylb)Y[g)t]`ggjYlkZ\[Pb+]$gjbjp+POXZ^`bjKMPOXbjKMPp+PvmnfZYlp+PONQPOXub+g©]7 U~d7\n]_X~bjKMP¥\[POXZ^`bjKMg ]7ZbjKMPpjfZXMgh]7Mav]_XMg+PvaOfZbjYl P©Y[xZPOXnbjY[aed7\1kZYlb+ge¢(KM]_p bjKMP 2 ¡fZXMaObjY[]_X²±vEPEaed7X bd7¦ P$d_gg+POb]7 xY[gjbjYlXZ^`fZY[g+KMPvxt]_YlXub+gd7\l\bjKMP 42k;d_xaed_g+Pvge¢JKMPOp+P$d7p+P$d7tZtZp+]v¤YlNd7b+PO\li 2 ]7¥bjKMPON d7XMxIbjKMPNd¤1YlNfZN xY[g+bd7XMavPk?PObPvPOX b]®av]_XMgPvaOfZbjYl P]_XMPvgY[gk?PO\[]e 2 ± gPvPr1PvaObjY[]_X {¢ EKM]1]`gjYlXZ^gjfMa+K dgPObs]7©xY[gjbjYlXZ^`fZY[gjKMPvx`d7\lfMPvgsxZPvaOp+Ped_gPvgsbjKMPPOpjp+]_pNd_xZP ]_X 2 YlX kZfZbg+YlXMavP$bjKMPWNd¤1YlNfZN xY[gjbd7XMavP~kPO2b«EPv=POX32 b«E·]2x,Y[gjbjYlXZ^`fZY[gjKMPvx`d7\lfMPvgY[gXM]_bcxZPvaOp+Ped_gPvxD± bjKMPPOpjp+]_pSNd_xZPKZYl\[PPO`d7\lf;d7bjYlXZÎbjKMPt]_\li1XM]_NY¨d7\d7tZtZp+]e¤1YlNd7bjYlXZ^ 2 ¬]_p h aO\[]`g+PQb+] 0 Y[gp+]_fZ^`KZ\li9bjKMP®g d7NQP`± Y¢P`¢ gjYlXMavP®bjKMPav]uP<aOY[POXub+gQ]7cbjKMP®t]_\li1XM]_NY¨d7\d7p+PxZ]_f kZ\[Pvg/d7XMxg+YlXMavPQbjKZY[g~Y[g/d7\[g+]3≈bjKM2PQE]_pj¦1YlXZ^tZp+PvaOY[gjY[]_X²±²bjKMP®xZPO^`p+PvP< 1 av]1P<aOY[POXnbY[gav]_pjp+PvaOb YlbjK³POpjp+]_pk]_fZXMxZPvx9kni ≈ 2 ¢ gd3av]_XMg+PvmufMPOXMavP`± bjKMPPOpjp+]_pNd_xZPYlX bjKMPmufZY[a+¦ tZK;d_gPSY[gp+]_fZ^`KZ\li¯bjKMP/gd7NQP`¢ fZbWd_gEPYl\l\g+PvPYlXIbjKMPXMP<¤1bWgjfZkMg+PvaObjY[]_X²±bjKZY[gWa+KM]_Y[avP/]7 xY[g+bjYlXZ^`fZY[gjKMPvxt?]_YlXnb+gNd7¦ PvgcbjKMP~d_avaOfZpd7b+P~tZK;d_g+P~NQ]_p+P~P<aOY[POXnbe¢ KM]_psbjKMP 5fZXMaObjY[]_X²±?gjYlXMavPSbjKMPOp+Pd7p+Pb]p+PO\¨d7b+Pvx3¡fZXMaObjY[]_XMgsYlXIbjKMPgPvav]_XMx3pd7XZ^ P p+PvxfMaObjY[]_X 5X;sind7NQxPO\li sin x d7XMx cos x<±DEPaed7XZXM]_bP<¤tPvaOb$pjfZXMg]7av]_XMg+PvaOfZbjYl PQY[xZPOXubjY[aed7\ kZYlb+gd_g)\[]_XZ^~d_g ¡]_p 2 Z¢ =XMxZPvPvxD±7bjKMPk?PvgjbaKM]_Y[avP]7!xY[gjbjYlXZ^`fZY[gjKMPvx/`d7\lfMPvg)Y[gNd_xZP]7d7\l\bjKMP g)bjK;d7bEd7p+PgjYlNSfZ\lbd7XMPv]_fMg+\li 21k;d_xaed_g+Pvg)¬]_p sin x d7XMx cos x ¢`JKMPOp+Psd7p+Pd7tZtZp+]v¤YlNd7b+PO\li x ]7 bjKMPON¯±Zd7XMxbjKMP$dv POpd7^ P$xY[g+bd7XMavPsk?PObPvPOX¯b]av]_XMg+PvaOfZbjYl P$]_XMPvgY[gkPO\[] 2 2 d_gEgjKM]eXkni/bjKMPcP<¤t?POpjYlNQPOXnb+g]7hr1PvaObjY[]_X{¢ JKZY[g©NQPed7XMg©bjK;d7bEbjKMPcPOpjp+]_p+g¥Nd_xZPs]_X sin x d7XMx cos x YlX d7p+P$xZPvaOp+Ped_gPvx®5p+]_N ≈sin2 x =b+] sin≈ x2 · cos¢¶ hP$+d7pjcos ^`fMP$xbjK;·d7sinbbjKMhP$POpjp+(1)]_p+gNd_xZPWYlXbjKMPWt]_\li1XM]7 NY¨d7\PO_d7\lf;d7bjY[]_XMgWav]_pjp+Pvgjt?]_XMxYlXZ^®b+] d7XMx aed7Xd7\[g]Qk?P~Nd_xZP~bjK;d7bgjNd7\l\±!YlXd POpjiP<aOY[POXnbsdei ¢M¶ P~xZP<µMXMP cos h sin h d7XMx C(h) = 1 − 1 h + a h , S(h) = h − a h + a h 2 KMPOp+P a Y[g©bjKMPxZ]_fZkZ\[PsaO\[]`gPvgjb©b+] ¬]_p i ∈ {3, 4, 5} ¢ gjYlXZ^bjKMPd_aOb©bjK;d7b |h| ≤ 2 d7\[]_XZ^YlbjK d7^`pd7XZ^ P`gk?]_fZXMxD±ZP~]_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!.

(75)

(76)

(77) 3 5

(78)

(79)

(80) h h

(81) +

(82) a3 − | sin h − S(h)| ≤

(83)

(84) sin h − h − + 6 120

(85)

(86). ¶0¶¸·¹Lº»º¼. 2. 4. 4. −10.977.

(87)

(88)

(89) 1

(90)

(91) 3

(92)

(93) 1

(94)

(95) 5 |h| +

(96) a5 − |h | 6

(97) 120

(98).

(99) He. aVb';XZ

(100) j ')V$VWVXLX ≤ 2−76.839 · 2−12.299 + 2−56.584 · 2−32.931 + 2−62.906 · 2−54.885. ≤ 2−88.314 ,

(101)

(102)

(103)

(104) 4

(105) 2

(106)

(107)

(108) h 1 h

(109) +

(110) a4 −

(111) |h|4 + | cos h − C(h)| ≤

(112)

(113) cos h − 1 −

(114)

(115) 2 24 24

(116) ≤ 2−65.862 · 2−9.491 + 2−58.584 · 2−43.908. E]£Nd_a+KZYlXMP uX¶ fZP®NSXMk?]POp+gEd7P<¤tZtZtZ\¨p+d7]eYl¤1X³YlNKMd7]e´ bjYlXZb+^ ]£cosPO`d7x\lf;d7d7b+XMP®x Pvsinmuf;d7xbjY[]_¢ X i/(1)av]_¢ XMgjbjPOpjb fMaOcbjY[]_d7XXMx PsK;dvk? PP |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]_b X²av ]_NtZfZb+Pd7tZtZp+]v¤YlNd7bjY[]_XMg]7 S (h) d7XMx C (h) YlbjKfO]_pjXMPOpeg NQPObjKM]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]7 sin x POXMxZgYlbjK¯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 YlXd7X P<¤1b+POXMxZPvx tZp+PvaOY[g+Y[]_X²±?¡]_p~P<¤Md7NtZ\[PkuiIfMgjYlXZ^d muf;d_xpjfZtZ\[P`±Md/xZ]_fZkZ\[P<xZ]_fZkZ\[P`±;]_pkui®g+YlNSfZ\¨d7bjYlXZ^QbjKMP$P<¤b+POXMxZPvxtZp+PvaOY[gjY[]_X¯YlbjKbjKMPWfMg+PW]7 d¡Nd/]_t?POpd7bjY[]_X²¢YKM]_pcP<¤Md7NtZ\[P$P~xZ]/bjKZY[gYlbjKId¡Nd¢ h ← o (s +¢ c · h) ¢ t ← o(h − s ) ¢ l ← o (t − c · h) ruYlXMavP ±ZP~K;de P h ≥ d7XMxbjKMPSg+Pvav]_XMx3]_t?POpd7bjY[]_X3Y[gP<¤Md_aObe¢MJKMPOp+P<¬]_p+P~P K;de P` |c · h| ≤ d7XMx |(h + l) − (sin x + cos x · h)| ≤ 2 + 2 h +l = s +c ·h ≤2 . av]_pjp+¶ PvaOPbj\liXM]bjKMP$avp+]_PvNg+fZtZ\lfZbb+NQP]`g+]_gjbcNQ]7PS)\[bjPvKMgPWgg+bjYlYlNQ^`XZP`Y µ;¢ aed7Xnb$kZYlb+ge±!KZY[a+K YlX£t;d7pjbjY[aOfZ\¨d7p~gjfavPb+]®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. ¶ .

(117) .

(118) "!$#%&(')'*!. HH. C ∗ ← o(S ∗ + C ∗ ) |C ∗ − (cos x0 · S ∗ (h) + sin x0 · C ∗ (h))| ≤ 2−86.631 + 2−74.610 ≤ 2−74.609 C ∗ ← o(l + C ∗ ) |(h0 + C ∗ ) − sin x| ≤ 2−74.609 + 2−73.999 + 2−1−106 ≤ 2−73.271 .. ¶ KMPOX bjKMP¯xY[g+bd7XMavP¯k?PObEPvPOX 2 · sin x d7XMx Z Y[gQKZYl^`KMPOpbjK;d7X 2 · 2 = §IKMPOp+P e Y[g 1 YlX bjKMPaed_g+P]7cd®p+]_fZXMxYlXZ^3b+]bjKMPXMPed7p+Pvg+b~NQ]uxZP`± d7XMx 0 ¡]_p 2 ^`Yl Pvg®bjKMP£av]_pjp+PvaOb]_fZbjtZfZbe¢JKZY[g d9xYlp+PvaOb+Pvx p+]_fZXMxYlXZ^³NQ]uxZP¯§³bjKMP d_xZxYlbjY[]_X YlNtZ\lY[PvgbjK;d7bkni®fMgjYlXZ^QgjfMa+K3d/ga+KMPONQP$bjKMPWp+PvgjfZh\lb+Y[gCav]_pjp+PvaObj\lip+]_fZXMxZPvxYlbjKtZp+]_k;d7kZYl\lYlb«i d7b©\[Ped_gjb 1 − 2 ± KZY[a+KNd7¦ PvgEbjKMP 99.9% Pvg+bjYlNd7b+Pc]7 U$d7\;YlXMaOp+Ped_gPb+] ≈ 99.9998% ¢ ` ¼ Z h ¼ ¸

(119) ¸ Z ¸ ½ ¼ ® l ¸ ¼ ½ =«X bjKZY[ggjfZkMg+PvaObjY[]_XPxZPvg+aOpjYlk?PQKM] ]_XMPav]_fZ\[xPO`d7\lf;d7b+P YlX xZ]_fZkZ\[P tZp+PvaOY[g+Y[]_XYlbjKav]_pjp+PvaObp+]_fZXMxYlXZ^knifMgjYlXZ^S]_XZ\liU~d7\g¥NQPObjKM2]1xD±n: Yl[XQ,bjKM1[→ Psaed_[1,g+Pc]72[hd$xYlp+PvaOb+Pvx p+]_fZXMxYlXZ^NQ]1xZP`¢ g~fMgjf;d7\±hbjKMPOp+P®d7p+Pb]3tZK;d_gPvge²bjKMP®mnfZY[a¦ tZK;d_g+Pd7XMx9bjKMPd_avaOfZpd7b+P tZK;d_gP`¢ ¸ 5 Z ¸ ¶ PSfMg+PQU$d7\gNQPObjKM]1x3YlbjK£bjKMPbd7kZ\[PNd_xZP/]7©bjKMP 42k;d_x aed_g+Pvg¢ JKZY[gYlXMxfMavPvgd7XPOpjp+]_pk?]_fZXMxZPvxkni ¡]_pbjKMPsb+POpjN ±1d7XMxbjKMPstZp+]_kZ\[PON Y[gp+PvxfMavPvx b+]av]_NtZfZbjYlXZ^d7tZtZp+]v¤YlNd7b+PO\li 2 2KMPOp+P |h| Y[gg+Nd7\l\[POp2 bjK;d7X 2 ¡g+PvPr1PvaObjY[]_X 3{ <¢ ¶ P/xZ]®bjKZY[gWkni£PO`d7\lf;d7bjYlXZ^bjKMPt?]_\li1XM]_NY¨d7\ P (h) = 1 + a h + . . . + a h ±KMPOp+P a Y[g bjKMPsxZ]_fZkZ\[PcbjK;d7bY[g©aO\[]`gPvgjb©b+] j¢ =b¥Y[gEt?]`g+gjYlkZ\[Pb+]a+KMPva¦QbjK;d7b P (h) aed7XkPsPO_d7\lf;d7b+Pvx YlbjK ]_XZ\li ]_t?POpd7bjY[]_XMg~]_X xZ]_fZkZ\[Pvg~b+]¯d7tZtZp+]e¤1YlNd7b+P 2 YlbjKPOpjp+]_p~k?]_fZXMxZPvx kni ≈ 2 KMPOX |h| ≤ 2 ¢JKZY[gYlNtZ\lY[PvgbjK;d7bY x Y[gXM]_bsd 10k;d_x¯aed_g+P`±bjKMPW`d7\lfMP$aed7\[aOfZ\¨d7b+Pvx g]¬d7p~¡]_p Y[g~av]_pjp+PvaObe¢ ]_p+Pv]e POpe±)d_gd¯gjY[xZPQP DPvaObe±²Y Y[gd k;d_x9aed_g+P`±hbjKMPp+PvgjfZ\lb aed7X d7\[g+]k2Pav]_pjp+PvaObj\li3p+]_fZXMxZPvxD¬]_pWPed_a+K POXnbjpji (x , 2 )x]7©bjKMP42bd7kZ\[P`±EPd_xZx£dkZYlbW]7 YlX¬]_pjNd7bjY[]_Xb+PO\l\lYlXZ^KMPObjKMPOpbjKMPg+b+]_p+Pvx`d7\lfMPc¡]_p 2 Y[ggj\lYl^`Kubj\li\¨d7pj^ POp]_pgjNd7\l\[POpbjK;d7X Ylb+gp+Ped7\²_d7\lfMP`¢ ¸ Z h7 ¸ Z ¸ rufZtZt?]`g+PsXM]e bjK;d7bEbjKMPmnfZY[a¦QtZK;d_g+Psd_gXM]_bgjfaOY[POXnbb+]^`f;d7p, d7Xub+PvP®dIav]_pjp+PvaObp+]_fZXMxYlXZ^£]7bjKMP®]_fZbjtZfZbe¢¶ P¦ PvPOt bjKMPg d7NQPt;d7Ylp d7XMx ]_XZ\li aK;d7XZ^ PcbjKMPct]_\li1XM]_NY¨d7\?PO_d7\lf;d7bjY[]_XgjfZktZK;d_g+P`¢u¶ PcPO`d7\lf;d7b+PcYlX®mnf;d_xpj(xfZtZ\[,Ps2tZp+)PvaOY[gjY[]_XbjKMP t?]_\li1XM]_NY¨d7\ Q(h) = 1 + b h + . . . + b h KMPOp+P b Y[gbjKMPmuf;d_xpjfZtZ\[PtZp+PvaOY[gjY[]_X Nd aKZYlXMP$XnfZNkPOpsbjK;d7bY[gaO\[]`g+Pvgjbb+] _¢ =baed7Xk?P$a+KMPva¦ Pvx¯bjK;d7b Q(h) aed7Xk?P$PO_d7\lf;d7b+Pvx YlbjK ]_tPOpd7bjY[]_XMg]_X9muf;d_xpjfZtZ\[PvgSb+]Id7tZtZp+]v¤YlNd7b+P 2 YlbjK9POpjp+]_pSk?]_fZXMxZPvx kui 2 KMPOX |h| ≤ 2 ¢ POb Q(h) kP~bjKMP$_d7\lfMPav]_NtZfZb+Pvx¬]_p Q(h) ±d7XMx y k?P$bjKMP$`d7\lfMP ¯ ¬]_p 2 bjK;d7bY[ggjb+]_p+Pvx¯YlXbjKMP$bd7kZ\[P`¢Z¶ PWµMX;d7\l\li¯d7tZtZp+]v¤YlNd7b+P 2 kui y · Q(h) ¯ ±ZbjKMP~POpjp+]_p k?POYlXZ^k?]_fZXMxZPvxkni? 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.

(120) HeR. aVb';XZ

(121) j ')V$VWVXLX. JKZY[gcYlNtZ\lY[PvgcbjK;d7bsY Y[gXM]_bWd k;d_x aed_g+P`±MbjKMPOX3bjKMP$p+PvgjfZ\lbsY[gp+]_fZXMxZPvxIav]_pjp+PvaObj\li ¢ J KMPOp+P<¬]_p+P`±_EPK;de Pd7xXYlNtZ\[PONQPO41.999 Xnbd7bjY[]_XQp+PObjfZpjXZYlXZ^~bjKMPav]_pjp+PvaObp+]_fZXMxYlXZ^W¬]_p¥d7XniSYlXZtZfZb x P<¤ZavPOtZb©bjKM]`gPcbjK;d7bd7p+P k;d_xaed_g+Pvg©kZfZbXM]_b k;d_xaed_g+Pvge¢n¶ PµZ¤QbjKZY[gEt]_YlXubEYlXbjKMP ¬]_\l\[]eYlXZ^$dvi ¢_¶ Pav]_XM41.999 gjY[xZPOp©dsbd7kZ\[PNd_xZP]_XZ\li]7 4242.00072k;d_xaed_g+PvgYlXMgjb+Ped_x]7 42k;d_x aed_gPvge±²KZY[aK9NQPed7XMgbjK;d7bPp+PONQ] P 6543825232174115/2 ¡p+]_N bjKMP\[]1]_¦nfZt9bd7kZ\[P d7XMx9av]_XMgjY[xZPOpYlbd_gd3gjt?PvaOY¨d7\Eaed_g+P`¢ JKZY[gSp+PONQ]_d7\xZ]uPvgSXM]_baK;d7XZ^ PbjKMPk]_fZXMx9]_X ¢ ]_p+Pv]e POpe±YlbY[gPed_gjiQb+]/g+PvPbjK;d7bbjKMPWPOpjp+]_pd7X;d7\li1gjY[gq,fMgjbd7k] PXM]e

(122) ^`Yl Pvgd7X®µMX;d7\DPOpj|h|p+]_p k?]_fZXMxZPvx£kni 2 ¢?JKMPOp+P<¬]_p+PbjKMP/p+PvgjfZ\lb~Y[g$av]_pjp+PvaObj\li£p+]_fZXMxZPvx k?Pvaed7fMg+P/EP/NfMgjb~k?P/YlX ]_XMP]7bjKMP/bjKZp+PvP¡]_\l\[]YlXZ^3gjYlbjf;d7bjY[]_XMge Y[gWXM]_bSd k;d_x aed_gP`±?bjKMPPOpjp+]_pSk]_fZXMx£^`Yl Pvg bjK;d7bSbjKMP/_d7\lfMPp+PObjfZpjXMPvx ¡]_p 2 Y[g~av]_pjp+PvxaObj\li p+]_fZXMxZ42Pvx x Y[gSd 42.000072k;d_xªaed_gP`±?bjKZP ]_fZbjtZfZbcY[ggjb+]_p+PvxYlXbjKMP$bd7kZ\[P ;]_bjKMPOpjY[g+P x Y[g 6543825232174115/2 ¢ 53. −94. . x. . . 53. c2 . 96A/

(123) /3:2<-s2/ . .

(124). . . . r1]®d7pe±DPkZfZYl\lbS]_fZp$¡fZXMaObjY[]_XPO`d7\lf;d7bjY[]_Xg+a+KMPONQP®d_g$Y EP¦1XMPOyP<¤tZ\lY[aOYlbj\libjKMP/bd7kZ\[Pvg P/xZPvg+aOpjYlkPvxD¢ PO POpjbjKMPO\[Pvg+ge±bjKMPbd_gj¦3]7av]_NtZfZbjYlXZ^bjKMPbd7kZ\[PvgWY[gkui¯d7psXM]_XubjpjYluY¨d7\?Y PSPOp+PSfMg+YlXZ^®d7XIP<¤K;d7fMgjbjYl Pg+Ped7p+a+K d_g$U$d7\ xZ]1Pvge±MbjKMPav]`gjbs]7©av]_NtZfZbjYlXZ^bjKMPSbd7kZ\[P]7 POYlbjKMPOp ]_p ]_fZ\[x¯k?P aed7\l\[gcb+] 5p+Pvgjt²¢ EYlXIP<¤1b+POXMxZPvx3tZp+PvaOY[gjY[]_X²L¢ KM]_p bjKMPWtZp+]_2kZ\[PON sinY[gPexd_gjYl\lig+]_\l Pvx2]_XMavPWYlbY[gXM2]_b+PvxbjK;d7bcsinbjKMP x42k;d_xIaed_g+Pvgaed7XkPWaed7\[aOfZ\¨d7b+Pv2x kui P<¬LOup+P`g$d7\l^ ]_pjYlbjKZN FIHe|^± HOZ J]_psbjKMPS\¨d7bjbjY[avP<k;d_gPvx£d7\l^ ]_pjYlbjKZN ]7rub+POKZ\[V`± P<¬LOup+Pd7XMx !YlNNQPOpjNd7XZ X F R`R±+R`.| J¢ =«X3bjKZY[gWg+PvaObjY[]_X²±Dd¡b+POps^`Yl1YlXZ^®g]_NQPSk;d_a¦u^`p+]_fZXMx ]_X\¨d7bjbjY[avPvg$d7XMx \¨d7bjbjY[avP~p+PvxfMaObjY[]_Xd7\l^ ]_pjYlbjKZNQge±;EP~xZPvg+aOpjYlk?PSd/NQPObjKM]1xb+]Qav]_XMg+bjpjfMaObcgjYlNSfZ\lbd7XMPv]_fMg+\lik;d_x aed_gPvg~¬]_pb«E]I¡fZXMaObjY[]_XMgen¢ KM]_p/d3^`Yl POX³tZp+PvaOY[gjY[]_X ±²bjKZY[gd7\l^ ]_pjYlbjKZN d_xNYlb+gd¯KMPOfZpjY[gjbjY[a av]_NtZ\[P<¤YlbiI]7 ≈ 2 §YlXMgjb+Ped_x]7 ≈ 2 ¬]_pcbjKMPnP<¤1K;d7fMgjbjYl Pg+Ped7p+aK²±d7XMx ≈ 2 YlbjK P<¡LO1p+P`gcd7\l^ ]_pjYlbjKZN¯¢ «¸ »®¸

(125) ½ ¸ ¼ ©v A ¸ =«X~bjKZY[g gjfZkMg+PvaObjY[]_XSP¥kZpjY[P DMi$^`Yl PEg+]_NQPXMPvavPvg+gd7pji$k;d_a¦u^`p+]_fZXMx]_X~\¨d7bjbjY[avPvg)d7XMxS]_X\¨d7bjbjY[avP p+PvxfMaObjY[]_X³d7\l^ ]_pjYlbjKZNQge¢²¶ PQp+P<¡POp~b+}] FIHH`^± H(.JE¡]_pSNQ]_p+PQxZPObd7Yl\[gS]_X k?]_bjK bjKMPd7\l^ ]_pjYlbjKZNY[a d7XMx¯Nd7bjKMPONd7bjY[aed7\d_gjt?PvaOb+ge¢ U*'U i Y[gd xY[g+aOp+POb+P£gjfZkZ^`p+]_fZt]7 ±©]_pPvmnfZYl`d7\[POXnbj\li°bjKMP3g+POb]7d7\l\\lYlXMPed7p YlXub+PO^`pd7\ av]_NkZLYlX;d7bjY[]_XMgs]7 ` ≤ n \lYlXMPed7pj\liYlXMxZPORt?POXMxZPOXnbc PvaOb+]_p+g ] POp R ±bjK;d7bY[ge X x L={ | x ∈ Z}. ]_bjY[avPbjK;d7bbjKMPpd7XZ¦ ` ]7WbjKMP\¨d7bjbjY[avP¯aed7XªkPg+Nd7\l\[POpbjK;d7XªbjKMP¯xYlNQPOXMgjY[]_X n ]7sbjKMP PONkPvxZxYlXZ^ g+t;d_avP`¢ g/g+]1]_Xªd_g ` ≥ 2 ±dI^`Yl POX³\¨d7bjbjY[avP L d_xNYlb+gd7X³YlXµMXZYlbi³]7ck;d_g+Pvg x. 52. x. n/2. . x. n. 2n/3. . n. i. `. i i. i. i=1. ¶ .

(126) He|. .

(127) "!$#%&(')'*!. p+PO\¨d7b+Pvx°b+] Ped_a+K]_bjKMPOpkui fZXZYlNQ]uxfZ\¨d7pbjpd7XMg,¬]_pjNd7bjY[]_XMge¢E¶ PIxZP<µMXMPIbjKMP ! V$')XXY ]7/bjKMP \¨d7bjbjY[avP L d_g det(L) = Q || || ±KMPOp+P , . . . , Y[gbjKMP Upd7N/ r1a+KZNY[xb ]_pjbjKM]_^ ]_X;d7\lYId7bjY[]_X']7/d9\¨d7bjbjY[avPk;d_gjY[g , . . . , ¢EJKZY[gmuf;d7XnbjYlb«i xZ]1PvgXM]_bxZPOtPOXMx]7 bjKMP/aKM]_Y[avPQ]7©bjKMP/k;d_gjY[ge¢ ]`gjb$]7EbjKMPbjYlNQP`±D]_XZ\li3k;d_g+PvgWKZY[aK av]_XMgjY[gjb$]7g+KM]_pjb PvaOb+]_p+g d7p+Ps]7DYlXnb+POp+Pvgjbe¢1JKMP i kCWV$')XL'V$

(128) jV ]7?bjKMPc\¨d7bjbjY[avP L Y[g¥bjKMPsgjNd7\l\[Pvgjb r g+fMa+KbjK;d7bEbjKMPk;d7\l\ avPOXub+POp+PvxYlX ¯d7XMx]7hpd_xYlfMg av]_Xnbd7YlXMgcd7b\[Ped_gjb \lYlXMPed7pj\li®YlXMxZPOt?POXMxZPOXnb\¨d7bjbjY[avPW PvaOb+]_p+ge¢ K;]_pP<¤Zd7NtZ\[P¯bjKMPµMp+gjbQNYlXZYlNr fZN λ (L) Y[gbjKMPi\[POXZ^`bjK ]7Sd£gjKM]_pjb+Pvg+bQXM]_X; ePOp+] \¨d7bjbjY[avP PvaOb+]_pe¢ ruYlXMavPWYlbav]_pjp+Pvg+t]_XMxZgb+]/]_fZpgjYlbjf;d7bjY[]_XYlXbjKMP¡]_\l\[]YlXZ^QgjfZkMg+PvaObjY[]_XEPWXM] gjfZtZt?]`g+P bjK;d7b ` = 4 d7XMx L ⊂ Z j¢ =b©Y[gEaO\¨d_g+g+Y[aed7\ FIHHJbjK;d7bEYlXbjKZY[gEaed_g+PcbjKMPOp+Pd7\ldvigP<¤1Y[gjb+gd$k;d_g+Y[g p+Ped_aKZYlXZ^³bjKMP¯¬]_fZpµMp+g+bNYlXZYlNd±gjfMa+K

(129) √dk;d_gjY[gk?POYlXZ^³aed7\l\[Pvx 'X_lgnil"'k !Z

(130) _ i! ±sd7XMx bjK;d7bbjKMP$µMp+gjbNYlXZYlNSfZN ]7 L Y[gk?PO\[]e 2 · det(L) ¢ ]_p+Pv] POpe±MbjKMPWEPO\l\ ¦1XM]eX d7\l^ ]_pjYlbjKZN FIHe.{ J¥aed7X£kPfMg+PvxIb+]QµMXMx d PvaOb+]_pWKZY[a+K Y[gXM]_bsNfMa+K \[]_XZ^ POpWbjK;d7X£bjKMPµMp+gjb NYlXZYlNfZN¯¢ d\WEIU*'U i L ⊂ Z ' \ M = VN( || || % ¸u¼ ¸

(131) ')&ZX 't [ , . . . , ] ` i=1. ∗ i. 1. ∗ 1. ∗ `. `. 1. 5. 1/4. Zz3Z'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¥]_tZbjYlNd7\MkPvaed7fMgPEbjKMP]_fZbjtZfZbk;d_g+Y[g)Y[g)XM]_bXMPvavPvg+gd7pji YlXZ¦ ]eg+¦uY p+PvxfMavPvxD± d7XMx~bjKMPOp+P<¡]_p+P©bjKMP¥ PvaOb+]_p+g²Ndei$XM]_b²k?P©d_ghgjKM]_pjbd_g²t?]`g+gjYlkZ\[P`¢eJKMP^`p+PvPvxiSd7\l^ ]_pjYlbjKZN ]7 FIHe}.J ]_fZbjtZfZb+gWdk;d_g+Y[gp+Ped_a+KZYlXZ^bjKMP~µMp+gjb¬]_fZpcNYlXZYlNd±?d7XMx¯K;d_gsbjKMPd_xZxYlbjY[]_X;d7\d_x_d7Xubd7^ P]7 d_xNYlbjbjYlXZ^d O(log M ) av]_NtZ\[P<¤Ylbik?]_fZXMxD¢ ¼ ¨

(132) «¼ Q ¸ sin xº Z ¸ JKMPg+Ped7p+a+K] POp [ , 1[ Y[g~xYl1Y[xZPvx YlXnb+] mufZY[a+¦ g+Ped7p+aKMPvgS] POp~YlXnb+POpj`d7\[gS]7\[POXZ^`bjK ¢ JKMPvgP$mnfZY[a¦gPed7p+a+KMPvgsd7p+P~tPOp,¬]_pjNQPvxkni®fMgjYlXZ^QbjKMP~d7\l^ ]_pjYlbjKZN EP~xZPvgaOpjYlkP$k?PO\[]e~¢ UYl POX3b«E]5fZXMaObjY[]_XMg d7XMx ±!dtZp+PvaOY[g+Y[]_X ±;dQk;d_x1aed_g+PSk?]_fZXMx d7XMxdg+Ped7p+a+K k?]_fZXMx T ±_bjKMP¡]_\l\[]YlXZ^d7\lf^ ]_pjYlbjKZNyf bjpjY[Pvg©b+]WµMXMxd7\ln\MbjKMPNd_a+KZYlXMPsXnfZNSk?POp+mg x ∈ [− , ] g+fMa+K¯bjK;d7be aONQ]1x 1| ≤ 2 ¬]_p i ∈ {1, 2}. |2 · f (x) =«XI¬d_aObWYlb$g+]_\l PvgbjKZY[gWtZp+]_kZ\[PON ¡]_p$d7Xni N d7XMx M YlX£tZ\¨d_avP]7 2 d7XMx 2 R¢ =«bWgjbd7pjb+gkni d7tZtZp+]e¤1YlNd7bjYlXZ^Sk]_bjK d7XMx kuiSt?]_\liuXM]_NY¨d7\[g±_d7XMxbjKMPOX/bjpjY[Pvgb+]Wg+]_\l P©bjKMPtZp+]_kZ\[PON ¬]_p bjKMPvgPt]_\li1XM]_NY¨d7\[gEYlXMfgjb+Ped_xQ]7fDbjKMPYlXZYlbjY¨d7\;5fZXMaObjY[]_XMge± knifMgjYlXZ^ ]_tZt?POp+gjNYlbjK²gENQPObjKM]uxb+] µMXMxgjNd7\l\ p+]1]_b+gh]7ZNSfZ\lbjYl`d7pjY¨d7b+PNQ]uxfZ\¨d7p t]_\li1XM]_NY¨d7\[|g F |±,Z J¢7sfZpd7\l^ ]_pjYlbjKZN Y[g²KMPOfZpjY[g+bjY[a` d7\l\Ylb+g$]_fZbjtZfZb+gd7p+Pav]_pjp+PvaObe±DkZfZbWYlb$Ndvi£PO POXnbjf;d7\l\li¬d7Yl\¢PO POpjbjKMPO\[Pvg+ge±?bjKMP/KMPOfZpjY[g+bjY[a/Y[g 2. . 1 2. N 2T. 1. n. T N. 2. T T 2n 2n. −m. i. n. 1. ¶0¶¸·¹Lº»º¼. 2. m.

(133) HO. aVb';XZ

(134) j ')V$VWVXLX. um fZYlb+Pp+Ped_g]_X;d7kZ\[Psd7XMxQbjKMPd7\l^ ]_pjYlbjKZN E]_pj¦ Pvx POpjiPO\l\!YlX®]_fZpEP<¤tPOpjYlNQPOXub+gExZPvgaOpjYlkPvxYlX r1PvaObjY[]_X{ZEPW¬]_\l\[]e'bjKMP~gjbjpd7b+PO^ì]7K;d7\luYlXZ^ T fZXubjYl\hbjKMPSd7\l^ ]_pjYlbjKZN xZ]uPvgXM]_b¬d7Yl\±Md7XMx YlbK;d7tZt?POXMgbjK;d7bcbjKMP~dv POpd7^ P T av]_pjp+Pvg+t]_XMxZgcb+]/K;d7bbjKMP$bjKMPv]_pjitZp+PvxY[aOb+ge¢ =X bjKMPQd7\l^ ]_pjYlbjKZN PfMg+PQd®p+]_fZbjYlXMP d7bjbjY[avPvwcPvxfMavP`±²KZY[aKaed7XPOYlbjKMPOp$k?P/bjKMP d7\l^ ]_pjYlbjKZN FIHe.{ J©]_pbjKMP/^`p+PvPvxi d7\l^ ]_pjYlbjKZN ]7 FIHe.} J¢DJKMP/YlXZtZfZbW5fZXMaObjY[]_XMgd7p+P/Nd_xZP/YlXMxZP< t?POXMxZPOXnbs]7 N F (t) = N f (t/N ) ¢ i. i.

(135) ªC¯(§"µ i°"¨ ¦ §«(«¨ ¨¥d !#"%$&('()*,+.-· /0 ·21 F ·54 F ·547 8)*:91;< >=?@ · ??0 1 M T A "%$B#"($& C > t ∈ [−T, T ] 1;3 -2/D 7 65 3 |F (t) /

(136) E4 1| < +. i ∈ 6 {1, 2} 3 1. 2. 1 M. F :G ?0 P (t), P (t) H ?I 2?@4? ?0?0J 2 ' >C?0L9 · 1;·21M+ F (t), F (t) 3K O

(137) E921- ? 21;-2/D 7 . +P 3 |t| ≤ T ·24 1 i ∈ {1,2 2} NE <ε 3 » :G ?0 0 ε 1/2 3 |Pi (t) −0 Fi·5(t)| 42Q P˜ (τ ) = bC · P (T +. i ∈ {1, 2} τ )e M = b 1/M +ε c 6 C = 3M 653 i i R :G ?0. 2 5 = φ º :G ?0 eg11 == 1C6 6 eg22==τC6 e·3τ =6 gτ3 =6 eP˜41=(τ υ) +6 e3υ ˜ ) + 3φ. 2?V/?0WE/?d· +X ?I

(138) E·

(139) E e · g S EO ? ?I 2? 4 × 5 · ? !

(140) ; <L L )M 2?T 6 ? gL4 = P1U2 (τ 3 3 3 3 l k k,l Y [, , [ H ?I ?V ?0?I1; 2; ?1\ C=?/0 1C+ V ·54 Q (τ ) ? 1;1;/0 ?B4]9 ·2

(141) E 1 Z :VG ?0 M←[ LatticeReduce(L) K 3 ¼ ^+X 2?T1 ?I?02L1\ 31 i ∈ {1, 2, 3} 1;-2/D 7 3 ||[ i ||1 ≥ C 6 65 3 ?0 - ;·(_ai` G!bT 3 3 E G * Fc ?0 Q(τ ) H ? ·2? d/0

(142) H · ·e+f 2? Q 1h)M 2/D i>17·54 ?09?d·54?d· 7+ υ ·54 φ j ?] =? ig 3 3 3 3 4? ≤ 1 3 2 ` C ? D /. · 4 FF EG ?0 Qq(t) Q( Tt ) <+ |F =. 2C? ·k[−T, - 92T-]) t /

(143) E4 1| <3 1 t+.0 IntegerRoots(q, (t i 0) 0 3 i ∈ {1, 2} 6 M i. ¼ 7 ¸ ` ¸ e ¼ ¸ ¼ A

(144) JKMP$¡]_\l\[]YlXZ^QbjKMPv]_p+PON ^`Yl PvgbjKMPSav]_pjp+PvaObjXMPvg+gc]7)bjKMPSd7\l^ ]_pjYlbjKZN¯¢ ¸u¼ ¸

(145) xX ^vZz3Z'V !3" X *

(146) X x '*A Z&Z iZ ^*~[5'Uw(wR'*

(147) r

(148) b3 ^*[EZ')XUz t ∈ [−T, T ]

(149) _ |F (t) VN(! 1| < \^Z i ∈ {1, 2} w d(\ Pvaed7fMg+P]7?bjKMPµMX;d7\;a+KMPva¦YlXrub+POt ± EP]_XZ\li/K;dv Pb+]$a+KMPva+¦bjK;d7bXM]~E]_p+gjb©aed_gP Y[g²NY[g+g+PvxD¢rufZtZt?]`g+PbjKMPOp+PY[g t ∈ [−T, T ] 11YlbjK |F (t ) aONQ]1x 1| < ¡]_p i ∈ {1, 2} ¢ KZp+]_N bjKMPSxZP<µMXZYlbjY[]_X]7 ± aONQ]1x ¢!ruYlXMavP ¬]_p |τ | ≤ 1 ±knia+KMP]1]`gjYl|PXZ^ (tτ )= P$1|^ <POb |P˜ +(τε)≤aONQ]1x C| < 3|C¢ · P (T τ ) − P˜ (τ )| ≤ ¶°KMPOXMavPbjKMP gK;dv Pdav]_NNQ]_Xp+]1]_b ±!NQ]1xfZ\[] ¢?ruYlXMavP bjKMP Q gd7p+P\lYlXMPegd7pYlXub+PO^ POpav]_NkZYlX;d7bjY[]_XMg]7(tbj/T, KMP gυ ,geφ±²bj)KMPO∈i[−1, d_xN1]Ylb/d¯av]_NNQ]_X³Cp+]u]_bYlX NQ]uxfZ\[] C ±?d7XMx PO POX9]e POp~bjKMP/p+Ped7\[g~g+YlXMavP || || , || || , || || ≤ C ¢ KYlX;d7\l\li [−1, 1] [ Y c g 7 d ¯ X l Y Xub+PO^ POpcp+]1]_b]7 q(t) d7XMxYl\l\hkP¬]_fZXMx3d7bsrub+POt 11 ¢ t . . . . 1 M. i. 0. i. i. 0. 0. i. i 3. i. t0 T. 1 M. 1 M. 0. 1 2M 0. i. i. 0. 0. 0. 3 2. i. 3. 0. i. 1 1. 2 1. 3 1. 0. Q *' ?<· . >· bC · P (T τ )e

(150) E? ·21l m)n?C -·24, o ? 2· ? ?1\ *? /T p/?TWE/?d· *+ 3 3 3 3 >=?1 ·k3 ?0>?0

(151) E?d· M+ iZ[τ ] 3. C · Pi (T τ ). '* 1. ¶ .

(152) He{. .

(153) "!$#%&(')'*!. ¼ ~ ¸ 5 ¼ =«Xrub+POt H`±MEPaed7X3fMg+PD;] d7bjYlXZ^_t]_YlXub$av]uP<aOY[POXub+gcYlXIbjKMPSJ deiu\[]_p P<¤t;d7XMgjY[]_X P (t) YlXMgjb+Ped_x]7©gjiuNk]_\lY[aav]1P<aOY[POXnb+ge±Dd_gs\[]_XZ^d_gsYlbcYlXubjp+]uxfMavPvgWXM]POpjp+]_pYlX rub+POt³|KZYl\[Pav]_NtZfZbjYlXZ^ P˜ (τ ) ±²¡]_p i ∈ {1, 2} ¢ POb a k?PbjKMP j bjK Jdvi1\[]_pav]1P<aOY[POXnb ]7 f ¢ =«X]_p+xZPOpWb+]®^ PObSdav]_pjp+PvaOb P˜ (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+favPvg©b+]av]_NtZfZb+P a YlbjK log (2CN ) ≤ 2n kZYlb+gd5b+POpEbjKMPkZYlX;d7pjit?]_YlXnbe¢ 5¶ PcYl\l\g+PvP k?PO\[]e bjK;d7b C = O(N ) ¢ ¼

(154) ¸ A ¹ h ¹ 5 ¼ ¸ ¼ A

(155) ]ebjK;d7bbjKMPWav]_pjp+PvaObjXMPvg+g]7hbjKMP$d7\l^ ]_pjYlbjKZNÝ[gtZp+]e PvxD±PPvg+bjYlNd7b+PWYlb+gav]_NtZ\[P<¤Ylbi ¢ POb k?PbjKMP~J deiu\[]_pav]uP<aOY[POXub+gc]7 f ¡]_p i ∈ {1, 2} Y¢ OPOp+PWEPd_g+gjfZNQP$bjK;d7b¬]_pcd7Xni a ,a ,... dk 7XMx i ± |a | = O(1) ¢ ¹ ¼ l /¼ h½ ruYlXMavP®EP®XMPO^`\[PvaObJ dvi1\[]_pQav]1P<aOY[POXnb+g]7sxZPO^`p+PvP®bjKZp+PvPd7XMx KZYl^`KMPOpe± bjKMPSPOpjp+]_pcNd_xZP~YlX3bjKMPd7tZtZp+]v¤YlNd7bjY[]_Xb+] kui Y[g ¢ruYlXMavP P d7p+P£\[]u]_¦1YlXZ^³¬]_pg+YlNSfZ\lbd7XMPv]_fMgj\lik;d_x Naed_·g+Pvfg(Yl)bjK |PP(t)(t)aONQ]1≈x a1| · <T · N±P£d7Xnb ¢ ±Y¢P`¢ T ·N =O T =O ¸¼ ¸ ¼D¸ ¸ e ¼ ¸ P˜ l JKMPxZPO^`p+PvP< 2 t]_\li1XM]_NY¨d7\[g P˜ (τ ) = p˜ + ¸ av]_NtZfZb+Pvxd7brub+POt|S]7²bjKMPd7\l^ ]_pjYlbjKZN´gd7bjY[g,¡i p˜ = O(M · T · N ) ¢ p˜ τ + p˜ τ ∈ Z[τ ] =«XMxZPvPvxD±`gjYlXMavP P (t) = p +p t+p t Y[g)bjKMPxZPO^`p+PvP< 2 Jdvi1\[]_p©P<¤t;d7XMgjY[]_XQ]7 N ·f ±_P K;de P p = O N ¢ ]_p+Pv] POpe± P˜ Y[g xZP<µMXMPvxkni P˜ (τ ) = p˜ +p˜ τ +p˜ τ = bC ·P (T τ )e ¢ JKMP$¬d_aObbjK;d7b C = Θ(M ) av]_XMaO\lfMxZPvgsbjKMPWtZp+]1]7,¢ ¸ ¶ PK;dv PWb+]/p+PvxfMavPbjKMPW\¨d7bjbjY[avP$gjt;d7XZXMPvx®knibjKMP ¸ e ¼

(156) ² ¸ ½ª p+]gc]7)bjKMP¡]_\l\[]YlXZ^Nd7bjpjY ¤D¢ . . . i. i j. i. 1 2. i j 1/2. i 1. N j T. 1 2C·N. i j. i j. 2. . i 0. T j N. i. i. . . i. i k. . t i N. i 3. i. i. 3. 1 M. −2. . −2. 1 M. N2. 3. . 3. M. . . i. i. i 1. i 2 2. i 0. i. i 2. i 1. i 2 2. −1. . i 2. i. i. i 0. i. i 1. i 2 2. . .  L= . C p˜10 p˜20. .  T ·C . 1 1 p˜1 p˜2 3  p˜21 p˜22 3. =bY[gPed_g+ib+]av]_NtZfZb+PSbjKMP$xZPOb+POpjNYlX;d7XnbW]7bjKZY[g\¨d7bjbjY[avP` | det(L)| = 3C 2 · T ·. ¶0¶¸·¹Lº»º¼. q p22 )2 + 3 = O(M 3 · T 3 · N −1 ). (˜ p12 )2 + (˜. i 0 −1. 2. t N. i.

(157) aVb';XZ