Fast computation of some asymptotic functional inverses
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(76) vF = ¡ GH8 "I"E93#J(*& |P M!K4OSK4UI_GBRD:P 4BR?aD K³P;FIFSBED!OS?-OIBb[4PGM!K4WRWj KMLN DYC_ K4US? ` USBE_G?:=O - EO$P +9 Ng PQ C -l/n / hv { - k v:¡6R Fgv
(77) C = GS 7T13#UV& jDTaK4UIFSBR?-DM!K g t q r K4O g WE?a^ t g P8Y _G?8Z-K4DM!K]FDC?-_0A USK4F³M\K ! OIBRUSWEBRD ^Q= h! * WXO 6
(78) ZY +9 P [ kE 6 Ng-E-_ 8 \^] h(v { k v { RFv { 3 = GS 7T13#5U&3+ ( h! A * E/4 =2>@K4BbM!K]W °?aUvM!USKN[SHO6 v = %$. . . . . J. $. . J. .. .. 3. . . . .. $1. E%. %. . %$. .. . %. . %. . 6%$. )%. J. .. . . . J$. %.
(79) l NcE / n 9 ; 4=2°?8T-K]UN v cv = ¤USK4` USBED!O@?;eOIH KOIHCBEUVM ?-UIOSH ?-WRWbP8D:M KNM!BROSBE?aD v { h :UIFSOxKNM\BEOSBR?-D v ak =
(80) ;f h 6- h ( kA 8 D:MK6M=R8T-?aW= \K4_ BRD!Y _ K]UIBb[4P8W WR^-?-USBROIH _GFN= M M\BEFS?-D! K4FSWEK]Z v v = . ª T-P8WRY:P8OSBR?-DP;FIZ!_ `COI?-OSBb-YCK#M\K Wj ?aUvM!USK _P c BE_Y _
(81) M Y D K4ª W6K4ª _GK4D!O«M!Yw^aUI?-YC`:KFIZ!_K4ª OIUSBuaY Ka= 9/ ; & E h(v ak
(82) cv , = x´|K4US_P8DCK4D:[VK M!K USK4WbP8OSBE?aD FxM!K U KNª [VY USUIK4DQ[VK M P8DCF[VK4USOvP;BEDCFMLK4ª TaK4WR?-` `QK4_GK4D!OIFxP8FSZ!_ ` OS?-OIBbaY K4F6= E n 6 n( 6 x 6N ( - G(h v ak v n =. 0 B1. /0 =H !23# @& . ;,2= J. 3# @& .& . . . +. . ! . . H823#. . '+. %. $E. . D. . G. 4. .3. E. ( S . ) . +. +. . $. . 4 H823#. . &. J . . . F. 3. $. &. F%. . . 9 R !H89 # & 6.& # SH 7"I9 # 7&)UV&8#)92:. 4 . SC. ! . J. . (. &3+. . 2. Y. +. H. C. %[ . ) ¼ 0 3¼ 9 , 9 ) ?. (. C%. ) . %. MD. .J. . 3. J. F. .. ¼4¼x3 2 $ 53 !. H K
(83) !ef?a#"WE%WR?8$%J@&(' BRD )*^ &(',`C+-/.UI?-0^aUv 21#P;_&234'BR_ 56',78` '9WR+K4:;_G&=<>76K4D!5#OI1?F $67A @?H &=<K4 !?a7BUIK4 21D_GCFF EHGJv I=K6P8L9D:MONPM 5F <>Q;= ',7SRHN
(84) &=3T'D"6 !UV'9+#.W',+-<X',.VY <Z7&234'QW:F+-<2:O[]\^'D_9N :F7A`)Ha. bcKJd efhg]ikjlI2mFn^oTpAmTqOrPq%s]qut8LwvZikjOxTvcbhq%yHq6z!{ |^} ~]%- I2rPq s!q K6LeKDW 4wVVIW-!O6O%!OO%FOP-q rALJV{ z]d e ~] t4 ? s!I2z ?Z z!I=m;n oTpAmO!n F IFVs-Fb L- z(yT4z!I2s,es-%I=Os-Fb
(85) L,q z ?^ z!IZn F I2rTL-q s-q t8DOL9L9q s!q tJDOL?q o nZ ut ;¡ L?{ yHd e ~!6? I2z]q s!q K6L?{ ]i¢t¤£Hi¢yHd ez -9Z z!I=z(yTAz]I2b e yHq z-L9q s!q tJDOL iA£ 4 £ d
(86) ¦¥D§
(87) !#"%$%&('&234' )¤¨w+-.-&PEHG©.]:F&2<Z.Z1?ªW<>765DE P e¬«OE H P *I9)®¯-L(E H Nh5F<ZQO'97E J :F7A`°&=3T'V5%'978'9+9:F&2<Z765.W',+-<X',.±FI2_L/ X1CFEHGJI=K%LM%a ,!~ d eDfHg!iAj¢I>oAKTqOmTq%Hq ~ q6z!q6s!q6t8Lwv>iAjOxTvwbHH:q Qc{ R bV³ KF´d e oAKT ~!6? I2z]q s!q KOL-{ } iAg P} g]iAµ¶K/!i¢tH £hilb³ W´d eVz F s]I2m6Fb³ ´=IZ8 ~ LuI t6s!I=b³ ´=q s-L9 ~!O- IZz!q s]q L;sL-q s-LkiA£ { Vw· ¸VV¹I³ z ]º I2bV³ ´= ~]%- I=z!q s!q L9q eKT»^» tJLX´=L 4 £ d
(88) ¼+,'(@
(89) !#"%$%&(' )*&(',+-/.0 21#&234':F.-ªF#"%& !&2<=@0'½]"u:F7T.-<X W7¾ 21¿6NUw34'9+-'VÀ!ÁO¿6Â YIXO;¿6LHe*ÃAN ¦5F<ZQO'97R¬:F7A`0&234'h1W W+-®:O\k"6 !UV'9+#.W',+-<X',.VYI=ÃAL?a sp !F--¡ %d eDfHg!iAjI2mFn^oTpAmTqFrPqTÄAqut8LwvZiAjuxTvcz!q { z]d ez(yTAz]I2Ä4e n O IZn O IZÄ4L9L9q -]~ I2mFn^o%pAm;;ÄAwn O I ~]O? I=rPq ÄTq K%LL-q m;n oTpAm%q^ mFn^oTpTmTq K%q bcKTI>mFn^o6pAmJq rPq ÄTq t8L?q ÄAq tALL?{ Vw· ¸VV¹I ~! tT - s]I9³ n O I2Ä6L?q z !º I o8I q zL9!n F I2Ä6L]Å;I=% L-q eD%» » t8DOL?q I>n O I^n O I2Ä4L9L-ÅFI>tAuL!n O I2Ä6L!Å!IZt8DOL9L>´qÆPL9L 4 £ d
(90) ¼&234'9 !+,',®È
(91) !#"%$%&(' )*&(',+-/.0 21#&234':F.-ªF#"%& !&2<=@0'½]"u:F7T.-<X W7¾ 21 n O ¿6N Uc34',+,'DÀ Á ¿  YIOF¿%LHe*ÃAN ¦5F<ZQO'97R¬:F7A`0&234'h1W W+-®:O\k"6 !UV'9+#.W',+-<X',.VYI=ÃAL?a sp !F--¡¤É n O d eDfHg!iAj¢I2mFn^oTpAmTqFrPq6Äkq6t8Lwv>iAjOxTvhz]q { z]d ez(yTAz]I2Ä4e n O IZn O IZÄ4L9L9q -]~ I2KTq m!n^oTp4mJq^
(92) m;n^o6pAm%q KTq z 9Z z!I>n O IXOÄT!bcKTI2m!n^oTp4mJq rAq Äkq t8VOL9L?q Äkq t8DOL9q Äkq tJL9L-{ Vw· ¸VV¹I ~! tT - s]I9³ n O I>n F I2Ä6LL?q z !º I oHI q z,L9?n O I=ÄTL-ÅFIu L-q eD%»^» tADOL-q I>n O IZn F I2Ä6L9L]Å!I>t8DOL?n O I2Ä4L?ÅFIZtADOL9L2´q2ÆHL9L 4 £ d
(93) ¼&234'9 !+,',/Ê \^ 5
(94) !#"%$%&(' )*&(',+-/.0 21#&234':F.-ªF#"%& !&2<=@0'½]"u:F7T.-<X W7¾ 21DÀ]ËuÁ%¿OÌ8ÍIOF¿6L9NwUw34'9+,'DÀ!ÁO¿JÎ ÂPYIOF¿%LHe*ÃANc5;<ZQ;',7¤RHNhÏHNkÐ8N Ò:F7A`0&234'h1W !+-0:O\A"% !UV'9+#.W',+-<X'9.YÓ:;7A`®Ía sp !F--¡¤É oAm s É d eDfHg!ikj0I=m;n^oTpAmTqF smTq m ¡0¡ mTqFrPq q%ÄAq%t8LwvZikjOxTvHbhq4z!q { z]d ez(yTAz]I2Ä4e n O IZn O IZÄ4L9L9q -]~ I ~]O? I q ÄTq K6L?q mFn^o%pAmTq^ mFn oTpAm%q mFn^o6pAmO%I2m!n^oTp4m%! s9mO m ¡/¡ mOL?q z -9Z z!I=z(yTAz!I=b ebcKTI2mFn oTpAm%q rPq Äkq tJL?q Ä;oHIZ smO%I2b
(95) n F I ~]O? I=rPq ÄJq K6LL9LL9 ~! t4 - s]I2z ?Z z!I9IXFÄ4;b
(96) L!Å m ¡¡ m6;zy%Az]I>Ä4eÄT%I=OÄTFb
(97) L?q L-q ÄJq t8uL9q o nZ ut F¡ L9L-q ÄTq t8DOL?q ÄAq tALL?{ Vw· ¸VV¹II2ÄT ~]%- I=rPq Äkq K6LL!Å9 s9mO%I>n O IZÄ4L9L-ÅFI2m!n^oTp4m%! s9mO m ¡0¡ m%L ~] tT 9 s]I³ z !º I oHI q z,L9!n O IZÄTL]Å!I% L-q eD%»^» t8DOL9q I>n O IZn O I2Ä4L(L]ÅFIÔt8uL!n O I>Ä4L!Å!IZt8DOLLX´q2ÆHLL 4 £ d
(98) ¼&234'9 !+,',/Ê "u:F+-&2Ê. J. ². Ç. Ñ.
(99) !X ?s¼ #,#%$ 5 3Z"¼ )1 /! $-,. /!. I. K^-BRT-KOSH K:USFIOxOSK4UI_GF@?8e2OIHCK KVc `:P8DCFIBR?-D F@_GK4D!OIBR?-DCKNMGBEDG?-Y U[V?-US?-WRWbP8USBEK]FNh= \BEDQ[VK BRDw_ ?-FSO~?;eOSH BRF[4P;FIK4F6:OSH K:USFIO efK4J OSK4UI_GFHQP6T-KxP8WRUSKNP-M!ZA:K4K]D ` Y ACWEBRFSH KNMaOIH K]FIKxKdc\`QP8D FSBE?aD F2H K]WE`0T!BRD:M!Bb[4P;OIKLA:?aOIH0?aY U2OIHCK4?-USK4_ F~P;D:M?-Y U2`CUI?-^aUvP;_=.
(100) ¢ OSH K4?aUIK4_ v8h(v v c ak. Ç Ñ n^tHI=ÃALk ^n t8I>n^t8I2ÃAL9LA n^tPn^I>t8n^t8I2ÃAI2ÃAL LL "! #%$&! #*$(')(n^tP) I2ÃA L * n^tHI^n^tHI2ÃALL ! #($&! #,$(')+) .-6 É n^n^tHtHI2IÔÃAn^tHL ,I=ÃAL9L * n^tHI^n^tHI2ÃALL ! #($&! #0$(')()+/ OO21Pn^t8I>n^tPI=n^ÃAtHL9I2L ÃA, L 0.- n^tHIZn^tPI=ÃAL9L * n^tHI>n^t8I2ÃAL9L ! #%$&! #36$(')+)54 *7R 6* n^tHI^n^tHI=ÃAL9L 0 8 -:9u3 n^1Pt8n^I2tPÃAIZLn^tP6 I=ÃAL9L , ;9kn^tHI^n^tHI2ÃALL * n^tHIÔn^tHI=ÃAL9L " ! #($&! #=$(')()+< R = ,J > n^t8I>n^tHI2ÃALL 6 @ ?A9O3 Bkn^tHIZn^tPI2ÃALn^L tP0 I=8ÃAL =BC9O3 1Pn^tPIZn^tPI2ÃALL , 9O É n^tHI^n^tHI2ÃAL9L * n^tHI^n^tHI2ÃALL ! #%$&! #3$(> ')+)5D 07* EJ n^tHI^n^tHI=ÃAL9L = * R J 6 , n^tPIZn^tPI=ÃAL9L 6 3É F 9O3 Bkn^tPn^t8IZI2n^tPÃAI2L ÃA> LL 0 ?9O2 1An^tPI>n^t8I2ÃALL , É O É n^tHI>n^tPI=ÃAL9L * n^tHI^n^tHI2ÃALL ;G n^tPn^I>tHn^t8I=ÃAI2ÃAL HLIL H J. *. +. 7J KML
(101) ONPQLR SQ TU
(102) V" TOW.
(103) XS ¢ } [V?-D!T-K]UIO h FIKN h B ,Y £¬ Bj B };{ =R= v { k I[Z Z k ¢ OSH K4?aUIK4_ `:P;UIO \h(v v { ~ v £ -kA n^tHIÔn^tHII%\ LL n^tHI^n^tHI%%\ LL É "! #%$&! #3*$ G )+) Ç 8-n^t8I>n^tPII\%LLhOu É \hn^t8I%\%L n^tHI%6\ L n^tHI%6\ L * n^tPII\%L , Ñ G * ! #($&! #,$ )() @O? É n^tHI^n^tHIIn^%\ tHL9I%L 6\ L 0 F n^tPIZn^tPII\%L9LPB2-6O31 G , * "! #%$&! #0$ )() / É %- 3k1 n^tHI^n^tHII%\ L9L F] n^tP I%É 6\ n^t8L 6 I>n^tHII%\ L9L 9 ] É n^tHIZn^tPI%\6L9LP R ,7R ,7* , G ! #($&! #6$ )() 4 07R E* n^t8I>n^tPII%\ LL 0 ;?36- É n^tHI>n^tPII%\ LL ,n^t8.I%%\ -L 1:= u? É n^t8I>n^t8I%%\ LL * 86- F %- 3D1 n^tPIZn^tPII\%L9LH R ,7* 6 J H E G 6 0 , * "! #%$&! #=$ )()^< *7= 6J > n^tHI^n^tHII%\ L9L R *7R 0E7, n^tHIZn^tPII%\ L9L FK É O? n^3tPk- IIn^%\ tPL IZ> n^tPII%\ L9L R >_R E * R > n^tPIZn^tPII%\ L9L 0> R J * 67, n^tPI>n^tHII\%LL R 7E ,70 *7J *7, `G n^t8n^I>t8n^t8I%%\I%%\ L HLL(H &K. . . J. *. +. a KML
(104) b2 Q Uc_dfehgji3kml ¢ OSH K4?aUIK4_ `:P;UIO \h(v vC
(105) n { ~ v D0 k Ç 8 -6 F n^t8I>n^tPI=)PLLh ] 3B ! #+$&! #A$ H )() Ñ R R ^n tPIZn^tPI=)PLL * 7, E n^tHI^n^tHI2)8L9LP 76 , H H É ! ( # & $ ! # $ + ) ) u " ! % # & $ ! # $ + ) ) * R = ^n tPI2)8L , R = R= R= H ^n tHI=)PL * )Vn^tPI2)8L"o-V n^tHI2)8L .
(106) X 9Ou É Bkn^tHI^n^tHI2)8L9L 0 =E7=R n^t8I>n^t8I2)8L9L , n^t8I2)8*7=7,7L 0 E 0 n^tPI>n^t8I2)8L9L * ,7,707* , n^t8I>n^t8I2)8L9Lh 6 , J H > 0 R ?u É 931kn^tPIZn^tPI=)PLL 6 07>_6= H> ^n t8I>n^tPI=)PLL 0 =7R ,7* EH n^t8I>n^t8n^t8I2)8I2)8L9L L , 6 * R H * >_H E n^tPI>n^tHI=)PLL * 07,>_=>_*7H E n^tHI^n^tHI=)PLL R =*7H67=7= , R4J* *7R 0 n^tPIZn^tPI=)PLL = *7R 07670= JR n^t8I>n^tHI2)8L9L 6 R , E J J >_H * R n^tHIZn^tPI=)PLL 0 E7*7n^E t867R =I26 )8n^L t8= I>n^tPI=)PLL , R 67R4J07*7* 0R E n^tHIZn^tPI=)PLL * 6707R *67,7>_67= E n^tPIZn^tHI2)8L9LP * *J 6 R R * * R J >7> G n^t8n^I>tHn^t8I=)PI2)8L >L9L > *. +. KML
(107) b2 Q i gji k l:l ¢ OSH K4?-USK4_ `QP8UIO !hv v:6n v ~ FIY ACF h D } D k D| k É # n O) ) ! #%$&! #*n^$ tPH I=)()P) L 86- É 6- 3B n^t8I>n^t8I2)8L9L * n^tPOOI=)P FL * n^t8I>n^tPI=)PLLP;99O2B XO9 O2P1 n^tHI^n^tHI2)8L9L , =R =R n^t8n^I>t8n^tHI2)8I=)PL , LL * , R R = E n^t8I>n^tPI=)PLLP > R R = R R ,7* 6 H n^tPIZn^tPI=)PLL 0 07E76= > n^t8I>n^tPI=)PLL , *7n^0=7tP0>7I=> )Pn^L t80 I>n^tHI=)PLL * 67,7= *R 6 n^tHI^n^tHI2)8L9LA R 0 , R H E70 ,> *7=767, = n^tHI^n^tHI2)8L9L 6 R *707670 = R n^tPIZn^tPI=)PLL 0 *70, HR 0* R n^t8I>n^tHn^I2tH)8I2L9)8L , L 6 6767R 0* HH E n^tHIZn^tPI=)PLL * 0> *7J 67>_= , R n^tPIZn^tHI2)8L9LA 6 0 , R4*7J , *7R 0 n^tHI^n^tHI=)PLL = R *7=7670 = H J R n^tPI>n^t8I2)8L9L * H, *7J =7>_*7* , n^tPIZn^tPI2)8L9L =76 >_H = =7H *n^t8R I2n^)8tHL I^n^= tHI=)PLL 67=7R4* J R *7E70 67, n^tHI>n^tPI=)PLL * G n^t8n^I>tHn^tPI2)8I=)PL >LL > . * H E7>_07= =7H =7, ,707=7R 67=7,767==>_E n^tPIZn^tPI=)PLLk > J*RR ,776 E7,7*= J> H R. *. +. KML
(108)
(109) X
(110) XSQ .
(111). e~[V?-Y USFIK-Q?aY U~`CUI?-^aUvP;_ BRFLD ?-O@WRBR_ BROIK6MOI? [V?aD FIOV P8D!Ox[V?!KV[VBRK4D!OIF } u q u {i u i } q {i i ¢ OSH K4?aUIK4_ `:P;UIO \h w t t t t t i t -k. Ã Ë n^t8I2ÃAL ÂTË QPÌ o J R0IRDÏl Ð8L J n^tHI>n^t8I2ÃAL9LA R Ðn^tHI J L J Ë $ ² QHÂ ! #$ ) ) J
(112) n^tPI=ÃAL n^t8I2 ÃAL * I9 R¢R0IRDÏ0©Ð8L9L6R°É IRDÏ®BÐ8L J n^tHIÔn^tHI=ÃAL9L * R * IRDÏ® Ð8L J *I9VRlR°IXRDÏ0©Ð8L9L R Ðn^tHI J L J Ï0I R R n^tPI J L J L J n^tHI^n^tHI2ÃALL J
(113)
(114) Ð R BRSn^tHI J L J R n^tHI J LA * Фn^tPI J L R Ðn^t8I J L * É I>Ð0BOL J Ï/I R *Rn^tPI J L J LTI R Ðn^tHI J L J L J
(115) * * n^t8I Lh É n^tPI L Rn^tPI L * * É * Ï * É Ï R n^J tPI L ©Ï/R * n^tPI L * * É É Ï ! R. R * J R R J R J J R J J J J R J J J J J ! É J* ¢ . . . =. . :. . . =. G ^n t8^n I>tHn^tPI2ÃAI=ÃAL ,L9L , *. +. .
(116)
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