GPU Simulation of Rigid Fibers
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(3) GPU Simulation of Rigid Fibers. ERIC. WOLTER. Master’s Thesis in Scientific Computing (30 ECTS credits) Master's Programme in Applied and Computational Mathematics Royal Institute of Technology year 2015 Supervisor at KTH was Katarina Gustavsson Examiner was Michael Hanke. TRITA-MAT-E 2015:07 ISRN-KTH/MAT/E--15/07--SE. Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci.
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(5) Abstract h?2 K DQ` Q#D2+iBp2 Q7 i?Bb J bi2`Ƕb i?2bBb Bb iQ ++2H2` i2 b2`B H BKTH2K2Mi iBQM Q7 MmK2`B+ H H;Q`Bi?K 7Q` i?2 bBKmH iBQM Q7 bH2M/2` }#2` /vM KB+b #v mbBM; :` T?B+ H S`Q@ +2bbBM; lMBib U:SlVX q2 7Q+mb QM `B;B/ }#2`b b2/BK2MiBM; /m2 iQ ;` pBiv BM 7`22@bT +2 aiQF2b ~QrX h?2 #BHBiv iQ bBKmH i2 H `;2 MmK#2` Q7 }#2`b BM `2 bQM #H2 +QKTmi @ iBQM H iBK2 QM ?B;?@T2`7Q`K M+2 T ` HH2H TH i7Q`K QT2Mb 2t+BiBM; M2r `2b2 `+? QTTQ`imMBiB2bX h?2 T`2pBQmb b2`B H BKTH2K2Mi iBQM Bb `2r`Bii2M 7Q` T `@ HH2H 2t2+miBQMX h?2 H;Q`Bi?K Bb BKTH2K2Mi2/ BM bBM;H2 T`2+BbBQM mbBM; i?2 *QKTmi2 lMB}2/ .2pB+2 `+?Bi2+im`2 U*l.V QM LoA.A :SlbX AM //BiBQM- r2 /2p2HQT M PT2MJS p2`bBQM Q7 i?2 T ` HH2H BKTH2K2Mi iBQM iQ `mM QM KmHiB@+Q`2 *SlbX lbBM; #Qi? BKTH2K2Mi iBQMb- r2 T2`7Q`K MmK#2` Q7 #2M+?K `Fb iQ /2i2`KBM2 i?2 7 bi2bi p `B Mi Q7 i?2 H;Q`Bi?KX q2 Q#b2`p2 bT22/mT Q7 ϵЈ iQ ΚЈ QM i?2 LoA.A :hs Ndy +QKT `2/ iQ M AMi2H *Q`2 Bd 9ddyX h?2 :Sl BKTH2K2Mi iBQM + M bBKmH i2 mT iQ ϵЈЈЈ }#2`b QM /2bFiQT +QKTmi2` M/ Bi i F2b QMHv BM i?2 Q`/2` Q7 ΅ b2+QM/b iQ /p M+2 QM2 iBK2 bi2TX 6m`i?2`KQ`2- r2 ? p2 T2`7Q`K2/ MmK#2` Q7 bBKmH @ iBQMb Q7 FMQrM 2tT2`BK2Mib 7Q` b2/BK2MiBM; }#2`b iQ p HB/ i2 i?2 H;Q`Bi?K M/ iQ 2tTHQ`2 i?2 MmK2`B+ H T`2+BbBQM Q7 i?2 `2bmHibX h?2 `2bmHib b?Qr M 2t+2HH2Mi ;`22K2Mi rBi? `2bmHib 7`QK T`BQ` 2tT2`BK2Mib BM i?2 }2H/X.
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(7) Sammanfattning GPU simulering av stela fibrer >mpm/bv7i2i K2/ /2ii `#2i2 ` ii- K2/ ?D HT p ;` }F@ T`Q+2bbQ`2` U:Sl,2`V- ++2H2`2` 2M b2`B2HH BKTH2K2Mi iBQM p 2M MmK2`BbF H;Q`BiK 7ƺ` bBKmH2`BM; p bi2H }#`2` B 2M p ibF X 6B#`2`M ` KQ/2HH2` /2 T´ T `iBF2HMBp´ Q+? 2ii biQ`i Mi H }#`2` F` pb B #2` FMBM; `M 7ƺ` ii FmMM ;ƺ@ ` `2 HBbiBbF bBKmH2`BM; `X h`Qib 7ƺ`2MFHBM; ` B KQ/2HH2M #HB` 2Fp iBQM2`M b´ FQKTH2t ii /2 K´bi2 Hƺb b MmK2`BbFi Q+? /2i F` pb F` 7i7mHH K2iQ/2` 7ƺ` #2` FMBM; `M pBHF2i BMM2# ` ii bM ## Q+? 2z2FiBp H;Q`BiK2` K´bi2 mip2+FH bX li;´2M/2 7`´M /2M b2`B2HH BKTH2K2Mi iBQM2M ? ` 2M H;Q`BiK 7ƺ` T ` HH2HH 2t2Fp2`BM; T´ :Sl,2` mip2+FH ibX H@ ;Q`BiK2M ` BKTH2K2Mi2` / B 2MF2H T`2+BbBQM T´ LoA.A@ :Sl,2` ULoA.A :hs NdyV K2/ ?D HT p *QKTmi2 lMB}2/ .2pB+2 `+?Bi2+im`2- U*l.VX 6ƺ` ii FmMM ;ƺ` 2M ` ii@ pBb T`2bi M/ D K7ƺ`2Hb2 K2HH M :Sl@BKTH2K2Mi iBQM2M Q+? 2M *Sl@BKTH2K2Mi2`BM; ? ` /2bbmiQK 2M PT2MJS@p2`bBQM p H;Q`BiK2M BKTH2K2Mi2` ib 7ƺ` 2M *Sl UAMi2H *Q`2 Bd 9ddyV K2/ ~2` F `MQ`X "´/ p2`bBQM2`M ? ` QTiBK2` ib K2/ pb22M/2 T´ `2bT2FiBp2 `FBi2Fim`X 1ii Mi H T`2bi M/ @ i2bi2` pBb ` ii /2M Mv :Sl@p2`bBQM2M p H;Q`BiK2M ;2` 2M mTTbM ##MBM; p 2t2Fp2`BM;biB/2M T´ ky iBHH 9y ;´M;2` D K7ƺ`i K2/ *Sl@PT2MJS@p2`bBQM2MX .2ii ;ƺ` ii pB Mm F M bBKmH2` mTT iBHH kyyy }#`2` T´ 2M #Q`/b/ iQ` K2/ 2M 2t2Fp2`BM;biB/ T´ + 3 b2FmM/2` T2` iB/bbi2;X oB ? ` p2M mi7ƺ`i 2ii Mi H bBKmH2`BM; ` p `2/ M F M/ 2tT2`BK2Mi K2/ }#2`bmbT2MbBQM2` 7ƺ` ii p HB/2` H;Q`Bi@ K2M b Ki 7ƺ` ii mip `/2` /2M MmK2`BbF T`2+BbBQM2M B `2bmHi i2MX ƹp2`2Mbbi KK2Hb2M K2/ iB/B; `2 ` TTQ`i2` /2 `2bmHi i ` p H/B;i ;Q/X.
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(9) Acknowledgements h?Bb J bi2`Ƕb i?2bBb T`QD2+i r b + ``B2/ Qmi i i?2 LmK2`B+ H M HvbBb :`QmT Q7 i?2 .2T `iK2Mi Q7 J i?2K iB+b i i?2 _Qv H AMbiBimi2 Q7 h2+?MQHQ;v BM ar2/2MX 6B`bi M/ 7Q`2KQbi- A 2bT2+B HHv r Mi iQ i? MF Kv bmT2`pBbQ` E i `BM :mbi pbbQM 7Q` ;BpBM; K2 i?2 QTTQ`imMBiv iQ +QK2 iQ ar2/2M M/ +QMi`B#mi2 iQ i?2 }2H/ Q7 T `iB+H2 bmbT2MbBQMbX >2` ;mB/ M+2 M/ bmTTQ`i r b BMp Hm #H2 iQ i?2 bm++2bb Q7 i?Bb i?2bBbX //BiBQM HHv- A K ;` i27mH iQ JB+? 2H a+?HB2T? F2 7Q` ?Bb BMbB;?i7mH +QKK2Mib M/ /Bb+mbbBQM QM :Sl T`Q;` KKBM;X 6BM HHv- A r Mi iQ i? MF JB+? 2H > MF2 7Q` #2BM; Kv 2t KBM2` M/ ?Bb +QMiBMm2/ bmTTQ`iX.
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(11) Contents GBbi Q7 6B;m`2b GBbi Q7 h #H2b GBbi Q7 GBbiBM;b R AMi`Q/m+iBQM. R. k h?2Q`2iB+ H 7QmM/ iBQM. 8. kXR. aiQF2b ~Qr X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. 8. kXk. "QmM/ `v BMi2;` H 7Q`KmH iBQM. X X X X X X X X X X X X X X X X X X X X X. e. kXkXR. 6mM/ K2Mi H bQHmiBQMb X X X X X X X X X X X X X X X X X X X X X X. e. kXkXk. 6Q`KmH iBQM 7Q` Q#D2+ib BKK2`b2/ BM ~mB/ X X X X X X X X X X. d. aH2M/2` }#2`b X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. 3. kXj. j LmK2`B+ H H;Q`Bi?K M/ b2`B H BKTH2K2Mi iBQM. Rj. jXR. .Bb+`2iBx iBQM X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. Rj. jXk. bb2K#H2 avbi2K X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. R9. jXj. aQHp2 bvbi2K X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. R8. jX9. lT/ i2 p2HQ+BiB2b X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. Re. jX8. lT/ i2 }#2`b X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. Re. jXe. H;Q`Bi?K bmKK `v X X X X X X X X X X X X X X X X X X X X X X X X X X X. Re. jXd. _2K `Fb +QM+2`MBM; :Sl BKTH2K2Mi iBQM X X X X X X X X X X X X X X X. Rd. 9 :Sl T`Q;` KKBM;. RN. 9XR. :2M2` H Tm`TQb2 +QKTmiBM; QM :Slb. X X X X X X X X X X X X X X X X X. RN. 9Xk. *l. pbX PT2M*G X X X X X X X X X X X X X X X X X X X X X X X X X X X X. kR. 9Xj. *l. T`Q;` KKBM; KQ/2H X X X X X X X X X X X X X X X X X X X X X X X. kk.
(12) 8 S ` HH2H BKTH2K2Mi iBQM. kN. 8XR. .2p2HQTK2Mi 2MpB`QMK2Mi X X X X X X X X X X X X X X X X X X X X X X X X. kN. 8Xk. GBM2 ` bQHp2`b X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. jy. 8Xj. E2`M2Hb X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. jR. 8X9. PTiBKBx iBQMb X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. je. 8X9XR. LmK2`B+ pbX M HviB+ BMi2;` iBQM X X X X X X X X X X X X X X X X. jd. 8X9Xk. a? `2/ K2KQ`v X X X X X X X X X X X X X X X X X X X X X X X X X X. jd. 8X9Xj. h?`2 / #HQ+F /BK2MbBQM X X X X X X X X X X X X X X X X X X X X X. j3. PT2MJS X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. 9y. 8X8. e "2M+?K `Fb eXR. 9j. J2i?Q/QHQ;v X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. 9j. eXRXR. > `/r `2 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. 9j. eXRXk. "2M+?K `F b+?2K2 X X X X X X X X X X X X X X X X X X X X X X X X. 99. PTiBKBx iBQMb X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. 9e. eXkXR. LmK2`B+ pbX M HviB+ AMi2;` iBQM X X X X X X X X X X X X X X X X. 9e. eXkXk. a? `2/ K2KQ`v X X X X X X X X X X X X X X X X X X X X X X X X X X. 93. eXkXj. h?`2 / #HQ+F /BK2MbBQM X X X X X X X X X X X X X X X X X X X X X. 9N. GBM2 ` bQHp2`b X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. 8R. eXjXR. .B`2+i bQHp2` pb Bi2` iBp2 bQHp2` QM *Sl X X X X X X X X X X X X. 8R. eXjXk. 6B#2` +QM+2Mi` iBQM 2z2+i QM :J_1a Bi2` iBQMb X X X X X X X. 8R. eXjXj. .B`2+i bQHp2` pbX Bi2` iBp2 bQHp2` QM :Sl. X X X X X X X X X X X. 8j. eX9. AM/BpB/m H bi2Tb Q7 i?2 H;Q`Bi?K X X X X X X X X X X X X X X X X X X X X. 88. eX8. :Sl pbX *Sl X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. 8d. eXk. eXj. d LmK2`B+ H 2tT2`BK2Mib. eR. dXR. hmK#HBM; Q`#Bib X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. eR. dXk. a2/BK2MiBM; Q7 bT?2`B+ H +HQm/ X X X X X X X X X X X X X X X X X X X X. ej. dXkXR. aT?2`B+ H +HQm/ #`2 F@mT X X X X X X X X X X X X X X X X X X X X X. e9. dXkXk. 6B#2` +QM+2Mi` iBQM 2z2+i QM #`2 F@mT iBK2 X X X X X X X X X X. e3. dXkXj. LmK#2` Q7 }#2`b 2z2+i QM #`2 F@mT iBK2 X X X X X X X X X X X X. e3. *HQm/ Q7 }#2`b rBi? /Bz2`2Mi /2MbBiB2b X X X X X X X X X X X X X X X X X. eN. dXj. 3 *QM+HmbBQM. dj. "B#HBQ;` T?v. d8. aBKmH iBQM S ` K2i2`b. dN.
(13) List of Figures kXR. hrQ BKK2`b2/ Q#D2+ib BM aiQF2b ~QrX. X X X X X X X X X X X X X X X X X X X. 3. kXk. AHHmbi` iBQM Q7 i?2 bH2M/2`M2bb T ` K2i2`bX X X X X X X X X X X X X X X X X X. N. kXj. aH2M/2` }#2` TT`QtBK iBQMX. X X X X X X X X X X X X X X X X X X X X X X X X. Ry. kX9. hrQ BMi2` +iBM; }#2`b rBi? }p2 [m /` im`2 TQBMibX X X X X X X X X X X X X. RR. 9XR. h?2Q`2iB+ H :6GPSfbX X X X X X X X X X X X X X X X X X X X X X X X X X X X X. ky. 9Xk. Pp2`pB2r Q7 i?2 *l. TH i7Q`KX X X X X X X X X X X X X X X X X X X X X X X. kk. 9Xj. miQK iB+ b+ HBM; Q7 #HQ+Fb +`Qbb M `#Bi` `v MmK#2` Q7 ai`2 KBM; JmHiBT`Q+2bbQ`bX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. k9. 9X9. *l. #HQ+F //`2bbBM; mbBM; k. ;`B/ rBi? k. i?`2 / #HQ+FbX X X X X X X. ke. 9X8. *l. K2KQ`v ?B2` `+?vX X X X X X X X X X X X X X X X X X X X X X X X X X X. k3. 8XR. AHHmbi` iBQM Q7 R. M/ k. i?`2 / #HQ+F /BK2MbBQMb 7Q` i?2 bvbi2K K i`BtX jN. eXR. "2M+?K `F +QKTmiBM; BMM2` BMi2;` H QM *SlX X X X X X X X X X X X X X X X. 9d. eXk. "2M+?K `F +QKTmiBM; BMM2` BMi2;` H QM :SlX X X X X X X X X X X X X X X. 9d. eXj. "2M+?K `FBM; i?`2 / #HQ+F /BK2MbBQMbX X X X X X X X X X X X X X X X X X X. 8y. eX9. "2M+?K `F HBM2 ` bQHp2`b QM *SlX X X X X X X X X X X X X X X X X X X X X X. 8k. eX8. 1z2+i Q7 }#2` +QM+2Mi` iBQM QM :J_1a Bi2` iBQMbX X X X X X X X X X X X X. 8j. eXe. "2M+?K `F HBM2 ` bQHp2`b QM :SlX X X X X X X X X X X X X X X X X X X X X X. 89. eXd. "2M+?K `F BM/BpB/m H bi2Tb QM *SlX X X X X X X X X X X X X X X X X X X X X. 8e. eX3. "2M+?K `F BM/BpB/m H bi2Tb QM :SlX X X X X X X X X X X X X X X X X X X X X. 8e. eXN. "2M+?K `F Qp2` HH 2t2+miBQM iBK2X X X X X X X X X X X X X X X X X X X X X X. 83. dXR. oBbm HBx iBQM Q7 imK#HBM; Q`#BibX X X X X X X X X X X X X X X X X X X X X X X. ek. dXk. *QKT `BbQM Q7 b2/BK2Mi iBQM p2HQ+Biv 7Q` bBM;H2@ M/ /Qm#H2@T`2+BbBQM. dXj. bBKmH iBQMX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X. ej. oBbm HBx iBQM Q7 b2/BK2MiBM; bT?2`B+ H +HQm/X. e8. X X X X X X X X X X X X X X X.
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(15) List of Tables eXR. "2M+?K `F ? `/r `2 bT2+B}+ iBQMX X X X X X X X X X X X X X X X X X X X X X. 99. eXk. q `T 1t2+miBQM 1{+B2M+v Q7 LmK2`B+ H pbX M HviB+ H AMi2;` iBQMX X X X. 93. eXj. iQKB+ i` Mb +iBQMb Q7 k. pbX j. i?`2 / #HQ+F /BK2MbBQMbX X X X X X X X. 8y. eX9. aT22/mT 7 +iQ`b 7Q` Qp2` HH 2t2+miBQM iBK2X. 8N. X X X X X X X X X X X X X X X X.
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(17) List of Listings 9XR. Sb2m/Q+Q/2 7Q` *l. p2+iQ` //BiBQMX X X X X X X X X X X X X X X X X X. kj. 9Xk. Sb2m/Q+Q/2 7Q` *l. K i`Bt //BiBQM- BHHmbi` iBM; k. i?`2 / #HQ+FbX k8. 8XR. Sb2m/Q+Q/2 7Q` T ` HH2H H;Q`Bi?K QM i?2 ?QbiX. X X X X X X X X X X X X. jk. 8Xk. Sb2m/Q+Q/2 7Q` i?2 bb2K#H2 bvbi2K bi2T rBi? R. i?`2 / #HQ+FX X X. jj. 8Xj. Sb2m/Q+Q/2 7Q` i?2 mT/ iBM; p2HQ+BiB2b bBKmH iBQM bi2TX X X X X X X X X. j8. 8X9. Sb2m/Q+Q/2 7Q` i?2 mT/ iBM; }#2`b bBKmH iBQM bi2TX X X X X X X X X X X. je. eXR. Sb2m/Q+Q/2 7Q` #2M+?K `F b+?2K2X X X X X X X X X X X X X X X X X X X X. 98.
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(19) Chapter 1. Introduction S`2/B+iBM; i?2 T?vbB+ H #2? pBQ` Q7 T `iB+H2b bmbT2M/2/ BM ~mB/b Bb Q7 ;`2 i BMi2`2bi BM p `B2iv Q7 /Bz2`2Mi }2H/bX h?2 #BHBiv iQ ++m` i2Hv KQ/2H M/ bBKmH i2 /Bz2`2Mi + i2;Q`B2b Q7 T `iB+H2 bmbT2MbBQMb HHQrb i?2B` T`QT2`iB2b iQ #2 M Hvx2/ M/ QTiBKBx2/ 7Q` H `;2 MmK#2` Q7 TTHB+ iBQMbX 1t KTH2b BM+Hm/2 K2/B+ H TTHB+ iBQMb r2`2 i?2 /2HBp2`v M/ /Bbi`B#miBQM Q7 i?2 +iBp2 ;2Mi ? b iQ #2 KQ/2H2/- r bi2 K M ;2K2Mi iQ 2{+B2MiHv 2ti` +i r bi2 7`QK r i2` M/ i?2 T T2` BM/mbi`v r?B+? Bb i`vBM; iQ BKT`Qp2 i?2 +? ` +i2`BbiB+b Q7 i?2B` K i2`B HX 1p2M BM i?2 bBKTH2bi ~Qr + b2b- i?2 ~Qr Q7 T `iB+H2 bmbT2MbBQM 2t?B#Bib p2`v +QKTH2t M/ +QKTHB+ i2/ /vM KB+ H #2? pBQ`X h?2 `?2QHQ;B+ H T`QT2`iB2b Q7 i?2 bmbT2MbBQM /2T2M/ bi`QM;Hv QM 72 im`2b bm+? b i?2 +QM+2Mi` iBQM Q7 T `iB+H2b- T `iB+H2 b? T2b M/ T `iB+H2 BMi2` +iBQMbX AM Q`/2` iQ ++m` i2Hv + Tim`2 i?2 +QKTH2t /vM KB+b Q7 bmbT2MbBQM mbBM; MmK2`B+ H bBKmH iBQMb- H `;2 MmK#2` Q7 T `iB+H2b `2 `2[mB`2/ BM i?2 bBKmH iBQMX >2M+2- i?2 #BHBiv Q7 i?2 MmK2`B+ H H;Q`Bi?K iQ 2{+B2MiHv ? M/H2 H `;2 KQmMi Q7 T `iB+H2b Bb Q7 +`m+B H BKTQ`i M+2X h?2 rQ`F BM i?Bb i?2bBb 7Q+mb2b QM BM+`2 bBM; M/ QTiBKBxBM; i?2 2{+B2M+v Q7 MmK2`B+ H H;Q`Bi?K 7Q` i?2 bBKmH iBQM Q7 b2/BK2MiBM; `B;B/ M/ bH2M/2` }#2`b BM pBb+Qmb ~mB/X h?2 }#2`b `2 KQ/2H2/ M/ bBKmH i2/ QM T `iB+H2@H2p2H BM j. 7`22@bT +2 aiQF2b ~mB/X h?2 K i?2K iB+ H KQ/2H Bb # b2/ QM #QmM/ `v BMi2;` H 7Q`KmH iBQM M/ MQM@HQ+ H bH2M/2` #Q/v TT`QtBK iBQM #v hQ`M#2`; M/ :mbi pbbQM- (jR)X h?2`2 `2 K Mv MmK2`B+ H bim/B2b Q7 }#2` bmbT2MbBQM M/ b2p2` H /Bz2`2Mi K2i?Q/b ? p2 #22M /2p2HQT2/ 7Q` #Qi? `B;B/ M/ ~2tB#H2 }#2`bX PM2 TT`Q +? Bb i?2 bQ@+ HH2/ #2 /b@KQ/2H- r?2`2 i?2 }#2`b `2 KQ/2H2/ b b2i Q7 +QMM2+i2/ bT?2`B+ H. R.
(20) *>Sh1_ RX ALh_P.l*hAPL. #2 /b U2X;X (9- RR- kN- jj)VX h?2 BKK2`b2/ #QmM/ `v K2i?Q/ /Bb+`2iBx2b i?2 }#2`b rBi? G ;` M;B M K `F2`b M/ /Bbi`B#mi2b i?2 7Q`+2 QMiQ # +F;`QmM/ ;`B/ r?B+? Bb i?2M mb2/ iQ KQ/B7v i?2 ~mB/ ~Qr U2X;X (k9- jy)VX MQi?2` TT`Q +? Bb # b2/ QM bH2M/2` #Q/v i?2Q`v r?B+? mb2b i?2 H `;2 bT2+i ` iBQ Q7 i?2 }#2`b iQ bBKTHB7v i?2 mM/2`HvBM; KQ/2H U2X;X (N- jR- jk)VX +QKT`2?2MbBp2 `2pB2r Q7 i?2 MmK2`B+ H bim/B2b Q7 }#2` bmbT2MbBQM + M #2 7QmM/ BM :m xx2HHB M/ >BM+?- (3)X AM i?Bb i?2bBb r2 /2p2HQT ?B;? T2`7Q`K M+2 :Sl BKTH2K2Mi iBQM Q7 i?2 MmK2`@ B+ H H;Q`Bi?K 7Q` bBKmH iBM; }#2` bmbT2MbBQMX "v i FBM; /p Mi ;2 Q7 i?2 K bbBp2Hv T ` HH2H `+?Bi2+im`2 Q7 KQ/2`M :Slb K Mv KQ`2 }#2`b + M #2 bBKmH i2/ +QKT `2/ iQ i?2 T`2pBQmb *Sl # b2/ BKTH2K2Mi iBQM Q7 i?2 H;Q`Bi?KX h?2 K DQ` ;Q H Bb iQ 2 bBHv M/ 2{+B2MiHv T2`7Q`K bBKmH iBQMb QM ?B;?@2M/ /2bFiQT +QKTmi2` Q` rQ`Fbi iBQM `2 /BHv ++2bbB#H2 #v i?2 `2b2 `+?2`X h?Bb rBHH HHQr i?2 `2b2 `+?2` iQ 2tTHQ`2 i?2 ?m;2 T`Q#H2K bT +2 M/ bBKmH i2 H `;2 MmK#2` Q7 }#2`b BM b?Q`i KQmMi Q7 iBK2X h?mb Bi 2M? M+2b i?2 + T +Biv iQ ` TB/Hv Bi2` i2 M/ /Bb+Qp2` BMi2`2biBM; i2bi + b2bX h?2b2 + b2b + M i?2M #2 mb2/ b bi `iBM; TQBMi 7Q` H `;2 b+ H2 bBKmH iBQMb mbBM; +QKTmiBM; +Hmbi2`bX h?2 KQbi +QbiHv T `i Q7 i?2 MmK2`B+ H H;Q`Bi?K Bb iQ ++QmMi 7Q` i?2 BMi2` +iBQMb #2ir22M HH }#2`b BM i?2 bvbi2KX 6Q` i?2 + H+mH iBQMb r2 +?Qb2 i?2 ǴM Bp2Ǵ H;Q`Bi?Kr?2`2 i?2 BMi2` +iBQMb #2ir22M 2p2`v T B` Q7 }#2`b Bb +QKTmi2/X h?2 /2p2HQTK2Mi Q7 bm+? M Bp2 H;Q`Bi?K Bb 2 bB2`- KQ`2 ++m` i2 M/ BM bQK2 + b2b KQ`2 +Qbi 2z2+iBp2 i? M M Hi2`M iBp2 7 bi bmKK iBQM TT`Q +?X 7 bi bmKK iBQM TT`Q +?HBF2 i?2 7 bi KmHiBTQH2 K2i?Q/- Bb #Qi? +QKTH2t iQ BKTH2K2Mi M/ BMi`Q/m+2b TQi2MiB HHv H `;2 T2`7Q`K M+2 Qp2`?2 /- (j8)X h?2`27Q`2 Bi KB;?i MQi `2bmHi BM i?2 /2bB`2/ T2`7Q`K M+2 BM+`2 b2 +QKT `2/ iQ i?2 M Bp2 H;Q`Bi?KX Pm` +?QB+2 Q7 +QMbmK2`@;` /2 :Sl ? `/r `2 HBKBib i?2 bBKmH iBQM iQ bBM;H2 T`2+BbBQM- i?mb K BMi BMBM; i?2 ++m` +v Bb p2`v BKTQ`i MiX "v mbBM; i?2 M Bp2 H;Q`Bi?K QM i?2 :Sl i?2 ;Q H Bb iQ bi`BF2 ;QQ/ # H M+2X h?Bb i?2bBb Bb /BpB/2/ BMiQ i?2 i?`22 K DQ` T `ib, S`2pBQmb qQ`F- :Sl AK@ TH2K2Mi iBQM M/ _2bmHibX 6B`bi r2 BMi`Q/m+2 i?2 i?2Q`2iB+ H 7QmM/ iBQM Q7 i?2 MmK2`B+ H K2i?Q/ BM *? Ti2` k M/ `272` iQ i?2 Q`B;BM H T T2` #v hQ`M#2`; M/ :mbi pbbQM- (jR)- 7Q` BM@/2Ti? /2i BHbX h?2 MmK2`B+ H H;Q`Bi?K M/ i?2 b2`B H BKTH2K2Mi iBQM QM i?2 *Sl /2p2HQT2/ BM i?2 Q`B;BM H T T2`- (jR)- Bb T`2b2Mi2/ BM *? Ti2` jX hQ;2i?2` i?2b2 irQ +? Ti2`b 7Q`K i?2 S`2pBQmb qQ`F T `i Q7 i?2 i?2bBbX " b2/ QM i?Bb i?2 :Sl AKTH2K2Mi iBQM T `i 7QHHQrbX *? Ti2` 9 #`B2~v BMi`Q/m+2b ;2M2` H Tm`TQb2 +QKTmiBM; QM i?2 :SlX *QK#BMBM; i?2 T`2pBQmb rQ`F M/ :Sl. k.
(21) +QKTmiBM;- r2 i?2M /2b+`B#2 Qm` M2r T ` HH2H BKTH2K2Mi iBQM Q7 i?2 MmK2`B+ H H;Q`Bi?K QM :Slb BM *? Ti2` 8X h?Bb `2T`2b2Mib i?2 # b2 7Q` i?2 2{+B2M+v M/ i?2 T2`7Q`K M+2 BKT`Qp2K2Mib M/ Bb i?2`27Q`2 i?2 +Q`2 Q7 i?2 i?2bBbX 7i2`r `/b- i?2 }M H _2bmHib `2 T`2b2Mi2/X h?2 #2M+?K `Fb BM *? Ti2` e BHHmbi` i2 i?2 +?B2p2/ T2`7Q`K M+2 BM+`2 b2 Q7 i?2 T ` HH2H :Sl AKTH2K2Mi iBQMX 6BM HHv- r2 T2`7Q`K M/ +QKT `2 Qm` bBKmH iBQM `2bmHib iQ 72r FMQrM 2tT2`BK2Mib BM *? Ti2` dX. j.
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(23) Chapter 2. Theoretical foundation h?2 BMi`Q/m+iBQM /Bb+mbb2/ /Bz2`2Mi TTHB+ iBQMb Q7 `B;B/ }#2` bBKmH iBQMbX Ai 2bT2+B HHv bi`2bb2/ i?2 BKTQ`i M+2 Q7 #2BM; #H2 iQ bBKmH i2 H `;2 MmK#2` Q7 }#2`b BM Q`/2` iQ ;2M2` i2 i?2 p `BQmb T ii2`Mb 7QmM/ BM `2 H rQ`H/ 2tT2`BK2MibX AM i?Bb +? Ti2`- r2 rBHH T`2b2Mi i?2 i?2Q`2iB+ H 7QmM/ iBQM Q7 i?2 T?vbB+b M/ i?2 K i?2K iB+ H KQ/2H i?2 bBKmH iBQMb `2 # b2/ QMX h?Bb Bb `2[mB`2/ iQ #2 #H2 iQ mM/2`bi M/ i?2 MmK2`B+ H K2i?Q/ mb2/ i?`Qm;?Qmi i?2 `2bi Q7 i?2 i?2bBbX >Qr2p2`i?2 KQ/2H Bb QMHv T`2b2Mi2/ BM `2/m+2/ bmKK `v- M/ 7Q` KQ`2 /2i BHb r2 `272` iQ hQ`M#2`; M/ :mbi pbbQM- (jR)X. 2.1. Stokes flow. q2 `2 BMi2`2bi2/ BM KQ/2HBM; i?2 #2? pBQ` Q7 Q#D2+ib BKK2`b2/ BM M BM+QKT`2bbB#H2 ~mB/X q2 `2bi`B+i Qm`b2Hp2b iQ `B;B/ #Q/B2b- r?B+? `2 bHQrHv b2/BK2MiBM; BM i?2 ~mB/ /m2 iQ ;` pBivX AM i?2 KQbi ;2M2` H + b2- i?2 ~Qr + M #2 KQ/2H2/ #v i?2 L pB2`@aiQF2b 2[m iBQMbᅶ. ᆄ
(24) ਼ ও ਲԟ
(25) ᅲɞ ֦
(26) ᆄԣ. UkXRV. ਼ r?2`2 א /2MQi2b i?2 p2HQ+Biv }2H/- ԟ אi?2 T`2bbm`2 }2H/ M/ ֦ אi?2 7Q`+2 +iBM; QM i?2 ~mB/ i i?2 HQ+ iBQM אԧ Ԩ ԩ ୩ නɘ X h?2 +QMbi Mi ᅲ Bb i?2 pBb+QbBiv Q7 i?2 ~mB/ M/ ᅶ Bb i?2 ~mB/ /2MbBivX aQHpBM; i?2b2 2[m iBQMb Bb [mBi2 +? HH2M;BM; /m2 iQ i?2B` iBK2 /2T2M/2M+2 M/ MQM@HBM2 `BivX >Qr2p2`- BM i?Bb rQ`F r2 `2 QMHv +QM+2`M2/ rBi? bHQrHv KQpBM; Q#D2+ib M/ i?2 _2vMQH/b MmK#2`- ԇԔ . ᆠՀԷ ᆜ -. Bb. bbmK2/ iQ #2 p2`v bK HHX :Bp2M i?2 +QMbi` BMi i? i ԇԔ ୁ - i?2 BM2`iB H M/ 8.
(27) *>Sh1_ kX h>1P_1hA*G 6PlL.hAPL. ++2H2` iBQM i2`Kb BM i?2 L pB2`@aiQF2b 1[MbX UkXRV + M #2 M2;H2+i2/ M/ r2 ``Bp2 i i?2 HBM2 ` M/ iBK2 BM/2T2M/2Mi aiQF2b 2[m iBQMbԟ ਲ ᅲ֦ . BM ဂ-. ਼ . BM ဂX. UkXkV. AM //BiBQM iQ i?2b2 2[m iBQMb r2 M22/ irQ #QmM/ `v +QM/BiBQMb iQ bQHp2 i?2 bvbi2KX h?2 }`bi #QmM/ `v +QM/BiBQM ++QmMib 7Q` i?2 BM~m2M+2 QM i?2 ~mB/ 7`QK i?2 T`2b2M+2 Q7 i?2 Q#D2+ibX >2`2- r2 7Q`+2 i?2 ~mB/ p2HQ+Biv i i?2 #QmM/ `v Q7 i?2 Q#D2+i iQ #2 2[m H iQ i?2 p2HQ+Biv Q7 i?2 Q#D2+i Bib2H7X h?2b2 MQ@bHBT +QM/BiBQMb QM i?2 bm`7 +2 Q7 i?2 Q#D2+ib `2 /2}M2/ b QM က-. ယ. UkXjV. r?2`2 က /2MQi2b i?2 mMBQM Q7 HH Q#D2+i bm`7 +2b M/ ယ i?2 +Q``2bTQM/BM; bm`7 +2 p2HQ+BivX h?Bb +QM/BiBQM +QMbi` BMb i?2 ~mB/ iQ ? p2 x2`Q p2HQ+Biv `2H iBp2 iQ i?2 Q#D2+i bm`7 +2bX h?2 b2+QM/ #QmM/ `v +QM/BiBQM KQ/2Hb i?2 7 +i i? i i?2 ~mB/ p2HQ+Biv 7 ` 7`QK i?2 Q#D2+i Bb MQi z2+i2/ #v Bib T`2b2M+2X h?Bb Bb KQ/2H2/ #v bi iBM; i? i i?2 p2HQ+Biv }2H/ b?QmH/ #2 2[m H iQ # +F;`QmM/ p2HQ+Biv ɱ i BM}MBiv ܽ ɱ. b. ]] ܽ ]]אX. UkX9V. AM Qm` bBKmH iBQMb i?Bb # +F;`QmM/ p2HQ+Biv Bb Hr vb b2i iQ X h?`Qm;? i?2 KQiBQM Q7 i?2 BKK2`b2/ Q#D2+ib M/ i?2 MQ@bHBT #QmM/ `v +QM/BiBQMb /2}M2/ BM 1[MX UkXjV i?2 /2T2M/2M+v QM iBK2 Bb `2BMi`Q/m+2/ BM i?2 aiQF2b 1[MbX UkXkVX. 2.2. Boundary integral formulation. h?2 aiQF2b 2[m iBQMb `2 HBM2 ` BM #Qi? p2HQ+Biv M/ T`2bbm`2- r?B+? HHQrb i?2K iQ #2 bQHp2/ mbBM; MmK#2` Q7 /Bz2`2Mi K2i?Q/b 7Q` HBM2 ` T `iB H /Bz2`2MiB H 2[m iBQMbX h?2 TT`Q +? mb2/ BM i?Bb i?2bBb Bb i?2 #QmM/ `v BMi2;` H K2i?Q/- b22 2X;X SQx`BFB/Bb (k8)X. 2.2.1. Fundamental solutions. ai `iBM; 7`QK #QmM/ `v BMi2;` H 7Q`KmH iBQM Q7 i?2 aiQF2b 2[m iBQMb- M HviB+ H bQHmiBQMb + M #2 /2`Bp2/- bQ@+ HH2/ 7mM/ K2Mi H bQHmiBQMbX PM2 bm+? bQHmiBQM Bb i?2 e.
(28) kXkX "PlL._u ALh1:_G 6P_JlGhAPL. aiQF2bH2iX A7 ֦ BM 1[MX UkXkV Bb ;Bp2M #v TQBMi 7Q`+2 +iBM; i גrBi? bi`2M;i? ֥- BX2X ֦ ֥ ਼ ᅩ אਲ ג- r?2`2 ᅩ אਲ גBb i?2 .B` + /2Hi 7mM+iBQM- i?2 p2HQ+Biv }2H/ ԤՎ א. Ԉ ג אӻՏ ᅵᅲ ՎՏ. Ԙ ԙ -. UkX8V. r?2`2 i?2 i2MbQ` T`Q/m+i ԈՎՏ ӻՏ ԈՎȯ ӻȯ
(29) ԈՎɞ ӻɞ
(30) ԈՎɘ ӻɘ -. UkXeV. Bb bQHmiBQM iQ i?2 aiQF2b 2[m iBQMbX h?2 i2`K ԈՎՏ Bb i?2 aiQF2bH2i M/ Bb ;Bp2M #v ԈՎՏ אਲ ג. ᅩՎՏ ԧՎ ਲ ԨՎ ԧՏ ਲ ԨՏ.
(31) X ] אਲ ]ג ] אਲ ]גɘ. UkXdV. G i2`- r2 rBHH M22/ ?B;?2` Q`/2` 7mM/ K2Mi H bQHmiBQMb iQ i?2 aiQF2b 1[MbX UkXkVX h?2b2 + M #2 Q#i BM2/ #v bBKTHv /Bz2`2MiB iBM; i?2 aiQF2bH2iX PM2 2t KTH2 Bb i?2 bQ@+ HH2/ /Qm#H2i ԧՎ ਲ ԨՎ ԧՏ ਲ ԨՏ. ɞ ֭ ӹՎՏ אਲ גԈՎՏ אਲ ג া ਲ ি X UkX3V ᅵᅲ ] אਲ ]גɘ ] אਲ ]גȁ. 2.2.2. Formulation for objects immersed in a fluid. lbBM; i?2 7mM/ K2Mi H bQHmiBQM BM 1[MX UkX8V r2 + M MQr KQ/2H i?2 KQiBQM Q7 BKK2`b2/ Q#D2+ib BM ~mB/X bbmK2 i? i r2 ? p2 iQi H Q7 Ԃ BKK2`b2/ `B;B/ Q#D2+ib BM i?2 ~mB/X 1 +? Q#D2+i Ԝ- 7Q` Ԝ w Ԃ - Bb +2Mi2`2/ i אՈՒ rBi? M. bbQ+B i2/ Q`i?QMQ`K H # bBb ׆Ւ M/ bm`7 +2 ကՒ X :Bp2M `B;B/ #Q/v KQiBQM M/ i?2 MQ@bHBT #QmM/ `v +QM/BiBQM BM 1[MX UkXjV r2 + M KQ/2H i?2 p2HQ+Biv }2H/ א i 2 +? bm`7 +2 TQBMi א୩ ကՒ Q7 i?2 Q#D2+i b א Ւ
(32) ረՒ אਲ אՈՒ -. UkXNV. r?2`2 Ւ Bb i?2 i` MbH iBQM H p2HQ+Biv M/ ረՒ Bb i?2 `Qi iBQM H p2HQ+Biv Q7 i?2 Q#D2+iX h?Bb b2imT Bb BHHmbi` i2/ 7Q` irQ Q#D2+ib BM 6B;X kXRX "v +QK#BMBM; `B;B/ #Q/v KQiBQM- 1[MX UkXNV- rBi? #QmM/ `v BMi2;` H 7Q`KmH iBQM Q7 i?2 aiQF2b 2[m iBQMb `2H iBQMb?BT #2ir22M i?2 Q#D2+ib p2HQ+BiB2b M/ i?2 7Q`+2 /Bbi`B#miBQM- ԕ- +iBM; QM i?2 bm`7 +2 Q7 i?2 Q#D2+i Bb ;Bp2M #vՒ Ւ אਲ אՈՒ. Վ Վ
(33) ረ. Ը ௲ Ԉ ג אԕՏՑ גԓԈ՞ ᅵᅲ Ց ȯ ယՑ ՎՏ d. Ԙ ԙ X UkXRyV.
(34) *>Sh1_ kX h>1P_1hA*G 6PlL.hAPL. 6B;m`2 kXR, hrQ BKK2`b2/ Q#D2+ib BM aiQF2b ~Qr- mbBM; `B;B/ #Q/v KQiBQMX 1 +? }#2` Bb +? ` +i2`Bx2/ #v i?2 TQbBiBQMb אȯ אɞ M/ i?2 Q`B2Mi iBQMb ׆ȯ ׆ɞ X h?2 p2HQ+Biv Q7 2 +? TQBMi i?2 bm`7 +2b ကȯ ကɞ Bb ;Bp2M #v i?2 `B;B/ #Q/v KQiBQM /2b+`B#2/ BM 1[M UkXNVX 6Q` b2/BK2MiBM; Q#D2+i- i?2 i` MbH iBQM H M/ `Qi iBQM H p2HQ+BiB2b Ւ M/ ረՒ b r2HH b i?2 7Q`+2 /Bbi`B#miBQM ֦ Ւ `2 mMFMQrMX AM Q`/2` iQ #2 #H2 iQ bQHp2 i?2 bvbi2K r2 mb2 i?2 //BiBQM H +QMbi` BMib Ւ ֥Ւ Q#D2+i ௲ ֦ גԓԈ՞ ယՒ. Ւ Ւ ׅՒ Q#D2+i ௲ אਲ אՈ ֦ גԓԈ՞ -. UkXRRV. ယՒ. bi iBM; i? i i?2 BMi2;` i2/ 7Q`+2 M/ iQ`[m2 Qp2` 2 +? Q#D2+i Kmbi #2 2[m H iQ i?2 2ti2`M HHv TTHB2/ 7Q`+2 M/ iQ`[m2 QM i?2 Q#D2+iX :Bp2M i?2b2 //BiBQM H +QMbi` BMib r2 `2 MQr #H2 iQ bQHp2 1[MbX UkXRyV M/ UkXRRV 7Q` ֦ Ւ - Ւ M/ ረՒ 7Q` HH }#2`b Ԝ w Ԃ X PM+2 r2 ? p2 i?2 p2HQ+BiB2b- Ւ M/ ረՒ - r2 + M mT/ i2 i?2 TQbBiBQM M/ Q`i?QMQ`K H # bBb Q7 2 +? Q#D2+i #v bQHpBM;ԓ Ւ אՒ ԓԣ Ո. ԓ Ւ ׆ ׆Ւ ረՒ X ԓԣ. UkXRkV. h?2 BMi2;` H TT2 `BM; BM i?2 `B;?i@? M/ bB/2 Q7 1[MX UkXRyV Kmbi BM KQbi + b2b #2 2p Hm i2/ #v MmK2`B+ H [m /` im`2X. 2.3. Slender fibers. h?2 7Q`KmH iBQM /2p2HQT2/ BM i?2 T`2pBQmb b2+iBQM rBHH MQr #2 / Ti2/ 7Q` `B;B/ }#2` bmbT2MbBQMX *QMbB/2` bi` B;?i- `B;B/ #Q/v Q7 H2M;i? ԁ M/ ` /Bmb Ԑ- M/ H2i ᅪ Ԑԁ /2MQi2 bH2M/2`M2bb T ` K2i2`X A7 ᅪ ୁ i?2 #Q/v Bb `272``2/ iQ b bH2M/2` #Q/v UbH2M/2` }#2`V b BHHmbi` i2/ BM 6B;X kXkX Pm` ;Q H Bb iQ bBKmH i2 K Mv bH2M/2` }#2`bX >Qr2p2`- mbBM; i?2 k. #QmM/ `v BMi2;` H 7Q`KmH iBQM rBHH #2 p2`v 2tT2MbBp2 iQ bQHp2 MmK2`B+ HHvX b i?2 bT2+i ` iBQ ᅪ Q7 i?2 }#2` BM+`2 b2b- i?2 MmK#2` Q7 [m /` im`2 TQBMib Kmbi BM+`2 b2 BM Q`/2` 3.
(35) kXjX aG1L.1_ 6A"1_a. U V ᆾ φΚ. U#V ᆾ φφЈ. U+V ᆾ φϵЈ. U/V ᆾ φφЈЈ. 6B;m`2 kXk, AHHmbi` iBQM Q7 i?2 bH2M/2`M2bb T ` K2i2`bX q?2M ᅪ #2+QK2b bK HH2` i?2 b? T2 Q7 i?2 #Q/v bvKTiQiB+ HHv TT`Q +?2b i? i Q7 M BM}MBi2bBK H bH2M/2` }#2`X iQ ++m` i2Hv `2bQHp2 i?2 ~Qr }2H/ `QmM/ i?2 }#2`bX >Qr2p2`- 7Q` bH2M/2` #Q/B2b bH2M/2` #Q/v TT`QtBK iBQM + M #2 mb2/ BMbi2 /X h?2 bH2M/2` #Q/v TT`QtBK iBQM Bb /2`Bp2/ 7`QK i?2 #QmM/ `v BMi2;` H 7Q`Km@ H iBQM Q7 i?2 aiQF2b 1[MbX UkXRyVX lbBM; bvKTiQiB+ M HvbBb i?2 ;Qp2`MBM; bm`7 +2 BMi2;` Hb `2 `2/m+2/ iQ R. BMi2;` H 2[m iBQMb HQM; +2Mi2`HBM2 Q7 i?2 }#2` #v K i+?BM; i?2 ~mB/ p2HQ+Biv i pB`im H #QmM/ `v Q7 bH2M/2` 2HHBTbQB/ iQ i?2 }#2` +2Mi2`HBM2 p2HQ+Biv- b22 :ƺix- (e)X h?2 ++m` +v Q7 i?Bb TT`QtBK iBQM Bb BM Q`/2` Ԅ ᅪ h?2 `2/m+iBQM BM /BK2MbBQM HBiv 7`QK k. #QmM/ `v BMi2;` H 2[m iBQMb iQ R. BMi2;` H 2[m iBQMb Bb +`m+B H 7Q` i?2 #BHBiv iQ BM+Hm/2 H `;2 MmK#2` Q7 }#2`b BM i?2 bBKmH iBQMbX h?2 bH2M/2` #Q/v TT`QtBK iBQM vB2H/b +QmTH2/ bvbi2K Q7 R. BMi2;` H 2[m iBQMb Qp2` irQ 7mM/ K2Mi H bQHmiBQMb Q7 i?2 aiQF2b 2[m iBQMb- i?2 aiQF2bH2i- 1[MX UkXdV M/ i?2 .Qm#H2i- 1[MX UkX3VX h?2 +QmTH2/ bvbi2K `2H i2b i?2 7Q`+2b 2t2`i2/ QM i?2 }#2`b iQ i?2B` p2HQ+BiB2b M/ + Tim`2b i?2 MQM@HQ+ H BMi2` +iBQM Q7 i?2 }#2` rBi? Bib2H7 U b K2/B i2/ #v i?2 ~mB/V- b r2HH b rBi? Mv Qi?2` bi`m+im`2b rBi? i?2 ~mB/- bm+? b Qi?2` }#2`b Q` 2ti2`M H #QmM/ `B2bX h?2 bvbi2K Q7 BMi2;` H 2[m iBQMb Bb bQHp2/ mbBM; #QmM/ `v BMi2;` H K2i?Q/X .2i BHb Q7 i?2 KQ/2H `2 ;Bp2M BM hQ`M#2`; M/ :mbi pbbQM- (jR)X N.
(36) *>Sh1_ kX h>1P_1hA*G 6PlL.hAPL. fiber. fiber. 6B;m`2 kXj, aH2M/2` }#2` TT`QtBK iBQMX 1 +? }#2` Bb /2b+`B#2/ #v i?2 +2Mi2` TQBMib אՒ אՑ i?2 Q`B2Mi iBQMb ;Bp2M #v i?2 mMBi i M;2Mi p2+iQ`b ׆Ւ ׆Ց M/ i?2 +2Mi2` HBM2 T ` K2i2`Bx2/ #v Ԣ M/ Ԣ X h?2 TQBMib QM 2 +? +2Mi2` HBM2 Bb i?mb ;Bp2M #v אՒ Ԣ אՒ
(37) Ԣ׆Ւ X bbmK2 i? i r2 ? p2 Ԃ }#2`b BKK2`b2/ BM i?2 ~mB/X b b?QrM BM 6B;X kXj 2 +? }#2` Bb MQr /2}M2/ #v Bib +2Mi2`HBM2 M/ T ` K2i2`Bx2/ #v i?2 `+@H2M;i?- Ԣ ୩ <ਲԁ ԁ>X 6Q` }#2` Ԝ i?2 +QQ`/BM i2b Q7 i?2 +2Mi2`HBM2 `2 ;Bp2M #v אՒ Ԣ ԣ אՒ ԣ
(38) Ԣ׆Ւ ԣ -. r?2`2 אՒ Bb i?2 +2Mi2` TQBMi M/ ׆Ւ i?2 mMBi i M;2Mi p2+iQ` Q7 i?2 }#2` M/ Ԝ w Ԃ X. A7 i?2 ~mB/ 2t2`ib 7Q`+2 T2` mMBi H2M;i?- ֦Ւ QM }#2` Ԝ- i?2 bH2M/2` #Q/v TT`Qt@. BK iBQM 7Q` i?2 p2HQ+Biv Q7 i?2 +2Mi2`HBM2 Q7 }#2` Ԝ + M #2 bi i2/ BM MQM@/BK2MbBQM H 7Q`K b. ળ ۾ीԓ ֭
(39) ׆Ւ ׆ળ ԓ ۾אՒ
(40) Ԣ׆Ւ Ւ
(41) ֭ ਲ ׆Ւ ׆Ւ ु ֦Ւ Ԣ. ۽
(42) ֭
(43) ׆Ւ ׆ળ Ւ ֯<֦Ւ > Ԣ
(44) Ւ Ԣ X. UkXRjV. >2`2 ԓ Bb ;2QK2i`v T ` K2i2`ԓ ਲ MPH ᅪɞ Ԕ -. UkXR9V. ۽Ւ > Ԣ Bb M BMi2;` H QT2` iQ` ;Bp2M #v M/ ֯<֦ ȯ ۽Ւ > Ԣ ௲ ֦ Ԣ ਲ ֦ Ԣ ԓԢ X ֯<֦ ]Ԣ ਲ Ԣ] ਲȯ. UkXR8V. h?2 +QMi`B#miBQM iQ i?2 p2HQ+Biv Q7 }#2` Ԝ 7`QK i?2 Qi?2` }#2`b BM i?2 bvbi2K Bb ++QmMi2/ 7Q` BM Ւ Ԣ b Ը. ȯ. Ւ Ԣ ௲ ֩ ֿՑՒ Ԣ Ԣ. ֦Ց Ԣ ԓԢ -. UkXReV. Ց ȯ ਲȯ ՑଃՒ. r?2`2 ֿՑՒ Ԣ Ԣ אՒ
(45) Ԣ׆Ւ ਲ אՑ
(46) Ԣ ׆Ց Bb i?2 /Bbi M+2 #2ir22M QM2 TQBMi QM }#2` Ԝ M/ QM2 TQBMi QM }#2` ԛ b BHHmbi` i2/ BM 6B;X kX9X Ry.
(47) kXjX aG1L.1_ 6A"1_a. 6B;m`2 kX9, hrQ }#2`b rBi? }p2 [m /` im`2 TQBMib 2 +?X h?2 BMi2` +iBQM #2ir22M irQ }#2`b Bb ;Bp2M #v i?2 i2`K Ւ M/ /2T2M/b bQH2Hv QM i?2 /Bbi M+2 #2ir22M i?2 irQX h?2 BMi2;` H BM 1[MX UkXReV + M MQi #2 2p Hm i2/ M HviB+ HHv M/ rBHH #2 TT`QtBK i2/ #v [m /` im`2 `mH2X h?2 :`22MǶb 7mM+iBQM BM i?Bb + b2 Bb HBM2 ` +QK#BM iBQM Q7 irQ 7mM/ K2Mi H bQHmiBQMb M/ `2 /b ֩ ֿ ৎ. ׁ ֿ
(48) ᅪɞ ֞ ֿ B7 ԛ ଃ Ԝ B7 ԛ Ԝ. . UkXRdV. rBi? ׁ M/ ֞ b /2}M2/ BM 1[MbX UkXdV M/ UkX3VX AM i?2 MQM@/BK2MbBQM HBx iBQM i?2 ? H7@H2M;i? Q7 i?2 }#2` ? b #22M mb2/ b +? ` +i2`BbiB+ H2M;i?- ԁՈ ԁX h?Bb rBHH ;Bp2 mb +? ` +i2`BbiB+ p2HQ+Biv M/ iBK2 b ԊԮ . ԓᅶԖԋ ᅵᅲՋ ԁ. ԉԮ . ᅵᅲՋ ԁɞ X ԓᅶԖԋ. UkXR3V. h?2 mMFMQrMb BM 1[MbX UkXRjV `2 i?2 i` MbH iBQM H M/ `Qi iBQM H p2HQ+BiB2b- ۾אՒ ۾M/ i?2 7Q`+2 /Bbi`B#miBQM HQM; i?2 }#2` ֦Ւ Ԣ X hQ +HQb2 i?2 7Q`KmH iBQM M/ ׆Ւ. r2 mb2 i?2 //BiBQM H +QM/BiBQMb bi iBM; i? i i?2 BMi2;` i2/ 7Q`+2 M/ iQ`[m2 QM 2 +? }#2` Kmbi # H M+2 i?2 2ti2`M H 7Q`+2b M/ iQ`[m2b TTHB2/ iQ i?2 }#2`bȯ. ֥Ւ ௲ ֦Ւ Ԣ ԓԢ ֥Ռ ਲȯ. ȯ. ׅՒ ௲ Ԣ ׆Ւ ֦Ւ Ԣ. ԓԢ X. UkXRNV. ਲȯ. 6Q` Qm` bBKmH iBQM i?2 2ti2`M H 7Q`+2 Bb Dmbi ;` pBiv M/ i?2 2ti2`M H iQ`[m2 Bb b2i iQ x2`QX. RR.
(49)
(50) Chapter 3. Numerical algorithm and serial implementation AM i?2 T`2pBQmb +? Ti2`- r2 T`2b2Mi2/ i?2 i?2Q`2iB+ H 7QmM/ iBQM Q7 i?2 T?vbB+b M/ K i?2K iB+b BMpQHp2/ BM bBKmH iBM; `B;B/ }#2`bX " b2/ QM i?2 aiQF2b 2[m iBQMr2 BMi`Q/m+2/ i?2 7` K2rQ`F Q7 #QmM/ `v BMi2;` H 7Q`KmH iBQMb M/ i?2 bH2M/2` #Q/v TT`QtBK iBQM iQ 2{+B2MiHv KQ/2H i?2 #2? pBQ` Q7 `B;B/ }#2`bX lbBM; i?Bb # +F;`QmM/ r2 rBHH MQr `2pB2r i?2 MmK2`B+ H TT`Q +? mb2/ 7Q` i?2 bBKmH iBQMX q2 rBHH b2T ` i2 i?2 Qp2` HH H;Q`Bi?K BMiQ 7Qm` bi2Tb M/ /Bb+mbb 2 +? bi2T BM/BpB/m HHvX //BiBQM HHv- r2 rBHH iQm+? mTQM BKTH2K2Mi iBQM /2i BHb mb2/ BM i?2 Q`B;BM H b2`B H p2`bBQMX q2 +HQb2 i?2 +? Ti2` rBi? #`B27 `2~2+iBQM Q7 i?2 T2`7Q`K M+2 +? ` +i2`BbiB+b Q7 i?2 b2`B H BKTH2K2Mi iBQM iQ ;mB/2 i?2 T ` HH2H BKTH2K2Mi iBQM QM i?2 :SlX. 3.1. Discretization. AM a2+X kXj r2 /2`Bp2/ +HQb2/ bvbi2K Q7 2[m iBQMb /2b+`B#BM; i?2 KQiBQM Q7 bH2M/2` }#2`b BM ~mB/- b22 1[Mb UkXRjV M/ UkXRNVX AM Q`/2` iQ bQHp2 i?2b2 2[m iBQMb r2 ? p2 iQ /Bb+`2iBx2 i?2KX 6Q` i?Bb r2 bi `i #v 2tT M/BM; i?2 7Q`+2 b bmK Q7 ԃ
(51) G2;2M/`2 TQHvMQKB Hb- ԅՓ Ԣ - b Թ Փ ֦Ւ ֥Ռ
(52) ֙Ւ ԅՓ Ԣ Փ ȯ. UjXRV. Փ r?2`2 i?2 +Q2{+B2Mib ֙Ւ `2 mMFMQrM p2+iQ`b rBi? i?`22 +QKTQM2Mib- 7Q` 2 +?. /B`2+iBQM BM bT +2X h?2 MmK#2` Q7 G2;2M/`2 TQHvMQKB Hb- ԃ - mb2/ BM i?2 7Q`+2 2tT MbBQM Bb MmK2`B+ H T ` K2i2`X Rj.
(53) *>Sh1_ jX LlJ1_A*G G:P_Ah>J L. a1_AG AJSG1J1LhhAPL. "v H;2#` B+ K MBTmH iBQM Q7 i?2 1[MbX UkXRjV M/ UkXRNV M/ i?2 mb2 Q7 i?2 Q`i?Q;QM HBiv T`QT2`iB2b Q7 i?2 G2;2M/`2 TQHvMQKB Hb- HBM2 ` bvbi2K Q7 2[m iBQMb Փ 7Q` ֙Ւ 7Q` Ԝ w Ԃ M/ ԝ w ԃ r?B+? QMHv BM+Hm/2b +QKTmi iBQM H. [m MiBiB2b- + M #2 Q#i BM2/X 6m`i?2`KQ`2- irQ b2T ` i2 2[m iBQMb 7Q` i?2 i` MbH iBQM H M/ `Qi iBQM H p2HQ+BiB2b Q7 i?2 }#2`b `2 Q#i BM2/X h?2v `2 ;Bp2M #v ۾אՒ ۾ ׆Ւ. ȯ <ԓ ֭
(54) ׆Ւ ׆Ւ
(55) ֭ ਲ ׆Ւ ׆Ւ > ֥Ւ
(56) ௲ Ւ Ԣ ԓԢԓ ԓ ਲȯ ȯ ֭ ਲ ׆Ւ ׆Ւ ௲ ԢՒ Ԣ ԓԢX ԓ ਲȯ. UjXkV UjXjV. Փ PM+2 i?2 HBM2 ` bvbi2K Q7 2[m iBQMb ? b #22M bQHp2/ 7Q` ֙Ւ - i?2 7Q`+2b QM. i?2 }#2`b + M #2 +QKTmi2/ mbBM; 1[MX UjXRVX lbBM; i?2 7Q`+2b- i?2 TQbBiBQM M/ Q`B2Mi iBQM Q7 i?2 }#2`b + M #2 mT/ i2/ #v BMi2;` iBM; 2[m iBQMb UjXkV M/ UjXjV BM iBK2X 6Q` KQ`2 /2i BHb M/ BM@/2Ti? /Bb+mbbBQMb TH2 b2 `272` iQ i?2 Q`B;BM H T T2` #v hQ`M#2`; M/ :mbi pbbQM- (jR)X "2HQr r2 rBHH /2b+`B#2 i?2 MmK2`B+ H H;Q`Bi?K /2p2HQT2/ iQ bQHp2 i?2 T`Q#H2KX. 3.2. Assemble System. h?2 }`bi bi2T Q7 i?2 H;Q`Bi?K Bb iQ +QKTmi2 M/ bb2K#H2 i?2 HBM2 ` bvbi2K Q7 2[m iBQMbX AM i?2 bb2K#H2 avbi2K bi2T HH BMi2` +iBQMb #2ir22M i?2 }#2`b `2 +QKTmi2/X h?Bb Bb p2`v iBK2 +QMbmKBM; T `i Q7 i?2 H;Q`Bi?K bBM+2 i?2 +QKTmi iBQM H +Qbi Bb Q7 Ԅ Ԃ ɞ X q`BiBM; i?2 bvbi2K BM bi M/ `/ 7Q`K ֘֙; ֛ ۽Bp2b i?2 7QHHQrBM; bi`m+im`2 Q7 i?2 /2Mb2 Ԃ ԃ Ԃ ԃ @K i`Bt ֘ M/ i?2 `B;?i@? M/ bB/2 ֛۽ ۽য় Ӷȯɞ ӶȯԸ ড় ֭ ۽ ۽ Ӷɞȯ ֭ ӶɞԸ ֘ ੂ ੂ ੂ ۽ ۽ Ӷ Ӷ ֭ Ըɞ Ըȯ ঢ়. ড় ԑ۽ȯ য় ԑ۽ɞ ֛ X ੂ ۽ ԑԸ ঢ়. UjX9V. ۽/2b+`B#2b i?2 ԃ ԃ bm#K i`Bt 2M+ TbmH iBM; i?2 +QMi`B#miBQM AM i?Bb MQi iBQM ӶՒՑ 7`QK i?2 7Q`+2 +Q2{+B2Mib QM }#2` ԛ QMiQ i?2 7Q`+2 +Q2{+B2Mib 7Q` }#2` ԜX R9.
(57) jXjX aPGo1 auah1J. AMM2` BMi2;` H. ۽- K i`Bt- ဇՐՓ - r?2`2 6Q` 2 +? ӶՒՑ ՑՒ ȯ. ȯ. ਲȯ. ਲȯ. ဇՐՓ ՑՒ ௲ ম௲ ֩ ֿ Ԣ Ԣ. ԅՐ Ԣ ԓԢ য ԅՓ Ԣ ԓԢ-. UjX8V. ? b iQ #2 2p Hm i2/ 7Q` 2 +? 7Q`+2 BM/2t Ԛ ԝ w ԃ X ֩ M/ ֿ `2 /2}M2/ b BM 1[MbX UkXRdV M/ UkXReV- `2bT2+iBp2HvX bBKBH ` i2`K ? b iQ #2 2p Hm i2/ 7Q` i?2 `B;?i@? M/ bB/2 ֛X PM2 TT`Q +? 7Q` 2p Hm iBM; i?2 BMi2;` Hb Bb iQ mb2 bi M/ `/ : mbbB M [m /` im`2 7Q` #Qi? i?2 BMM2` M/ Qmi2` BMi2;` HX MQi?2` QTiBQM Bb iQ mb2 M M HviB+ H bQHmiBQM 7Q` i?2 BMM2` BMi2;` H M/ QMHv bQHp2 i?2 Qmi2` BMi2;` H MmK2`B+ HHvX q2 rBHH MQi /Bb+mbb i?2 /2i BH2/ /2`Bp iBQM Q7 i?2 M HviB+ H bQHmiBQM- 7Q` M BM@/2Ti? /Bb+mbbBQM TH2 b2 b22 hQ`M#2`; M/ :mbi pb@ bQM- (jR)X AM i?2Q`v i?Bb TT`Q +? HHQrb 7Q` T2`72+i ++m` +v 7Q` i?2 BMM2` BMi2;` H?Qr2p2` BM T` +iB+2 i?Bb Bb HBKBi2/ #v i?2 MmK2`B+ H T`2+BbBQM Q7 i?2 bBKmH iBQMX h?2 Q#i BM2/ 7Q`KmH b `2 `2+m`bBp2 M/ b2MbBiBp2 iQ `QmM/ Qz 2``Q`bX hQ KBMBKBx2 i?2 ++mKmH iBQM Q7 `QmM/ Qz 2``Q`b- i?2 Q`B;BM H b2`B H BKTH2K2Mi iBQM mb2b i`B+F M/ brBi+?2b i?2 /B`2+iBQM Q7 i?2 `2+m`bBQM- /2T2M/BM; QM ?Qr 7 ` T `i i?2 }#2`b `2X h?Bb BKT`Qp2b i?2 T` +iB+ H ++m` +v M/ /Q2b MQi ? p2 M2; iBp2 2z2+i QM i?2 T2`7Q`K M+2X *?QQbBM; #2ir22M #Qi? QTiBQMb `2[mB`2b + `27mH 2t KBM iBQM Q7 i?2 ++m` +v M/ T2`7Q`K M+2 i` /2@QzX 6Q` i?2 Q`B;BM H b2`B H BKTH2K2Mi iBQM i?2 +QK#BM2/ MmK2`B+ H M/ M HviB+ H TT`Q +? 7Q` 2p Hm iBQM T`Qp2/ iQ #2 i?2 7 bi2bi M/ r b i?mb +?Qb2M b i?2 /27 mHiX q2 rBHH H i2` 2tTHQ`2 ?Qr Bb TTHB2b iQ i?2 M2r T ` HH2H :Sl BKTH2K2Mi iBQMX q?2M2p2` r2 2p Hm i2 i?2 BMi2;` Hb mbBM; MmK2`B+ H [m /` im`2- r2 mb2 i?2 b K2 TT`Q +? b i?2 Q`B;BM H T T2`X q2 /BpB/2 2 +? }#2` BMiQ 3 bm#BMi2`p Hb M/ mb2 i?`22@TQBMi ; mbbB M [m /` im`2 QM 2 +? BMi2`p HX h?Bb `2bmHib BM iQi H Q7 [m /` im`2 TQBMib T2` }#2`- r?B+? `2T`2b2Mib ;QQ/ i` /2@Qz #2ir22M ++m` +v M/ T2`7Q`K M+2X. 3.3. Solve system. 7i2` ? pBM; bb2K#H2/ i?2 HBM2 ` bvbi2K ֘֙ ֛ ۽i?2 M2ti bi2T Bb iQ bQHp2 BiX h?Bb + M #2 /QM2 mbBM; bi M/ `/ HBM2 ` 2[m iBQM bQHp2`bX h?2 HBM2 ` bvbi2K + M 2Bi?2` #2 bQHp2/ mbBM; /B`2+i bQHp2` Q` M Bi2` iBp2 K2i?Q/ HBF2 :J_1aX b HQM; b i?2 }#2`b `2 MQi iQQ +HQb2 iQ 2 +? Qi?2` i?2 K i`Bt Bb r2HH@+QM/BiBQM2/ M/ :J_1a Bb #H2 iQ bQHp2 i?2 bvbi2K BM H2bb i? M Bi2` iBQMbX R8.
(58) *>Sh1_ jX LlJ1_A*G G:P_Ah>J L. a1_AG AJSG1J1LhhAPL. h?Bb Bb i?2 `2 bQM r?v i?2 Q`B;BM H b2`B H BKTH2K2Mi iBQM mb2b :J_1a #v /27 mHiX >Qr i?2 /Bz2`2Mi bQHp2`b T2`7Q`K QM i?2 :Sl rBHH #2 +QKT `2/ BM a2+X eXjX. 3.4. Update velocities. h?2 7Q`+2 +Q2{+B2Mib Q#i BM2/ #v bQHpBM; i?2 HBM2 ` bvbi2K + M MQr #2 mb2/ iQ ۾- 1[MbX UjXkV M/ UjXjVX + H+mH i2 i?2 `B;?i ? M/ bB/2 BM i?2 2[m iBQMb 7Q` ۾אՒ M/ ״׆ >2`2- i?2 `2[mB`2/ BKTH2K2Mi iBQM Bb bBKBH ` iQ i?2 BKTH2K2Mi iBQM Q7 i?2 bb2K#H2. avbi2K bi2TX. 3.5. Update fibers. h?2 }M H bi2T i F2b + `2 Q7 /p M+BM; i?2 }#2`b 7Q`r `/ BM iBK2 #v bQHpBM; 1[MbX UjXkV M/ UjXjVX h?2b2 2[m iBQMb /Q MQi BKTQb2 Mv bi`B+i bi #BHBiv `2bi`B+iBQMb- bQ M 2tTHB+Bi iBK2@bi2TTBM; b+?2K2 + M #2 mb2/X q2 mb2 i?2 b K2 b2+QM/ Q`/2` KmHiB@bi2T K2i?Q/ b mb2/ BM i?2 Q`B;BM H T T2`X h?2 mT/ i2 7Q` i?2 TQbBiBQM Q7 i?2 +2Mi2` +QQ`/BM i2 אՒ Bb ;Bp2M #v i?2 7QHHQrBM; /Bb+`2iBx iBQM BM iBK2 Վ
(59) ȯ Վ Վਲȯ אՒ ਲ אՒ
(60) אՒ Վ Վਲȯ ۾אՒ ਲ ۾אՒ. ԣ. UjXeV. r?2`2 i?2 iBK2 bi2T Bb /2MQi2/ #v ԣ M/ bmT2`b+`BTib /2MQi2 i?2 MmK2`B+ H TT`Qt@ BK iBQM Q7 אՒ ԣՎ X AM Q`/2` iQ +QKTmi2 i?2 M2ti bi i2 i?Bb K2i?Q/ `2[mB`2b #Qi? i?2 T`2pBQmb M/ i?2 +m``2Mi bi i2X b i?2`2 Bb MQ T`2pBQmb iBK2 bi2T 7Q` ԣɱ - i?2 ȯ }`bi bi2T אՒ Bb +QKTmi2/ #v }`bi Q`/2` 7Q`r `/ 1mH2` K2i?Q/X aQHpBM; 7Q` i?2. Q`B2Mi iBQM p2+iQ`- ׆Ւ - Bb /QM2 mbBM; i?2 b K2 /Bb+`2iBx iBQMX //BiBQM HHv- r2 Kmbi `2MQ`K HBx2 i?2 Q`B2Mi iBQM p2+iQ` bQ i? i Bi K BMi BMb Bib mMBi H2M;i?X. i i?2 2M/ Q7 i?Bb bi2T i?2 bi i2 Q7 i?2 }#2`b + M QTiBQM HHv #2 r`Bii2M iQ M 2ti2`M H }H2 7Q` TQbi T`Q+2bbBM; M/ pBbm HBx iBQM BM Qi?2` iQQHbX 7i2` +QKTH2iBM; i?2 lT/ i2 6B#2`b bi2T- i?2 H;Q`Bi?K bi `ib ; BM 7`QK i?2 iQT rBi? i?2 bb2K#H2 avbi2K bi2TX h?Bb +v+H2 `2T2 ib mMiBH bT2+B}2/ MmK#2` Q7 iBK2 bi2Tb ? p2 #22M 2t2+mi2/X. 3.6. Algorithm summary. h?2 Q`B;BM H T T2` BKTH2K2Mi2/ i?Bb H;Q`Bi?K mbBM; 6Q`i` M NyX HH +QKTmi iBQMb r2`2 T2`7Q`K2/ BM /Qm#H2 T`2+BbBQM M/ 2t2+mi2/ mbBM; bBM;H2 i?`2 / QM i?2 *SlX AM bmKK `v i?2 7Qm` bi2Tb Q7 i?2 MmK2`B+ H H;Q`Bi?Kb `2, Re.
(61) jXdX _1J_Ea *PL*1_LAL: :Sl AJSG1J1LhhAPL. Փ i 2 +? iBK2 bi2T ԣՓ - ;Bp2M i?2 }#2` TQbBiBQMb- אՒ - M/ Q`B2Mi iBQM ׆Փ Ւ. RX bb2K#H2 bvbi2K h?Bb bi2T bb2K#H2b i?2 K i`Bt- ֘- M/ i?2 `B;?i ? M/ bB/2- ֛ Q7 i?2 HBM2 ` bvbi2K Q7 2[m iBQMbX 6Q` i?2 bb2K#Hv r2 ? p2 iQ 2p Hm i2 ဇՐՓ ՑՒ BM 1[MX UjX8V 7Q` HH }#2`b BM i?2 bvbi2KX kX aQHp2 bvbi2K h?Bb bi2T bQHp2b i?2 HBM2 ` bvbi2K Q7 2[m iBQMb 7Q` i?2 +Q2{+B2Mib BM i?2 7Q`+2 2tT MbBQM- 1[MX UjXRVX jX lT/ i2 p2HQ+BiB2b lbBM; i?2 7Q`+2b 7`QK i?2 T`2pBQmb bi2T- i?Bb bi2T +QKTmi2b i?2 p2HQ+BiB2b Q7 i?2 }#2`b #v +QKTmiBM; i?2 `B;?i ? M/ bB/2 Q7 1[MbX UjXkV M/ UjXjVX 9X lT/ i2 }#2` TQbiBQMb M/ Q`B2Mi iBQMb AM i?Bb bi2T i?2 }#2` TQbBiBQMb M/ Q`B2Mi iBQMb `2 mT/ i2/ #v BMi2;` iBM; 1[MbX UjXkV M/ UjXjV BM iBK2X h?Bb bi2T vB2H/b i?2 }#2` +QM};m` iBQM i iBK2 Փ
(62) ȯ bi2T ԣՓ
(63) ȯ - BX2X אՒ - M/ ׆Փ
(64) ȯ Ւ X. 6Q` i?2 MmK#2` Q7 }#2`b mb2/ BM i?Bb rQ`F UnV- 2KTB`B+ H `2bmHib b?Qr i? i i?2 K DQ`Biv Q7 i?2 `2[mB`2/ +QKTmi iBQM iBK2 Bb bT2Mi QM i?2 bb2K#H2 avbi2K bi2T M/ QM i?2 lT/ i2 o2HQ+BiB2b bi2TX h?2 iBK2 `2[mB`2/ 7Q` /p M+BM; i?2 bBKmH iBQM bi i2 BM i?2 lT/ i2 6B#2`b bi2T Bb +QKTH2i2Hv M2;HB;B#H2X AM *? Ti2` e- r2 rBHH b22 i? i i?2 b K2 ?QH/b i`m2 7Q` i?2 M2r T ` HH2H BKTH2K2Mi iBQMX. 3.7. Remarks concerning GPU implementation. h?2`2 `2 bQK2 `2K `Fb iQ #2 K /2 #27Q`2 r2 im`M Qm` ii2MiBQM iQ i?2 :Sl BKTH2K2Mi iBQM Q7 i?2 H;Q`Bi?KX h?2 KQbi BKTQ`i Mi bi2T iQ QTiBKBx2 Bb i?2 bb2K#H2 avbi2K bi2T- bBM+2 Bi Bb i?2 KQbi iBK2 +QMbmKBM; bi2TX 6Q`imM i2Hv Bi Bb r2HH bmBi2/ 7Q` T ` HH2HBx iBQMX h?2 }#2`b + M #2 T `iBiBQM2/ M im` HHv +`Qbb i?2 +QKTmi2 mMBib- r?2`2 2 +? mMBi Bb `2bTQMbB#H2 7Q` bm#b2i Q7 }#2`bX b i?2 `2[mB`2/ +QKTmi iBQMb 7Q` i?2 lT/ i2 o2HQ+BiB2b bi2T `2 bBKBH ` iQ i?2 +QKTmi iBQMb K /2 BM i?2 bb2K#H2 avbi2K bi2T- Bi rBHH HbQ #2M2}i 7`QK i?2 QTiBKBx iBQMbX //BiBQM HHvr2 rBHH HQQF i ?Qr i?2 irQ /Bz2`2Mi QTiBQMb 7Q` bQHpBM; i?2 BMi2;` H BM 1[MX UjX8VRd.
(65) *>Sh1_ jX LlJ1_A*G G:P_Ah>J L. a1_AG AJSG1J1LhhAPL. 2Bi?2` +QK#BM2/ M HviB+ H M/ MmK2`B+ HHv Q` Tm`2Hv MmK2`B+ H- T2`7Q`K BM i?2 T ` HH2H 2MpB`QMK2MiX aBM+2 r2 rBHH MQi r`Bi2 Qm` QrM BKTH2K2Mi iBQM Q7 HBM2 ` bQHp2`b QM i?2 :Sl r2 ? p2 QMHv HBKBi2/ BM~m2M+2 QM i?2 T2`7Q`K M+2 Q7 i?2 aQHp2 avbi2K bi2TX b r2 BMbi2 / i`2 i Bi b #H +F #Qt r2 ? p2 iQ `2Hv QM i?2 2{+B2M+v Q7 T`2@2tBbiBM; HB#` `B2bX h?2 QMHv i?BM; r2 + M +QMi`QH Bb i?2 +?QB+2 Q7 r?B+? HB#` `v M/ bQHp2` iQ mb2X 6Q` /B`2+i bQHp2`b i?2 +QKTmi iBQM H iBK2 QMHv /2T2M/b QM i?2 MmK#2` Q7 mMFMQrMb M/ Bb TT`QtBK i2Hv +QMbi Mi i?`Qm;?Qmi bBKmH iBQMX h?2 T2`7Q`K M+2 Q7 Bi2` iBp2 bQHp2`b QM i?2 Qi?2` ? M/ /2T2M/b QM K Mv 7 +iQ`b M/ Bb i?mb mMT`2/B+i #H2X AM HBM2 rBi? i?2 Q`B;BM H T T2` r2 rBHH #Qi? i2bi /B`2+i bQHp2` M/ Bi2` iBp2 bQHp2`b iQ ;2i #2ii2` mM/2`bi M/BM; Q7 i?2B` `2bT2+iBp2 T2`7Q`K M+2 #2? pBQ`X. R3.
(66) Chapter 4. GPU programming AM i?2 T`2pBQmb +? Ti2` i?2 MmK2`B+ H H;Q`Bi?K M/ Bib b2`B H BKTH2K2Mi iBQM r b T`2b2Mi2/X Ai /Bb+mbb2/ p `BQmb BKTH2K2Mi iBQM /2i BHb r?B+? ? p2 iQ #2 +QMbB/2`2/ iQ ``Bp2 i i?2 KQbi 2{+B2Mi BKTH2K2Mi iBQMX hQ 7m`i?2` BM+`2 b2 i?2 2{+B2M+v KQ/2`M :Slb Qz2` K bbBp2Hv T ` HH2H `@ +?Bi2+im`2 iQ ++2H2` i2 K Mv /Bz2`2Mi TTHB+ iBQMbX q2 rBHH #2;BM rBi? b?Q`i BMi`Q/m+iBQM iQ ;2M2` H Tm`TQb2 +QKTmiBM; QM :SlbX 7i2`r `/b- /Bz2`2Mi T@ THB+ iBQM S`Q;` KKBM; AMi2`7 +2b USAV 7Q` T`Q;` KKBM; QM :Slb `2 /Bb+mbb2/ `2; `/BM; i?2B` /p Mi ;2b M/ /Bb /p Mi ;2bX h?2 +?Qb2M SA- *l.- rBHH #2 T`2b2Mi2/ rBi? Bib T`Q;` KKBM; KQ/2H i i?2 2M/ Q7 i?Bb +? Ti2`X. 4.1. General purpose computing on GPUs. AM i?2 #2;BMMBM; :` T?B+b S`Q+2bbBM; lMBib r2`2 ?B;?Hv bT2+B HBx2/ TB2+2b Q7 ? `/r `2 /2p2HQT2/ iQ 2t+HmbBp2Hv BKT`Qp2 i?2 T2`7Q`K M+2 Q7 `2 H@iBK2 j. ;` T?B+bX >Qr2p2`BM `2+2Mi v2 `b :Slb ? p2 bi `i2/ iQ #2 mb2/ 7Q` `mMMBM; `#Bi` `v +Q/2 BMbi2 / Q7 #2BM; HBKBi2/ iQ ;` T?B+b `2H i2/ +QKTmi iBQMbX h?Bb HHQrb 7Q` BKT`2bbBp2 T2`7Q`K M+2 BM+`2 b2b +`Qbb rB/2 ` M;2 Q7 /Bz2`2Mi ;2M2` H Tm`TQb2 TTHB+ iBQMbX ;QQ/ Qp2`pB2r Q7 i?2 2pQHmiBQM Q7 :Sl +QKTmiBM; + M #2 7QmM/ BM Pr2Mb 2i HX- (kk)X h?2 /2+B/BM; 7 +iQ` 7Q` i?2 T2`7Q`K M+2 Bb ?Qr r2HH i?2 TTHB+ iBQM + M #2 T ` HH2HBx2/ BM Q`/2` iQ i F2 /p Mi ;2 Q7 i?2 K bbBp2Hv T ` HH2H `+?Bi2+im`2 Q7 :SlbX h?Bb K bbBp2 T ` HH2HBbK ? b H2 / iQ TQi2MiB HHv H `;2 T2`7Q`K M+2 /p Mi ;2b Q7 :Slb Qp2` *Slb b BHHmbi` i2/ BM 6B;X 9XRX Ai b?Qrb i?2 v2 ` Qp2` v2 ` BM+`2 b2 BM i?2 i?2Q`2iB+ H ~Q iBM; TQBMi QT2` iBQMb T2` b2+QM/ U6GPSfbVX h?2 6GPSfb MmK#2` Bb + H+mH i2/ #v +QK#BMBM; BM7Q`K iBQM Q7 i?2 MmK#2` Q7 +QKTmi2 mMBibRN.
(67) *>Sh1_ 9X :Sl S_P:_JJAL:. 7`2[m2M+v M/ K2KQ`v # M/rB/i? 7Q` #Qi? i?2 *Sl M/ :Sl KQ/2HbX h?Bb /Q2b MQi M2+2bb `BHv i` MbH i2 iQ /B`2+i `2 H rQ`H/ T2`7Q`K M+2 BM+`2 b2b #mi i`B2b iQ pBbm HBx2 i?2 TQi2MiB H :Slb ? p2X AMi2H *Sl .Qm#H2 S`2+BbBQM LoA.A :Sl .Qm#H2 S`2+BbBQM AMi2H *Sl aBM;H2 S`2+BbBQM LoA.A :Sl aBM;H2 S`2+BbBQM. 8-yyy. :6GPSfb. 9-yyy. :26Q`+2 :hs hBi M. :26Q`+2 :hs e3y. j-yyy. k-yyy :26Q`+2 :hs 83y. R-yyy. h2bH Eky. :26Q`+2 :hs k38 :26Q`+2 :hs k3y. :26Q`+2 33yy :ha. y kyye. h2bH Ekys. h2bH *kyNy h2bH *ky8y s2QM 18keNd. s2QM s893k. kyy3. s2QM q88Ny. kyRy u2 `. s2QM s8eNy. kyRk. kyR9. 6B;m`2 9XR, h?2Q`2iB+ H :6GPSfbX u2 ` Qp2` v2 ` BM+`2 b2 BM i?2Q`2iB+ H ~Q iBM;@TQBMi QT2` iBQMb T2` b2+QM/ 7Q` *Slb M/ :Slb- (ke)X h?2 ?m;2 /Bz2`2M+2 BM T2`7Q`K M+2 #2ir22M :Sl M/ *Sl K BMHv /2`Bp2b 7`QK i?2 MmK#2` Q7 BM/2T2M/2Mi +QKTmi2 mMBibX >Qr +QKTmi2 mMBi Bb 2t +iHv /2}M2/ /Bz2`b #2ir22M `+?Bi2+im`2b M/ p2M/Q`bX PM *Sl i?2 MmK#2` Q7 +QKTmi2 mMBib mbm HHv `272`b iQ i?2 MmK#2` Q7 *Sl +Q`2bX 1 +? BM/BpB/m H *Sl +Q`2 Bb p2`v 7 bi- ?Qr2p2`- *Slb mbm HHv QMHv ? p2 7Qm`- 2B;?i Q` K v#2 bBti22M +Q`2bX AM +QMi` bi iQ i?Bb- :Slb + M ? p2 b2p2` H ?mM/`2/b Q7 BM/2T2M/2Mi +QKTmi2 mMBibX 1 +? :Sl +QKTmi2 mMBi + M T2`7Q`K + H+mH iBQMb BM T ` HH2H M/ i?mb T`QpB/2b i?2 QTTQ`imMBiv iQ vB2H/ #B; T2`7Q`K M+2 BKT`Qp2K2Mib 7Q` ?B;?@i?`Qm;?Tmi ivT2 +QKTmi iBQMbX h?Bb Bb QM2 `2 bQM r?v ;2M2` H Tm`TQb2 +QKTmiBM; QM :Slb r b BMi`Q/m+2/ iQ i?2 rQ`H/ Q7 bmT2`+QKTmi2`bX Pp2` iBK2- ;`QrBM; MmK#2` Q7 bmT2`+QKTmi2`b bi `i2/ bmTTH2K2MiBM; i?2B` +QKTmiBM; TQr2` rBi? :Slb M/ bQK2 2p2M `2Hv 2t+HmbBp2Hv QM :Slb 7Q` i?2B` +QKTmi iBQMbX ky.
(68) 9XkX *l. oaX PS1L*G. AM Q`/2` iQ i F2 /p Mi ;2 Q7 i?2b2 M2r K bbBp2Hv T ` HH2H `+?Bi2+im`2b M2r SAb ? / iQ #2 /2p2HQT2/X h?2 irQ K DQ` T`QTQb2/ SAb `2 PT2M*G M/ *l.X PT2M*G Bb M QT2M M/ +`Qbb TH i7Q`K bi M/ `/ K BMi BM2/ #v i?2 E?`QMQb :`QmT- (R9)X h?2 b K2 ;`QmT Bb HbQ `2bTQMbB#H2 7Q` Bib ;` T?B+b 7Q+mb2/ +QmMi2`T `i PT2M:GX PT2M*G Bb MQi 2t+HmbBp2 iQ :Slb- #mi BMbi2 / i`B2b iQ #2 ;2M2` H #bi` +i H v2` 7Q` /Bz2`2Mi T ` HH2H `+?Bi2+im`2bX h?Bb HHQrb PT2M*G +Q/2 iQ #2 `mM MQi QMHv QM :Slb Q7 /Bz2`2Mi p2M/Q`b #mi HbQ QM *SlbX *l. QM i?2 Qi?2` ? M/ Bb /2p2HQT2/ #v LoA.A 2t+HmbBp2Hv 7Q` i?2B` HBM2 Q7 :SlbX. 4.2. CUDA vs. OpenCL. *?QQbBM; #2ir22M PT2M*G M/ *l. Bb i?2 }`bi /2+BbBQM iQ #2 K /2 r?2M bi `iBM; iQ BKTH2K2Mi M2r T`QD2+i QM :SlbX h?2 K BM /p Mi ;2 Q7 PT2M*G Bb i?2 #BHBiv iQ `mM QM K Mv /Bz2`2Mi /2pB+2bX HH K DQ` TH v2`b BM i?2 +QKTmiBM; bT +2 T`QpB/2 M BKTH2K2Mi iBQM QM iQT Q7 i?2B` TH i7Q`KbX "Qi? AMi2H M/ J. T`QpB/2 i?2 SA 7Q` i?2B` *Sl M/ #Qi? J. M/ LoA.A ? p2 /`Bp2`b p BH #H2 7Q` i?2B` :SlbX >Qr2p2`- i?Bb /p Mi ;2 + M HbQ #2 /Bb /p Mi ;2 b i?2 +?B2p #H2 T2`7Q`K M+2 KB;?i bmz2` 7`QK i?2 #bi` +iBQM +`Qbb HH TH i7Q`KbX h?2 PT2M*G 7` K2rQ`F Bb TQi2MiB HHv MQi QTiBKBx2/ 7Q` T `iB+mH ` /2pB+2 bT2+B}+ `+?Bi2+im`2X *l. QM i?2 +QMi` `v Bb BM i?2Q`v ?B;?Hv QTiBKBx2/ iQ +?B2p2 i?2 #2bi TQbbB#H2 T2`7Q`K M+2 QM LoA.AǶb :SlbX AM T` +iB+2 i?2 /Bz2`2M+2 + M TQbbB#Hv #2 KBiB; i2/ #v bT2M/BM; i?2 2ti` iBK2 iQ }M2@imM2 i?2 PT2M*G BKTH2K2Mi iBQM iQ i?2 ? `/r `2Ƕb bT2+B}+ M22/bX MQi?2` /Bb /p Mi ;2 Q7 PT2M*G Bb i?2 TQi2MiB HHv Qmi/ i2/ M/ BM+QMbBbi2Mi /`Bp2` bmTTQ`i 7Q` i?2 p `BQmb /2pB+2bX h?Bb Bb 2bT2+B HHv i`m2 7Q` LoA.A r?Q b22K iQ ? p2 biQTT2/ mT/ iBM; PT2M*G- biBHH QMHv bmTTQ`iBM; PT2M*G RXR r?B+? r b `2H2 b2/ # +F BM kyRyX h?2B` K BM 7Q+mb Bb QM Tmb?BM; *l. M/ mT/ iBM; Bi iQ bmTTQ`i HH i?2 72 im`2 BM i?2B` M2r :SlbX 6Q` i?Bb i?2bBb r2 +?Qb2 iQ ;Q rBi? LoA.AǶb *l. 7` K2rQ`F K BMHv #2+ mb2 Q7 i?2 p BH #H2 ? `/r `2 #Qi? i i?2 rQ`Fbi iBQM +QKTmi2`b M/ i i?2 HQ+ H +QKTmiBM; +Hmbi2`X //BiBQM HHv- i?Bb T`QD2+i /Q2b MQi M22/ i?2 +`Qbb@TH i7Q`K + T #BHBiv b i?2 K BM 7Q+mb Bb QM Tm`2 T2`7Q`K M+2 BM ?B;?Hv bT2+B HBx2/ b2imT M/ bBKmH iBQM b+2M `BQX h?2 TTHB+ iBQM rBHH MQi #2 rB/2Hv /Bbi`B#mi2/ M/ QMHv mb2/ 7Q` BMi2`M H Tm`TQb2bX kR.
(69) *>Sh1_ 9X :Sl S_P:_JJAL:. 4.3. CUDA programming model. h?2 ##`2pB iBQM *l. bi M/b 7Q` *QKTmi2 lMB}2/ .2pB+2 `+?Bi2+im`2 M/ r b BMi`Q/m+2/ #v LoA.A BM kyye b ;2M2` H Tm`TQb2 T ` HH2H +QKTmiBM; TH i7Q`KX Ai H2p2` ;2b i?2 ?B;?Hv T ` HH2H `+?Bi2+im`2 Q7 KQ/2`M LoA.A :Slb iQ bQHp2 K Mv /Bz2`2Mi +QKTmi iBQM H T`Q#H2Kb- r?B+? + M H2 / iQ TQi2MiB HHv H `;2 T2`7Q`K M+2 BKT`Qp2K2Mib +QKT `2/ iQ i` /BiBQM H *SlbX. 6B;m`2 9Xk, Pp2`pB2r Q7 i?2 *l. TH i7Q`K- (kR)- /2b+`B#BM; i?2 p BH #H2 HB#` `B2b- T`Q;` KKBM; 2MpB`QMK2Mib M/ TTHB+ iBQM Q7 ;2M2` H Tm`TQb2 +QKTmiBM; mbBM; *l.X h?2 *l. TH i7Q`K HHQrb /2p2HQT2`b iQ mb2 p `B2iv Q7 /Bz2`2Mi QTiBQMb iQ T`Q;` K i?2 :SlX h?2 2 bB2bi r v Bb iQ HBMF iQ Mv *l.@ ++2H2` i2/ HB#` `v M/ bBKTHv mb2 i?2 HB#` `B2b BMi2`7 +2b 7`QK Mv bQ7ir `2 2MpB`QMK2MiX 6Q` KQ`2 /p M+2/ mb2b 2ti2MbBQMb iQ p `BQmb T`Q;` KKBM; H M;m ;2b 2tBbi HBF2 *f*YY- 6Q`i` M M/ 2p2M K2KQ`v K M ;2/ H M;m ;2b HBF2 C p - Svi?QM M/ K Mv KQ`2X h?Bb HHQrb 7Q` 2 bv M/ 7 bi BMi2;` iBQM BMiQ Mv bQ7ir `2 2MpB`QMK2Mi i?2 /2p2HQT2` Bb +QK7Q`i #H2 rBi?X 6B;X 9Xk BHHmbi` i2b i?2 /Bz2`2Mi +QKTQM2Mib Q7 i?2 Qp2` HH *l. TH i7Q`KX kk.
(70) 9XjX *l. S_P:_JJAL: JP.1G. h?2 # bB+ #mBH/BM; #HQ+Fb Q7 i?2 *l. S`Q;` KKBM; JQ/2H 7`QK /2p2HQTK2Mi T2`bT2+iBp2 `2 F2`M2HbX *l. F2`M2Hb `2 i?2 2[mBp H2Mi Q7 MQ`K H * 7mM+iBQMbX >Qr2p2`- i?2 K DQ` /Bz2`2M+2 Bb i? i BMbi2 / Q7 #2BM; 2t2+mi2/ Dmbi QM+2- F2`M2Hb `2 2t2+mi2/ BM T ` HH2H #v ԃ /Bz2`2Mi i?`2 /bX h?2b2 *l. i?`2 /b `2 /Bbi`B#mi2/ M/ `mM +`Qbb i?2 p BH #H2 +QKTmi2 mMBib Q7 i?2 :SlX hQ BHHmbi` i2 ?Qr p2`v # bB+ F2`M2H BMpQ+ iBQM HQQFb- GBbiBM; 9XR b?Qrb +Q/2 b KTH2 7Q` p2`v bBKTH2 p2+iQ` //BiBQMX R k j 9 8 e d 3 N Ry RR Rk Rj R9. ff E2`M2H /27BMBiBQM nn;HQ# Hnn pQB/ o2+//U7HQ i & BMi B 4 i?`2 /A/tXtc *(B) 4 (B) Y "(B)c '. - 7HQ i. "- 7HQ i. *V. BMi K BMUV & XXX ff E2`M2H BMpQ+ iBQM rBi? L i?`2 /b o2+//IIIR-L===U-"-*Vc XXX '. GBbiBM; 9XR, Sb2m/Q+Q/2 7Q` *l. p2+iQ` //BiBQMX. *l. F2`M2Hb Ai Bb BKTQ`i Mi iQ `2K2K#2` i? i 2 +? F2`M2H BMpQ+ iBQM Bb 2t2@ +mi2/ BM/2T2M/2MiHv M/ MQ Q`/2`BM; Bb ;m ` Mi22/X Ai Bb i?2`27Q`2 2bb2MiB H iQ pQB/ Mv Q`/2`@/2T2M/2Mi QT2` iBQMb Q` b? `2/ K2KQ`v ++2bbX h?2`2 `2 ?Qr2p2`- r vb iQ HHQr 7Q` b? `2/ K2KQ`v ++2bb r?B+? rBHH #2 #`B2~v iQm+?2/ mTQM H i2` BM i?2 T` +iB+ H BKTH2K2Mi iBQM Q7 i?2 bBKmH iBQMX h?`2 / ?B2` `+?v AM Q`/2` iQ 2{+B2MiHv /Bbi`B#mi2 i?2 /Bz2`2Mi i?`2 /b +`Qbb i?2 +QKTmi2 mMBib Q7 i?2 :Sl- *l. /2}M2b i?`2 / ?B2` `+?vX b /Bb+mbb2/ T`2pBQmbHv :Sl +QMbBbib Q7 K Mv BM/2T2M/2Mi +QKTmi2 mMBibX PM LoA.A :Slb i?2b2 mMBib `2 `272``2/ iQ b ai`2 KBM; JmHiBT`Q+2bbQ`b UaJbVX .m`BM; 2t2+miBQM Q7 i?2 TTHB+ iBQM 2 +? aJ Bb i bF2/ rBi? `mMMBM; /BbiBM+i b2i Q7 i?`2 /bX AM *l. i?2b2 b2ib Q7 i?`2 /b `2 + HH2/ i?`2 / #HQ+FbX 1 +? i?`2 / #HQ+F Bb /Bbi`B#mi2/ iQ HH i?2 p BH #H2 aJb- r?B+? HHQrb 7Q` miQK iB+ b+ H #BHBiv /2T2M/BM; QM i?2 MmK#2` Q7 aJb p BH #H2 b BHHmbi` i2/ BM 6B;X 9XjX h?mb i?2 /2p2HQT2` QMHv ? b iQ /BpB/2 i?2 rQ`FHQ / BMiQ TT`QT`B i2Hv bBx2/ #HQ+Fb Q7 i?`2 /b M/ BMpQF2 i?2 F2`M2HX kj.
(71) *>Sh1_ 9X :Sl S_P:_JJAL:. Mu lt it hreaded CUDA Pr ogr am Block 0. Block 1. Block 2. Block 3. Block 4. Block 5 Block 5. Block 6 Block 6. Block 7. GPU w it h 2 SMs. GPU w it h 4 SMs. SM 0. SM 1. SM 0. SM 1. SM 2. SM 3. Block 0. Block 1. Block 0. Block 1. Block 2. Block 3. Block 2. Block 3. Block 4. Block 5. Block 6. Block 7. Block 4. Block 5. Block 6. Block 7. 6B;m`2 9Xj, miQK iB+ b+ HBM; Q7 i?`2 / #HQ+Fb +`Qbb M `#Bi` `v MmK#2` Q7 ai`2 KBM; JmHiBT`Q+2bbQ`b- (kR)X. >Qr iQ +?QQb2 i?2 QTiBK H bBx2 Q7 #HQ+F BM Q`/2` iQ K tBKBx2 i?2 T2`7Q`K M+2 Bb MQi M 2 bv [m2biBQM iQ Mbr2` M/ Bi Bb ?B;?Hv /2T2M/2Mi QM i?2 T `iB+mH ` i bF M/ BKTH2K2Mi iBQMX AM T` +iB+2 i?2 bBx2 Bb Q7i2M +?Qb2M #v `mMMBM; #2M+?K `Fb rBi? p `BQmb bBx2b iQ /2i2`KBM2 i?2 QTiBK H +QM};m` iBQMX AM Q`/2` iQ K F2 T`Q;` KKBM; 2 bB2`- *l. #HQ+Fb + M #2 //`2bb2/ mbBM; 2Bi?2` QM2@/BK2MbBQM H- irQ@/BK2MbBQM H- Q` i?`22@/BK2MbBQM H i?`2 / BM/2tX 6Q` 2t KTH2 BM i?2 + b2 Q7 + H+mH iBQM BMpQHpBM; K i`B+2b- Bi Bb KQ`2 M im` H iQ i?BMF #Qmi T ` HH2HBxBM; 2 +? 2H2K2Mi ;Bp2M #v i?2 `Qr M/ +QHmKM BM/2t BMbi2 / Q7 bBM;H2 QM2@/BK2MbBQM H BM/2tX h?Bb Bb BHHmbi` i2/ BM i?2 +Q/2 b KTH2 BM GBbiBM; 9XkX k9.
(72) 9XjX *l. S_P:_JJAL: JP.1G R k j 9 8 e d 3 N Ry RR Rk Rj R9 R8 Re Rd. ff E2`M2H /27BMBiBQM nn;HQ# Hnn pQB/ J i//U7HQ i (L)(L)- 7HQ i "(L)(L)- 7HQ i *(L)(L)V & BMi B 4 i?`2 /A/tXtc BMi D 4 i?`2 /A/tXvc *(B)(D) 4 (B)(D) Y "(B)(D)c ' BMi K BMUV & XXX ff E2`M2H BMpQ+ iBQM rBi? QM2 #HQ+F Q7 L L R i?`2 /b BMi MmK"HQ+Fb 4 Rc /BKj i?`2 /bS2`"HQ+FUL- LVc J i//IIIMmK"HQ+Fb- i?`2 /bS2`"HQ+F===U-"-*Vc XXX '. GBbiBM; 9Xk, Sb2m/Q+Q/2 7Q` *l. K i`Bt //BiBQM- BHHmbi` iBM; k. i?`2 / #HQ+FbX 6BM HHv- b i?2 `2bQm`+2b Q7 2 +? ai`2 KBM; JmHiBT`Q+2bbQ` `2 HBKBi2/- i?2`2 Bb M mTT2` #QmM/ Q7 ?Qr K Mv i?`2 /b #HQ+F + M +QMi BMX *m``2MiHv i?Bb K tBKmK MmK#2` Q7 i?`2 /b Bb X h?Bb K2 Mb i? i i?2 K tBKmK bBx2 Q7 K i`B+2b- i? i + M #2 //2/ rBi? i?2 irQ@/BK2MbBQM H i?`2 / #HQ+F +Q/2 b KTH2 BM GBbiBM; 9Xk Bb X hQ bQHp2 i?Bb T`Q#H2K *l. BMi`Q/m+2b MQi?2` H v2` #Qp2 #HQ+Fb + HH2/ ;`B/X ;`B/ Q`; MBx2b i?`2 / #HQ+Fb ; BM BMiQ 2Bi?2` QM2- irQ- Q` i?`22 /BK2MbBQMbX h?2 MmK#2` Q7 i?`2 / #HQ+Fb BM ;`B/ Bb mMHBKBi2/ M/ i?mb bQH2Hv /2T2M/2Mi QM i?2 bBx2 Q7 i?2 rQ`FHQ /X 6B;X 9X9 b?Qrb M 2t KTH2 +QM};m` iBQM Q7 k. ;`B/ rBi? k. #HQ+FbX J2KQ`v ?B2` `+?v AM //BiBQM iQ i?2 h?`2 / >B2` `+?v *l. HbQ BKTH2K2Mib j@H v2` K2KQ`v ?B2` `+?v b BHHmbi` i2/ BM 6B;X 9X8X 1 +? H2p2H Q7 i?Bb ?B2` `+?v /Bz2`b BM bBx2- H i2M+v M/ b+QT2X h?2 bBx2b `272`b iQ i?2 KQmMi Q7 #vi2b i? i + M #2 biQ`2/ BM i?2 H2p2H M/ i?2 H i2M+v `272`b iQ i?2 iBK2 Bi i F2b 7`QK `2[m2biBM; / i iQ #2BM; #H2 iQ +im HHv mb2 BiX //BiBQM HHv- 2 +? H2p2H Bb +QM}M2/ iQ bT2+B}+ b+QT2- `2bi`B+iBM; 7`QK r?2`2 i?2 / i + M #2 ++2bb2/X h?2 }`bi H2p2H +QMi BMb i?2 7 bi2bi M/ bK HH2bi K2KQ`vX h?Bb Bb T`Bp i2 K2KQ`v r?2`2 2 +? i?`2 / ? b Bib QrM `2 M/ QMHv i?Bb i?`2 / + M `2 / 7`QK M/ r`Bi2 iQ i?Bb `2 X >Qr Km+? bT +2 2 +? i?`2 / ? b Bb /2i2`KBM2/ #v ?Qr K Mv i?`2 /b `2 2t2+mi2/ QM 2 +? aJX h?2 KQ`2 K2KQ`v M BM/BpB/m H i?`2 / k8.
(73) *>Sh1_ 9X :Sl S_P:_JJAL:. Gr id Block ( 0, 0). Block ( 1, 0). Block ( 2, 0). Block ( 0, 1). Block ( 1, 1). Block ( 2, 1). Block (1, 1) Thr ead ( 0, 0) Thr ead ( 1, 0). Thr ead ( 2, 0) Thr ead ( 3, 0). Thr ead ( 0, 1) Thr ead ( 1, 1). Thr ead ( 2, 1) Thr ead ( 3, 1). Thr ead ( 0, 2). Thr ead ( 1, 2) Thr ead ( 2, 2) Thr ead ( 3, 2). 6B;m`2 9X9, *l. #HQ+F //`2bbBM; mbBM; k. ;`B/ rBi? k. i?`2 / #HQ+Fb- (kR)X. `2[mB`2b i?2 72r2` i?`2 /b + M `mM BM T ` HH2H QM aJX PTiBKBxBM; i?Bb i` /2@Qz Bb BKTQ`i Mi 7Q` +?B2pBM; ;QQ/ T2`7Q`K M+2X PM i?2 M2r2bi :Slb 2 +? aJ ? b ਗ਼ @#Bi E" iQ /BpB/2 KQM; i?2 i?`2 /bX h?2 b2+QM/ H2p2H Bb + HH2/ b? `2/ K2KQ`v- Q` bQK2iBK2b HQ+ H K2KQ`vX Hi?Qm;? Bi Bb bHQr2` i? M i?2 T`Bp i2 K2KQ`v- Bi ? b i?2 /p Mi ;2 i? i HH i?`2 /b 2t2+miBM; QM M aJ ? p2 bBKmHi M2Qmb ++2bb iQ BiX h?Bb HHQrb /Bz2`2Mi i?`2 /b iQ +QQT2` i2 M/ b? `2 rQ`FX .2T2M/BM; QM i?2 +? ` +i2`BbiB+b Q7 i?2 H;Q`Bi?K- i?Bb + M b p2 HQi Q7 +QKTmiBM; iBK2X h?2`27Q`2 Bi Bb BKTQ`i Mi iQ M Hvx2 r?2i?2` b? `2/ K2KQ`v + M +im HHv BKT`Qp2 i?2 T2`7Q`K M+2X h?2 bBx2 Q7 i?Bb K2KQ`v H2p2H QM +m``2Mi :Slb Bb `QmM/ E" iQ E"X h?2 i?B`/ H2p2H +QMi BMb i?2 H `;2bi K2KQ`v `2 r?B+? Bb + HH2/ i?2 ;HQ# H K2KQ`vX h?Bb Bb i?2 KQmMi Q7 K2KQ`v mb2/ QM i?2 T +F ;BM; Q7 i?2 :Slb iQ /p2`iBb2 i?2 T`Q/m+iX LQr / vb Bi Bb mbm HHv :" Q` :"X Ai Bb ++2bbB#H2 #v HH i?`2 /b M/ biQ`2b HH / i `2H iBM; iQ i?2 +m``2Mi TTHB+ iBQMX >Qr2p2`- ++2bbBM; Bi ke.
(74) 9XjX *l. S_P:_JJAL: JP.1G. + M #2 [mBi2 bHQr +QKT `2/ iQ i?2 Qi?2` H2p2Hb BM i?2 K2KQ`v ?B2` `+?vX //BiBQM HHvQM2 ? b iQ #2 + `27mH iQ pQB/ K2KQ`v +QM~B+ib r?2M ++2bbBM; i?2 b K2 K2KQ`v HQ+ iBQM 7Q`K /Bz2`2Mi i?`2 /b i i?2 b K2 iBK2X h?Bb Bb 2bT2+B HHv BKTQ`i Mi bBM+2 i?Bb + M 7m`i?2` `2/m+2 i?2 T2`7Q`K M+2X h FBM; 2p2`vi?BM; BMiQ ++QmMi- Bi + M #2 b B/ i? i Bi Bb M BKTQ`i Mi QTiBKBx iBQM bi2T iQ K F2 ++2bb iQ ;HQ# H K2KQ`v b 2{+B2Mi b TQbbB#H2X h?Bb + M #2 +?B2p2/ #v BMi2HHB;2MiHv T +FBM; / i - Q` + +?BM; / i BM HQr2` H2p2H Q7 i?2 ?B2` `+?vX MQi?2` QTiBQM iQ pQB/ i?2 bHQr T2`7Q`K M+2 Q7 i?2 ;HQ# H K2KQ`v Bb iQ K F2 bm`2 i? i i?2 / i - r?B+? 2 +? BM/BpB/m H i?`2 / `2[mB`2b 7`QK ;HQ# H K2KQ`v- Bb bK HHX h?2 bbQ+B i2/ H i2M+v r?2M `2 /BM; / i 7`QK ;HQ# H K2KQ`v + M #2 ?B//2M r?2M 2MQm;? + H+mH iBQMb `2 T2`7Q`K2/ QM i?Bb / i X h?2 :Sl Bb i?2M #H2 iQ [mB+FHv brBi+? iQ MQi?2` b2i Q7 i?`2 /b M/ +QMiBMm2 i?2 +QKTmi iBQM r?BH2 r BiBM; 7Q` / i 7`QK ;HQ# H K2KQ`vX. kd.
(75) *>Sh1_ 9X :Sl S_P:_JJAL:. Thr ead Per -t hread local m em or y. Thread Block Per -block shared m em or y. Gr id 0 Block ( 0, 0). Block ( 1, 0). Block ( 2, 0). Block ( 0, 1). Block ( 1, 1). Block ( 2, 1). Gr id 1. Global m em or y. Block ( 0, 0). Block ( 1, 0). Block ( 0, 1). Block ( 1, 1). Block ( 0, 2). Block ( 1, 2). 6B;m`2 9X8, *l. K2KQ`v ?B2` `+?v- (kR)X h?2 ?B2 `+?v +QMbBbib Q7 i?`22 H v2`b UT`Bp i2- b? `2/- ;HQ# HVX 1 +? H v2` /Bz2`b BM bBx2- H i2M+v M/ b+QT2X. k3.
(76) Chapter 5. Parallel implementation " b2/ QM i?2 T`2pBQmb irQ +? Ti2`b /2b+`B#BM; i?2 b2`B H 6Q`i` M BKTH2K2Mi iBQM M/ :Sl T`Q;` KKBM; r2 rBHH MQr HQQF i i?2 H;Q`Bi?K BM KQ`2 /2i BH M/ b?Qr ?Qr Bi Bb / Ti2/ iQ i F2 /p Mi ;2 Q7 KmHiB@+Q`2 `+?Bi2+im`2bX h?2 K BM 7Q+mb Q7 i?Bb i?2bBb Bb i?2 BKTH2K2Mi iBQM Q7 i?2 H;Q`Bi?K QM KQ/2`M K bbBp2Hv T ` HH2H LoA.A :Slb mbBM; *l.X AM Q`/2` iQ ; BM #2ii2` mM/2`bi M/BM; Q7 i?2 +?B2p #H2 T2`7Q`K M+2 BKT`Qp2K2Mib r2 //BiBQM HHv # +F@TQ`i2/ i?2 }MBb?2/ :Sl +Q/2 Q7 i?2 H;Q`Bi?K iQ KmHiB@+Q`2 *Slb mbBM; i?2 PT2MJS 7` K2rQ`FX q2 rBHH #2;BM #v BMi`Q/m+BM; i?2 bQ7ir `2 iQQHb M/ HB#` `B2b miBHBx2/X 7i2`r `/br2 rBHH 2tTH BM i?2 T` +iB+ H BKTH2K2Mi iBQM Q7 i?2 H;Q`Bi?K BM *l.X h?Bb Bb 7QHHQr2/ #v /Bb+mbbBQM Q7 b2p2` H TQi2MiB H QTiBKBx iBQM TT`Q +?2b iQ 7m`i?2` BKT`Qp2 i?2 T2`7Q`K M+2 Q7 i?2 BKTH2K2Mi iBQMX h?2 +? Ti2` 2M/b rBi? #`B27 2tTH M iBQM Q7 PT2MJS M/ ?Qr i?2 +Q/2 r b T ` HH2HBx2/ QM i?2 *SlX. 5.1. Development environment. h?2 `B;B/ }#2` bBKmH iBQM H;Q`Bi?K /2p2HQT2/ b T `i Q7 i?Bb i?2bBb Bb QMHv HQQb2Hv # b2/ QM i?2 Q`B;BM H b2`B H 6Q`i` M BKTH2K2Mi iBQMX h?2 `2 bQM 7Q` i?Bb Bb iQ 2Mbm`2 +H2 M bi `iBM; TQBMi M/ iQ pQB/ /B{+mHiB2b BM / TiBM; i?2 2tBbiBM; +Q/2 7Q` T ` HH2H 2t2+miBQM- b Bi r b M2p2` BMi2M/2/ iQ #2 `mM +`Qbb KmHiBTH2 +Q`2bX h?Bb HbQ T`QpB/2b M QTTQ`imMBiv iQ H2 `M 7`QK i?2 b?Q`i+QKBM;b Q7 i?2 QH/ +Q/2- M/ MQi QMHv T ` HH2HBx2 Bi #mi HbQ BKT`Qp2 i?2 2{+B2M+v BM ;2M2` HX h?2 /2p2HQTK2Mi Bb /QM2 2t+HmbBp2Hv QM GBMmt rQ`Fbi iBQM `mMMBM; l#mMim b i?Bb rBHH HbQ #2 i?2 2t +i b K2 `mMiBK2 2MpB`QMK2Mi mb2/ BM H i2` 2tT2`BK2MibX h?2 #mBH/ bvbi2K 7Q` +QKTBHBM; M/ HBMFBM; i?2 }M H TTHB+ iBQM Bb *J F2- (R8)X Ai Bb kN.
(77) *>Sh1_ 8X S_GG1G AJSG1J1LhhAPL. +?Qb2M #2+ mb2 Bi Bb rB/2Hv mb2/ QT2M@bQm`+2 M/ +`Qbb@TH i7Q`K #mBH/ bvbi2K- r?B+? HHQrb 7Q` 2 bv BMi2;` iBQM Q7 i?2 p `BQmb `2[mB`2/ HB#` `B2b BM r2HH /Q+mK2Mi2/ M/ bi` B;?i7Q`r `/ K MM2`X lM/2` i?2 ?QQ/ i?2 #mBH/ bvbi2K mb2b LoA.AǶb *l. TH i7Q`K iQQHb iQ +QKTBH2 i?2 +Q/2X 6Q` i?Bb LoA.A BM+Hm/2b i?2 *l. +QKTBH2`+ HH2/ Mp++- + T #H2 Q7 +QKTBHBM; *f*YY +Q/2 iQ;2i?2` rBi? i?2 *l. bT2+B}+ 2ti2MbBQMbX AM Q`/2` iQ 7 +BHBi i2 2 bB2` mb ;2 Q7 i?2 TTHB+ iBQM #Qi? /m`BM; /2p2HQTK2Mi M/ H i2` `2 H@rQ`H/ mb ;2 Svi?QM r` TT2` b+`BTi Bb HbQ p BH #H2X h?2 b+`BTi +QKTH2i2Hv miQK i2b i?2 #mBH/BM; T`Q+2bb M/ /vM KB+ HHv +mbiQKBx2b i?2 TTHB+ iBQM +Q/2 iQ bmTTQ`i i?`22 /Bz2`2Mi KQ/2b Q7 QT2` iBQMX h?2 }`bi Bb bBKTH2 `mM KQ/2 r?B+? i F2b i?2 bmTTHB2/ T ` K2i2`b M/ 2t2+mi2b i?2 bBKmH iBQMX h?2 b2+QM/ KQ/2 Bb p HB/ i2- r?B+? HHQrb 7Q` 7mHHv miQK i2/ r v iQ i2bi M/ p HB/ i2 /Bz2`2Mi H;Q`Bi?K p2`bBQMb ; BMbi FMQrM +Q``2+i bBKmH iBQM `mMX h?2 p HB/ i2 KQ/2 miQK iB+ HHv +QKTmi2b i?2 2``Q` M/ K F2b /2#m;;BM; Q7 +? M;2b 2 bB2`X h?2 H bi KQ/2 Bb #2M+?K `F r?B+? 2t2+mi2b T `iB+mH ` bBKmH iBQM KmHiBTH2 iBK2b M/ +QHH2+ib M/ ;;`2; i2b i?2 iBKBM;b 7Q` 2 +? bBKmH iBQM bi2T b r2HH b i?2 iQi H iBK2X. 5.2. Linear solvers. AM //BiBQM iQ i?2 *l. TH i7Q`K- i?2 TTHB+ iBQM HbQ `2[mB`2b bmTTQ`i HB#` `B2b 7Q` /Bz2`2Mi HBM2 ` bQHp2`bX .m`BM; bBKmH iBQM- HBM2 ` bvbi2K Q7 2[m iBQMb BM i?2 7Q`K Q7 1[MX UjX9V ? b iQ #2 bQHp2/ BM 2 +? iBK2bi2TX h?Bb + M #2 QM2 Q7 i?2 KQbi iBK2 +QMbmKBM; T `ib Q7 i?2 H;Q`Bi?KX >2M+2- i?2 Qp2` HH bBKmH iBQM iBK2 `2HB2b ?2 pBHv QM i?2 #BHBiv iQ bQHp2 H `;2 M/ /2Mb2 bvbi2Kb b 7 bi b TQbbB#H2X h?2`27Q`2 Bi Bb p2`v BKTQ`i Mi iQ r2B;?i HH i?2 p BH #H2 QTiBQMb + `27mHHv r?2M +?QQbBM; T `iB+mH ` HBM2 ` bQHp2` H;Q`Bi?K Q` HB#` `vX AM i?Bb rQ`F r2 ? p2 +QKT `2/ i?2 T2`7Q`K M+2 Q7 /B`2+i bQHp2` iQ +2`i BM Bi2`@ iBp2 bQHp2`b 7Q` bQHpBM; i?2 bvbi2KX 6Q` i?2 /B`2+i bQHp2`- i?2 J:J HB#` `v- (Re)Bb mb2/ M/ 7Q` i?2 Bi2` iBp2 r2 mb2 i?2 oB2MM *G HB#` `v- (kd)X "Qi? HB#` `B2b rBHH #2 BMi`Q/m+2/ #2HQrX. J:J f *m"Ga f PT2M"Ga. J:J Bb /2Mb2 HBM2 ` H;2#` HB#` `v. i? i T`QpB/2b 72 im`2b bBKBH ` iQ bi M/ `/ GS*E 7mM+iBQMb #mi 7Q` KmHiB+Q`2 `+?Bi2+im`2bX Ai HbQ ? b 72 im`2b iQ bmTTQ`i ?v#`B/ H;Q`Bi?Kb 7Q` 2t2+miBM; +Q/2 +`Qbb KmHiBTH2 :Slb Q` *Slb i i?2 b K2 iBK2X >Qr2p2`- i?2b2 72 im`2b `2 MQi jy.
(78) 8XjX E1_L1Ga. 2tTHQ`2/ BM i?Bb i?2bBbX AMbi2 / i?2 7Q+mb Bb QM ?B;? T2`7Q`K M+2 bBM;H2 :Sl BKTH2K2Mi iBQM Q7 /B`2+i HBM2 ` bvbi2K bQHp2`X J:J T`QpB/2b ++2bb iQ p `BQmb ?B;?@H2p2H H;2#` `QmiBM2b- #mi i?2 mM/2`Hv@ BM; K i? 7mM+iBQMb miBHBx2 i?2 TH i7Q`K bT2+B}+ BKTH2K2Mi iBQMb Q7 i?2 "Ga H2p2HbX 6Q` *l. i?Bb Bb BKTH2K2Mi2/ /B`2+iHv #v LoA.A BM i?2 7Q`K Q7 i?2 *m"Ga HB#` `B2b- (RN)X //BiBQM HHv- J:J i`B2b iQ 7m`i?2` BKT`Qp2 i?2B` T2`7Q`K M+2 #v +QK#BMBM; :Sl rBi? *Sl # b2/ H;Q`Bi?KbX h?mb *Sl # b2/ "Ga BKTH2@ K2Mi iBQM Bb HbQ M22/2/X 6Q` i?Bb i?2 PT2M"Ga HB#` `v- (j9)- Bb +?Qb2M r?B+? Bb i?2 KQbi mT@iQ@/ i2 M/ ?B;? T2`7Q`K M+2 HB#` `v p BH #H2 T `i 7`QK AMi2HǶb J i? E2`M2H GB#` `v- (Ry) M/ hGa- (R)X PT2M"Ga i F2b 7mHH /p Mi ;2 Q7 KmHiB+Q`2 bvbi2Kb M/ Bb HbQ mb2/ 7Q` i?2 +QKT `BbQM Q7 Qm` 6Q`i` M *Sl BKTH2K2Mi iBQM ; BMbi i?2 *l. :Sl BKTH2K2Mi iBQMX oB2MM *G. oB2MM *G Bb M QT2M@bQm`+2 HBM2 ` H;2#` HB#` `v /2p2HQT2/ i i?2. lMBp2`bBiv Q7 oB2MM X h?2 HB#` `v T`QpB/2b M #bi` +iBQM H v2` +`Qbb K Mv /Bz2`2Mi T ` HH2HBx iBQM K2i?Q/b BM Q`/2` iQ 7 +BHBi i2 +QMbBbi2Mi M/ 2 bv iQ mb2 bmTTQ`i 7Q` "Ga H2p2H R@j M/ Bi2` iBp2 bQHp2`bX h?Bb mMB[m2 72 im`2 HHQrb i?2 /2p2HQT2` iQ 2 bBHv brBi+? #2ir22M /Bz2`2Mi SAb M/ `+?Bi2+im`2b 7Q` T ` HH2HBx iBQMX *m``2MiHvi?2 HB#` `v bmTTQ`ib PT2MJS- PT2M*G M/ KQbi BKTQ`i MiHv 7Q` i?Bb i?2bBb @ *l.X oB2MM *GǶb 7Q+mb Bb QM bQHpBM; bT `b2 K i`B+2b rBi? i?2 BKTH2K2Mi2/ Bi2` iBp2 bQHp2`bX >Qr2p2`- Bi HbQ ? b # bB+ bmTTQ`i 7Q` bQHpBM; /2Mb2 K i`B+2b mbBM; p `B2iv Q7 /Bz2`2Mi Bi2` iBp2 bQHp2`bX b i?2 `B;B/ }#2` bBKmH iBQM 2t+HmbBp2Hv `2HB2b QM /2Mb2 K i`B+2b i?Bb K F2b Bi M B/2 H + M/B/ i2 7Q` #2M+?K `FBM;X 6Q` i?Bb i?2bBb #Qi? TTHB+ #H2 Bi2` iBp2 bQHp2`b T`QpB/2/ #v oB2MM*G- "B*:ah" M/ :J_1a- `2 mb2/ M/ i2bi2/X. 5.3. Kernels. h?2 Qp2` HH T ` HH2H H;Q`Bi?K Bb p2`v bBKBH ` iQ i?2 b2`B H p2`bBQM- ?Qr2p2`- 2 +? bBKmH iBQM bi2T Bb b2T ` i2/ BMiQ /Bz2`2Mi F2`M2HbX 1 +? F2`M2H Bb BMpQF2/ BM b2`B H K MM2`- i?Bb K2 Mb i? i *l. ;m ` Mi22b i? i HH / i KQ/B}2/ BM F2`M2H Bb p BH #H2 #27Q`2 i?2 M2ti F2`M2H Bb 2t2+mi2/X h?2b2 F2`M2Hb `2 i?2M /Bbi`B#mi2/ +`Qbb i?2 :SlX HH + H+mH iBQMb `2 /QM2 mbBM; bBM;H2 T`2+BbBQM ~Q iBM; TQBMi MmK#2`b- b LoA.A HBKBib ?B;? T2`7Q`K M+2 /Qm#H2 T`2+BbBQM +QKTmi iBQM iQ i?2B` b2`p2` +H bb :SlbX h?2 *l. Tb2m/Q+Q/2 7Q` i?2 H;Q`Bi?K Bb BHHmbi` i2/ BM GBbiBM; 8XRX Ai #2;BMb #v T `bBM; i?2 bBKmH iBQM T ` K2i2`b M/ i?2 BMBiB H }#2` +QM};m` iBQMX 7i2` jR.
(79) *>Sh1_ 8X S_GG1G AJSG1J1LhhAPL. i? i- i?2 `2[mB`2/ K2KQ`v Bb HHQ+ i2/ QM i?2 :SlX 6BM HHv- bBKTH2 HQQT 2t2+mi2b HH iBK2 bi2Tb M/ i?2 7Qm` K BM H;Q`Bi?K bm# bi2Tb ě RX bb2K#H2 avbi2K- kX aQHp2 avbi2K- jX lT/ i2 o2HQ+BiB2b M/ 9X lT/ i2 6B#2`b ě b /2b+`B#2/ BM a2+X jXe `2 `mM QM i?2 :SlX R k j 9 8 e d. BMi K BMUV & ff S `bBM; H;Q`Bi?K T ` K2i2`b M/ BMBiB H 7B#2` TQbBiBQMb `2 /S ` K2i2`bUVc `2 /6B#2`*QM7B;m` iBQMUVc HHQ+ i2:SlJ2KQ`vUVc XXX. 3 N Ry RR Rk Rj R9 R8 Re Rd. '. 7Q` UBMi bi2T 4 yc bi2T I K tniBK2bi2Tc bi2TYYV & bb2K#H2avbi2KIIIMmK"HQ+Fb- i?`2 /bS2`"HQ+F===UXXXVc aQHp2avbi2KIIIMmK"HQ+Fb- i?`2 /bS2`"HQ+F===UXXXVc lT/ i2o2HQ+BiB2bIIIMmK"HQ+Fb- i?`2 /bS2`"HQ+F===UXXXVc lT/ i26B#2`bIIIMmK"HQ+Fb- i?`2 /bS2`"HQ+F===UXXXVc ' XXX. GBbiBM; 8XR, Sb2m/Q+Q/2 7Q` T ` HH2H H;Q`Bi?K QM i?2 ?QbiX h?2 TTHB+ iBQM `2[mB`2b irQ ;2M2` H +QM};m` iBQM }H2b b M BMTmiX h?2 }`bi }H2 Bb `272``2/ iQ b i?2 T ` K2i2`b }H2 r?B+? +QMi BMb i?2 /Bz2`2Mi +QM};m` iBQM p `B #H2b M/ +QMbi Mib mb2/ i?`Qm;?Qmi i?2 H;Q`Bi?KX h?2b2 BM+Hm/2- 7Q` 2t KTH2 i?2 MmK#2` M/ bBx2 Q7 i?2 iBK2 bi2Tb b r2HH b i?2 MmK#2` Q7 7Q`+2 2tT MbBQM i2`Kb M/ [m /` im`2 TQBMibX //BiBQM HHv- i?Bb }H2 Bb HbQ mb2/ iQ +QM};m`2 i?2 Bi2` iBp2 bQHp2`b HBF2 bT2+B7vBM; i?2 MmK#2` Q7 `2bi `ib 7Q` :J_1a Q` i?2 bQHmiBQM iQH2` M+2 7Q` "B*:ai # M/ :J_1aX 1 +? Q7 i?2 T ` HH2HBx2/ bm# bi2Tb `2 MQr /Bb+mbb2/ BM KQ`2 /2i BHX h?2 Tm`TQb2 Q7 2 +? F2`M2H b r2HH b i?2 `2[mB`2/ BMTmi M/ QmiTmi `2 /2b+`B#2/ #2HQrX bb2K#H2 avbi2K. h?2 bb2K#H2 avbi2K F2`M2H Bb i?2 KQbi BKTQ`i Mi bi2T Q7. i?2 H;Q`Bi?K- b Bi Bb i?2 KQbi iBK2 +QMbmKBM; bi2TX h?Bb r b H`2 /v /Bb+mbb2/ BM a2+X jXe M/ Bi rBHH HbQ #2 b?QrM BM i?2 #2M+?K `F i2bib T`2b2Mi2/ BM *? Ti2` eX h?2 ;Q H Q7 i?2 bb2K#Hv bi2T Bb iQ #mBH/ i?2 K i`Bt M/ `B;?i@? M/ bB/2 7Q` i?2 HBM2 ` bvbi2K Q7 2[m iBQMb BM i?2 K2KQ`vX b M 2t KTH2 GBbiBM; 8Xk b?Qrb Tb2m/Q+Q/2 7Q` i?2 bb2K#H2 avbi2K bi2TX AM i?Bb 2t KTH2 i?2 +Q/2 Bb T ` HH2HBx2/ mbBM; QM2@/BK2MbBQM H i?`2 / #HQ+F b /2b+`B#2/ BM a2+X 9XjX h?Bb K2 Mb i? i i?2 jk.
(80) 8XjX E1_L1Ga R k j 9 8 e d. nn;HQ# Hnn pQB/ bb2K#H2avbi2KR.U BM 7HQ i TQbBiBQMbBM 7HQ i Q`B2Mi iBQMbQmi 7HQ i nK i`BtQmi 7HQ i #np2+iQ`V & +QMbi BMi B 4 #HQ+FA/tXt #HQ+F.BKXt Y i?`2 /A/tXtc. 3. B7 UB =4 LlJ"1_nP6n6A"1_aV `2im`Mc. N Ry. 7Q` UBMi D 4 yc D I LlJ"1_nP6n6A"1_a YYDV & 7Q` UBMi 7Q`+2nBM/2tnD 4 yc 7Q`+2nBM/2tnD I LlJ"1_nP6nh1_JanALn6P_*1n1sSLaAPLc YY7Q`+2nBM/2tnDV & +QKTmi2AMM2`AMi2;` HUXXXVc. RR Rk Rj R9 R8 Re Rd R3 RN ky kR kk kj k9 k8 ke kd k3 kN jy jR. '. '. '. 7Q` UBMi 7Q`+2nBM/2tnB 4 yc 7Q`+2nBM/2tnB I LlJ"1_nP6nh1_JanALn6P_*1n1sSLaAPLc YY7Q`+2nBM/2tnBV & ff PMHv R. i?`2 / #HQ+F ff 1 +? i?`2 / mT/ i2b mMB[m2 K2KQ`v HQ+ iBQMb- i?mb ff MQ M22/ 7Q` iQKB+b b2iJ i`BtUXXXV b2io2+iQ`UXXXV '. GBbiBM; 8Xk, Sb2m/Q+Q/2 7Q` i?2 bb2K#H2 bvbi2K bi2T rBi? R. i?`2 / #HQ+FX. +Q/2 Bb T ` HH2HBx2/ 7Q` 2 +? }#2` M/ i? i 2 +? i?`2 / + H+mH i2b i?2 +QMi`B#miBQMb iQ i?Bb }#2` 7`QK HH i?2 Qi?2` }#2`bX GQQFBM; i i?2 K i`Bt BM 1[MX UjX9V 2 +? i?`2 / Bb i?mb `2bTQMbB#H2 7Q` ਗ਼ ԃ `Qrb Q7 i?2 K i`Bt- r?2`2 ԃ Bb i?2 iQi H MmK#2` Q7 i2`Kb BM i?2 7Q`+2 2tT MbBQM BM 1[MX UjXRVX HQM;bB/2 i?2 QM2@/BK2MbBQM H i?`2 / BKTH2K2Mi iBQM Qi?2` QTiBQMb 2tBbiX h?2v rBHH #2 /Bb+mbb2/ BM i?2 QTiBKBx iBQM a2+X 8X9X h?2 F2`M2H `2[mB`2b irQ BMTmib- i?2 +m``2Mi TQbBiBQM Q7 2 +? }#2` M/ Bib Q`B2Mi @ iBQMX lbBM; i?2b2 +QK#BM2/ rBi? i?2 2[m iBQMb QmiHBM2/ BM *? Ti2` k M/ *? Ti2` j i?2 K i`Bt M/ `B;?i@? M/ bB/2 2H2K2Mib `2 +QKTmi2/ M/ mb2/ BM i?2 M2ti bi2T iQ bQHp2 i?2 HBM2 ` bvbi2K i?2v /2}M2X jj.
(81) *>Sh1_ 8X S_GG1G AJSG1J1LhhAPL. aQHp2 avbi2K. b i?Bb i?2bBb /Q2b MQi BK iQ BKTH2K2Mi ;2M2`B+ HBM2 ` bQHp2`b- i?Bb. bi2T Bb i`2 i2/ b #H +F #QtX AM i?2 bb2K#H2 avbi2K F2`M2H irQ `` vb +QMi BMBM; i?2 K i`Bt M/ `B;?i@? M/ bB/2 Q7 i?2 HBM2 ` bvbi2K ? p2 #22M +QKTmi2/X h?2b2 irQ `` vb `2 MQr T bb2/ iQ i?2 HB#` `v i? i BKTH2K2Mib i?2 HBM2 ` bQHp2`X AM + b2 Q7 i?2 /B`2+i bQHp2` Bi Bb i?2 J:J HB#` `v M/ 7Q` i?2 irQ i2bi2/ Bi2` iBp2 bQHp2`b- "B*:ai # M/ :J_1a- Bi Bb i?2 oB2MM *G HB#` `vX "Qi? HB#` `B2b `2 #H2 iQ /B`2+iHv mb2 i?2 H`2 /v HHQ+ i2/ K2KQ`v `2;BQMb M/ MQ //BiBQM H HHQ+ iBQMb ? p2 iQ #2 T2`7Q`K2/X AM Q`/2` iQ +QMb2`p2 K2KQ`v bT +2 i?2 `2bmHiBM; bQHmiBQM p2+iQ` Bb biQ`2/ BM i?2 b K2 K2KQ`v HQ+ iBQM b i?2 `B;?i@? M/ bB/2 M/ Bb T bb2/ QM iQ i?2 bm#b2[m2Mi bi2TbX lT/ i2 o2HQ+BiB2b. "v bQHpBM; i?2 HBM2 ` bvbi2K Q7 2[m iBQMb r2 Q#i BM i?2. +Q2{+B2Mib BM i?2 7Q`+2 2tT MbBQM b b?QrM BM a2+X jXRX h?2 lT/ i2 o2HQ+BiB2b F2`M2H ++mKmH i2b i?2 2t2`i2/ 7Q`+2b 7Q` HH }#2`b M/ mT/ i2b #Qi? i?2 i` MbH iBQM H M/ i?2 `Qi iBQM H p2HQ+BiB2b Q7 i?2 }#2`b bBKmHi M2QmbHv #v +QKTmiBM; i?2 `B;?i@? M/ bB/2 BM 1[MbX UjXkV M/ UjXjVX irQ@/BK2MbBQM H i?`2 / #HQ+F p2`bBQM Q7 i?2 F2`M2H Bb BHHmbi` i2/ BM GBbiBM; 8XjX AM irQ@/BK2MbBQM H i?`2 / #HQ+F p2`bBQM 2 +? BM/BpB/m H F2`M2H BMpQ+ iBQM Bb `2bTQMbB#H2 7Q` bBM;H2 T B` Q7 }#2` BMi2` +iBQMbX AM + b2 Q7 irQ /Bz2`2Mi i?`2 /b + H+mH i2 i?2 BMi2` +iBQMb 7Q` i?2 b K2 }#2`- Bi KB;?i ? TT2M i? i i?2v i`v iQ r`Bi2 i?2B` `2bmHib iQ i?2 b K2 K2KQ`v HQ+ iBQMX q?2M i?Bb ? TT2Mb- Bi +QmH/ H2 / iQ BM+Q``2+i `2bmHib 7Q` i?2 p2HQ+BiB2bX 6Q`imM i2Hv *l. T`QpB/2b iQKB+ 7mM+iBQMb iQ +B`+mKp2Mi i?Bb Bbbm2 r?B+? rBHH #2 /Bb+mbb2/ BM KQ`2 /2i BH BM a2+X eXkXjX lT/ i2 6B#2`b. h?2 }M H bBKmH iBQM bi2T i F2b + `2 Q7 /p M+BM; i?2 TQbBiBQM. M/ Q`B2Mi iBQM Q7 i?2 }#2`b BM iBK2X h?2 Tb2m/Q+Q/2 BM GBbiBM; 8X9 BKTH2K2Mib i?2 b2+QM/@Q`/2` KmHiB@bi2T K2i?Q/- 1[MX UjXeV- BMi`Q/m+2/ BM a2+X jX8X b Bi rBHH #2 b?QrM /m`BM; #2M+?K `FBM; BM *? Ti2` e- i?2 `2[mB`2/ iBK2 7Q` i?Bb F2`M2H Bb KBMmb+mH2 +QKT `2/ iQ i?2 Qi?2` bi2TbX h?2 F2`M2H b+ H2b HBM2 `Hv rBi? i?2 MmK#2` Q7 }#2`b M/ BM //BiBQM Bi ? b T2`72+iHv HB;M2/ K2KQ`v ++2bb `2bmHiBM; BM +HQb2 iQ QTiBK H mb ;2 Q7 i?2 :Sl ? `/r `2X. j9.
(82) 8XjX E1_L1Ga. R k j 9. nn;HQ# Hnn pQB/ lT/ i2o2HQ+BiB2bk.UXXXV & +QMbi BMi B 4 #HQ+FA/tXt #HQ+F.BKXt Y i?`2 /A/tXtc +QMbi BMi D 4 #HQ+FA/tXv #HQ+F.BKXv Y i?`2 /A/tXvc. 8. B7 UB =4 LlJ"1_nP6n6A"1_aV `2im`Mc B7 UD =4 LlJ"1_nP6n6A"1_aV `2im`Mc B7 UB44DV `2im`Mc. e d 3 N. 7Q` UBMi [m /` im`2nBM/2tnB 4 yc [m /` im`2nBM/2tnB I hPhGnLlJ"1_nP6nZl._hl_1nSPALhac YY[m /` im`2nBM/2tnBV & 7Q` UBMi [m /` im`2nBM/2tnD 4 yc [m /` im`2nBM/2tnD I hPhGnLlJ"1_nP6nZl._hl_1nSPALhac YY[m /` im`2nBM/2tnDV & 7Q`+2 4 +QKTmi26Q`+2U+Q277B+B2Mib- XXXV +QKTmi2.2Hi o2HQ+BiB2bU7Q`+2V ' '. Ry RR Rk Rj R9 R8 Re Rd R3 RN ky kR kk. ff k. i?`2 / #HQ+F ff 1 +? i?`2 / `2bTQMbB#H2 7Q` M BMi2` +iBQM T B`- i?mb ff `2bmHi Bb r`Bii2M iQ i?2 b K2 K2KQ`v HQ+ iBQM ff lbBM; iQKB+b iQ pQB/ +QM7HB+ib iQKB+//U Ui` MbH iBQM Hnp2HQ+BiB2b(B)XtV/2Hi ni` MbH iBQM Hnp2HQ+BivXtVc iQKB+//U Ui` MbH iBQM Hnp2HQ+BiB2b(B)XvV/2Hi ni` MbH iBQM Hnp2HQ+BivXvVc iQKB+//U Ui` MbH iBQM Hnp2HQ+BiB2b(B)XxV/2Hi ni` MbH iBQM Hnp2HQ+BivXxVc. kj k9 k8 ke kd k3 kN jy jR jk jj j9 j8 je jd j3 jN 9y. '. iQKB+//U U`Qi iBQM Hnp2HQ+BiB2b(B)XtV/2Hi n`Qi iBQM Hnp2HQ+BivXtVc iQKB+//U U`Qi iBQM Hnp2HQ+BiB2b(B)XvV/2Hi n`Qi iBQM Hnp2HQ+BivXvVc iQKB+//U U`Qi iBQM Hnp2HQ+BiB2b(B)XxV/2Hi n`Qi iBQM Hnp2HQ+BivXxVc. GBbiBM; 8Xj, Sb2m/Q+Q/2 7Q` i?2 mT/ iBM; p2HQ+BiB2b bBKmH iBQM bi2TX. j8.
(83) *>Sh1_ 8X S_GG1G AJSG1J1LhhAPL R k j. nn;HQ# Hnn pQB/ lT/ i26B#2`bUXXXV & BMi B 4 #HQ+FA/tXt #HQ+F.BKXt Y i?`2 /A/tXtc. 9. B7 UB =4 LlJ"1_nP6n6A"1_aV `2im`Mc. 8 e. M2tinTQbBiBQMb(B) 4 9fj +m``2MinTQbBiBQMb(B) @ Rfj T`2pBQmbnTQbBiBQMb(B) Y kfj hAJ1ah1S Uk +m``2Mini` MbH iBQM Hnp2HQ+BiB2b(B) @ T`2pBQmbni` MbH iBQM Hnp2HQ+BiB2b(B)VV. d 3 N Ry RR Rk. M2tinQ`B2Mi iBQMb(B) 4 9fj +m``2MinQ`B2Mi iBQMb(B) @ Rfj T`2pBQmbnQ`B2Mi iBQMb(B) Y kfj hAJ1ah1S Uk +m``2Min`Qi iBQM Hnp2HQ+BiB2b(B) @ +m``2Min`Qi iBQM Hnp2HQ+BiB2b(B)VV. Rj R9 R8 Re Rd R3 RN ky. '. MQ`K HBx2UM2tinQ`B2Mi iBQMbV. GBbiBM; 8X9, Sb2m/Q+Q/2 7Q` i?2 mT/ iBM; }#2`b bBKmH iBQM bi2TX. 5.4. Optimizations. .m`BM; i?2 /2p2HQTK2Mi Q7 i?2 T ` HH2H :Sl BKTH2K2Mi iBQM ;`2 i + `2 r b i F2M iQ +QMiBMmQmbHv QTiBKBx2 i?2 +Q/2 #Qi? QM M H;Q`Bi?KB+ H2p2H b r2HH b QM M BKTH2K2Mi iBQM H2p2HX LmK2`Qmb bK HH +Q/2@H2p2H QTiBKBx iBQMb ? p2 #22M T2`7Q`K2/ # b2/ QM i?2 Q`B;BM H b2`B H +Q/2 HBF2 T`2+QKTmiBM; b Km+? / i b TQbbB#H2- pQB/BM; p `B #H2 HHQ+ iBQMb Q` mMM2+2bb `v +QTv QT2` iBQMb BM T2`7Q`K M+2 +`BiB+ H b2+iBQMb Q7 i?2 +Q/2X //BiBQM HHv- KQ`2 /p M+2/ QTiBKBx iBQMb Q7 i?2 +Q/2 r2`2 K /2 HBF2 `2 `` M;BM; + H+mH iBQMb BMbB/2 HQQTb iQ pQB/ 2t2+miBM; `2/mM/ Mi + H+mH iBQMb M/ +QMbQHB/ iBM; KmHiBTH2 HQQTb BMiQ QM2X 6BM HHv- i2+?MB[m2b bm+? b HQQT mM`QHHBM; M/ 7 bi2` K i? 7mM+iBQMb- HBF2 i?2 `2+BT`Q+ H Q7 i?2 b[m `2 `QQi- r?2`2 HbQ i2bi2/ M/ BM+Hm/2/X h?`Qm;?Qmi i?2 QTiBKBx iBQM T? b2 #2M+?K `F bmBi2 r b `mM 7i2` 2 +? bi2TX h?Bb 2Mbm`2b i? i QTiBKBx iBQMb r2`2 QMHv BM+Hm/2/ B7 i?2v ? / K2 bm` #H2 BKT +i QM i?2 Qp2` HH T2`7Q`K M+2 Q7 i?2 bBKmH iBQMX JQ`2Qp2`- rBi? i?Bb TT`Q +? TQi2MiB H T2`7Q`K M+2 `2;`2bbBQMb +QmH/ #2 B/2MiB}2/ 2 `Hv M/ #2 pQB/2/X J Mv QTiBKBx iBQMb T2`7Q`K2/ /m`BM; i?Bb T`Q+2bb `2 TTHB+ #H2 iQ #Qi? i?2 *Sl M/ :Sl BKTH2K2Mi iBQMb- bBM+2 i?2v b?Qr2/ T2`7Q`K M+2 BKT`Qp2K2Mib 7Q` #Qi?X >Qr2p2`- bQK2 QTiBKBx iBQMb M/ H;Q`Bi?K p `B iBQMb `2 mMB[m2Hv bmBi2/ je.
(84) 8X9X PShAJAwhAPLa. iQ i?2 :Sl ? `/r `2X "2HQr r2 rBHH HQQF BMiQ i?`22 /Bz2`2Mi QTiBKBx iBQMb QM i?2 :Sl BM KQ`2 /2i BHX h?2 T2`7Q`K M+2 `2bmHib 7Q` 2 +? rBHH i?2M H i2` #2 /Bb+mbb2/ BM *? Ti2` eX. 5.4.1. Numeric vs. analytic integration. AM i?2 Q`B;BM H T T2`- (jR)- Bi r b Q#b2`p2/ i? i i?2 M HviB+ H BMi2;` iBQM Q7 i?2 BMM2` BMi2;` H BM 1[MX UjX8V vB2H/2/ T2`7Q`K M+2 BM+`2 b2 +QKT `2/ iQ Tm`2Hv MmK2`B+ H BMi2;` iBQMX :2M2` HHv- mbBM; M HviB+ H BMi2;` iBQM b?QmH/ MQi QMHv #2 T`272``2/ #2+ mb2 Q7 #2BM; 7 bi2` #mi KQ`2 BKTQ`i MiHv HbQ #2+ mb2 Q7 #2BM; KQ`2 ++m` i2X >Qr2p2`- 7Q` MmK2`B+ H T`2+BbBQM `2 bQMb i?2 +im H BKTH2K2Mi iBQM Q7 i?2 M HviB+ H BMi2;` iBQM + M MQi +?B2p2 i?Bb i?2Q`2iB+ H H2p2H Q7 ++m` +vX 1bT2+B HHv 7Q` +QM};m` iBQMb rBi? }#2`b i? i `2 p2`v 7 ` T `i i?2 `2+m`bBp2 BKTH2K2Mi iBQM bmz2`b 7`QK `QmM/@Qz 2``Q`b M/ MmK2`B+ H BMbi #BHBiB2bX h?2 bi2Tb i F2M iQ KBMBKBx2 i?2b2 BMbi #BHBiB2b TQi2MiB HHv z2+i i?2 T2`7Q`K M+2X " b2/ QM i?2b2 +QMbB/2` iBQMb M/ i?2 7 +i i? i i?2 +QKTmi iBQM Q7 i?2 BMi2;` Hb Bb p2`v T2`7Q`K M+2 +`BiB+ H T `i Q7 i?2 BKTH2K2Mi iBQM- 2tTHQ`BM; i?2 T2`7Q`K M+2 BKTHB+ iBQMb Q7 #Qi? TT`Q +?2b QM i?2 :Sl Bb Q7 ;`2 i BMi2`2biX AM +QMi` bi iQ i?2 b2`B H *Sl BKTH2K2Mi iBQM Q7 i?2 Q`B;BM H T T2`- Qm` BKTH2K2Mi iBQM QM i?2 :Sl Bb +im HHv 7 bi2` r?2M bQHpBM; #Qi? BMi2;` Hb MmK2`B+ HHvX q2 rBHH /Bb+mbb i?2 `2 bQM M/ +QMb2[m2M+2b Q7 i?Bb BM KQ`2 /2i BH BM a2+X eXkXRX. 5.4.2. Shared memory. b /2b+`B#2/ BM a2+X 9Xj- i?2 *l. +Q/2 Bb bm#D2+i2/ iQ ?B;?Hv bT2+B HBx2/ K2KQ`v ?B2` `+?vX q?2`2 b i` /BiBQM H *Slb QMHv ? p2 bK HH + +?2b M/ H `;2 K BM K2KQ`v TQQH- *l. BMi`Q/m+2b i?2 +QM+2Ti Q7 b? `2/ HQ+ H K2KQ`v bT +2X h?2 ++2bb iBK2 iQ i?Bb HQ+ H K2KQ`v Bb Q`/2`b Q7 K ;MBim/2b 7 bi2` +QKT `2/ iQ ++2bbBM; i?2 ;HQ# H :Sl K2KQ`vX //BiBQM HHv- HQ+ H K2KQ`v + M #2 b? `2/ KQM; i?2 i?`2 /b `mMMBM; QM ai`2 KBM; JmHiBT`Q+2bbQ` M/ TQi2MiB HHv b p2 iBK2 #v pQB/BM; iQ +QMbi MiHv ++2bb i?2 bHQr ;HQ# H K2KQ`vX AM Q`/2` iQ i2bi i?Bb- b? `2/ K2KQ`v p2`bBQM r b BKTH2K2Mi2/ M/ i2bi2/ 7Q` i?2 bb2K#H2 avbi2K bi2T- bBM+2 Bi Bb i?2 KQbi T2`7Q`K M+2 +`BiB+ H F2`M2HX hQ mM/2`bi M/ i?2 B/2 - BK ;BM2 i?2 irQ@/BK2MbBQM H i?`2 / #HQ+F BKTH2K2M@ i iBQM Q7 i?2 F2`M2HX h?2M 2 +? i?`2 / #HQ+F Bb `2bTQMbB#H2 7Q` K Mv T B`b Q7 }#2` BMi2` +iBQMb- 2X;X }#2`b < w > 2 +? BMi2` +iBM; rBi? }#2`b < w >X AM iQi Hi?2b2 `2 BMi2` +iBQMbX 1 +? F2`M2H BMpQ+ iBQM Bb `2bTQMbB#H2 7Q` QM2 T B` jd.
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Common data computed during the run-time of the kernel and data that was to large to be put in constant memory were shared within each thread block using two synchronization points
Assessment proposed by the supervisor of Master ’s thesis: Excellent Assessment proposed by the reviewer of Master ’s thesis: Excellent?. Course of
visa kunskap och förståelse inom huvudområdet Human Resource Management and Development, inbegripet såväl brett kunnande inom området som väsentligt fördjupade kunskaper inom
When calculating the number of toilets needed with the figures available in the National guideline for Water, Sanitation and Hygiene for Tanzania schools (UNICEF, 2016b), the
