{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 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 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Warni ng" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 } {PSTYLE "_pstyle1" -1 214 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle6" -1 217 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle14" -1 219 1 {CSTYLE " " -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle24" -1 223 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 } {PSTYLE "_pstyle26" -1 225 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle27 " -1 226 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle28" -1 227 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle31" -1 230 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_ps tyle32" -1 231 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle33" -1 232 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 61 "################# ############################################" }}}{EXCHG {PARA 214 "> \+ " 0 "" {MPLTEXT 1 -1 41 "# COSTA Hoffman Meeks genus one family " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "# This worksheet takes as input a balanced configu ration and outputs a minimal surface" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "# as explained in the paper : Exploring the space of embedded minimal surfaces of finite total curvature" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "# Experimental mathematics 17:2, page 205 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 86 "# Important note : if you change the configura tion, you won't be able to see pictures." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "# That's because the domains are triangulated externa ly using mathlab." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "# If y ou want to see pictures, leave all parameters as they are." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 61 "#################################################### #########" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 11 "Digits:=15: " }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 13 "with(linalg):" }}} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 27 "######################### ##" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 21 "# input configurat ion" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 27 "################# ##########" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 18 "# number o f layers" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 5 "M:=2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG\"\"#" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 27 "# number of necks per layer" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 16 "N[1]:=1;N[2]:=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"NG6#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>& %\"NG6#\"\"#F'" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 15 "# size of necks" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 19 "C[1]:=1.;C[ 2]:=1.; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6#\"\"\"$F'\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6#\"\"#$\"\"\"\"\"!" }}} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 19 "# position of necks" }}} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 15 "P:=[0.,1.,-1.];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG7%$\"\"!F'$\"\"\"F'$!\"\"F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "plot1:=complexplot(\{op(P[1..N[1]] )\},axes=none,style=point,scaling=constrained,symbolsize=20,symbol=CIR CLE,color=black):plot2:=complexplot(\{op(P[N[1]+1..N[1]+N[2]])\},axes= none,style=point,color=black,scaling=constrained,symbolsize=15,symbol= CROSS):display(plot1,plot2,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 276 278 278 {PLOTDATA 2 "6(-%'CURVESG6&X0%)anythingG6\"6\"[g l'\"%!!!##\"\"\"#!!!!00000000000000000000000000000000-%'COLOURG6&%$RGB G$\"\"!F/F.F.-%&STYLEG6#%&POINTG-%'SYMBOLG6$%'CIRCLEG\"#?-F$6&X0F'F(F) [gl'\"%!!!#%\"#\"#!!!!BFF00000000000003FF00000000000000000000000000000 0000000000000000F*F0-F56$%&CROSSG\"#:-%(SCALINGG6#%,CONSTRAINEDG-%*AXE SSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!F)FK-%%FONTG6#%(DEFAULTG-%%VIEWG6$FO FO" 1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 64 "########### #####################################################" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 63 "# check that the configuration i s balanced and non-degenerate. " }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 65 "#################################################### #############" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 58 "maxN:=0 :for k from 1 to M do maxN:=max(maxN,N[k]):od:maxN;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 53 "if N[M]=1 then print(\"N[M] must be at least two\");fi;" }}} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 41 "N[0]:=0;N[M+1]:=0;NN:=sum (N[kk],kk=1..M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"NG6#\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"NG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NNG\"\"$" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 40 "psi:=(kk,ii)-> sum(N[ll],ll=1..kk-1)+ii;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiGf*6$%#kkG%#iiG6\"6$%)operatorG%&arrow GF),&-%$sumG6$&%\"NG6#%#llG/F4;\"\"\",&9$F7F7!\"\"F79%F7F)F)F)" }}} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 118 "invpsi:=proc(jj) global \+ N;k:=1;j:=jj;while true do if j<=N[k] then return (k,j) else j:=j-N[k] ;k:=k+1;fi;od;end proc: " }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `k` \+ is implicitly declared local to procedure `invpsi`\n" }}{PARA 7 "" 1 " " {TEXT -1 64 "Warning, `j` is implicitly declared local to procedure \+ `invpsi`\n" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 30 "qq:=vector (NN);FF:=vector(NN);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#qqG-%&array G6$;\"\"\"\"\"$7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FFG-%&arrayG6 $;\"\"\"\"\"$7\"" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 244 "for l from 1 to NN do (k,i):=invpsi(l):FF[l]:=sum(2*C[k]^2/(qq[l]-qq[psi( k,j)]),j=1..i-1)+sum(2*C[k]^2/(qq[l]-qq[psi(k,j)]),j=i+1..N[k])-sum(C[ k]*C[k+1]/(qq[l]-qq[psi(k+1,j)]),j=1..N[k+1])-sum(C[k]*C[k-1]/(qq[l]-q q[psi(k-1,j)]),j=1..N[k-1]):od:" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 55 "S:=\{\}:for l from 1 to NN do S:=\{op(S),qq[l]=P[l] \}:od:S;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/&%#qqG6#\"\"\"$\"\"!F*/ &F&6#\"\"#$F(F*/&F&6#\"\"$$!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# compute forces" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 17 "subs(S,eval(FF));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'vectorG6#7%$\"\"!F(F'F'Q(pprint06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "# compute jacobian matrix" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 30 "jaco:=subs(S,jacobian(FF,qq)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "# the following determinant should \+ be nonzero" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 37 "det(submat rix(jaco,1..NN-2,1..NN-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"# \"\"!" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 51 "#invjaco:=inver se(submatrix(jaco,1..NN-2,1..NN-2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 71 "#### ###################################################################" } }}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 71 "# Weierstrass data phi \+ and immersion X=Re\\int phi represented as series" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "# (see formula 3-6) " }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 71 "############################################ ###########################" }}}{EXCHG {PARA 214 "> " 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#>%$phiGf*6%%#nuG%\"kG%\"zG6\"6$%)oper atorG%&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])+sum(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 11 "" 1 "" {XPPMATH 20 "6#>%\"XG f*6%%#nuG%\"kG%\"zG6\"6$%)operatorG%&arrowGF*,(-%$sumG6$,&*&&%\"AG6&9$ 9%%\"jG\"\"\"F:-%$logG6#-%$absG6#,&9&F:&%\"aG6$F8F9!\"\"F:F:-%#ReG6#-F 06$*(&F56&F7F8F9%\"nGF:,&F:F:FOFFFF)FA,&FOF:F:FFFF/FO;\"\"#%%maxnGF:/F 9;F:&%\"NG6#F8F:-F06$,&*&&%\"BG6&F7,&F8F:F:FFF9F:F:-F<6#-F?6#,&FBF:&% \"bG6$F]oF9FFF:F:-FH6#-F06$*(&F[o6&F7F]oF9FOF:FPFF)FboFRFFFSF:/F9;F:&F Z6#F]oF:-FH6#*&&%'lambdaG6$F7F8F:FBF:F:F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 47 "###############################################" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 43 "# initialise parameters using configuratio n" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 47 "################### ############################" }}}{EXCHG {PARA 214 "> " 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$\"\"#\"\"\"$F(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6$\"\"#F'$!\"\"\"\"!" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 105 "# main parameter of the family. t=0 is the nod ed case. Algorithm converges quickly for small values of t." }}} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 6 "t:=.1;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"tG$\"\"\"!\"\"" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 67 "co:=proc(x,y) if type(x,even) then y else conjugate( y) fi end proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#coGf*6$%\"xG%\"y G6\"F)F)@%-%%typeG6$9$%%evenG9%-%*conjugateG6#F0F)F)F)" }}}{EXCHG {PARA 214 "> " 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 219 "" 0 "" {TEXT -1 0 "" }}{PARA 7 "" 1 "" {TEXT -1 67 "Warning, `k` \+ is implicitly declared local to procedure `initparam`\n" }}{PARA 7 "" 1 "" {TEXT -1 67 "Warning, `i` is implicitly declared local to procedu re `initparam`\n" }}{PARA 7 "" 1 "" {TEXT -1 67 "Warning, `f` is impli citly declared local to procedure `initparam`\n" }}}{EXCHG {PARA 214 " > " 0 "" {MPLTEXT 1 -1 12 "initparam();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"!F$" }}}{EXCHG {PARA 214 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 45 "# do not change the following no rmalisations." }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 268 "normal isation := 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 d o 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 219 "" 0 "" {TEXT -1 0 "" }}{PARA 7 "" 1 "" {TEXT -1 71 "Warning, `k` is impl icitly declared local to procedure `normalisation`\n" }}{PARA 7 "" 1 " " {TEXT -1 72 "Warning, `nu` is implicitly declared local to procedure `normalisation`\n" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 17 "no rmalisation ();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!\"!\"\"!" }}} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 45 "######################### ####################" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 34 " # solve linear system by iteration" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "# (see section 3.2)" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 45 "#############################################" }}} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 38 "# set order to truncate L aurent series" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 9 "maxn:=10 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%maxnG\"#5" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 13 "maxiter:=500;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(maxiterG\"$+&" }}}{EXCHG {PARA 214 "> " 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 214 "> " 0 "" {MPLTEXT 1 -1 180 "for nu from 1 to 3 do for k from 1 to M do for i from 1 to N[k] d o 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 214 "> \+ " 0 "" {MPLTEXT 1 -1 22 "# the Kronecker symbol" }}}{EXCHG {PARA 214 " > " 0 "" {MPLTEXT 1 -1 52 "delta:=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 214 "> " 0 "" {MPLTEXT 1 -1 1036 "build:=proc() global s,A,B:for nu from 1 to 3 do f or k from 1 to M do for i from 1 to N[k] do A[nu,k,i,1]:=c[nu,k,i];B[n u,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(BB[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: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 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 dec lared local to procedure `build`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "War ning, `k` is implicitly declared local 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` i s implicitly declared local to procedure `build`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `BB` is implicitly declared local to procedure ` build`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `AA` is implicitly d eclared 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 214 "> " 0 "" {MPLTEXT 1 -1 412 "Buil d:=proc () global s,A,B,c:iter:=0;for nu from 1 to 3 do for k from 1 t o 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(\"converging\"):break:fi:iter:=iter+1:if(iter>ma xiter)then print(\"failed to converge\"):break:fi:od:end proc:" }} {PARA 7 "" 1 "" {TEXT -1 66 "Warning, `iter` is implicitly declared lo cal to procedure `Build`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `n u` is implicitly declared local to procedure `Build`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `k` is implicitly declared local to procedu re `Build`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `i` is implicitl y declared local to procedure `Build`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `m` is implicitly declared local to procedure `Build`\n" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 -1 8 "Build();" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"0o+'eMvZI!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0'HJnV`+j!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0?a0ER)QR!#A " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0<8=2OKR'!#D" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"0Tq408Ia\"!#G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q +converging6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0u'o9:\"H\\(!#U" } }}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 60 "####################### #####################################" }}}{EXCHG {PARA 214 "> " 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 214 "> " 0 "" {MPLTEXT 1 -1 60 "################## ##########################################" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 104 "# compute residues of phi_1^2+phi_2^2+phi_3^2 by hand, since formal com putation of residues is very slow" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "# (see formula at the end of section 3.1)" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 46 "# residue of phi_1^2+phi_2^2+phi _3^2 at a[k,i]" }}}{EXCHG {PARA 214 "> " 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 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]*binomial(nn+mm-2,nn-1)*(-1)^(nn-1) *ss:od:od:od:s:end proc:" }}{PARA 219 "" 0 "" {TEXT -1 0 "" }}{PARA 7 "" 1 "" {TEXT -1 62 "Warning, `s` is implicitly declared local to proc edure `Resa`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `nu` is implic itly declared local to procedure `Resa`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `nn` is implicitly declared local to procedure `Resa`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `mm` is implicitly declared lo cal 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 declared local to procedure \+ `Resa`\n" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 46 "# residue of phi_1^2+phi_2^2+phi_3^2 at b[k,i]" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 380 "Resb:=proc(k,i) s:=0: for 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 m axn 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 declared local to procedure `R esb`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `nu` is implicitly dec lared local to procedure `Resb`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warn ing, `nn` is implicitly declared local to procedure `Resb`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `mm` is implicitly declared local to p rocedure `Resb`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `ss` is imp licitly declared local to procedure `Resb`\n" }}{PARA 7 "" 1 "" {TEXT -1 62 "Warning, `j` is implicitly declared local to procedure `Resb`\n " }}}{EXCHG {PARA 214 "> " 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 214 "> " 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]:for 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:=s s+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 local to procedure `Resav`\n" }} {PARA 7 "" 1 "" {TEXT -1 63 "Warning, `j` is implicitly declared local to procedure `Resav`\n" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 19 "Nparam:=9*NN-5*M-3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'NparamG \"#9" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 54 "# encode all par ameters into a vector v of size Nparam" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 455 "xatov:=(kk,ii)->ii+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):c3 tov:=(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 214 "> " 0 "" {MPLTEXT 1 -1 465 "paramtov:=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[c2tov(k,i)]:=c[2,k,i]:od:for i fr om 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 implici tly declared local to procedure `paramtov`\n" }}{PARA 7 "" 1 "" {TEXT -1 66 "Warning, `i` is implicitly declared local to procedure `paramto v`\n" }}}{EXCHG {PARA 214 "> " 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[yatov(k,i)]:od: for i from 1 to N[k] -1 do b[k,i]:=v[xbtov(k,i)]+I*v[ybtov(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[c2tov(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,i i],ii=1..N[k]-1):c[2,k,N[k]]:=-sum(c[2,k,ii],ii=1..N[k]-1):od:normalis ation(): end proc:" }}{PARA 7 "" 1 "" {TEXT -1 66 "Warning, `k` is imp licitly declared local to procedure `vtoparam`\n" }}{PARA 7 "" 1 "" {TEXT -1 66 "Warning, `i` is implicitly declared local to procedure `v toparam`\n" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 50 "# compute \+ all equations and put them in a vector w" }}}{EXCHG {PARA 214 "> " 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$%)operato rG%&arrowGF),(*$)-%$phiG6%\"\"\"9$9%\"\"#F3F3*$)-F16%F6F4F5F6F3F3*$)-F 16%\"\"$F4F5F6F3F3F)F)F)" }}}{EXCHG {PARA 214 "> " 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]+sq rt(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 -1 703 "equations:=proc(w) global N,M,p hi,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[xatov(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[xbto v(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*period(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[c3tov(M,N[M])]:=1/t^2*evalc(Re (b[M,1]*Rb[M,1]+b[M,2]*Rb[M,2])):end proc:" }}{PARA 7 "" 1 "" {TEXT -1 67 "Warning, `k` is implicitly declared local to procedure `equatio ns`\n" }}{PARA 7 "" 1 "" {TEXT -1 67 "Warning, `i` is implicitly decla red local to procedure `equations`\n" }}{PARA 7 "" 1 "" {TEXT -1 68 "W arning, `Ra` is implicitly declared local to procedure `equations`\n" }}{PARA 7 "" 1 "" {TEXT -1 68 "Warning, `Rb` is implicitly declared lo cal to procedure `equations`\n" }}{PARA 7 "" 1 "" {TEXT -1 69 "Warning , `RRa` is implicitly declared local to procedure `equations`\n" }}} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 13 "equations(w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0u0b\\'=-Z!#?" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 32 "# compute (approximate) jacobian" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 1903 "jacob:=proc() global N,M,phi,X ,A,B:vvv:=vector(Nparam):vv:=vector(Nparam):w:=vector(Nparam):paramtov (vv):vtoparam(vvv):for k from 1 to M do for i from 1 to N[k] do Ra[k,i ]:=2*c[1,k,i]+2*lambda[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*lambda[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]:=evalc(Re(lambda[nu,k]*(a[k,i]-a[k,1])+lambda[nu,k+1]* (b[k,1]-b[k,i]) +c[nu,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[n u,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,jj]*(log(b[k,1]-a[k+1,jj])-log(b[k,i]-a[k+1,jj])),j j=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[xatov(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: f or 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[c2tov(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[c3tov(M,N[M])]:=1/t^2*evalc(Re(b[M,1]*Rb[M,1]+b[M,2]* Rb[M,2])): vtoparam(vv):S:=\{\}:for i from 1 to Nparam do S:=\{op(S),v vv[i]=vv[i]\}:od:evalf(subs(S,jacobian(w,vvv))):end proc:" }}{PARA 7 " " 1 "" {TEXT -1 65 "Warning, `vvv` is implicitly declared local to pro cedure `jacob`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `vv` is impl icitly declared local to procedure `jacob`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `w` is implicitly declared local to procedure `jacob` \n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `k` is implicitly 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 declared local to proc edure `jacob`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `Rb` is impli citly declared local to procedure `jacob`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `nu` is implicitly declared local to procedure `jacob` \n" }}{PARA 7 "" 1 "" {TEXT -1 65 "Warning, `Per` is implicitly declar ed local to procedure `jacob`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warnin g, `S` is implicitly declared local to procedure `jacob`\n" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 30 "# solve equations by iteration" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "verbose:=false;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(verboseG%&falseG" }}}{EXCHG {PARA 214 "> \+ " 0 "" {MPLTEXT 1 -1 444 "iter:=0:v:=vector(Nparam):w:=vector(Nparam): while true do print(\"computing jacobian...\"):J:=jacob():print(\"inve rsing matrix...\"):Jinv:=inverse(J): print(\"computing equations...\") :s:=0:build():equations(w):print(\"done\"):s:=s+evalf(norm(w,1)):print (s):if (s<10^(4-Digits)) then print(\"converging after\",iter,\"loops \"):break:fi:paramtov(v):v:=evalm(v-Jinv &* w):vtoparam(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#$\"07&>)42 l7)!#;" }}{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#$\"06 AQEJP/#!#<" }}{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 #$\"00\\in1Na\"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q6computing~jaco bian...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#$\"0p=zd9$G;!#@" }}{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#$\"0f$QH>e'f$!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q 6computing~jacobian...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4inversi ng~matrix...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~equatio ns...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q%done6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0j;'=j8/a!#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #Q6computing~jacobian...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q4inver sing~matrix...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7computing~equat ions...6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q%done6\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"0!3R\"eloj)!#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%Q1converging~after6\"\"\"'Q&loopsF$" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 72 "# the following line is used to push the parameter \+ t by smallincrements." }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 -1 512 "#incr:=1.05:ok:=true:while ok do iter:=0:v:=vector(Nparam):w:=vec tor(Nparam):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():Ji nv:=inverse(J): s:=0: Build() :equations(w):s:=s+evalf(norm(w,1)):prin t(s):if (s<10^(6-Digits)) then print(\"converging after\",iter,\"loops \"):save t,a,b,c,r,\"costa.sav\":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 223 "" 0 "" {TEXT -1 0 "" }}{PARA 223 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 44 "############################################" }}} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 31 "# Check equations are sat isfied" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 44 "############## ##############################" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 19 "Order:=max(maxn,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OrderG\"#5" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 42 "res: =(f,z0)->residue(series(f,z=z0),z=z0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$resGf*6$%\"fG%#z0G6\"6$%)operatorG%&arrowGF)-%(residueG6$-%'s eriesG6$9$/%\"zG9%F4F)F)F)" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 16 "# period problem" }}}{EXCHG {PARA 214 "> " 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 214 "> " 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 #$!0xb6G(Hia!#G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0O`D6!4$p\"!#H" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 20 "# residues are zero " }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 120 "for k from 1 to M do for i from 1 to N[k] do p rint(k,i,\" \",res(phi(1,k,z)^2+phi(2,k,z)^2+phi(3,k,z)^2,a[k,i])):o d:od: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"\"F#Q$~~~6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#\"\"\"Q$~~~6\"^$$!0mllt'=_&)!#I$ \"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#F#Q$~~~6\"^$$\"0#GMk8 \\os!#I$\"\"!F+" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 126 "for \+ k from 1 to M do for i from 1 to N[k] do print(k,i,\" \",res(phi(1,k +1,z)^2+phi(2,k+1,z)^2+phi(3,k+1,z)^2,b[k,i])):od:od: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"\"F#Q$~~~6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#\"\"\"Q$~~~6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#F#Q$~~~6\"^$$\"0MWBMOL;%!#J$\"\"!F+" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 133 "for k from 1 to M do for i from 1 to N[k] d o print(k,i,\" \",res((phi(1,k,z)^2+phi(2,k,z)^2+phi(3,k,z)^2)*(z-a[ k,i]),a[k,i])):od:od: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"\"F#Q$~~ ~6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#\"\"\"Q$~~~6\"^$$!0& y,,\\B37!#H$\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#F#Q$~~~6\" ^$$!0\\&*[6B^B\"!#H$\"\"!F+" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 22 "# residues at the ends" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 126 "for k from 1 to M+1 do for nu from 1 to 3 do print(k,nu,\" \+ \", sum(c[nu,k-1,ii],ii=1..N[k-1])-sum(c[nu,k,ii],ii=1..N[k])):od:od: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"\"F#Q\"~6\"\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&\"\"\"\"\"#Q\"~6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"\"\"\"$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\"$!0j')*>wP)p*!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6&\"\"$\"\"\"Q\"~6\"$\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"$ \"\"#Q\"~6\"$\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"$F#Q\"~6\" $\"0m)*>wP)p>!#:" }}}{EXCHG {PARA 214 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 22 "# phi's are invariant;" }}} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 167 "for nu from 1 to 3 do fo r k from 1 to M do for i from 1 to N[k] do print(nu,k,i,\" \+ \",phi(nu,k,a[k,i]+sqrt(r[k,i]))+phi(nu,k+1,b[k,i]+sqrt(r[k,i])));od;o d;od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"F#F#Q-~~~~~~~~~~~~6\"^$ $!02(He>E%y&!#G$\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"\"\"# F#Q-~~~~~~~~~~~~6\"^$$\"0]Y'p!fuy\"!#G$\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"\"\"#F$Q-~~~~~~~~~~~~6\"^$$!0<*)\\V/o@\"!#F$\"\"! F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#\"\"\"F$Q-~~~~~~~~~~~~6\"^$ $\"\"!F)$!0!4%=i%)[R'!#G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#F#\" \"\"Q-~~~~~~~~~~~~6\"^$$\"\"!F)$\"0V!)pi1)>I!#G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#F#F#Q-~~~~~~~~~~~~6\"^$$\"\"!F($!0+,wNFa5(!#G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$\"\"\"F$Q-~~~~~~~~~~~~6\"^$$!00&Q qBm%*Q!#F$\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$\"\"#\"\"\"Q -~~~~~~~~~~~~6\"^$$\"0@!\\8L!*4:!#F$\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$\"\"#F$Q-~~~~~~~~~~~~6\"^$$\"0D%z2/7%z\"!#F$\"\"!F ," }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 -1 0 "" }}}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 -1 59 "###################################### #####################" }}}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 -1 50 "# graphics : use triangulation generated by matlab" }}}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 -1 59 "###################################### #####################" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "# read saved values of par ameters and triangulations of domains" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "read \"costa.data\":verbose:=true;Build();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(verboseG%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0MBkg3D,*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0 P0\">k*30(!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0*>\"3xzF\"\\!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0&)H6*f8VR!#=" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"0Ri2Qz:w%!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"0:0YWX]\"Q!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0i!zT'z]h%!#B" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0@*ycLm(p$!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0$*G7MMBZ%!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" 02AsO6-f$!#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\".XXm7,]$!#F" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#Q+converging6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0>#y7aJ6f!#E$!\"!\"\"!" }}}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 -1 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "# compute translation vector for each piece" }}} {EXCHG {PARA 227 "> " 0 "" {MPLTEXT 1 -1 137 "for nu from 1 to 3 do T[ nu,1]:=0:for k from 1 to M do T[nu,k+1]:=T[nu,k]+X(nu,k,a[k,1]+sqrt(r[ k,1]))-X(nu,k+1,b[k,1]+sqrt(r[k,1]));od;od:" }}}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 -1 61 "# generate data for matlab -- used to triangula te the domains" }}}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 -1 91 "# cent res:=[0,0,0.5*(b[2,1]+b[2,2])];rayons:=[1.5,2.,1.5]; # big circles to \+ cutoff the ends" }}}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 -1 281 "# fo r k from 1 to 3 do print(\"k=\",k):print(\"grand disque:\",Re(centres[ k]),Im(centres[k]),rayons[k]):for i from 1 to N[k] do print(\"a-disque \",Re(a[k,i]),Im(a[k,i]),abs(sqrt(r[k,i]))):od:for i from 1 to N[k-1] \+ do print(\"b-disque\",Re(b[k-1,i]),Im(b[k-1,i]),abs(sqrt(r[k-1,i]))):o d:od:" }}{PARA 226 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 225 "> " 0 " " {MPLTEXT 1 -1 0 "" }}}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 -1 17 "# display domains" }}}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 -1 59 "for \+ k from 1 to 3 do domain[k]:=vector(nops(point1[k]));od;" }}{PARA 226 " " 0 "" {TEXT -1 0 "" }}{PARA 226 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'domainG6#\"\"\"-%&arrayG6$;F'\"%#>\"7\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'domainG6#\"\"#-%&arrayG6$;\"\"\"\" %[<7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'domainG6#\"\"$-%&arrayG6 $;\"\"\"\"%C57\"" }}}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 -1 198 "for k from 1 to 3 do for i from 1 to nops(point1[k]) do domain[k][i]:=[[x [k][point1[k][i]],y[k][point1[k][i]]],[x[k][point2[k][i]],y[k][point2[ k][i]]],[x[k][point3[k][i]],y[k][point3[k][i]]]]:od:od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "PLOT(POLYGONS(op(convert(domain[1], list))),STYLE(PATCH));" }}{PARA 13 "" 1 "" {GLPLOT2D 529 529 529 {PLOTDATA 2 "6$-%)POLYGONSG6deo7%7$$!&uR\"!\"%$\"&FX&!\"&7$$!&DW\"F*$ \"&>6%F-7$$!&\\J\"F*$\"&fq%F-7%7$$!&[*p!\"'$!%u7!\"$7$$!&'e;F-$!&bP\"F *7$$!&u\\$F<$!%(Q\"F?7%7$$\"&\"f7F*$!&C[\"F-7$$\"&jN\"F*$!&af#F-7$$\"& #y8F*$!&#*>\"F-7%7$$\"&.B#F-$\"&0C\"F*7$$\"&(4MF-$\"&VL\"F*7$$\"&&H?F- $\"&YO\"F*7%7$$!&,j$F-$!&OW%!#@7$$!&**>%F-$!&d!eF<7$$!&.c$F-$!&33(F<7% 7$$!&AW'F-$\"&YN\"F*7$$!&)*o(F-$\"&zG\"F*7$$!&'>mF-$\"&#>7F*7%7$$\"&Zy )Fjo$!&&GOF-7$$\"&ZN&F<$!&K0%F-7$$\"&V;'F<$!&bd$F-7%7$$!%*Q\"F*$!&FN$F -7$$!&m,#F-$!&v,$F-7$$!&2)>F-$!&(>OF-7%7$$\"&\\_\"!\"($!&t]&F-7$$\"&Xi (!\")$!&zc%F-7$$!&sn'F<$!&W/&F-7%7$$!%.8F?$!&9V(F-7$$!&HB\"F*$!&Ta)F-7 $$!&C<\"F*$!&@K(F-7%7$$!&2N\"F-$!&:e%F-7$$!&Mv'F<$!%0TF*7$$!&*p8F-$!&r 'RF-7%7$$\"&MW&F-$!&;7#F-7$$\"&U+%F-$!&KM#F-7$$\"&ew%F-$!&y6$F-7%7$$!& vc#F-$\"&bc#F-7$$!&0R\"F-$\"&@N$F-7$$!&pQ#F-$\"&Jj$F-7%7$$!&d=\"F*$\"& )[@F-7$$!&.L\"F*$\"&;W#F-7$$!&>D\"F*$\"&*)\\\"F-7%7$$!&\\Z\"F*$\"&Xt#F -7$$!&J\\\"F*$\"&@V\"F-7$$!&lR\"F*$\"&=z\"F-7%7$$!#:!\"\"$\"&o=\"F<7$$ !&_\\\"F*$!&c>\"F-7$$!&*)R\"F*$!&P\\%F<7%7$$\"&i>)F-$!&(e(*F-7$$\"&47) F-$!&47\"F*7$$\"&pB*F-$!&U.\"F*7%7$$!&zH\"F*$!&u,\"F-7$$!&%)Q\"F*$!&\" 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