あまぁいRcpp生活

Rの関数をC++で実装することを可能にするパッケージRcppの紹介です。

Transcript

1. 55 R 08/06 2 # # #
2. 3DQQͱ͸ 3
   ͷؔ਺Λ
   $
   Ͱ࣮૷͢Δ͜ͱΛ
   Մೳʹ͢Δύοέʔδ
3. 3DQQΛར༻͍ͯ͠Δ
   ύοέʔδ਺
   ύʔηϯςʔδ       
   
   
             Ҏ্ͷύοέʔδͰར༻͞Ε͍ͯΔ ύοέʔδ਺ 
4. 1 RcppArmadillo 353992 0.0235 2 Rcpp 254575 0.0169 3 ggplot2 219765 0.0146 4 digest 179117 0.0119 5 stringr 171733 0.0114 6 plyr 170953 0.0113 7 stringi 170182 0.0113 8 scales 167722 0.0111 9 magrittr 159638 0.0106 10 gtable 150171 0.0099 ࠷΋ଟ͘μ΢ϯϩʔυ͞Ε͍ͯΔ
   ύοέʔδͷͻͱͭ %-਺ ׂ߹ IUUQDSBOMPHTSTUVEJPDPN
   ͔Β
   ೥݄ʹμ΢ϯϩʔυ͞ΕͨύοέʔδͷϥϯΩϯά
6. )BEMFZWFSTF HHQMPU
   SFTIBQF
   QMZS
   EQMZS
   UJEZS
   TUSJOHS
   MVCSJEBUF
   SFBES
   SFBEYM
   IBWFO
   IUUS
   SWFTU
   YNM T
7. )BEMFZWFSTF HHQMPU
   SFTIBQF
   QMZS
   EQMZS
   UJEZS
   TUSJOHS
   MVCSJEBUF
   SFBES
   SFBEYM
   IBWFO
   IUUS
   SWFTU
   YNM 3DQQ
   ʹґଘ T
8. 3DQQ
   
   5IF
   %SJWJOH
   'PSDF
   PG
   )BEMFZWFSTF %JSL
   &EEFMCVFUUFM
   
   3PNBJO
   'SBOÇPJT )BEMFZ
   8JDLIBN
9. Ͱ΋ʜ
   $ ͬͯʜ
10. ࠇຐज़ݴޠ$ ˠ·ͣ͸ศརͳͱ͜Ζ͚ͩษڧ͢Ε͹͍͍ ˠ34UVEJP
    ͕ʢଟগʣΘ͔Γ΍͘͢දࣔͯ͘͠ΕΔ ˠ
    ·͊ɺ൱Ίͳ͍Ͱ͢Ͷʜ ݴޠ࢓༷͕ڊେʹͳͨΊशಘʹֻ͕͔࣌ؒΔ ίϯύΠϧΤϥʔ͕Θ͔Γʹ͍͘ ม਺ͷܕ͕ෳࡶ $ ͷΤϥʔͷଟ͘͸
    ʲม਺ͷܕʳͱʲ୅ೖ͢Δ஋ͷܕʳͷෆҰகʹΑΔ
11. ˞͋͘·Ͱ΋ΠϝʔδͰ͢ E C
12. E 55
13. 3DQQͷΠϯετʔϧ ࣗ࡞ͷ3DQQؔ਺Λ࡞੒͢ΔͨΊʹ͸
    $ ͷίϯύΠϥΛΠϯετʔϧ͢Δඞཁ͕͋Δ 8JOEPXT
    3
    ͷόʔδϣϯʹ߹ͬͨ
    3UPPMT
    ΛΠϯετʔϧ͢Δ .BD
    λʔϛφϧͰ࣍ͷίϚϯυΛଧͭ IUUQTDSBOSQSPKFDUPSHCJOXJOEPXT3UPPMTJOEFYIUNM iUaVW)dW WUe ))[ deS 8JOɺ.BD
    ڞʹ
    3
    Ͱ [ deS *bSU]S Wd “IUbb”
14. 3UPPMTͷΠϯετʔϧ جຊతʹ͸ରԠ͢ΔҰ൪৽͍͠
    'SP[FO
    όʔδϣϯΛ
    Πϯετʔϧ͢Ε͹ྑ͍
15. 3UPPMTͷΠϯετʔϧ Πϯετʔϧ࣌ʹ
    1"5)
    ͷઃఆʹνΣοΫ͢Δ ʢίϯύΠϥ͕Πϯετʔϧ͞Εͨ৔ॴΛ
    3
    ʹڭ͑ΔͨΊʣ ͜͜ʂ
16. 3UPPMTͷΠϯετʔϧ : jd* WeW g #H; B# P- !=6ooHca cS_ [ WdooIooI)/*/*-ooT[ ooi207 U6ooIeaa dooT[ 7 U6ooIeaa do o_[ hR20ooT[ 7v wu : djdeW_ # '' )g# Ld[ Tf[ e)[ dbWUd* =GDD = RA==9U6oIeaa do_[ hR20oT[ oARRn-* O =GDD = RD GRNI;HH I9U6+Ieaa d+_[ hR20+T[ +**+ [TWiWU+ UU+i42R20)h20) _[ h/.+0*5*/+ ea)hcSbbWc*WiW v w cWSV _aVW 6 bad[i UU gWcd[a 0*5*/ i42R20)bad[i)dW ( <f[ e Tj E[ AN)N20 bca WUe ֬ೝͷͨΊ
    3
    Ͱ࣍ͷίϚϯυΛೖྗ͠
    ԼͷΑ͏ͳग़ྗ͕ಘΒΕͨΒ0,
17. E 55 3DQQ
    ͷίʔυΛॻ͘৔߹ʹ͸
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    ͷར༻Λڧ͘ਪ঑͠·͢ ίʔυิ׬΍Τϥʔද͕ࣔ਌੾ͳͷͰ
18. E 55 ϑΝΠϧ
    
    /FX
    'JMF
    
    $
    'JMF ద౰ͳ໊લͰϑΝΠϧʹอଘͯ͠
    4PVSDF
    Λԡ͢ʂ
19. E 55 : IUbb66dafcUW=bb # a]jaI*Ubb# : e[_Wd ha 0. P- 40 : e[_Wd ha U -(.(/ P- . 0 2 ϑΝΠϧ
    
    /FX
    'JMF
    
    $
    'JMF ద౰ͳ໊લͰϑΝΠϧʹอଘͯ͠
    4PVSDF
    Λԡ͢ʂ 3DQQͰ࡞੒ͨؔ͠਺͸௨ৗͷ3ؔ਺ͱಉ༷ʹ࢖༻Ͱ͖Δ
20. E 55 // Rcpp Λར༻͢ΔͨΊʹඞཁͳϔομϑΝΠϧ #include <Rcpp.h> // ͜ΕΛهड़͠ͳ͍৔߹͸ Rcpp::NumericVector // ͱॻ͔ͳ͍ͱ͍͚ͳ͍ using namespace Rcpp; // ͜ͷهड़ͷ௚Լͷؔ਺͕R͔Βར༻ՄೳʹͳΔ // [[Rcpp::export]] // ؔ਺ͷఆٛ NumericVector timesTwo(NumericVector x) { return x * 2; } /* C++ ίʔυதʹRͷίʔυΛຒΊࠐΊΔ ίϯύΠϧ௚ޙʹࣗಈͰ࣮ߦ͞ΕΔ */ /*** R timesTwo(42) */
21. 3
    ͱ
    $
    ͷॻ͖ํͷҧ͍ ϕΫτϧͳͲͷཁૉ൪߸͸
    
    ͔Β ม਺΍ؔ਺ʢͷฦ஋ʣͷσʔλܕΛࢦఆ͢Δඞཁ͕͋Δ จͷ࠷ޙʹ
    
    ඞਢ ม਺໊΍ؔ਺໊ʹ
    
    ϐϦΦυ͸࢖͑ͳ͍ ஋ͷ୅ೖ͸
    
    ͷΈ ؔ਺ͷฦ஋ʹ
    SFUVSO
    ඞਢ
22. df_I 8) Xf Ue[a i l df_ 8) , Xac ii [ i l df_ 8) df_ ' ii m cWefc df_ m ϕϯνϚʔΫ ࣮਺ϕΫτϧͷ߹ܭ஋ΛٻΊΔ
23. ϕϯνϚʔΫ ++ PPIUbb66Wibace VafT W df_IUbb Ff_Wc[UMWUeac i l [ e 9 i* W e 7 VafT W df_ 9 ,*,7 Xac [ e [9,7 [ 8 7 ''[ l df_ 9 df_ ' iP[ 7 m cWefc df_7 m ࣮਺ϕΫτϧͷ߹ܭ஋ΛٻΊΔ
24. ϕϯνϚʔΫ ࣮਺ϕΫτϧͷ߹ܭ஋ΛٻΊΔ ++ PPIUbb66b f [ d Ubb-- ++ PPIUbb66Wibace VafT W df_IUbb=bb-- Ff_Wc[UMWUeac i l VafT W df_ 9 ,*,7 Xac VafT W ii 6 i l df_ 9 df_ ' ii7 m cWefc df_7 m
25. ϕϯνϚʔΫ ࣮਺ϕΫτϧͷ߹ܭ஋ΛٻΊΔ ++ PPIUbb66Wibace VafT W df_IUbb f_ Ff_Wc[UMWUeac i l cWefc df_ i 7 m
26. ϕϯνϚʔΫ : cTW U _Sc]66TW U _Sc] ' df_I i ( ' df_ i ( ' df_IUbb i ( ' df_IUbb=bb-- i ( ' df_IUbb f Sc i ( ' Ua f_ d9U !eWde!( !cWb [USe[a d!( !W SbdWV!( !cW Se[gW! ( ' acVWc 9 FLDD( ' cWb [USe[a d9-,, eWde cWb [USe[a d W SbdWV cW Se[gW - df_I i -,, 1,*045 .30*/53 . df_ i -,, ,*.0/ -*/.- / df_IUbb i -,, ,*-5. -*,0/ 0 df_IUbb=bb-- i -,, ,*.,. -*,54 1 df_IUbb f_ i -,, ,*-40 -*,,, 3DQQ͸$ݴޠͰ࣮૷͞Εͨ3ͷؔ਺ΑΓ଎͍৔߹͕͋Δ
27. 3DQQͷσʔλߏ଄ %BUBGSBNFɹ-JTUɹ4 7FDUPSɹ.BUSJY ୯ʹಉ͡Α͏ͳσʔλߏ଄͕ఆٛ͞Ε͍ͯΔͷͰ͸ͳ͘
    3ͷϝϞϦʹ͋ΔΦϒδΣΫτΛ௚઀ૢ࡞Ͱ͖Δ
28. 3DQQͷσʔλܕ ̋̋.BUSJY ஋ܕ 3
    ϕΫτϧ 3DQQ
    ϕΫτϧ 3DQQ
    ߦྻ 3DQQ
    εΧϥʔ $
    εΧϥʔ ࿦ཧ MPHJBM -PHJDBM7FDUPS -PHJDBM.BUSJY  CPPM ੔਺ JOUFHFS *OUFHFS7FDUPS *OUFHFS.BUSJY  JOU ࣮਺ OVNFSJD /VNFSJD7FDUPS /VNFSJD.BUSJY  EPVCMF ෳૉ਺ DPNQMFY $PNQMFY7FDUPS $PNQMFY.BUSJY 3DPNQMFY DPNQMFY จࣈྻ DIBSBDUFS $IBSBDUFS7FDUPS $IBSBDUFS.BUSJY 4USJOH TUSJOH ೔෇ %BUF %BUF7FDUPS  %BUF  ೔࣌ 104*9DU %BUFUJNF7FDUPS  %BUFUJNF UJNF@U
29. @ 3DQQͰ͸
    ͍ۤ$ Λ؁͘͢ΔͨΊʹ
    3ͱಉ༷ʹ࢖͑Δ
    ؔ਺΍ԋࢉࢠ͕ఆٛ͞Ε͍ͯΔ
30. 4VHBS
    ؔ਺ ベクター関係 head, tail, rep_each, rep_len, rep, rev, seq_along, seq_len, cumsum, diff, pmin, pmax 確率分布 d/q/p/r 全てのRの提供する確率分布 数学関数 abs, acos, asin, atan, beta, ceil, ceiling, choose,cos, cosh, exp, expm1, factorial, floor, gamma, lbeta, lchoose, lfactorial, lgamma, log, log10, log1p, pentagamma, psigamma, round, signif, sin, sinh, sqrt, tan, tanh, tetragamma, trigamma, trunc ཁ໿౷ܭྔ
    NFBO
    NJO
    NBY
    TVN
    TE
    WBS
    ஋ͷݕࡧ
    NBUDI
    TFMG@NBUDI
    XIJDI@NBY
    XIJDI@NJO
    ॏෳͷ஋
    EVQMJDBUFE
    VOJRVF
    BQQMZ
    ؔ਺
    MBQQMZ
    TBQQMZ
    NBQQMZ
    ࿦ཧ஋
    JGFMTF
    BMM
    BOZ
31. 4VHBS
    ԋࢉࢠ ϕΫτϧ
    
    ϕΫτϧ
    i'jpi)jpi jpi+j ϕΫτϧ
    
    εΧϥʔ
    .'ip.)ip. ipi+. ࢛ଇԋࢉ
32. ϕΫτϧ
    
    ϕΫτϧ
    i99jpi 9jpi:jpi:9j ϕΫτϧ
    
    εΧϥʔ
    .99ip. 9ip.:ipi:9. 4VHBS
    ԋࢉࢠ ൺֱԋࢉ
33. ϕΫτϧͷ࡞੒ ++ g 8) cWb ,( -, Ff_Wc[UMWUeac g -, 7 ++ g 8) U -(.(/ ++ PPIUbb66b f [ d Ubb-- Ff_Wc[UMWUeac g 9 l-(.(/m7 ++ g 8) U i9-( j9.( k9/ Ff_Wc[UMWUeac g 9 Ff_Wc[UMWUeac66UcWSeW FS_WV "i", - ( FS_WV "j" 9. ( RP"k" 9/ 7
34. ϕΫτϧཁૉ΁ͷΞΫηε ࣮਺ɾ੔਺ɾจࣈྻɾ࿦ཧϕΫτϧͰΞΫηε Ff_Wc[UMWUeac [ 9 l,(.(0m7 C eW WcMWUeac [ 9 l,(.(0m7 = ScSUeWcMWUeac [ 9 l!i!(!j!(!k!m7 Da [US MWUeac [ 9 g:,7 Ff_Wc[UMWUeac c 9 gP[ 7 gP[ 9 gS 7
35. ߦྻͷ࡞੒ ++ _ 8) _Sec[i ,( cah 9 .( Ua 9 / Ff_Wc[UESec[i _ .( / 7 ++ _ 8) _Sec[i g( cah 9 .( Ua 9 / Ff_Wc[UESec[i _ .( / ( g*TW [ 7
36. ߦྻͷཁૉ΁ͷΞΫηε ++ , . _ , ( . 7 ++ , _ , ( R 7 ++ . _ R ( . 7 ++ ,s- .s/ _ IS W ,(- ( IS W .(/ 7
37. σʔλϑϨʔϜͷ࡞੒ >SeS cS_W VX 9 >SeS cS_W66UcWSeW g-(g.(g/ 7 ++ { >SeS cS_W66UcWSeW FS_WV ! -!( g- ( FS_WV ! .! 9g. ( RP! /! 9g/ 7
38. σʔλϑϨʔϜͷཁૉ΁ͷΞΫηε ࣮਺ɾ੔਺ɾจࣈྻɾ࿦ཧϕΫτϧͰΞΫηε >SeS cS_W VX- 9 VXPg 7 VXPg 9 VX.7 Ff_Wc[UMWUeac g- 9 VXP, 7 VXP!i! 9 g.7 ࣮਺ɾ੔਺ɾจࣈྻεΧϥʔͰΞΫηε ΧϥϜͷ஋ʹ͸
    g-
    Λհͯ͠ΞΫηε͢Δ
    VXP(-
    ΍
    VXP- P-
    ͳͲ͸ବ໨
39. D[de D 9 D[de66UcWSeW g-(g.(g/ 7 ++ { D[de66UcWSeW FS_WV ! -!( g- ( FS_WV ! .! 9g. ( RP! /! 9g/ 7 ˞
    W
    W
    W
    ͷܕ΍αΠζ͸ԿͰ΋ྑ͍ Ϧετͷ࡞੒
40. Ϧετͷཁૉ΁ͷΞΫηε ࣮਺ɾ੔਺ɾจࣈྻɾ࿦ཧϕΫτϧͰΞΫηε D[de D- 9 DPg 7 DPg 9 D.7 Ff_Wc[UMWUeac g- 9 DP, 7 DP!i! 9 g.7 ࣮਺ɾ੔਺ɾจࣈྻεΧϥʔͰΞΫηε Ϧετཁૉͷཁૉʹ͸
    g-
    Λհͯ͠ΞΫηε͢Δ
41. MWUeac M M* W e ++ M* S_Wd ++ ESec[i E E* cah ++ E* Ua ++ cah S_Wd E ++ Ua S_Wd E ++ >SeS cS_W > > * W e ++ > * S_Wd ++ > *Seec !cah* S_Wd! ++ D[de D D* W e ++ D* S_Wd ++ ϕΫτϧͷ௕͞΍ཁૉ໊ͳͲ
42. ଐੑ஋ MWUeac M M*Seec ! S_Wd! ++ ESec[i E7 E*Seec !V[_! 9 C eW WcMWUeac66UcWSeW ( 7 E*Seec !V[_ S_Wd! 9 D[de66UcWSeW ( 7 >SeS cS_W > 7 > *Seec ! S_Wd! ++ > *Seec !cah* S_Wd! ++ D[de D7 D*Seec ! S_Wd! ++
43. 3ͷؔ਺Λݺͼग़͢ f Ue[a X !c ac_! 7 Ff_Wc[UMWUeac cWd 9 X 1( FS_WV !_WS !(- ( FS_WV !dV!(. 7 g[ca _W e W g 9 g[ca _W e66 S_WdbSUWRW g ! r ! 7 f Ue[a X 9 W gP! ! 7 ͜ͷํ๏Ͱύοέʔδؔ਺Λݺͼग़͢৔߹͸
    
    MJCSBSZ ύοέʔδ໊
    ͢Δඞཁ͕͋Δ ͪ͜Β͸
    MJCSBSZ ύοέʔδ໊
    ͢Δඞཁ͕ͳ͍
44. 55 R 3DQQ
    Λ࢖͑͹
    3
    ͷίʔυΛ
    $
    ʹஔ͖׵͑Δ͜ͱ΋Մೳ
    ࣮ͨͩ૷͕໘౗ͳͷ͸ࣄ࣮ͳͷͰ
    ॲཧͷϘτϧωοΫʹͳ͍ͬͯΔ෦෼ʹ
    ద༻͢Δͷ͕ޮՌత
45. 3DQQֶशͷͨΊʹ 3DQQ
    (BMMFSZ
    IUUQHBMMFSZSDQQPSH 3DQQ
    /PUF
    IUUQTUBUSNFSDQQOPUF *OUSPEVDUJPO
    UP
    3DQQʢ೔ຊޠʣ
    IUUQTXXXHJUCPPLDPNCPPLUFVEFSJOUSPEVDUJPOUPSDQQEFUBJMTKB 3ݴޠపఈղઆ
    ʢڞཱग़൛ʣ 4MBDL
    ͷ
    SXBLBMBOH
    ʹ
    3DQQ
    ࣭໰෦԰͋Γ·͢ IUUQTSXBLBMBOHTMBDLDPNNFTTBHFTSDQQ
46. $ ֶशͷͨΊʹ 1SPHSBNNJOH
    1MBDF
    1MVT
    IUUQQQQMBCTBLVSBOFKQ1SPHSBNNJOH1MBDF1MVTDQQJOEFYIUNM ୔ࢁ͋Γ͗͢ΔͷͰ͚̍ͭͩ
47. + 1
49. 3DQQͷ஫ҙ఺ ίϯύΠϧͨ͠
    3DQQ
    ؔ਺͸
    3%BUB
    ʹอଘͰ͖ͳ͍
    ৽͍͠3ͷηογϣϯͷ౓ʹ࠶ίϯύΠϧ͢Δ ίϯύΠϧࡁΈؔ਺Λอଘ͍ͨ͠৔߹ʹ͸
    ࣗ࡞ύοέʔδʹ͢Δඞཁ͕͋Δ
50. 3DQQͷ஫ҙ఺ ++ g- ~ Ff_Wc[UMWUeac g- 9 l-(.(/m7 ++ g- g. ~ Ff_Wc[UMWUeac g. 9 g-7 ++ g. ~ g.P, 9 57 ++ g- ~ IUafe 88 g- 88 !o !7 ++ g- z } \ x t ++ 5 . /
51. 3DQQͷ஫ҙ఺ ++ g- ~ Ff_Wc[UMWUeac g- 9 l-(.(/m7 ++ g- g. U a W ~ Ff_Wc[UMWUeac g. 9 U a W g- 7 ++ g. ~ g.P, 9 57 ++ g- ~ IUafe 88 g- 88 !o !7 ++ g- x ++ 5 . /
52. 3DQQͷ஫ҙ఺ ϝϞϦʔ ++ g- ~ Ff_Wc[UMWUeac g- 9 l-(.(/m7 g- -(.(/ ୯७ͳ୅ೖͷϝΧχζϜ
53. ++ g- g. ~ Ff_Wc[UMWUeac g. 9 g-7 3DQQͷ஫ҙ఺ ϝϞϦʔ g- -(.(/ g. ୯७ͳ୅ೖͷϝΧχζϜ
54. ++ g. ~ g.P, 9 57 3DQQͷ஫ҙ఺ ϝϞϦʔ g- 5(.(/ g. ୯७ͳ୅ೖͷϝΧχζϜ
55. 3DQQͷ஫ҙ఺ ϝϞϦʔ ++ g- ~ Ff_Wc[UMWUeac g- 9 l-(.(/m7 g- -(.(/ DMPOF Λ࢖ͬͨ୅ೖͷϝΧχζϜ
56. ++ g- g. U a W ~ Ff_Wc[UMWUeac g. 9 U a W g- 7 o 3DQQͷ஫ҙ఺ ϝϞϦʔ g- -(.(/ g. -(.(/ DMPOF Λ࢖ͬͨ୅ೖͷϝΧχζϜ
57. ++ g. ~ g.P, 9 57 3DQQͷ஫ҙ఺ ϝϞϦʔ g- -(.(/ g. 5(.(/ DMPOF Λ࢖ͬͨ୅ೖͷϝΧχζϜ
58. 3DQQͷ஫ҙ఺ ϝϞϦʔ ++ g- ~ Ff_Wc[UMWUeac g- 9 l-(.(/m7 g- -(.(/ ؔ਺ͷҾ਺ͷҰ෦ͷཁૉΛมߋ͢Δέʔε
59. ++ g- IUbb X } y ++ ga[V X Ff_Wc[UMWUeac g. l tttt m : X g- 3DQQͷ஫ҙ఺ ϝϞϦʔ g- -(.(/ g. ؔ਺ͷҾ਺ͷҰ෦ͷཁૉΛมߋ͢Δέʔε
60. ++ X g. ~ ++ ga[V X Ff_Wc[UMWUeac g. l g.P, 957 m : X g- : bc[ e g- P- 5 . / 3DQQͷ஫ҙ఺ ϝϞϦʔ g- 5(.(/ g. ؔ਺ͷҾ਺ͷҰ෦ͷཁૉΛมߋ͢Δέʔε
61. 3
    ͷ
    DPQZ
    PO
    NPEJGZ
    ϝϞϦʔ " g- ~ g- 8) U -(.(/ g- -(.(/
62. ϝϞϦʔ g- -(.(/ g. 3
    ͷ
    DPQZ
    PO
    NPEJGZ
    " g- g. ~ g. 8) g-
63. ϝϞϦʔ g- -(.(/ g. 5(.(/ " g. ~ g.P- 8) 5 3
    ͷ
    DPQZ
    PO
    NPEJGZ
    
64. /"ͷѻ͍ 4VHBSؔ਺ɾԋࢉࢠ͸ศརͳ͚ͩͰ͸ͳ҆͘શͰ͋Δ ඪ४
    $
    ʹ͸
    /"
    ͸ఆٛ͞Ε͍ͯͳ͍ͷͰ
    3
    ͷ
    /"
    ஋Λ͏·͘ѻ͑ͳ͍
    4VHBSؔ਺ɾԋࢉࢠ͸
    3
    ͱಉ༷ͷ݁ՌΛฦ͢
65. ඪ४
    $
    ͷ
    JOU
    ಉ࢜ͷԋࢉࢠ ++ PPIUbb66Wibace C eW WcMWUeac SVVR[ eW Wc C eW WcMWUeac i( C eW WcMWUeac j l [ e 9 i* W e 7 C eW WcMWUeac k 7 Xac [ e [9,7 [8 7 ''[ l kP[ 9 iP[ ' jP[ 7 ++ [ e ' [ e m cWefc k7 m : i'j P- F; . : SVVR[ eW Wc i(j P- ).-0304/203 . : i 8) U - ( - : j 8) U F;( -