{VERSION 3 0 "SGI MIPS UNIX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{PSTYLE " Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "with(MS):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Consider the example of the Euler numbers , given by the exponential generating function of " }{XPPEDIT 18 0 "2/ (exp(x)+exp(-x));" "6#*&\"\"#\"\"\",&-%$expG6#%\"xGF%-F(6#,$F*!\"\"F%F ." }{TEXT -1 336 ". The denominator of this has a symmetry of order 2 . Below are two methods to compute the recurrence for the denominato r, when multisectioned by 8. The first method does not take into acco unt the symmetry, where as the second does. Also demonstrated in this section is the code egf/strip, which will strip away the useless ze ros." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "botNoSym := egf/st rip(bottom/ms/linalg/fft2(exp(x)+exp(-x),f,x,8,[2,2,2]), 8, 0);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%)botNoSymG6&/-%\"fG6#%\"xG,6-F(6#,&F *\"\"\"!#!)F/!CO>ZI(p)>K58u4)ebqJ)-F(6#,&F*F/!#sF/\"D[WY:g.)4zvQ7:y-IB P-F(6#,&F*F/!#kF/\"Cs'[2:&ooK\\P@'R.)ym6-F(6#,&F*F/!#cF/!@?()>n&yAq3%> r4P*fG-F(6#,&F*F/!#[F/!=s)>J9>h%*-F(6#,&F*F/!#SF/\":cm#R())3?Lw o\"*Q#-F(6#,&F*F/!#KF/\"3[oW)4&)3'Ra-F(6#,&F*F/!#CF/\"1+w$\\tbfj$-F(6# ,&F*F/!#;F/\"-sup!fe\"-F(6#,&F*F/!\")F/!'#R$GF(F*7-/-F(6#\"\"!\"$c#/-F (6#\"\")!'cqb/-F(6#\"#;\".%Q)3:8!R/-F(6#\"#C!3?zD%*p'3Go*/-F(6#\"#K\"9 /.m+<.]..'*))/-F(6#\"#S!?wX]wyPq$yLB*yo7R/-F(6#\"#[\"E?RF7Jq#G,1$>%*o Q>W%[#/-F(6#\"#c!KOz[!)e;#[<(*3%>Bhl\"pI@H\"/-F(6#\"#k\"PkGT8#)*G()\\ T^!f " 0 "" {MPLTEXT 1 0 85 "botSym := egf/strip(bottom/ms/linalg/fft2(exp(x)+ exp(-x),f,x,8,[2,2,2],2), 8, 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% 'botSymG6&/-%\"fG6#%\"xG,&-F(6#,&F*\"\"\"!#;F/!%'4%-F(6#,&F*F/!\")F/!% w@F(F*7$/-F(6#\"\"!\"#;/-F(6#\"\")!&3u\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "BotNoSym := egf/makeproc(botNoSym):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "BotSym := egf/makeproc(botSym):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 232 "Next consider the top recurren ce, determined by the bottom recurrence and the definition of the Eule r numbers, when multisectioning by 8 at 0. Again, the first method do es not take into account symmetries, where as the second does." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "topNoSym := egf/strip(top /ms/linalg/know(BotNoSym, euler, f, x, 8, 0, 30, 2),8,0);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)topNoSymG6&/-%\"fG6#%\"xG,X-F(6#,&F*\"\" \"!$7\"F/!jn?ESk4@yRo[Y8^iR-%f20O^Le5AGf-#R%-F(6#,&F*F/!$G\"F/\"ao8J \\Alsa.b:T4bw!G_pum/)QzJt*yL@PGx$R;-F(6#,&F*F/!$/\"F/!fng=K)4aBy)y^.< U0sr(fHOBmYA?.8-F(6#,&F*F/!#;F/!-6 ,:nY8-F(6#,&F*F/!#[F/!@e?SPY.o?lZae9u^*-F(6#,&F*F/!#kF/!I.Q0cV9+zXPNF/!pT5s%))f>+U0vim9)[-F(6#,&F*F/!$3#F/!_pr4H&GSKBJit9cir@-&>3 ld(=rf_T))3***Q&=P<#p0q))R;-F(6#,&F*F/!$;#F/\"^pD1R]iaWZ'*z_cf#**f^.I (3.$\\)4R1Y:sd,Q!fqXQ\"*38EB-F(6#,&F*F/!$%=F/\"_pepus(QdM6WT#Ra.!Q7B% 4og+Axq^wXQ;=,hyw))4K-F(6#,&F*F/!$o\"F/!]piMR)*e6jEfX\\2)yj%HT%3B(G%yr ZDD*yLyJB.\"Q7^v74@#-F(6#,&F*F/!$g\"F/!\\p=Fg!*f%*>>>Q**\\__ytl()psD[* HqAy\"o61v2)*Hs^ngL\"F(F*7=/-F(6#\"\"!\"$c#/-F(6#\"\")!''\\-#/-F(6#\" #;!.'p\\J&R1\"/-F(6#\"#C\"2!)[^6,tqX'/-F(6#\"#K\"9Wd@&Q#3GT3aZ6/-F(6# \"#S!>wdv*pTu(*e\")\\!)yh7/-F(6#\"#[!D!o*RZ/EaF9k5H(\\d(eN\"/-F(6#\"#c \"I%=&)y\"Hg5A#QL@4(*e!)>1<#/-F(6#\"#k\"OWthQgt%[]_%))Q$>FBdw'p/\"*e: /-F(6#\"#s!TS%*=%[,p6f(odc\"[,a!4%fT58d$)QL/-F(6#\"#!)!ZwTfkU(pSI*o(> P]<'o<&)Ro?XkSGmv\"/-F(6#\"#))\"inkc#eH;Pnlgn%))>1^-p!pDqD,!RmIuml\\[/ -F(6#\"#'*\"_o+'H:O3EOUw#\\A\"4j$[$\\r0=H8vaV31aHp%3K>/-F(6#\"$/\"!doO BUT@*)**4CjQ&G>8pVDE*fqf(*QDpezL!ye:iOsn/-F(6#\"$7\"!jo;[ZDKe\\P*Qmt!Q Z(o*>,d!)fjR^8_#okkVxEXsrExh?/-F(6#\"$?\"\"_pS9im\\'oC<)QW5.p')4OI]i\" eTh,y[=z(=?'zUYex[yUv<*/-F(6#\"$G\"\"ep/2gU'\\\\ln]=wF%=#G>*Q7%zOe^#y Y\"y()o*fF9\")*>D+-*=zP9@/-F(6#\"$O\"![q'4Ga(ya;!owN<^J#[z,SMp\"o_^> 7'G/UzLRt.(3xw&faZ7?!Q@\"/-F(6#\"$W\"!q?*pr**e9L\"*p-)GUz4q!o?:rnJ_Ld \"znLi44z\"R9+EVV\\gup&RkLM%[?/-F(6#\"$_\"\"fqW\\F;rj-bX=s)yQ_C&>F'3_V *=ths$3[O%zTdBD5l<R=iaCd\"/-F(6#\"$g\"\"[rC=U4x_13Si1opj@IF_ 8jZ>o)fsmt#\\(G!QZa=5\"e2Yv(e/&)ou*o<)*)z$z,\"=/-F(6#\"$o\"!ar![?)GZp ()3nonQr/yw'R%>\")=J0=(yb6$Q>HM^DWSa'y\"3*))pfJ#)pE8Cq]O%Q$z')***>/-F( 6#\"$w\"!fr;'*z]&zEHqX!o%y![fOdI$\\&edw)e3$RnO2q?Jj='>J8/-F(6#\"$%=\"\\sW4+.F\"3\"\\8=V'ff7*3?T6U3%))=9Am**=M()Q Cbz4$et#e]kw*yz^%os1_!\\.h6&y.KcG0,D/-F(6#\"$#>\"sgPP7%>]\\Lx![\\Yw$* 3,!f'))[gH&3#plnr'R18GvmbvTgEuv7J%[wI\$pP#H#3t@JO \\l " 0 "" {MPLTEXT 1 0 82 "topSym := egf/strip(top/ms/linalg/know`(BotSym, eu ler, f, x, 8, 0, 30, 2),8,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'to pSymG6&/-%\"fG6#%\"xG,*-F(6#,&F*\"\"\"!#KF/!%hl-F(6#,&F*F/!#CF/\"(G9d( -F(6#,&F*F/!#;F/!&)zX-F(6#,&F*F/!\")F/\"%)=\"F(F*7&/-F(6#\"\"!\"#;/-F( 6#\"\")\"%_Z/-F(6#FF\"(#**y_/-F(6#\"#C\"+OvmWh" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "So both the top and the bottom recurrences are smal ler when the symmetries of the denominator are taken into account." }} }}{MARK "9 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }