{VERSION 6 0 "IBM INTEL LINUX" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{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 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning " -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }3 3 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 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 }{PSTYLE "Maple Plo t" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "_pstyle2" -1 230 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle5" -1 232 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }1 0 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle15" -1 236 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }1 0 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle16 " -1 237 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle18" -1 239 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle24" -1 245 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }1 0 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle25" -1 246 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }1 0 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE " _pstyle26" -1 247 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }1 0 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle27" -1 248 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle28" -1 249 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }1 0 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE " _pstyle31" -1 251 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }1 0 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle32" -1 252 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }3 0 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle37" -1 255 1 {CSTYLE "" -1 -1 "Courier " 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }1 0 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle40" -1 256 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }1 0 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle43 " -1 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }3 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle45" -1 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }1 0 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle50" -1 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }1 0 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle52" -1 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle54" -1 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle63" -1 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 61 "################# ############################################" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "# Minimal surface from a configuration of type 1 ,3,3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "# This worksheet takes as input a balance d configuration and outputs a minimal surface" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "# as explained in the paper : Exploring the spa ce of embedded minimal surfaces of finite total curvature" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "# Experimental mathematics 17:2, pa ge 205" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "# Important note : if you change th e configuration, you won't be able to see pictures." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "# That's because the domains are triangul ated externaly using mathlab." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "# If you want to see pictures, leave all parameters as they are. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 230 " > " 0 "" {MPLTEXT 1 -1 61 "########################################### ##################" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 11 "Di gits:=15:" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 13 "with(linalg ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "##################### ########" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# input configu ration" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "################# ############" }}}{EXCHG {PARA 232 "> " 0 "" {MPLTEXT 1 -1 18 "# number of layers" }}}{EXCHG {PARA 232 "> " 0 "" {MPLTEXT 1 -1 5 "M:=3;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG\"\"$" }}}{EXCHG {PARA 232 "> " 0 "" {MPLTEXT 1 -1 27 "# number of necks per layer" }}}{EXCHG {PARA 232 "> " 0 "" {MPLTEXT 1 -1 24 "N[1]:=1;N[2]:=3;N[3]:=3;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"NG6#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"NG6#\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"NG6# \"\"$F'" }}}{EXCHG {PARA 232 "> " 0 "" {MPLTEXT 1 -1 15 "# size of nec ks" }}}{EXCHG {PARA 232 "> " 0 "" {MPLTEXT 1 -1 35 "C[1]:=1.5950;C[2]: = 1;C[3]:= 1.35;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6#\"\"\"$ \"&]f\"!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6#\"\"#\"\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6#\"\"$$\"$N\"!\"#" }}}{EXCHG {PARA 232 "> " 0 "" {MPLTEXT 1 -1 19 "# position of necks" }}}{EXCHG {PARA 232 "> " 0 "" {MPLTEXT 1 -1 198 "P:=[0., -1.3244558618145532783- .21269832293518539768*I,\n-1.3244558618145532783+.21269832293518539768 *I, .67930686040722205806,\n-1.5240875571628029414, -.8850998648061221 6290, 1.4091874219689251043];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"P G7)$\"\"!F'^$$!5$yKb9='eXC8!#>$!5o(R&=NHK)p7#!#?^$F)$\"5o(R&=NHK)p7#F. $\"51e?A2/'oIz'F.$!59%H!G;dv3C:F+$!5!H;Ah![')*4&))F.$\"5V5D*o>U(=49F+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 430 "plot1:=complexplot(\{op(P[1 ..N[1]])\},axes=none,style=point,scaling=constrained,symbolsize=20,sym bol=CIRCLE,color=black):plot2:=complexplot(\{op(P[N[1]+1..N[1]+N[2]]) \},axes=none,style=point,color=black,scaling=constrained,symbolsize=15 ,symbol=CROSS):plot3:=complexplot(\{op(P[N[1]+N[2]+1..N[1]+N[2]+N[3]]) \},axes=none,style=point,color=black,scaling=constrained,symbolsize=15 ,symbol=BOX):display(plot1,plot2,plot3,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 529 529 529 {PLOTDATA 2 "6)-%'CURVESG6&7#7$$\"\"! F)F(-%'COLOURG6&%$RGBGF)F)F)-%'SYMBOLG6$%'CIRCLEG\"#?-%&STYLEG6#%&POIN TG-F$6&7%7$$!3A`X\"='eXC8!#<$!3(R&=NHK)p7#!#=7$$\"3-@A2/'oIz'F@F(7$F;$ \"3(R&=NHK)p7#F@F*-F/6$%&CROSSG\"#:F3-F$6&7%7$$!3(H!G;dv3C:F=F(7$$\"35 D*o>U(=49F=F(7$$!3F@71[')*4&))F@F(F*-F/6$%$BOXGFJF3-%*AXESSTYLEG6#%%NO NEG-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q!6\"F_o-%%FONTG6#%(DEFA ULTG-%%VIEWG6$FdoFdo" 1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 64 "############################################### #################" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 63 "# c heck that the configuration is balanced and non-degenerate. " }}} {EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 65 "######################### ########################################" }}}{EXCHG {PARA 230 "> " 0 " " {MPLTEXT 1 -1 58 "maxN:=0:for k from 1 to M do maxN:=max(maxN,N[k]): od:maxN;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 53 "if N[M]=1 then print(\"N[M] must be at least two\");fi;" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 41 "N[0 ]:=0;N[M+1]:=0;NN:=sum(N[kk],kk=1..M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"NG6#\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"NG6#\" \"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NNG\"\"(" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 40 "psi:=(kk,ii)-> sum(N[ll],ll=1..k k-1)+ii;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiGf*6$%#kkG%#iiG6\"6$ %)operatorG%&arrowGF),&-%$sumG6$&%\"NG6#%#llG/F4;\"\"\",&9$F7F7!\"\"F7 9%F7F)F)F)" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 118 "invpsi:=p roc(jj) global N;k:=1;j:=jj;while true do if j<=N[k] then return (k,j) else j:=j-N[k];k:=k+1;fi;od;end proc: " }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `k` is implicitly declared local to procedure `invpsi`\n " }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `j` is implicitly declared l ocal to procedure `invpsi`\n" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 30 "qq:=vector(NN);FF:=vector(NN);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#qqG-%&arrayG6$;\"\"\"\"\"(7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FFG-%&arrayG6$;\"\"\"\"\"(7\"" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 244 "for l from 1 to NN do (k,i):=invpsi(l):FF[l]:=sum(2 *C[k]^2/(qq[l]-qq[psi(k,j)]),j=1..i-1)+sum(2*C[k]^2/(qq[l]-qq[psi(k,j) ]),j=i+1..N[k])-sum(C[k]*C[k+1]/(qq[l]-qq[psi(k+1,j)]),j=1..N[k+1])-su m(C[k]*C[k-1]/(qq[l]-qq[psi(k-1,j)]),j=1..N[k-1]):od:" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 55 "S:=\{\}:for l from 1 to NN do S: =\{op(S),qq[l]=P[l]\}:od:S;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<)/&%#q qG6#\"\"#^$$!5$yKb9='eXC8!#>$!5o(R&=NHK)p7#!#?/&F&6#\"\"$^$F*$\"5o(R&= NHK)p7#F//&F&6#\"\"\"$\"\"!FU(=49 F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# compute forces" }}} {EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 17 "subs(S,eval(FF));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'vectorG6#7)^$$!\"\"!#9$\"\"!F,^$$\" \"#F*$\"#6!#:^$F.$!#6F2^$$!#AF2F+F'^$F+F+^$$\"\"%F2F+Q(pprint16\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "# compute jacobian matrix" } }}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 30 "jaco:=subs(S,jacobian(F F,qq)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "# the following \+ determinant should be nonzero" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 37 "det(submatrix(jaco,1..NN-2,1..NN-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0;R%o1c5D!#6$\"0W:![Y(Gr'!#D" }}}{EXCHG {PARA 230 " > " 0 "" {MPLTEXT 1 -1 71 "########################################### ############################" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 71 "# Weierstrass data phi and immersion X=Re\\int phi represente d as series" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "# (see formu la 3-6)" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 71 "############# ##########################################################" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 129 "phi:=(nu,k,z)->sum(sum(A[nu,k,j ,n]/(z-a[k,j])^n,j=1..N[k])+sum(B[nu,k-1,j,n]/(z-b[k-1,j])^n,j=1..N[k- 1]),n=1..maxn)+lambda[nu,k];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phi Gf*6%%#nuG%\"kG%\"zG6\"6$%)operatorG%&arrowGF*,&-%$sumG6$,&-F06$*&&%\" AG6&9$9%%\"jG%\"nG\"\"\"),&9&F=&%\"aG6$F:F;!\"\"F " 0 "" {MPLTEXT 1 -1 246 "X:=(nu,k,z)->sum(A[nu,k,j,1]*log(abs(z-a[k,j]) )+Re(sum(A[nu,k,j,n]/((1-n)*(z-a[k,j])^(n-1)),n=2..maxn)),j=1..N[k])+s um(B[nu,k-1,j,1]*log(abs(z-b[k-1,j]))+Re(sum(B[nu,k-1,j,n]/((1-n)*(z-b [k-1,j])^(n-1)),n=2..maxn)),j=1..N[k-1])+Re(lambda[nu,k]*z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"XGf*6%%#nuG%\"kG%\"zG6\"6$%)operatorG%&a rrowGF*,(-%$sumG6$,&*&&%\"AG6&9$9%%\"jG\"\"\"F:-%$logG6#-%$absG6#,&9&F :&%\"aG6$F8F9!\"\"F:F:-%#ReG6#-F06$*(&F56&F7F8F9%\"nGF:,&F:F:FOFFFF)FA ,&FOF:F:FFFF/FO;\"\"#%%maxnGF:/F9;F:&%\"NG6#F8F:-F06$,&*&&%\"BG6&F7,&F 8F:F:FFF9F:F:-F<6#-F?6#,&FBF:&%\"bG6$F]oF9FFF:F:-FH6#-F06$*(&F[o6&F7F] oF9FOF:FPFF)FboFRFFFSF:/F9;F:&FZ6#F]oF:-FH6#*&&%'lambdaG6$F7F8F:FBF:F: F*F*F*" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 47 "############## #################################" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 43 "# initialise parameters using configuration" }}} {EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 47 "######################### ######################" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 44 "for l from 1 to NN do p[invpsi(l)]:=P[l];od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6$\"\"\"F'$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6$\"\"#\"\"\"^$$!5$yKb9='eXC8!#>$!5o(R&=NHK)p7#!#?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6$\"\"#F'^$$!5$yKb9='eXC8!#>$\" 5o(R&=NHK)p7#!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6$\"\"#\"\" $$\"51e?A2/'oIz'!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6$\"\"$ \"\"\"$!59%H!G;dv3C:!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6$\" \"$\"\"#$!5!H;Ah![')*4&))!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"p G6$\"\"$F'$\"5V5D*o>U(=49!#>" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 105 "# main parameter of the family. t=0 is the noded case. Algor ithm converges quickly for small values of t." }}}{EXCHG {PARA 230 "> \+ " 0 "" {MPLTEXT 1 -1 6 "t:=.1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" tG$\"\"\"!\"\"" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 67 "co:=pr oc(x,y) if type(x,even) then y else conjugate(y) fi end proc;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#coGf*6$%\"xG%\"yG6\"F)F)@%-%%typeG6 $9$%%evenG9%-%*conjugateG6#F0F)F)F)" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 440 "initparam:=proc() global a,b,c,r ;for k from 1 to M do for i from 1 to N[k] do a[k,i]:=co(k,p[k,i]);b[k,i]:=co(k+1,p[k,i] );c[3,k,i]:=C[k]*t;r[k,i]:=c[3,k,i]^2/4;od;od;for k from 1 to M do for i from 1 to N[k] do f[k,i]:=sum(c[3,k,i]*c[3,k,j]/(a[k,i]-a[k,j]),j=1 ..i-1)+sum(c[3,k,i]*c[3,k,j]/(a[k,i]-a[k,j]),j=i+1..N[k])-sum(c[3,k,i] *c[3,k-1,j]/(a[k,i]-b[k-1,j]),j=1..N[k-1]);c[1,k,i]:=-Re(f[k,i]):c[2,k ,i]:=(-1)^k*Im(f[k,i]);od;od; end proc:" }}{PARA 237 "" 0 "" {TEXT -1 0 "" }}{PARA 7 "" 1 "" {TEXT -1 67 "Warning, `k` is implicitly declare d local to procedure `initparam`\n" }}{PARA 7 "" 1 "" {TEXT -1 67 "War ning, `i` is implicitly declared local to procedure `initparam`\n" }} {PARA 7 "" 1 "" {TEXT -1 67 "Warning, `f` is implicitly declared local to procedure `initparam`\n" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 12 "initparam();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"!F$" }}} {EXCHG {PARA 230 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 45 "# do not change the following normalisations." }}} {EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 268 "normalisation := proc() \+ global a,b,c,lambda; for k from 1 to M+1 do lambda[1,k]:=1;lambda[2,k] :=(-1)^(k+1)*I;lambda[3,k]:=0;od:for k from 1 to M do b[k,N[k]]:=evalc (conjugate(a[k,N[k]])):for nu from 1 to 2 do c[nu,k,N[k]]:=-sum(c[nu,k ,jj],jj=1..N[k]-1):od:od:end proc:" }}{PARA 237 "" 0 "" {TEXT -1 0 "" }}{PARA 7 "" 1 "" {TEXT -1 71 "Warning, `k` is implicitly declared loc al to procedure `normalisation`\n" }}{PARA 7 "" 1 "" {TEXT -1 72 "Warn ing, `nu` is implicitly declared local to procedure `normalisation`\n " }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 17 "normalisation ();" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$!\"!\"\"!" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 45 "#############################################" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 34 "# solve linear system \+ by iteration" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "# (see sect ion 3.2)" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 45 "############ #################################" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 38 "# set order to truncate Laurent series" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 8 "maxn:=6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%maxnG\"\"'" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 13 "maxiter:=500;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(maxiterG \"$+&" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 76 "A:=array(1..3,0 ..M+1,1..maxN,1..maxn):B:=array(1..3,0..M+1,1..maxN,1..maxn):" }}} {EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 180 "for nu from 1 to 3 do fo r k from 1 to M do for i from 1 to N[k] do A[nu,k,i,1]:=c[nu,k,i];B[nu ,k,i,1]:=-c[nu,k,i];for m from 2 to maxn do A[nu,k,i,m]:=0:B[nu,k,i,m] :=0:od:od;od;od;" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 22 "# th e Kronecker symbol" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 52 "de lta:=proc(x,y) if (x=y) then 1 else 0 fi end proc;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&deltaGf*6$%\"xG%\"yG6\"F)F)@%/9$9%\"\"\"\"\"!F)F)F )" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 1036 "build:=proc() glo bal s,A,B:for nu from 1 to 3 do for k from 1 to M do for i from 1 to N [k] do A[nu,k,i,1]:=c[nu,k,i];B[nu,k,i,1]:=-c[nu,k,i]:od;od;od;for nu \+ from 1 to 3 do for k from 1 to M do for i from 1 to N[k] do for l from 1 to maxn-1 do BB[nu,k,i,l+1]:=(-1)^l*r[k,i]^l*sum( binomial(n+l-2,l- 1)*( sum(A[nu,k,j,n]/(a[k,i]-a[k,j])^(n+l-1),j=1..i-1)+sum(A[nu,k,j,n] /(a[k,i]-a[k,j])^(n+l-1),j=i+1..N[k])+sum(B[nu,k-1,j,n]/(a[k,i]-b[k-1, j])^(n+l-1),j=1..N[k-1]) ),n=1..maxn)-delta(l,1)*lambda[nu,k]*r[k,i]^ l:AA[nu,k,i,l+1]:=(-1)^l*r[k,i]^l*sum( binomial(n+l-2,l-1)*( sum(B[nu, k,j,n]/(b[k,i]-b[k,j])^(n+l-1),j=1..i-1)+sum(B[nu,k,j,n]/(b[k,i]-b[k,j ])^(n+l-1),j=i+1..N[k])+sum(A[nu,k+1,j,n]/(b[k,i]-a[k+1,j])^(n+l-1),j= 1..N[k+1]) ),n=1..maxn)-delta(l,1)*lambda[nu,k+1]*r[k,i]^l:s:=s+abs(B B[nu,k,i,l+1]-B[nu,k,i,l+1])+abs(AA[nu,k,i,l+1]-A[nu,k,i,l+1]):od:od:o d:od:for nu from 1 to 3 do for k from 1 to M do for i from 1 to N[k] d o for l from 1 to maxn-1 do B[nu,k,i,l+1]:=BB[nu,k,i,l+1]:A[nu,k,i,l+1 ]:=AA[nu,k,i,l+1]:od:od:od:od:end proc:" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `nu` is implicitly declared local to procedure `build`\n " }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `k` is implicitly declared l ocal to procedure `build`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, ` i` is implicitly declared local to procedure `build`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `l` is implicitly declared local to procedu re `build`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `BB` is implicit ly declared local to procedure `build`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `AA` is implicitly declared local to procedure `build`\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "verbose:=true;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(verboseG%%trueG" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 412 "Build:=proc () global s,A,B,c:iter:=0 ;for nu from 1 to 3 do for k from 1 to M do for i from 1 to N[k] do A[ nu,k,i,1]:=c[nu,k,i];B[nu,k,i,1]:=-c[nu,k,i];for m from 2 to maxn do A [nu,k,i,m]:=0:B[nu,k,i,m]:=0:od:od;od;od;while true do s:=0:build():if (verbose) then print(s):fi:if (s<10^(2-Digits)) then print(\"convergin g\"):break:fi:iter:=iter+1:if(iter>maxiter)then print(\"failed to conv erge\"):break:fi:od:end proc:" }}{PARA 7 "" 1 "" {TEXT -1 66 "Warning, `iter` is implicitly declared local to procedure `Build`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `nu` is implicitly declared local to p rocedure `Build`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `k` is imp licitly declared local to procedure `Build`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `i` is implicitly declared local to procedure `B uild`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `m` is implicitly dec lared local to procedure `Build`\n" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 -1 8 "Build();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0[[cNZ O;\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0W'p!*HB6:!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0%HPrk^]?!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0r[@7'p1()!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" 0u8O))=vb\"!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"01:[pDv<'!#C" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0akI'4C_5!#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0zxGm%GpY!#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0 S^u4HF>)!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q+converging6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0p0t\\,9O&!#H$!\"!\"\"!" }}} {EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 60 "######################### ###################################" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 58 "# now solve B-Period problem and phi_1^2+phi_2^2+phi _3^2=0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "# (see section 3. 2)" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 60 "################## ##########################################" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 104 "# compute residues of phi_1^2+phi_2^2+phi_3^2 \+ by hand, since formal computation of residues is very slow" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "# (see formula at the end of sectio n 3.1)" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 46 "# residue of p hi_1^2+phi_2^2+phi_3^2 at a[k,i]" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 379 "Resa:=proc(k,i) s:=0: for nu from 1 to 3 do s:=s+2* lambda[nu,k]*A[nu,k,i,1]:for nn from 1 to maxn do for mm from 1 to max n do ss:=0:for j from 1 to N[k] do if j<>i then ss:=ss+A[nu,k,j,mm]/(a [k,i]-a[k,j])^(nn+mm-1):fi:od:for j from 1 to N[k-1] do ss:=ss+B[nu,k- 1,j,mm]/(a[k,i]-b[k-1,j])^(nn+mm-1):od: s:=s+2*A[nu,k,i,nn]*binomial(n n+mm-2,nn-1)*(-1)^(nn-1)*ss:od:od:od:s:end proc:" }}{PARA 237 "" 0 "" {TEXT -1 0 "" }}{PARA 7 "" 1 "" {TEXT -1 62 "Warning, `s` is implicitl y declared local to procedure `Resa`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `nu` is implicitly declared local to procedure `Resa`\n" }} {PARA 7 "" 1 "" {TEXT -1 63 "Warning, `nn` is implicitly declared loca l to procedure `Resa`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `mm` \+ is implicitly declared local to procedure `Resa`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `ss` is implicitly declared local to procedure ` Resa`\n" }}{PARA 7 "" 1 "" {TEXT -1 62 "Warning, `j` is implicitly dec lared local to procedure `Resa`\n" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 46 "# residue of phi_1^2+phi_2^2+phi_3^2 at b[k,i]" }}} {EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 380 "Resb:=proc(k,i) s:=0: fo r nu from 1 to 3 do s:=s+2*lambda[nu,k+1]*B[nu,k,i,1]:for nn from 1 to maxn do for mm from 1 to maxn do ss:=0:for j from 1 to N[k] do if j<> i then ss:=ss+B[nu,k,j,mm]/(b[k,i]-b[k,j])^(nn+mm-1):fi:od:for j from \+ 1 to N[k+1] do ss:=ss+A[nu,k+1,j,mm]/(b[k,i]-a[k+1,j])^(nn+mm-1):od:s: =s+2*B[nu,k,i,nn]*binomial(nn+mm-2,nn-1)*(-1)^(nn-1)*ss:od:od:od:s:end proc:" }}{PARA 7 "" 1 "" {TEXT -1 62 "Warning, `s` is implicitly decl ared local to procedure `Resb`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warni ng, `nu` is implicitly declared local to procedure `Resb`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `nn` is implicitly declared local to p rocedure `Resb`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `mm` is imp licitly declared local to procedure `Resb`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `ss` is implicitly declared local to procedure `Resb` \n" }}{PARA 7 "" 1 "" {TEXT -1 62 "Warning, `j` is implicitly declared local to procedure `Resb`\n" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 58 "# residue of (z-a[k,i])(phi_1^2+phi_2^2+phi_3^2) at a[k,i]" } }}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 397 "Resav:=proc(k,i) s:=0: for nu from 1 to 3 do s:=s+A[nu,k,i,1]^2+2*lambda[nu,k]*A[nu,k,i,2]:f or nn from 1 to maxn-1 do for mm from 1 to maxn do ss:=0:for j from 1 \+ to N[k] do if j<>i then ss:=ss+A[nu,k,j,mm]/(a[k,i]-a[k,j])^(nn+mm-1): fi:od:for j from 1 to N[k-1] do ss:=ss+B[nu,k-1,j,mm]/(a[k,i]-b[k-1,j] )^(nn+mm-1):od:s:=s+2*A[nu,k,i,nn+1]*binomial(nn+mm-2,nn-1)*(-1)^(nn-1 )*ss:od:od:od:s:end proc:" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `s` is implicitly declared local to procedure `Resav`\n" }}{PARA 7 "" 1 " " {TEXT -1 64 "Warning, `nu` is implicitly declared local to procedure `Resav`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `nn` is implicitly declared local to procedure `Resav`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `mm` is implicitly declared local to procedure `Resav`\n" }} {PARA 7 "" 1 "" {TEXT -1 64 "Warning, `ss` is implicitly declared loca l to procedure `Resav`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `j` \+ is implicitly declared local to procedure `Resav`\n" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 19 "Nparam:=9*NN-5*M-3;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%'NparamG\"#X" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 54 "# encode all parameters into a vector v of size Npar am" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 455 "xatov:=(kk,ii)->i i+sum(N[ll],ll=1..kk-1):yatov:=(kk,ii)->NN-2+ii+sum(N[ll],ll=1..kk-1): xbtov:=(kk,ii)->2*NN-4+ii+sum(N[ll]-1,ll=1..kk-1):ybtov:=(kk,ii)->3*NN -M-4+ii+sum(N[ll]-1,ll=1..kk-1):c3tov:=(kk,ii)->4*NN-2*M-4+ii+sum(N[ll ]-1,ll=1..kk-1):c1tov:=(kk,ii)->5*NN-3*M-3+ii+sum(N[ll]-1,ll=1..kk-1): c2tov:=(kk,ii)->6*NN-4*M-3+ii+sum(N[ll]-1,ll=1..kk-1):xrtov:=(kk,ii)-> 7*NN-5*M-3+ii+sum(N[ll],ll=1..kk-1):yrtov:=(kk,ii)->8*NN-5*M-3+ii+sum( N[ll],ll=1..kk-1):" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 465 "p aramtov:=proc(v) global a,b,c,r; for k from 1 to M do for i from 1 to \+ N[k] -2*delta(k,M) do v[xatov(k,i)]:=Re(a[k,i]):v[yatov(k,i)]:=Im(a[k, i]):od: for i from 1 to N[k]-1 do v[xbtov(k,i)]:=Re(b[k,i]):v[ybtov(k, i)]:=Im(b[k,i]):od: for i from 1 to N[k]-1+delta(k,M) do v[c3tov(k,i)] :=c[3,k,i]:od: for i from 1 to N[k]-1 do v[c1tov(k,i)]:=c[1,k,i]:v[c2t ov(k,i)]:=c[2,k,i]:od:for i from 1 to N[k] do v[xrtov(k,i)]:=Re(r[k,i] ):v[yrtov(k,i)]:=Im(r[k,i]):od:od: end proc:" }}{PARA 7 "" 1 "" {TEXT -1 66 "Warning, `k` is implicitly declared local to procedure `paramto v`\n" }}{PARA 7 "" 1 "" {TEXT -1 66 "Warning, `i` is implicitly declar ed local to procedure `paramtov`\n" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 523 "vtoparam:=proc(v) global a,b,c,r; for k from 1 to M do for i from 1 to N[k] -2*delta(k,M) do a[k,i]:=v[xatov(k,i)]+I*v[ya tov(k,i)]:od: for i from 1 to N[k]-1 do b[k,i]:=v[xbtov(k,i)]+I*v[ybto v(k,i)]:od: for i from 1 to N[k]-1+delta(k,M) do c[3,k,i]:=v[c3tov(k,i )]:od: for i from 1 to N[k]-1 do c[1,k,i]:=v[c1tov(k,i)]:c[2,k,i]:=v[c 2tov(k,i)]:od:for i from 1 to N[k] do r[k,i]:=v[xrtov(k,i)]+I*v[yrtov( k,i)]:od:c[1,k,N[k]]:=-sum(c[1,k,ii],ii=1..N[k]-1):c[2,k,N[k]]:=-sum(c [2,k,ii],ii=1..N[k]-1):od:normalisation(): end proc:" }}{PARA 7 "" 1 " " {TEXT -1 66 "Warning, `k` is implicitly declared local to procedure \+ `vtoparam`\n" }}{PARA 7 "" 1 "" {TEXT -1 66 "Warning, `i` is implicitl y declared local to procedure `vtoparam`\n" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 50 "# compute all equations and put them in a vecto r w" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 49 "Q:=(k,z)->phi(1,k ,z)^2+phi(2,k,z)^2+phi(3,k,z)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"QGf*6$%\"kG%\"zG6\"6$%)operatorG%&arrowGF),(*$)-%$phiG6%\"\"\"9$9%\" \"#F3F3*$)-F16%F6F4F5F6F3F3*$)-F16%\"\"$F4F5F6F3F3F)F)F)" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 138 "period:=(nu,k,i)->Re(X(nu,k,a[k ,i]+sqrt(r[k,i]))-X(nu,k,a[k,1]+sqrt(r[k,1]))+X(nu,k+1,b[k,1]+sqrt(r[k ,1]))-X(nu,k+1,b[k,i]+sqrt(r[k,i])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'periodGf*6%%#nuG%\"kG%\"iG6\"6$%)operatorG%&arrowGF*-%#ReG6#,*-% \"XG6%9$9%,&&%\"aG6$F69&\"\"\"-%%sqrtG6#&%\"rGF:F6#&FAFFF " 0 "" {MPLTEXT 1 -1 730 "equ ations:=proc(w) global N,M,phi,X,A,B:for k from 1 to M do for i from 1 to N[k] do Ra[k,i]:=Resa(k,i):Rb[k,i]:=Resb(k,i):RRa[k,i]:=Resav(k,i) :od:od:for k from 1 to M do for i from 1 to N[k] -2*delta(k,M) do w[xa tov(k,i)]:=1/t^2*Re(Rb[k,i]):w[yatov(k,i)]:=1/t^2*Im(Rb[k,i]):od: for \+ i from 1 to N[k]-1 do w[xbtov(k,i)]:=1/t^2*Re(Ra[k,i]):w[ybtov(k,i)]:= 1/t^2*Im(Ra[k,i]):od: for i from 1 to N[k]-1 do w[c3tov(k,i)]:=1/t*per iod(3,k,i+1):od: for i from 1 to N[k]-1 do w[c1tov(k,i)]:=period(1,k,i +1):w[c2tov(k,i)]:=period(2,k,i+1):od:for i from 1 to N[k] do w[xrtov( k,i)]:=1/t^2*Re(RRa[k,i]):w[yrtov(k,i)]:=1/t^2*Im(RRa[k,i]):od:od:w[c3 tov(M,N[M])]:=1/t^2*Re(sum(sum(p[kk,ii]*co(kk+1,Rb[kk,ii]),ii=N[kk]-1. .N[kk]),kk=M..M)):end proc:" }}{PARA 7 "" 1 "" {TEXT -1 67 "Warning, ` k` is implicitly declared local to procedure `equations`\n" }}{PARA 7 "" 1 "" {TEXT -1 67 "Warning, `i` is implicitly declared local to proc edure `equations`\n" }}{PARA 7 "" 1 "" {TEXT -1 68 "Warning, `Ra` is i mplicitly declared local to procedure `equations`\n" }}{PARA 7 "" 1 " " {TEXT -1 68 "Warning, `Rb` is implicitly declared local to procedure `equations`\n" }}{PARA 7 "" 1 "" {TEXT -1 69 "Warning, `RRa` is impli citly declared local to procedure `equations`\n" }}}{EXCHG {PARA 230 " > " 0 "" {MPLTEXT 1 -1 13 "equations(w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0uI**[ " 0 "" {MPLTEXT 1 -1 32 "# compute (approximate) jacobian" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 1937 "jacob:=proc() global N,M,phi,X,A,B:vvv:=vector(Npa ram):vv:=vector(Nparam):w:=vector(Nparam):paramtov(vv):vtoparam(vvv):f or k from 1 to M do for i from 1 to N[k] do Ra[k,i]:=2*c[1,k,i]+2*lamb da[2,k]*c[2,k,i]+2*sum(c[3,k,i]*c[3,k,jj]/(a[k,i]-a[k,jj]),jj=1..i-1)+ 2*sum(c[3,k,i]*c[3,k,jj]/(a[k,i]-a[k,jj]),jj=i+1..N[k])-2*sum(c[3,k,i] *c[3,k-1,jj]/(a[k,i]-b[k-1,jj]),jj=1..N[k-1]):Rb[k,i]:=-2*c[1,k,i]-2*l ambda[2,k+1]*c[2,k,i]+2*sum(c[3,k,i]*c[3,k,jj]/(b[k,i]-b[k,jj]),jj=1.. i-1)+2*sum(c[3,k,i]*c[3,k,jj]/(b[k,i]-b[k,jj]),jj=i+1..N[k])-2*sum(c[3 ,k,i]*c[3,k+1,jj]/(b[k,i]-a[k+1,jj]),jj=1..N[k+1]):od:od:for k from 1 \+ to M do for i from 2 to N[k] do for nu from 1 to 3 do Per[nu,k,i]:=eva lc(Re(lambda[nu,k]*(a[k,i]-a[k,1])+lambda[nu,k+1]*(b[k,1]-b[k,i]) +c[n u,k,i]*log(r[k,i])-c[nu,k,1]*log(r[k,1])+sum( c[nu,k,jj]*(log(a[k,i]-a [k,jj])+log(b[k,i]-b[k,jj])),jj=1..i-1) +sum( c[nu,k,jj]*(log(a[k,i]- a[k,jj])+log(b[k,i]-b[k,jj])),jj=i+1..N[k]) -sum(c[nu,k,jj]*(log(a[k,1 ]-a[k,jj])+log(b[k,1]-b[k,jj])),jj=2..N[k]) +sum(c[nu,k-1,jj]*(-log(a[ k,i]-b[k-1,jj])+log(a[k,1]-b[k-1,jj])),jj=1..N[k-1]) +sum( c[nu,k+1,j j]*(log(b[k,1]-a[k+1,jj])-log(b[k,i]-a[k+1,jj])),jj=1..N[k+1]) )) :od: od:od:for k from 1 to M do for i from 1 to N[k] -2*delta(k,M) do w[xat ov(k,i)]:=1/t^2*evalc(Re(Rb[k,i])):w[yatov(k,i)]:=1/t^2*evalc(Im(Rb[k, i])):od: for i from 1 to N[k]-1 do w[xbtov(k,i)]:=1/t^2*evalc(Re(Ra[k, i])):w[ybtov(k,i)]:=1/t^2*evalc(Im(Ra[k,i])):od: for i from 1 to N[k]- 1 do w[c3tov(k,i)]:=1/t*Per[3,k,i+1]: w[c1tov(k,i)]:=Per[1,k,i+1]:w[c2 tov(k,i)]:=Per[2,k,i+1]:od:for i from 1 to N[k] do w[xrtov(k,i)]:=1/t^ 2*evalc(Re(c[1,k,i]^2+c[2,k,i]^2+c[3,k,i]^2-4*r[k,i])):w[yrtov(k,i)]:= 1/t^2*evalc(Im(c[1,k,i]^2+c[2,k,i]^2+c[3,k,i]^2-4*r[k,i])):od:od:w[c3t ov(M,N[M])]:=1/t^2*evalc(Re(sum(sum(p[kk,ii]*co(kk+1,Rb[kk,ii]),ii=N[k k]-1..N[kk]),kk=M..M))): vtoparam(vv):S:=\{\}:for i from 1 to Nparam d o S:=\{op(S),vvv[i]=vv[i]\}:od:evalf(subs(S,jacobian(w,vvv))):end proc :" }}{PARA 7 "" 1 "" {TEXT -1 65 "Warning, `vvv` is implicitly declare d local to procedure `jacob`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning , `vv` is implicitly declared local to procedure `jacob`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `w` is implicitly declared local to proc edure `jacob`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `k` is implic itly declared local to procedure `jacob`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `i` is implicitly declared local to procedure `jacob` \n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `Ra` is implicitly declare d local to procedure `jacob`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning , `Rb` is implicitly declared local to procedure `jacob`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `nu` is implicitly declared local to pro cedure `jacob`\n" }}{PARA 7 "" 1 "" {TEXT -1 65 "Warning, `Per` is imp licitly declared local to procedure `jacob`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `S` is implicitly declared local to procedure `j acob`\n" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 30 "# solve equat ions by iteration" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "verbos e:=false;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(verboseG%&falseG" }}} {EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 444 "iter:=0:v:=vector(Nparam ):w:=vector(Nparam):while true do print(\"computing jacobian...\"):J:= jacob():print(\"inversing matrix...\"):Jinv:=inverse(J): print(\"compu ting equations...\"):s:=0:build():equations(w):print(\"done\"):s:=s+ev alf(norm(w,1)):print(s):if (s<10^(4-Digits)) then print(\"converging a fter\",iter,\"loops\"):break:fi:paramtov(v):v:=evalm(v-Jinv &* w):vtop aram(v):iter:=iter+1:if(iter>maxiter)then print(\"failed to converge\" );break:fi:od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q6computing~jacobian ...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4inversing~matrix...6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~equations...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q%done6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"0:<2YHK?(!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q6computing~jacobi an...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4inversing~matrix...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~equations...6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#Q%done6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0qdnvR5?#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q6co mputing~jacobian...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4inversing~ matrix...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~equations. ..6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q%done6\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"0.h<:KE$=!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q6 computing~jacobian...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4inversin g~matrix...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~equation s...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q%done6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0u$)ouc%RT!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# Q6computing~jacobian...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4invers ing~matrix...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~equati ons...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q%done6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0%oX#o')RT(!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q6computing~jacobian...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4inv ersing~matrix...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~equ ations...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q%done6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0[e4C.JG&!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q6computing~jacobian...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4 inversing~matrix...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~ equations...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q%done6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0M.Epov!Q!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q6computing~jacobian...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4inversing~matrix...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~equations...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q %done6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0oJQldq(G!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q6computing~jacobian...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4inversing~matrix...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~equations...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q%done6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0bXo6_b ;#!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q6computing~jacobian...6\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4inversing~matrix...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~equations...6\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#Q%done6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0.H([ >jP;!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q6computing~jacobian...6\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4inversing~matrix...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~equations...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q%done6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0U R8VC5C\"!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q6computing~jacobian... 6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4inversing~matrix...6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~equations...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q%done6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"0MBqDP)3%*!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q6computing~jacob ian...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4inversing~matrix...6\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~equations...6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#Q%done6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0rmmD498(!#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q6co mputing~jacobian...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4inversing~ matrix...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~equations. ..6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q%done6\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"0IQm#*=wi&!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%Q 1converging~after6\"\"#8Q&loopsF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "# the following line is used to push the parameter t \+ by smallincrements." }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 -1 505 " #incr:=1.05:ok:=true:while ok do iter:=0:v:=vector(Nparam):w:=vector(N param):t:=t*incr:print(\"t=\",t):for k from 1 to M do for i from 1 to \+ N[k] do c[3,k,i]:=c[3,k,i]*incr:od:od:while true do J:=jacob():Jinv:=i nverse(J): s:=0: Build() :equations(w):s:=s+evalf(norm(w,1)):print(s): if (s<10^(6-Digits)) then print(\"converging after\",iter,\"loops\"):s ave t,a,b,c,r,FILE:break:fi:paramtov(v):v:=evalm(v-Jinv &* w):vtoparam (v):iter:=iter+1:if(iter>maxiter)then print(\"failed to converge\"):ok :=false:break:fi:od:od:" }}{PARA 239 "" 0 "" {TEXT -1 0 "" }}{PARA 239 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 44 "############################################" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 31 "# Check equations are satisfied" }}} {EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 44 "######################### ###################" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 19 "O rder:=max(maxn,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OrderG\"\"'" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 42 "res:=(f,z0)->residue(s eries(f,z=z0),z=z0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$resGf*6$%\" fG%#z0G6\"6$%)operatorG%&arrowGF)-%(residueG6$-%'seriesG6$9$/%\"zG9%F4 F)F)F)" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 16 "# period probl em" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 62 "aa:=(k,i)->a[k,i]+ sqrt(r[k,i]);bb:=(k,i)->b[k,i]+sqrt(r[k,i]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#aaGf*6$%\"kG%\"iG6\"6$%)operatorG%&arrowGF),&&%\"aG6 $9$9%\"\"\"-%%sqrtG6#&%\"rGF0F3F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bbGf*6$%\"kG%\"iG6\"6$%)operatorG%&arrowGF),&&%\"bG6$9$9%\"\"\"- %%sqrtG6#&%\"rGF0F3F)F)F)" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 151 "for nu from 1 to 3 do for k from 1 to M do for i from 2 to N[k ] do print(X(nu,k,aa(k,i))-X(nu,k,aa(k,1))+X(nu,k+1,bb(k,1))-X(nu,k+1, bb(k,i))):od:od:od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!$4\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!#**!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!\")!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$A\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0LoL$3Gai!#G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/ x'z6Uvx$!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/\")R7`o]g!#I" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!#=!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!#8!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!\")!#:" }}}{EXCHG {PARA 230 "> " 0 " " {MPLTEXT 1 -1 20 "# residues are zero " }}}{EXCHG {PARA 230 "> " 0 " " {MPLTEXT 1 -1 120 "for k from 1 to M do for i from 1 to N[k] do prin t(k,i,\" \",res(phi(1,k,z)^2+phi(2,k,z)^2+phi(3,k,z)^2,a[k,i])):od:o d: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"\"F#Q$~~~6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#\"\"\"Q$~~~6\"^$$!$L\"!#<$!%$Q'!#>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#F#Q$~~~6\"^$$!$6\"!#<$\"%,o!#>" } }{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#\"\"$Q$~~~6\"^$$\"#K!#;$!0>*))pZ )e$=!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"$\"\"\"Q$~~~6\"^$$!$8\" !#;$\"0&>\"RGT<^)!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"$\"\"#Q$~~ ~6\"^$$\"$!=!#<$\"0)[(4g')3N\"!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6& \"\"$F#Q$~~~6\"^$$\"#Y!#;$!0V98!yG8m!#J" }}}{EXCHG {PARA 230 "> " 0 " " {MPLTEXT 1 -1 126 "for k from 1 to M do for i from 1 to N[k] do prin t(k,i,\" \",res(phi(1,k+1,z)^2+phi(2,k+1,z)^2+phi(3,k+1,z)^2,b[k,i]) ):od:od: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"\"F#Q$~~~6\"^$$!%vw!# >$!0#R%>T)3MU!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#\"\"\"Q$~~~6 \"^$$\"$l%!#<$!%7B!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#F#Q$~~~6 \"^$$\"$'[!#<$\"%sA!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#\"\"$Q$ ~~~6\"^$$!#W!#;$!0d&>@7!eQ\"!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\" \"$\"\"\"Q$~~~6\"^$$!#6!#;$!0i?PBkTE\"!#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"$\"\"#Q$~~~6\"^$$\"#f!#<$\"0c/qBFT')*!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"$F#Q$~~~6\"^$$\"\"#!#;$!0dnYf&[^)*!#J" } }}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 133 "for k from 1 to M do f or i from 1 to N[k] do print(k,i,\" \",res((phi(1,k,z)^2+phi(2,k,z)^ 2+phi(3,k,z)^2)*(z-a[k,i]),a[k,i])):od:od: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"\"F#Q$~~~6\"^$$!#G!#;$\"/-)f(3&yy%!#K" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&\"\"#\"\"\"Q$~~~6\"^$$!##)!#<$!$\\$!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#F#Q$~~~6\"^$$!#()!#<$\"$]$!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#\"\"$Q$~~~6\"^$$\"#!#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"$F#Q$~~ ~6\"^$$!#()!#<$\"/=(y@Dq0$!#L" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 21 "#residues at the ends" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 126 "for k from 1 to M+1 do for nu from 1 to 3 do print( k,nu,\" \", sum(c[nu,k-1,ii],ii=1..N[k-1])-sum(c[nu,k,ii],ii=1..N[k])) :od:od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"\"F#Q\"~6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"\"\"\"#Q\"~6\"\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&\"\"\"\"\"$Q\"~6\"$!&]f\"!\"&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&\"\"#\"\"\"Q\"~6\"$\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#F#Q\"~6\"$\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#\"\"$Q\"~6\"$!/85Sn=8!)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6& \"\"$\"\"\"Q\"~6\"$\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"$\" \"#Q\"~6\"$\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"$F#Q\"~6\"$! 0c&=;Z+p6!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"%\"\"\"Q\"~6\"$\" \"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"%\"\"#Q\"~6\"$\"\"!F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"%\"\"$Q\"~6\"$\"0p&>!RB`c$!#:" }}} {EXCHG {PARA 230 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 -1 22 "# phi's are invariant;" }}}{EXCHG {PARA 230 "> " 0 " " {MPLTEXT 1 -1 167 "for nu from 1 to 3 do for k from 1 to M do for i \+ from 1 to N[k] do print(nu,k,i,\" \",phi(nu,k,a[k,i]+sqrt(r [k,i]))+phi(nu,k+1,b[k,i]+sqrt(r[k,i])));od;od;od;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6'\"\"\"F#F#Q-~~~~~~~~~~~~6\"^$$!)Zp[:!#:$!.T,%>H(4$!#I " }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"\"\"#F#Q-~~~~~~~~~~~~6\"^$$! ($*RL\"!#:$\"*Olb:$!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"\"\"#F $Q-~~~~~~~~~~~~6\"^$$!(\"*RL\"!#:$!*Klb:$!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"\"\"#\"\"$Q-~~~~~~~~~~~~6\"^$$!)x>WA!#:$!-'\\>JSd '!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"\"\"$F#Q-~~~~~~~~~~~~6\" ^$$\")(o2)=!#:$!0B')fkZsX#!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\" \"\"\"$\"\"#Q-~~~~~~~~~~~~6\"^$$!'NL>!#:$\"0biP4C8/\"!#J" }}{PARA 11 " " 1 "" {XPPMATH 20 "6'\"\"\"\"\"$F$Q-~~~~~~~~~~~~6\"^$$\")nDxU!#:$\"/_ G#3VLD\"!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#\"\"\"F$Q-~~~~~~~~ ~~~~6\"^$$\".$RDWy$y%!#I$!'vd))!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6' \"\"#F#\"\"\"Q-~~~~~~~~~~~~6\"^$$\")6\\&*H!#;$\"&j>'!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6'\"\"#F#F#Q-~~~~~~~~~~~~6\"^$$!)5\\&*H!#;$\"&j>'! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#F#\"\"$Q-~~~~~~~~~~~~6\"^$$ \"-jxj4X**!#H$\"(*))z>!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#\"\" $\"\"\"Q-~~~~~~~~~~~~6\"^$$!.;Z)QbsI!#H$\"'F#3%!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6'\"\"#\"\"$F#Q-~~~~~~~~~~~~6\"^$$\".Z_9^QR*!#I$\"&UL)! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#\"\"$F$Q-~~~~~~~~~~~~6\"^$$ \"-of1[:Q!#H$!(`'[H!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$\"\"\"F $Q-~~~~~~~~~~~~6\"^$$!(7N+(!#9$\"0yO*\\-/Qf!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$\"\"#\"\"\"Q-~~~~~~~~~~~~6\"^$$\"(!>+5!#9$!)O#)\\% )!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$\"\"#F$Q-~~~~~~~~~~~~6\"^ $$\"(\">+5!#9$\")O#)\\%)!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$\" \"#F#Q-~~~~~~~~~~~~6\"^$$!(^h\"y!#9$\".1p^T)\\F!#H" }}{PARA 11 "" 1 " " {XPPMATH 20 "6'\"\"$F#\"\"\"Q-~~~~~~~~~~~~6\"^$$!)i&*QB!#9$\".59'>Qh =!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$F#\"\"#Q-~~~~~~~~~~~~6\"^ $$\"'2*)H!#9$\"-2[RB_W!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$F#F# Q-~~~~~~~~~~~~6\"^$$\");6< \+ " 0 "" {MPLTEXT 1 -1 59 "############################################# ##############" }}}{EXCHG {PARA 245 "> " 0 "" {MPLTEXT 1 -1 50 "# grap hics : use triangulation generated by matlab" }}}{EXCHG {PARA 245 "> \+ " 0 "" {MPLTEXT 1 -1 59 "############################################# ##############" }}}{EXCHG {PARA 245 "> " 0 "" {MPLTEXT 1 -1 12 "with(p lots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "# read saved valu es of parameters and triangulations of domains" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "read \"133.data\":verbose:=true;Build();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(verboseG%%trueG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"0%='Rw:oj#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"0(*zflZKc\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0l*Ge&GS?*!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0%o[oK)yx&!#=" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"0(G+/f^.O!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"0NO%p.in8!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0x@&3d,\\Q!#A" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0nyA3L;n\"!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0(G9_3t1g!#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0 O!=#)fr*R#!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0<%41:iju!#G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#Q+converging6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0\\;8I(3<:!#?$!0+Hj0er+#!#L" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 43 "# compute translation vector for each piece" } }}{EXCHG {PARA 232 "> " 0 "" {MPLTEXT 1 -1 137 "for nu from 1 to 3 do \+ T[nu,1]:=0:for k from 1 to M do T[nu,k+1]:=T[nu,k]+X(nu,k,a[k,1]+sqrt( r[k,1]))-X(nu,k+1,b[k,1]+sqrt(r[k,1]));od;od:" }}}{EXCHG {PARA 245 "> \+ " 0 "" {MPLTEXT 1 -1 61 "# generate data for matlab -- used to triangu late the domains" }}}{EXCHG {PARA 245 "> " 0 "" {MPLTEXT 1 -1 38 "#cen tres:=[0,0,0,0];rayons:=[2,2,2,2];" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 -1 123 "#for k from 1 to 4 do centres[k]:=(sum(a[k,ii],ii=1 ..N[k])+sum(b[k-1,ii],ii=1..N[k-1]))/(N[k]+N[k-1]):print(centres[k]):o d:" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 -1 281 "#for k from 1 to \+ 4 do rayons[k]:=0:for i from 1 to N[k] do rr:=abs(a[k,i]-centres[k])+a bs(sqrt(r[k,i])):if rr>rayons[k] then rayons[k]:=rr:fi:od:for i from 1 to N[k-1] do rr:=abs(b[k-1,i]-centres[k])+abs(sqrt(r[k-1,i])):if rr>r ayons[k] then rayons[k]:=rr:fi:od:print(rayons[k]):od:" }}}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 -1 49 "#for k from 1 to 4 do rayons[k]: =rayons[k]+2.;od;" }}}{EXCHG {PARA 245 "> " 0 "" {MPLTEXT 1 -1 272 "#f or k from 1 to 4 do print(\"k=\",k):lprint(pdecirc(Re(centres[k]),Im(c entres[k]),rayons[k])):for i from 1 to N[k] do lprint(pdecirc(Re(a[k,i ]),Im(a[k,i]),abs(sqrt(r[k,i])))):od:for i from 1 to N[k-1] do lprint( pdecirc(Re(b[k-1,i]),Im(b[k-1,i]),abs(sqrt(r[k-1,i])))):od:od:" }} {PARA 249 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 245 "> " 0 "" {MPLTEXT 1 -1 17 "# display domains" }}}{EXCHG {PARA 251 "> " 0 "" {MPLTEXT 1 -1 59 "for k from 1 to 4 do domain[k]:=vector(nops(point1[k ]));od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'domainG6#\"\"\"-%&array G6$;F'\"%k77\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'domainG6#\"\"#-% &arrayG6$;\"\"\"\"%WR7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'domain G6#\"\"$-%&arrayG6$;\"\"\"\"%)=&7\"" 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