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" }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives its location." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \"C:\\\\Maple/procdrs/b utcher.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 113 "Procedures related to the construction and drawing of \+ rooted trees along with their associated order conditions:\n" }{TEXT 0 118 "Trees,AllTrees,SimpleTrees,AllSimpleTrees,TreeOrderCondition,Tr eeOrderCondition0,TreeErrorTerm,TreeErrorTerm0,DrawTree" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "Ack nowledgement: These procedures are based on Maple code from \"Runge-Ku tta Methods, Trees, and Maple\", " }}{PARA 0 "" 0 "" {TEXT -1 92 "by F olkmar Bornemann, Selcuk Journal of Applied Mathematics, Vol. 2, No. 1 , pp. 3-15, 2001. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{URLLINK 17 "htt p://wwwzenger.informatik.tu-muenchen.de/selcuk/sjam012101.html" 4 "htt p://wwwzenger.informatik.tu-muenchen.de/selcuk/sjam012101.html" "" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 133 "Trees, AllTrees, SimpleTrees, AllSimpleTrees, TreeOrd erCondition, TreeOrderCondition0, TreeErrorTerm, TreeErrorTerm0, DrawT ree: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 273 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT -1 9 " type(" }{TEXT 35 4 "expr" }{TEXT -1 7 ", tree)" }{TEXT 274 2 " \n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 " Trees(order) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 " A llTrees(order)" }{TEXT 275 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 22 " S impleTrees(order)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 " AllSimpleTrees(order)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 28 " TreeOrderCondition(tree)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " TreeO rderCondition0(tree)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 28 " TreeErrorCondition(tree)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " TreeErrorCondition0(tree)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 " Dra wTree(tree)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }} {PARA 0 "" 0 "" {TEXT 23 10 " expr - " }{TEXT -1 14 "any expression " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 12 " or der - " }{TEXT -1 89 " a positive integer giving the order (or max imum order) of the trees to be constructed" }}{PARA 0 "" 0 "" {TEXT 277 1 " " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 23 12 " tree - " }{TEXT -1 11 " a tree " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 12 "Description: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The p rocedure " }{TEXT 0 14 "type(...,tree)" }{TEXT -1 11 " checks if " } {TEXT 35 4 "expr" }{TEXT -1 23 " is a tree; it returns " }{TEXT 35 4 " true" }{TEXT -1 4 " if " }{TEXT 35 4 "expr" }{TEXT -1 6 " is a " } {TEXT 35 11 "rooted tree" }{TEXT -1 6 ", and " }{TEXT 35 5 "false" } {TEXT -1 12 " otherwise. " }}{PARA 0 "" 0 "" {TEXT -1 72 "The unique t ree with one node (order 1) is represented by the list []. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The unique root ed tree of order 1: [ ]" }}{PARA 0 "" 0 "" {TEXT -1 44 "The unique ro oted tree of order 2: [ [ ] ] " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 241 72 72 {PLOTDATA 2 "6--%'CURVESG6#-%'LEGENDG6#QB__never_d isplay_this_legend_entry6\"-F$6&7$7$$\"#I\"\"!$!+++++5!\")7$F/$\"+++++ 5F4-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&%$RGBG$F1F1F@F@-%&STYLEG6#%&POINTG- F$6&F--F96#%(DIAMONDGFFBFF-F$6&F]pFLFBFF-F$6&F]pFQFBFF-F$6%7$F^p FapFWFen-F$6%7$F^pFdpFWFen-F$6%7$FdpFgpFWFen-F$6%7#7$F_pF;FboFho-F$6&7 &7$F/$!+++++SF=7$F/$F6F=7$$!+)*********F3$!+nmmmEF=7$$\"+)*********F3F drF>FBFF-F$6&F[rFLFBFF-F$6&F[rFQFBFF-F$6%7$F\\rF_rFWFen-F$6%7$F_rFarFW Fen-F$6%7$F_rFfrFWFen-F$6%7#7$F/$!+++++?F=FboFho-F$6&7&7$F_pF]r7$FbpF` r7$F_pF`r7$FepF`rF>FBFF-F$6&F^tFLFBFF-F$6&F^tFQFBFF-F$6%7$F_tF`tFWFen- F$6%7$F_tFatFWFen-F$6%7$F_tFbtFWFen-F$6%7#7$F_pFjsFboFho-%*AXESSTYLEG6 #%%NONEG" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cur ve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14 " "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "C urve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" }}{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "The two trees [ [ [ ] ], [ ] ] and [ [ ], [ [ ] ] ] are consi dered to be equivalent (They give rise to the same order condition.). \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 323 74 74 {PLOTDATA 2 "62-%'CURVESG6#-%'LEGENDG6#QB__never_display_this_legend_entry6\"-F$6& 7&7$$\"#I\"\"!$!+++++5!\")7$$\"+++++?F4$F1F17$$\"+++++SF4F87$F:$\"++++ +5F4-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&%$RGBGF8F8F8-%&STYLEG6#%&POINTG-F$ 6&F--F@6#%(DIAMONDGFCFG-F$6&F--F@6#%&CROSSGFCFG-F$6%7$F.F5-FD6&FF$\"\" \"F1F8F8-%*THICKNESSG6#\"\"#-F$6%7$F.F9FXFfn-F$6%7$F9F and press [Enter].\nYou c an then close up the subsections." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 155 "`type/tree`, Trees, AllTrees, SimpleTrees, AllSimpleTrees, TreeOr derCondition, TreeOrderCondition0, TreeErrorTerm, TreeErrorTerm0, Draw Tree: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 260 4 "Note" }{TEXT -1 45 ": There are also four auxiliary, pr ocedures: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 259 8 "TreeSort" } {TEXT -1 1 "," }{TEXT 259 8 " Addleaf" }{TEXT -1 2 ", " }{TEXT 259 9 " TreeOrder" }{TEXT -1 2 ", " }{TEXT 259 9 "TreeAlpha" }{TEXT -1 2 ", " }{TEXT 259 22 "TreeSum, TreeFactorial" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23837 "`typ e/tree` := list(tree):\n\n## gives a unique representative for equival ent trees \nTreeSort := proc(beta::tree)\n sort(map(TreeSort,beta)); \nend proc: # TreeSort\n\n# constructs all trees obtained by introduc ing a new edge at each node of the given tree\nAddLeaf := proc(beta::t ree)\n option remember;\n local replace,val,child,new;\n\n repla ce := proc(new::tree,old::posint,beta::tree)\n subsop(old=new,bet a);\n end proc; # replace\n\n val := [[[],op(beta)]];\n for chil d to nops(beta) do\n new := AddLeaf(beta[child]);\n val := L istTools[MakeUnique]([op(val),op(map(replace,new,child,beta))]);\n e nd do;\n map(TreeSort,val);\nend proc: # AddLeaf\n\nAllTrees := proc (order::posint)\n option remember;\n local all,trees;\n\n trees \+ := [[]];\n all := trees;\n to order-1 do\n trees := ListTools [MakeUnique](ListTools[FlattenOnce]([op(map(AddLeaf,trees))]));\n \+ all := [op(all),op(trees)];\n end do;\n sort([op(all)],TreeCompar e);\nend proc: # AllTrees\n\nTrees := proc(order::posint)\n option r emember;\n local trees,i;\n\n trees := [[]];\n to order-1 do\n \+ trees := ListTools[MakeUnique](ListTools[FlattenOnce]([op(map(AddL eaf,trees))]));\n end do;\n sort([op(trees)],TreeCompare)\nend pro c: # Trees\n\nAllSimpleTrees := proc(order::posint)\n option remembe r;\n local inc,all,trees;\n\n inc := proc(beta::tree) [beta],[[],o p(beta)] end proc:\n\n if order=1 then return([[]]) end if;\n\n tr ees := [[[]]];\n all := trees;\n to order-2 do\n trees := map (inc,trees);\n all := [op(all),op(trees)];\n end do;\n sort([ [],op(all)],TreeCompare)\nend proc: # AllSimpleTrees\n\nSimpleTrees := proc(order::posint)\n option remember;\n local inc,trees;\n\n i nc := proc(beta::tree) [beta],[[],op(beta)] end proc:\n\n if order=1 then return([[]]) end if;\n\n trees := [[[]]];\n to order-2 do\n \+ trees := map(inc,trees);\n end do;\n sort([op(trees)],TreeCom pare)\nend proc: # SimpleTrees\n\nTreeOrder := proc(beta::tree)\n op tion remember;\n 1+`+`(op(map(TreeOrder,beta)));\nend proc: # TreeOr der\n\nTreeFactorial := proc(beta::tree)\n option remember;\n Tree Order(beta)*`*`(op(map(TreeFactorial,beta)));\nend proc: # TreeFactori al\n\nTreeAlpha := proc(beta::tree)\n option remember;\n local n; \n n := nops(beta);\n combinat[numbperm](beta,n)/n!*`*`(op(map(Tre eAlpha,beta)));\nend proc: # TreeAlpha\n\nTreeSum := proc(beta::tree,n o::posint)\n option remember;\n local vars,val,son;\n\n vars := \+ [i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda,mu,nu,rho,sigma,tau ,upsilon,omega];\n val := 1;\n for son in beta do\n if son=[] then\n val := val*c[vars[no]]\n else \n val := v al*Sum(a[vars[no],vars[no+1]]*\n TreeSum(son,no+1),var s[no+1]=1..vars[no]-1)\n end if;\n end do;\n val;\nend proc: \+ # TreeSum\n\nTreeOrderCondition := proc(beta::tree)\n Sum(b[i]*TreeS um(beta,1),i=1.._S)=1/TreeFactorial(beta);\nend proc: # TreeOrderCondi tion\n\nTreeErrorTerm := proc(beta::tree)\n TreeAlpha(beta)*(Sum(b[i ]*TreeSum(beta,1),i=1.._S)-1/TreeFactorial(beta));\nend proc: # TreeEr rorTerm\n\nTreeSum0 := proc(beta::tree,no::posint)\n option remember ;\n local val,son;\n unassign('``');\n val := 1;\n for son in \+ beta do\n if son=[] then\n val := val*c\n else \n \+ val := val*``(a*TreeSum0(son,no+1))\n end if;\n end do;\n val;\nend proc: # TreeSum0\n\nTreeOrderCondition0 := proc(beta::tre e)\n if beta=[] then return b*e=1 end if;\n b*TreeSum0(beta,1)=1/T reeFactorial(beta);\nend proc: # TreeOrderCondition0\n\nTreeErrorTerm0 := proc(beta::tree)\n local ta,z;\n if beta=[] then return b*e-1 \+ end if;\n z := b*TreeSum0(beta,1)-1/TreeFactorial(beta);\n ta := T reeAlpha(beta);\n `if`(ta=1,z,ta*``(z));\nend proc: # TreeErrorTerm0 \n\nTreeCompare := proc(beta1::tree,beta2::tree)\n local i,j,u,w,d,d a,z;\n # sort by using 'abreviated' order term\n # first sort by t otal degree (corresponds to Taylor series order)\n for i to 2 do\n \+ u[i] := TreeSum0(args[i],1);\n w[i] := eval(subs(``=(_U->_U), u[i]));\n d[i] := degree(w[i]);\n end do;\n # sort by degree \+ in 'a'\n if d[1]d[2] then ret urn(false)\n else\n for i to 2 do\n w[i] := eval(subs(c =1,w[i]));\n d[i] := degree(w[i]);\n end do;\n # sor t by number of occurrences of 'a'\n if d[1]>d[2] then return(true )\n elif d[1]d[2] then return(true)\n elif d[1]d[2] then return(false)\n else\n # sort \+ by magnitude of the 'tree factorial'\n for i to 2 do\n \+ d[i] := TreeFactorial(args[i]);\n end do ;\n if d[1]>d[2] then return(true)\n elif \+ d[1]_U^j),u[i])) ;\n d[i] := degree(w[i]);\n \+ da[i] := degree(w[i],a);\n end do;\n \+ if d[1]d[2] then return(false) end if;\n if da [1]da [2] then return(false) end if; \n end do;\n \+ for i to 2 do\n w[i] := eval(subs(``=(_U-> _U^2+_U+1),u[i]));\n d[i] := eval(w[i],\{a=2,c=3\} );\n end do;\n if d[1]d[2] then return(false)\n \n \+ ## d[1]=d[2] should not occur\n else \n WARNING(\"equality in comparison\");\n \+ print(u[1],u[2]);\n print(args[1],args [2]); \n print(d[1]=d[2]);\n end \+ if; \n return(false)\n e nd if\n end if\n end if\n end if\n end if\ne nd proc:\n\nDrawTree := proc(beta::tree)\n local bi,bi2,bi3,bi4,bi5, bi6,bi7,bi8,bi9,bi_10,bi_11,bi_12,bi_13,bi_14,\n m,mi,mi2,mi3,mi4,mi 5,mi6,mi7,mi8,mi9,mi_10,mi_11,mi_12,mi_13,mi_14,\n i,i2,i3,i4,i5,i6, i7,i8,i9,i_10,i_11,i_12,i_13,i_14,i_15,\n s,s2,s3,s4,s5,s6,s7,s8,s9, s_10,s_11,s_12,s_13,s_14,s_15,\n k,k2,k3,k4,k5,k6,k7,k8,k9,k_10,k_11 ,k_12,k_13,k_14,k_15,edges,nodes,\n Options,temp,shwnds,shwrdrcn,clr ,thk,fntclr,lvls,ht,wd,scl,mxv,B,pp,p1,p2,\n toc,str,j,pw,L,M,K,dn,l dn,N,U,V,t1,t2,t3,t4,t5,pT,pL,pR,pOC;\n\n # Get the options.\n # S et the default value to start with.\n shwnds := true;\n shwrdrcn : = false;\n clr := [COLOR(RGB,1,0,0),COLOR(RGB,0,0,0)];\n thk := 2; \n ht := 1;\n wd := -1;\n fntclr := COLOR(RGB,0,0,1);\n scl := 1;\n if nargs>1 then\n Options:=[args[2..nargs]];\n if no t type(Options,list(equation)) then\n error \"each optional ar gument must be an equation\"\n end if;\n if hasoption(Option s,'show_nodes','shwnds','Options') or \n hasoption(Options,'sh ow_vertices','shwnds','Options') then\n if shwnds<>true then s hwnds := false end if;\n end if;\n if hasoption(Options,'sho w_ordercondition','temp','Options') then\n if member(shwrdrcn, \{true,false\}) then \n shwrdrcn := temp\n else \n \+ error \"\\\"show_ordercondition\\\" must be either true or \+ false\"\n end if;\n end if;\n if hasoption(Options,c olor,'temp','Options') or \n hasoption(Options,colour,'temp' ,'Options') then\n if type(temp,list) and nops(temp)=2 then\n \+ clr := temp;\n elif type(temp,name) or (type(temp,f unction) and \n member(op(0,temp),\{COLOR,COLOUR\})) then\n \+ clr[1] := temp;\n clr[2] := COLOR(RGB,0,0,0);\n \+ else\n error \"\\\"color\\\" or \\\"colour\\\" must be a single colour, or a list of two colours\"\n end if;\n \+ end if;\n if hasoption(Options,'thickness','thk','Options') th en\n if not type(thk,posint) then\n error \"\\\"thi ckness\\\" must be a positive integer\"\n end if;\n end i f;\n if hasoption(Options,font_color,'temp','Options') or \n \+ hasoption(Options,font_colour,'temp','Options') then\n i f type(temp,name) or (type(temp,function) and \n member(op(0 ,temp),\{COLOR,COLOUR\})) then\n fntclr := temp;\n \+ else\n error \"\\\"font_color\\\" or \\\"font_colour\\\" mu st be a colour\"\n end if;\n end if;\n if hasoption( Options,'height','ht','Options') then\n if not type(ht,posint) then\n error \"\\\"order\\\" must be a positive integer\" \n end if;\n end if;\n if hasoption(Options,'width', 'wd','Options') then\n if not type(wd,realcons) or evalf(wd)<= 0 or has(wd,infinity) then\n error \"\\\"width\\\" must be \+ a finite positive real constant\"\n end if;\n end if;\n \+ if hasoption(Options,'scale','scl','Options') then\n if no t type(scl,realcons) or evalf(scl)<=0 or has(scl,infinity) then\n \+ error \"\\\"scale\\\" must be a finite positive real constant\" \n end if;\n end if;\n if nops(Options)>0 then\n \+ error \"%1 is not a valid option for %2 .. the recognised options are \\\"show_nodes\\\",\\\"colour\\\", or (\\\"color\\\") and \\\"thi ckness\\\",\\\"height\\\", \\\"show_ordercondition\\\",\\\"font_color \\\" or \\\"font_colour\\\" and \\\"scale\\\" \",op(1,Options),procnam e;\n end if;\n end if;\n\n B := beta;\n for i while B<>[] d o\n B := ListTools[FlattenOnce](B);\n end do;\n if i>16 then \+ \n error \"the tree has more than 16 levels\";\n end if;\n \n \+ m := nops(beta);\n edges := []:\n nodes := [[0,0]];\n if m>0 th en for i to m do \n ## there are m nodes at the 1st level abov e the root \n ## i indexes the nodes at the 1st level\n i f m>1 then\n s := 1/(m-1); # spacing between nodes at the firs t level\n k := (i-(m+1)/2)*s;\n else\n s := 9/8; \n k := 0;\n end if;\n edges := [op(edges),[[0,0],[k ,1]]];\n nodes := [op(nodes),[k,1]];\n bi := op(i,beta);\n \+ mi := nops(bi); # number of nodes at 2nd level\n if mi>0 then for i2 to mi do\n if mi>1 then\n s2 := 2/3*s*1/(mi -1);\n k2 := k+(i2-(mi+1)/2)*s2;\n else\n \+ s2 := s;\n k2 := k;\n end if;\n edges := \+ [op(edges),[[k,1],[k2,2]]];\n nodes := [op(nodes),[k2,2]];\n \+ bi2 := op(i2,bi);\n mi2 := nops(bi2); # number of nodes at 3rd level \n if mi2>0 then for i3 to mi2 do\n i f mi2>1 then\n s3 := 2/3*s2*1/(mi2-1);\n k 3 := k2 +(i3-(mi2+1)/2)*s3;\n else\n s3 := s2 ;\n k3 := k2;\n end if; \n edg es := [op(edges),[[k2,2],[k3,3]]];\n nodes := [op(nodes),[k 3,3]]; \n bi3 := op(i3,bi2);\n mi3 := nops(bi3); \n if mi3>0 then for i4 to mi3 do\n if mi3>1 \+ then\n s4 := 2/3*s3*1/(mi3-1);\n k4 \+ := k3 +(i4-(mi3+1)/2)*s4;\n else\n s4 : = s3;\n k4 := k3;\n end if; \n \+ edges := [op(edges),[[k3,3],[k4,4]]];\n nodes := \+ [op(nodes),[k4,4]];\n bi4 := op(i4,bi3);\n \+ mi4 := nops(bi4);\n if mi4>0 then for i5 to mi4 do\n \+ if mi4>1 then\n s5 := 2/3*s4*1/(mi4 -1);\n k5 := k4 +(i5-(mi4+1)/2)*s5;\n \+ else\n s5 := s4;\n k5 := \+ k4;\n end if;\n edges := [op(edges), [[k4,4],[k5,5]]];\n nodes := [op(nodes),[k5,5]];\n \+ bi5 := op(i5,bi4);\n mi5 := nops(bi5); \n if mi5>0 then for i6 to mi5 do\n \+ if mi5>1 then\n s6 := 2/3*s5*1/(mi5-1);\n \+ k6 := k5 +(i6-(mi5+1)/2)*s6;\n \+ else\n s6 := s5;\n k6 \+ := k5;\n end if;\n edges := [o p(edges),[[k5,5],[k6,6]]];\n nodes := [op(nodes),[ k6,6]];\n bi6 := op(i6,bi5);\n \+ mi6 := nops(bi6);\n if mi6>0 then for i7 to mi6 d o\n if mi6>1 then\n s 7 := 2/3*s6*1/(mi6-1);\n k7 := k6 +(i7-(mi6+ 1)/2)*s7;\n else\n s7 := s6;\n k7 := k6;\n \+ end if;\n edges := [op(edges),[[k6,6],[k7,7]]] ;\n nodes := [op(nodes),[k7,7]];\n \+ bi7 := op(i7,bi6);\n mi7 := nops(bi7 );\n if mi7>0 then for i8 to mi7 do\n \+ if mi7>1 then\n s8 := 2/ 3*s7*1/(mi7-1);\n k8 := k7 +(i8-(mi7+1)/2 )*s8;\n else\n \+ s8 := s7;\n k8 := k7;\n \+ end if;\n edges := [op(edges),[[k7, 7],[k8,8]]];\n nodes := [op(nodes),[k8,8]]; \n bi8 := op(i8,bi7);\n \+ mi8 := nops(bi8);\n if mi8>0 then for \+ i9 to mi8 do \n \+ if mi8>1 then\n s9 := 2/3*s8*1/(mi 8-1);\n k9 := k8 +(i9-(mi8+1)/2)*s9;\n else\n s 9 := s8;\n k9 := k8;\n \+ end if;\n edges := [op(edge s),[[k8,8],[k9,9]]];\n nodes := [op(nodes ),[k9,9]];\n bi9 := op(i9,bi8);\n \+ mi9 := nops(bi9);\n \+ if mi9>0 then for i_10 to mi9 do\n if mi9>1 then\n s_10 := 2/3*s9*1/(mi9 -1);\n k_10 := k9 +(i_10-(mi9+1)/2) *s_10;\n else\n \+ s_10 := s9;\n k_10 := \+ k9;\n end if;\n \+ edges := [op(edges),[[k9,9],[k_10,10]]];\n \+ nodes := [op(nodes),[k_10,10]];\n \+ bi_10 := op(i_10,bi9);\n mi _10 := nops(bi_10);\n if mi_10>0 then \+ for i_11 to mi_10 do\n if mi_10>1 t hen\n s_11 := 2/3*s_10*1/(mi_10- 1);\n k_11 := k_10 +(i_11-(mi_10 +1)/2)*s_11;\n else\n \+ s_11 := s_10;\n \+ k_11 := k_10;\n end if;\n \+ edges := [op(edges),[[k_10,10],[k_1 1,11]]];\n nodes := [op(nodes),[k_1 1,11]];\n bi_11 := op(i_11,bi_10); \n mi_11 := nops(bi_11);\n \+ if mi_11>0 then for i_12 to mi_11 do\n \+ if mi_11>1 then\n \+ s_12 := 2/3*s_11*1/(mi_11-1);\n \+ k_12 := k_11 +(i_12-(mi_11+1)/2)*s_12;\n \+ else\n \+ s_12 := s_11;\n \+ k_12 := k_11;\n end if;\n \+ edges := [op(edges),[[k_11,11],[k_12,1 2]]];\n nodes := [op(nodes),[k_1 2,12]];\n bi_12 := op(i_12,bi_11 );\n mi_12 := nops(bi_12);\n \+ if mi_12>0 then for i_13 to mi_12 d o\n if mi_12>1 then\n \+ s_13 := 2/3*s_12*1/(mi_12-1);\n \+ k_13 := k_12 +(i_13-(mi_12+ 1)/2)*s_13;\n else\n \+ s_13 := s_12;\n \+ k_13 := k_12;\n \+ end if;\n edges \+ := [op(edges),[[k_12,12],[k_13,13]]];\n \+ nodes := [op(nodes),[k_13,13]];\n \+ bi_13 := op(i_13,bi_12);\n \+ mi_13 := nops(bi_13);\n \+ if mi_13>0 then for i_14 to mi_13 do\n \+ if mi_13>1 then\n \+ s_14 := 2/3*s_13*1/(mi_13-1);\n \+ k_14 := k_13 +(i_14-(mi_13+1)/2)*s_14;\n \+ else\n \+ s_14 := s_13;\n \+ k_14 := k_13;\n \+ end if;\n edges := [op(edges),[[k_13,13],[k_14,14]]];\n \+ nodes := [op(nodes),[k_14,14]];\n \+ bi_14 := op(i_14,bi_13);\n \+ mi_14 := nops(bi_14);\n \+ if mi_14>0 then for i_15 to mi_14 do\n \+ if mi_14>1 then\n \+ s_15 := 2/3*s_14*1/(mi_14-1);\n \+ k_15 := k_14 +(i_15-(mi_1 4+1)/2)*s_15;\n else\n \+ s_15 := s_14;\n \+ k_15 := k_14;\n \+ end if;\n \+ edges := [op(edges),[[k_14,14],[k_15,15]]]; \n nodes := [op(nodes), [k_15,15]];\n ##bi_15 : = op(i_15,bi_14);\n ##m i_15 := nops(bi_15);\n \+ ##if mi_15>0 then\n \+ ##error \"too many levels\";\n \+ ##end if;\n end do ; end if;\n end do; end if;\n end do; end if;\n \+ end do; end if;\n \+ end do; end if;\n end do; end if;\n \+ end do; end if;\n end do ; end if;\n end do; end if;\n end do; end if;\n end do; end if;\n end do; end \+ if;\n end do; end if;\n end do; end if; \n end do; end \+ if;\n\n mxv := max(op(map(_u->op(2,_u),nodes)));\n lvls := max(ht, mxv);\n pp := plot([[0,lvls]],style=line);\n if wd>0 then\n p L := plot([[-wd/2,0]],style=line);\n pR := plot([[wd/2,0]],style= line);\n pp := plots[display]([pp,pL,pR]); \n end if;\n\n \+ if edges<>[] then\n p1 := plot(edges,color=clr[1],thickness=thk) ;\n if shwnds then\n p2 := plot([nodes$3],style=point,sym bol=[circle,diamond,cross],color=clr[2]);\n if lvls>0 then \+ \n pT := plots[display]([p2,p1,pp],axes=none);\n el se\n pT := plots[display]([p2,p1],axes=none);\n end if; \n else\n if lvls>0 then\n pT := plot s[display]([p1,pp],axes=none);\n else\n pT := plots [display](p1,axes=none);\n end if;\n end if;\n else\n \+ p2 := plot([nodes$3],style=point,symbol=[circle,diamond,cross],col or=clr[2]);\n if lvls>0 then\n pT := plots[display]([p2,p p],axes=none);\n else\n pT := plots[display](p2,axes=none );\n end if;\n end if;\n\n if shwrdrcn then\n toc := Tre eOrderCondition0(beta);\n str := convert(lhs(toc),string);\n \+ str := StringTools[Remove](u_->`if`(u_=\"`\",true,false),str);\n \+ L := StringTools[Explode](str);\n j := 1;\n while j<= nops( L)-4 do\n if op(j,L)=\"c\" and op(j+1,L)=\"^\" and StringTools [IsDigit](op(j+2,L)) and \n op(j+3,L)=\"*\" and op(j+4,L)=\" a\" then\n pw := op(j+2,L);\n L := subsop(j=\"a \",L);\n L := subsop((j+1)=\"*\",L);\n L := subs op((j+2)=\"c\",L);\n L := subsop((j+3)=\"^\",L);\n \+ L := subsop((j+4)=pw,L);\n j := j+5;\n else\n \+ j := j+1;\n end if;\n end do; \n \+ M := []:\n for j to nops(L) do\n if op(j,L)=\"^\" then \+ M := [op(M),j] end if;\n end do;\n for j to nops(M) do\n \+ L := subsop(op(j,M)-(j-1)=NULL,L);\n end do:\n L := sub s(\"*\"=\" \",L);\n M := []:\n for j to nops(L) do\n \+ if StringTools[IsDigit](op(j,L)) then \n M := [op(M),j]; \+ \n end if;\n end do;\n if M<>[] then\n \+ K := [seq(\" \",j=1..nops(L))];\n for j to nops(M) do\n \+ K := subsop(op(j,M)=op(op(j,M),L),K);\n end do:\n \+ for j to nops(M) do\n L := subsop(op(j,M)=\" \",L);\n \+ end do:\n K := [\" \",op(K)];\n end if;\n dn := \+ denom(rhs(toc));\n if dn>1 then\n ldn := length(dn);\n \+ L := [op(L),\" \",\"=\",\" \"$ldn];\n N := [\" \"$(nops( L)+1),\"1\"];\n U := [\" \"$(nops(L)+1),\"_\"$ldn];\n \+ V := [\" \"$(nops(L)+1),op(StringTools[Explode](convert(dn,string)))]; \n L := StringTools[Implode](L); \n N := StringTools[I mplode](N);\n U := StringTools[Implode](U);\n V := Str ingTools[Implode](V); \n t1 := plots[textplot]([0,-0.5*scl,L ],font=[COURIER,10],color=fntclr);\n t2 := plots[textplot]([0, (-0.34)*scl,N],font=[COURIER,10],color=fntclr);\n t3 := plots[ textplot]([0,(-0.39)*scl,U],font=[COURIER,10],color=fntclr);\n \+ t4 := plots[textplot]([0,(-0.71)*scl,V],font=[COURIER,10],color=fntcl r);\n if M<>[] then \n K := [op(K),\" \",\" \",\" \+ \",\" \"$ldn];\n K := StringTools[Implode](K);\n \+ t5 := plots[textplot]([0,(-0.32)*scl,K],font=[COURIER,10],color=fntcl r);\n pOC := plots[display]([t1,t2,t3,t4,t5]);\n el se\n pOC := plots[display]([t1,t2,t3,t4]);\n end if ;\n else\n L := [op(L),\" \",\"=\",\" \",\"1\"];\n \+ L := StringTools[Implode](L);\n pOC := plots[textplot]([0,-0 .5*scl,L],font=[COURIER,10],color=fntclr);\n end if;\n plots [display]([pOC,pT],axes=none);\n else\n pT;\n end if;\nend pr oc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 39 "Examples are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "All tree s of order" }{XPPEDIT 18 0 "`` <= 5;" "6#1%!G\"\"&" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 431 "TT := AllTrees(5):\nm := 3: ## number of trees per row\nnn := nop s(TT): q := iquo(nn,m,'r'):\nfor i to nn do \n p||i := DrawTree(TT[i] ,height=`if`(i<=3,2,3),width=2,\n show_ordercondition=true,scale=`i f`(i<=3,1,1.7),\n color=COLOR(RGB,.4,0,1),font_color=black);\nend d o:\npp := plot([[1,1]],style=line,axes=none):\nplots[display](convert( [seq([p||((k-1)*m+1..m*k)],k=1..q),\n `if`(r>0,[p||(m*q+1..nn),pp$(m* (q+1)-nn)],NULL)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 716 1281 1281 {PLOTDATA 2 "6dy-%%TEXTG6&7$$\"#I\"\"!$!+,X=]%)!\"*Q'b~c~=~6\"-%' COLOURG6&%$RGBGF)F)F)-%%FONTG6$%(COURIERG\"#5-F$6&7$F'$!+$ps$psF,Q)~~~ ~~~~1F.F/F3-F$6&7$F'$!+$Qw$QwF,Q)~~~~~~~_F.F/F3-F$6&7$F'$!+**********F ,Q)~~~~~~~2F.F/F3-%'CURVESG6&7$7$F'$!+,w9gZF,7$F'$\"+*>E*>EF,-%'SYMBOL G6#%'CIRCLEG-%&COLORG6&F2$F)F)FenFen-%&STYLEG6#%&POINTG-FK6&FM-FU6#%(D IAMONDGFXFfn-FK6&FM-FU6#%&CROSSGFXFfn-FK6%FM-FY6&F2$\"\"%!\"\"Fen$\"\" \"F)-%*THICKNESSG6#\"\"#-FK6%7#7$F'$\"+**********F,-F06&F2$F7FjoFenFen -Fgn6#%%LINEG-FK6%7#7$$\"+++++?!\")FOFgpFjp-FK6%7#7$$\"+++++SFcqFOFgpF jp-F$6&7$$\"#gF)F*Q+b~(a~c)~=~F.F/F3-F$6&7$F]rF;Q-~~~~~~~~~~~1F.F/F3-F $6&7$F]rFAQ-~~~~~~~~~~~_F.F/F3-F$6&7$F]rFGQ-~~~~~~~~~~~6F.F/F3-FK6&7%7 $F]rFO7$F]rFR7$F]rFepFTFXFfn-FK6&F^sF\\oFXFfn-FK6&F^sFaoFXFfn-FK6%7$F_ sF`sFfoF]p-FK6%7$F`sFasFfoF]p-FK6%7#FasFgpFjp-FK6%7#7$$\"+++++]FcqFOFg pFjp-FK6%7#7$$\"+++++qFcqFOFgpFjp-F$6&7$F'$!+'GGIG\"!\"(Q/b~c~(a~c~)~= ~~F.F/F3-F$6&7$F'$!+du4q7F`uQ1~~~~~~~~~~~~~~~1F.F/F3-F$6&7$F'$!+T$QTF \"F`uQ2~~~~~~~~~~~~~~~__F.F/F3-F$6&7$F'$!+++++8F`uQ2~~~~~~~~~~~~~~~15F .F/F3-F$6&7$F'$!+.6[o7F`uQ1~~~~~~~~~2~~~~~~F.F/F3-FK6&7'7$F'$!+W%>EC\" F`u7$$\"+++++DFcq$!+H'z]>\"F`u7$$\"+++++NFcqFcw7$$\"+nmmmJFcq$!+:)Rv9 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"Curve 136" "Curve 137" "Curve 138" "Curve 139" "Curve 140" "Curve 141" "Curve 142" "Curve 143" "Curve 144" "Curve 145 " "Curve 146" "Curve 147" "Curve 148" "Curve 149" "Curve 150" "Curve 1 51" "Curve 152" "Curve 153" "Curve 154" "Curve 155" "Curve 156" "Curve 157" "Curve 158" "Curve 159" "Curve 160" "Curve 161" "Curve 162" "Cur ve 163" "Curve 164" "Curve 165" "Curve 166" "Curve 167" "Curve 168" "C urve 169" "Curve 170" "Curve 171" "Curve 172" "Curve 173" "Curve 174" "Curve 175" "Curve 176" "Curve 177" "Curve 178" "Curve 179" "Curve 180 " "Curve 181" "Curve 182" "Curve 183" "Curve 184" "Curve 185" "Curve 1 86" "Curve 187" "Curve 188" "Curve 189" "Curve 190" "Curve 191" "Curve 192" "Curve 193" "Curve 194" "Curve 195" "Curve 196" "Curve 197" "Cur ve 198" "Curve 199" "Curve 200" "Curve 201" "Curve 202" "Curve 203" "C urve 204" "Curve 205" "Curve 206" "Curve 207" "Curve 208" "Curve 209" "Curve 210" "Curve 211" "Curve 212" "Curve 213" "Curve 214" "Curve 215 " "Curve 216" "Curve 217" "Curve 218" "Curve 219" "Curve 220" "Curve 2 21" "Curve 222" "Curve 223" "Curve 224" "Curve 225" "Curve 226" "Curve 227" "Curve 228" "Curve 229" "Curve 230" "Curve 231" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "All trees of order" }{XPPEDIT 18 0 "`` <= 6;" "6#1%!G\"\"'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 414 "TT := AllTrees(6):\n m := 3: ## number of trees per row\nnn := nops(TT): q := iquo(nn,m,'r' ):\nfor i to 6 do \n p||i := DrawTree(TT[i],height=3,width=2,show_ord ercondition=true,\n scale=`if`(i<=6,1.5,2.5),color=COLOR(RGB,0,.7,0 ),font_color=black);\nend do:\npp := plot([[1,1]],style=line,axes=none ):\nplots[display](convert([seq([p||((k-1)*m+1..m*k)],k=1..q),\n `if` (r>0,[p||(m*q+1..nn),pp$(m*(q+1)-nn)],NULL)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 741 2682 2682 {PLOTDATA 2 "6_]m-%%TEXTG6&7$$\"\"!F($!+# 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s-F$6&7$Fd`lFhiqQ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "A n animation to show trees of order 8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "TT := Trees(8):\nfor i to nops(TT) do \n p||i := DrawTree(TT[i],height=5,show_nodes=true,show_ ordercondition=true,scale=1.3);\nend do:\nplots[display]([p||(1..nops( TT))],insequence=true);" }}{PARA 13 "" 1 "" {GLPLOT2D 383 343 343 {PLOTDATA 2 "6#-%(ANIMATEG6_r74-%%TEXTG6&7$$\"\"!F,$!#l!\"#QCb~(a~(a~( a~(a~(a~(a~c))))))~=~~~~~6\"-%&COLORG6&%$RGBGF+F+$\"\"\"F,-%%FONTG6$%( COURIERG\"#5-F(6&7$F+$!$U%!\"$QE~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1F1 F2F8-F(6&7$F+$!$2&FBQI~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_____F1F2F8-F (6&7$F+$!$B*FBQI~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~40320F1F2F8-%'CURVE SG6%7#7$$\"3-+++++++5!#>$\"\"(F,-%'COLOURG6&F5$Fb~(a~(a~c~(a~c~(a~c))))~=~~~~F1F2F8-F(6&F?Q@~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~1F1F2F8-F(6&FFQC~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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(6&F[dlQ1~~~~~~~~~5~~~~~~F1F2F8F\\_l-FQ6&7*F`oFa\\lFd\\lFj`mF[doFg\\lF ^doF[amFbpFfpFhp-FQ6&FbdtF]qFfpFhp-FQ6&FbdtFbqFfpFhpF`]lFc]lF`amFfdoFf ]lFidoFcamF_sFfsFjs75-F(6&F*Q/b~c~~(a~c)~=~~F1F2F8F`brFcbr-F(6&FLQ2~~~ ~~~~~~~~~~~~16F1F2F8-F(6&F[dlQ1~~~~5~~~~~~~~~~~F1F2F8F\\_l-FQ6&7*F`oFa \\l7$$!3))**************HFduF67$$!3/+++++++5FduF67$$\"3/+++++++5FduF67 $$\"3))**************HFduF6Fd\\lFg\\lFbpFfpFhp-FQ6&FcetF]qFfpFhp-FQ6&F cetFbqFfpFhpF`]l-FQ6%7$F`oFdetFhqFjq-FQ6%7$F`oFgetFhqFjq-FQ6%7$F`oFjet FhqFjq-FQ6%7$F`oF]ftFhqFjqFc]lFf]lF_sFfsFjs75-F(6&F*Q-b~(a~c~)~=~~F1F2 F8-F(6&F?Q/~~~~~~~~~~~~~1F1F2F8-F(6&FFQ0~~~~~~~~~~~~~__F1F2F8-F(6&FLQ0 ~~~~~~~~~~~~~56F1F2F8-F(6&F[dlQ/~~~~~~~6~~~~~~F1F2F8F\\_l-FQ6&7*F`oFao Fhz7$$!31++++++]AFduFco7$$!3s*************\\(FWFco7$$\"3s************* \\(FWFco7$$\"31++++++]AFduFcoFizFbpFfpFhp-FQ6&FbhtF]qFfpFhp-FQ6&FbhtFb qFfpFhpFeqF^[l-FQ6%7$FaoFchtFhqFjq-FQ6%7$FaoFfhtFhqFjq-FQ6%7$FaoFihtFh qFjq-FQ6%7$FaoF\\itFhqFjqFa[lF_sFfsFjs75-F(6&F*Q(b~c~~=~F1F2F8-F(6&F?Q *~~~~~~~~1F1F2F8-F(6&FFQ*~~~~~~~~_F1F2F8-F(6&FLQ*~~~~~~~~8F1F2F8-F(6&F [dlQ*~~~~7~~~~F1F2F8F\\_l-FQ6&7*F`oFa\\l7$$!3:LLLLLLLLFduF6Fj^qFaoF[_q 7$F\\doF6Fd\\lFbpFfpFhp-FQ6&Fa[uF]qFfpFhp-FQ6&Fa[uFbqFfpFhpF`]l-FQ6%7$ F`oFb[uFhqFjqF`_qFeqFc_q-FQ6%7$F`oFe[uFhqFjqFc]lF_sFfsFjs" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curv e 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 63 "Some trees of order 10 \+ and their corresponding order conditions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "T10 := Trees(10):\nno ps(T10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$>(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "In each example the erro r condition is given in both the abreviated form and the long summatio n form in terms of the number of stages " }{TEXT 279 1 "s" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "nn := 400:\nT10[nn];\nDrawTree(T10[nn]);\nTreeOrderC ondition0(T10[nn]);\nsubs(_S=s,TreeOrderCondition(T10[nn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$7#7$7\"F'7#F'F'F'" }}{PARA 13 "" 1 "" {GLPLOT2D 479 273 273 {PLOTDATA 2 "62-%'CURVESG6&7,7$$\"\"!F)F(7$$!3++ ++++++]!#=$\"\"\"F)7$$!3ImmmmmmmmF-$\"\"#F)7$F1$\"\"$F)7$$!3!zxxxxxxx( F-$\"\"%F)7$$!3!ebbbbbbb&F-F;7$$!3:LLLLLLLLF-F37$FAF67$F(F.7$$\"3+++++ +++]F-F.-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&%$RGBGF(F(F(-%&STYLEG6#%&POINT G-F$6&F&-FI6#%(DIAMONDGFLFP-F$6&F&-FI6#%&CROSSGFLFP-F$6%7$F'F*-FM6&FOF .F(F(-%*THICKNESSG6#F4-F$6%7$F*F0F[oF]o-F$6%7$F0F5F[oF]o-F$6%7$F5F8F[o F]o-F$6%7$F5F=F[oF]o-F$6%7$F*F@F[oF]o-F$6%7$F@FCF[oF]o-F$6%7$F'FDF[oF] o-F$6%7$F'FEF[oF]o-F$6%7#7$F(F;-%'COLOURG6&FO$\"#5!\"\"F(F(-FQ6#%%LINE G-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!6\"F\\r-%%FONTG6#%(DEFAULTG-% %VIEWG6$FarFar" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7 " "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" }}} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"bG\"\"\"-%!G6#*(%\"aGF&-F(6#*&F+ F&-F(6#*&)%\"cG\"\"#F&F+F&F&F&-F(6#*&F+F&F3F&F&F&F2F&#F&\"%!o\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"-F%6$*( &%\"aG6$F+%\"jGF,-F%6$*&&F16$F3%\"kGF,-F%6$*&&F16$F9%\"lGF,)&%\"cG6#F? \"\"#F,/F?;F,,&F9F,F,!\"\"F,/F9;F,,&F3F,F,FHF,-F%6$*&F7F,&FB6#F9F,FIF, /F3;F,,&F+F,F,FHF,)&FBF*FDF,/F+;F,%\"sG#F,\"%!o\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "nn := 657: \nT10[nn];\nDrawTree(T10[nn]);\nTreeOrderCondition0(T10[nn]);\nsubs(_S =s,TreeOrderCondition(T10[nn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$ 7%7$7\"F&F%F&F&" }}{PARA 13 "" 1 "" {GLPLOT2D 386 220 220 {PLOTDATA 2 "62-%'CURVESG6&7,7$$\"\"!F)F(7$$!3++++++++]!#=$\"\"\"F)7$$!3qLLLLLLL$) F-$\"\"#F)7$$!3?WWWWWWW%*F-$\"\"$F)7$$!35AAAAAAAsF-F87$F+F37$$!3h66666 66hF-F87$$!3'*))))))))))))))QF-F87$$!3emmmmmmm;F-F37$$\"3++++++++]F-F. -%'SYMBOLG6#%'CIRCLEG-%&COLORG6&%$RGBGF(F(F(-%&STYLEG6#%&POINTG-F$6&F& -FK6#%(DIAMONDGFNFR-F$6&F&-FK6#%&CROSSGFNFR-F$6%7$F'F*-FO6&FQF.F(F(-%* THICKNESSG6#F4-F$6%7$F*F0F]oF_o-F$6%7$F0F5F]oF_o-F$6%7$F0F:F]oF_o-F$6% 7$F*F=F]oF_o-F$6%7$F=F>F]oF_o-F$6%7$F=FAF]oF_o-F$6%7$F*FDF]oF_o-F$6%7$ F'FGF]oF_o-F$6%7#7$F(F8-%'COLOURG6&FQ$\"#5!\"\"F(F(-FS6#%%LINEG-%*AXES STYLEG6#%%NONEG-%+AXESLABELSG6%Q!6\"F^r-%%FONTG6#%(DEFAULTG-%%VIEWG6$F crFcr" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve \+ 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve \+ 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" }}}{PARA 11 " " 1 "" {XPPMATH 20 "6#/*(%\"bG\"\"\"-%!G6#*(%\"aGF&)-F(6#*&)%\"cG\"\"# F&F+F&F2F&F1F&F&F1F&#F&\"$?(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$Su mG6$*(&%\"bG6#%\"iG\"\"\"-F%6$*(&%\"aG6$F+%\"jGF,)-F%6$*&&F16$F3%\"kGF ,)&%\"cG6#F:\"\"#F,/F:;F,,&F3F,F,!\"\"F?F,&F=6#F3F,/F3;F,,&F+F,F,FCF,& F=F*F,/F+;F,%\"sG#F,\"$?(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "nn := 658:\nT10[nn];\nDrawTree(T10 [nn]);\nTreeOrderCondition0(T10[nn]);\nsubs(_S=s,TreeOrderCondition(T1 0[nn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7\"F$7$7$F$F$F&" }} {PARA 13 "" 1 "" {GLPLOT2D 448 185 185 {PLOTDATA 2 "62-%'CURVESG6&7,7$ $\"\"!F)F(7$$!3++++++++]!#=$\"\"\"F)7$F(F.7$$\"3++++++++]F-F.7$$\"3:LL LLLLLLF-$\"\"#F)7$$\"35AAAAAAAAF-$\"\"$F)7$$\"3?WWWWWWWWF-F<7$$\"3Immm mmmmmF-F77$$\"3!ebbbbbbb&F-F<7$$\"3!zxxxxxxx(F-F<-%'SYMBOLG6#%'CIRCLEG -%&COLORG6&%$RGBGF(F(F(-%&STYLEG6#%&POINTG-F$6&F&-FK6#%(DIAMONDGFNFR-F $6&F&-FK6#%&CROSSGFNFR-F$6%7$F'F*-FO6&FQF.F(F(-%*THICKNESSG6#F8-F$6%7$ F'F0F]oF_o-F$6%7$F'F1F]oF_o-F$6%7$F1F4F]oF_o-F$6%7$F4F9F]oF_o-F$6%7$F4 F>F]oF_o-F$6%7$F1FAF]oF_o-F$6%7$FAFDF]oF_o-F$6%7$FAFGF]oF_o-F$6%7#7$F( F<-%'COLOURG6&FQ$\"#5!\"\"F(F(-FS6#%%LINEG-%*AXESSTYLEG6#%%NONEG-%+AXE SLABELSG6%Q!6\"F^r-%%FONTG6#%(DEFAULTG-%%VIEWG6$FcrFcr" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve \+ 10" "Curve 11" "Curve 12" "Curve 13" }}}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/*(%\"bG\"\"\")%\"cG\"\"#F&-%!G6#*&%\"aGF&)-F+6#*&F'F&F.F&F)F&F&#F& \"$I'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\" \")&%\"cGF*\"\"#F,-F%6$*&&%\"aG6$F+%\"jGF,)-F%6$*&&F56$F7%\"kGF,)&F/6# F>F0F,/F>;F,,&F7F,F,!\"\"F0F,/F7;F,,&F+F,F,FEF,/F+;F,%\"sG#F,\"$I'" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "nn := 659:\nT10[nn];\nDrawTree(T10[nn]);\nTreeOrderCondition0(T 10[nn]);\nsubs(_S=s,TreeOrderCondition(T10[nn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7&7\"F%7#F%F&F%F%" }}{PARA 13 "" 1 "" {GLPLOT2D 441 187 187 {PLOTDATA 2 "62-%'CURVESG6&7,7$$\"\"!F)F(7$$!3++++++++]!#=$\" \"\"F)7$$!3ImmmmmmmmF-$\"\"#F)7$$!3!ebbbbbbb&F-F37$$!3?WWWWWWWWF-F37$F 9$\"\"$F)7$$!3:LLLLLLLLF-F37$F?F<7$F(F.7$$\"3++++++++]F-F.-%'SYMBOLG6# %'CIRCLEG-%&COLORG6&%$RGBGF(F(F(-%&STYLEG6#%&POINTG-F$6&F&-FG6#%(DIAMO NDGFJFN-F$6&F&-FG6#%&CROSSGFJFN-F$6%7$F'F*-FK6&FMF.F(F(-%*THICKNESSG6# F4-F$6%7$F*F0FinF[o-F$6%7$F*F5FinF[o-F$6%7$F*F8FinF[o-F$6%7$F8F;FinF[o -F$6%7$F*F>FinF[o-F$6%7$F>FAFinF[o-F$6%7$F'FBFinF[o-F$6%7$F'FCFinF[o-F $6%7#7$F(F<-%'COLOURG6&FM$\"#5!\"\"F(F(-FO6#%%LINEG-%*AXESSTYLEG6#%%NO NEG-%+AXESLABELSG6%Q!6\"Fjq-%%FONTG6#%(DEFAULTG-%%VIEWG6$F_rF_r" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9 " "Curve 10" "Curve 11" "Curve 12" "Curve 13" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"bG\"\"\"-%!G6#*(%\"aGF&)%\"cG\"\"#F&)-F(6#*&F+F&F -F&F.F&F&F,F&#F&\"$!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*(& %\"bG6#%\"iG\"\"\"-F%6$*(&%\"aG6$F+%\"jGF,)&%\"cG6#F3\"\"#F,)-F%6$*&&F 16$F3%\"kGF,&F66#F?F,/F?;F,,&F3F,F,!\"\"F8F,/F3;F,,&F+F,F,FEF,)&F6F*F8 F,/F+;F,%\"sG#F,\"$!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "nn := 660:\nT10[nn];\nDrawTree(T10 [nn]);\nTreeOrderCondition0(T10[nn]);\nsubs(_S=s,TreeOrderCondition(T1 0[nn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7$7\"F%F$F%7#F%" }} {PARA 13 "" 1 "" {GLPLOT2D 355 134 134 {PLOTDATA 2 "62-%'CURVESG6&7,7$ $\"\"!F)F(7$$!3++++++++]!#=$\"\"\"F)7$$!3h6666666hF-$\"\"#F)7$$!3'*))) )))))))))))QF-F37$$!3emmmmmmm;F-F.7$$!3!zxxxxxxx#F-F37$$!3Ebbbbbbbb!#> F37$$\"3emmmmmmm;F-F.7$$\"3++++++++]F-F.7$FFF3-%'SYMBOLG6#%'CIRCLEG-%& COLORG6&%$RGBGF(F(F(-%&STYLEG6#%&POINTG-F$6&F&-FJ6#%(DIAMONDGFMFQ-F$6& F&-FJ6#%&CROSSGFMFQ-F$6%7$F'F*-FN6&FPF.F(F(-%*THICKNESSG6#F4-F$6%7$F*F 0F\\oF^o-F$6%7$F*F5F\\oF^o-F$6%7$F'F8F\\oF^o-F$6%7$F8F;F\\oF^o-F$6%7$F 8F>F\\oF^o-F$6%7$F'FBF\\oF^o-F$6%7$F'FEF\\oF^o-F$6%7$FEFHF\\oF^o-F$6%7 #7$F(F3-%'COLOURG6&FP$\"#5!\"\"F(F(-FR6#%%LINEG-%*AXESSTYLEG6#%%NONEG- %+AXESLABELSG6%Q!6\"F]r-%%FONTG6#%(DEFAULTG-%%VIEWG6$FbrFbr" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "C urve 10" "Curve 11" "Curve 12" "Curve 13" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%\"bG\"\"\")-%!G6#*&)%\"cG\"\"#F&%\"aGF&F.F&F-F&-F)6 #*&F/F&F-F&F&#F&\"$!=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$**& %\"bG6#%\"iG\"\"\")-F%6$*&&%\"aG6$F+%\"jGF,)&%\"cG6#F4\"\"#F,/F4;F,,&F +F,F,!\"\"F9F,&F7F*F,-F%6$*&F1F,F6F,F:F,/F+;F,%\"sG#F,\"$!=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "n n := 1:\nT10[nn];\nDrawTree(T10[nn]);\nTreeOrderCondition0(T10[nn]);\n subs(_S=s,TreeOrderCondition(T10[nn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7#7#7#7#7#7#7#7#7\"" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "62-%'CURVESG6&7,7$$\"\"!F)F(7$F($\"\"\"F)7$F($\"\"#F)7$F( $\"\"$F)7$F($\"\"%F)7$F($\"\"&F)7$F($\"\"'F)7$F($\"\"(F)7$F($\"\")F)7$ F($\"\"*F)-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&%$RGBGF(F(F(-%&STYLEG6#%&POI NTG-F$6&F&-FF6#%(DIAMONDGFIFM-F$6&F&-FF6#%&CROSSGFIFM-F$6%7$F'F*-FJ6&F LF+F(F(-%*THICKNESSG6#F/-F$6%7$F*F-FhnFjn-F$6%7$F-F0FhnFjn-F$6%7$F0F3F hnFjn-F$6%7$F3F6FhnFjn-F$6%7$F6F9FhnFjn-F$6%7$F9F " 0 "" {MPLTEXT 1 0 108 "nn := 719:\nT10[nn]; \nDrawTree(T10[nn]);\nTreeOrderCondition0(T10[nn]);\nsubs(_S=s,TreeOrd erCondition(T10[nn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+7\"F$F$F$F $F$F$F$F$" }}{PARA 13 "" 1 "" {GLPLOT2D 837 122 122 {PLOTDATA 2 "62-%' CURVESG6&7,7$$\"\"!F)F(7$$!3++++++++]!#=$\"\"\"F)7$$!3+++++++]PF-F.7$$ !3++++++++DF-F.7$$!3+++++++]7F-F.7$F(F.7$$\"3+++++++]7F-F.7$$\"3++++++ ++DF-F.7$$\"3+++++++]PF-F.7$$\"3++++++++]F-F.-%'SYMBOLG6#%'CIRCLEG-%&C OLORG6&%$RGBGF(F(F(-%&STYLEG6#%&POINTG-F$6&F&-FG6#%(DIAMONDGFJFN-F$6&F &-FG6#%&CROSSGFJFN-F$6%7$F'F*-FK6&FMF.F(F(-%*THICKNESSG6#\"\"#-F$6%7$F 'F0FinF[o-F$6%7$F'F3FinF[o-F$6%7$F'F6FinF[o-F$6%7$F'F9FinF[o-F$6%7$F'F :FinF[o-F$6%7$F'F=FinF[o-F$6%7$F'F@FinF[o-F$6%7$F'FCFinF[o-F$6%7#F9-%' COLOURG6&FM$\"#5!\"\"F(F(-FO6#%%LINEG-%*AXESSTYLEG6#%%NONEG-%+AXESLABE LSG6%Q!6\"Fjq-%%FONTG6#%(DEFAULTG-%%VIEWG6$F_rF_r" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "C urve 11" "Curve 12" "Curve 13" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&% \"bG\"\"\")%\"cG\"\"*F&#F&\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$ SumG6$*&&%\"bG6#%\"iG\"\"\")&%\"cGF*\"\"*F,/F+;F,%\"sG#F,\"#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 63 "Some trees of order 12 \+ and their corresponding order conditions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "T12 := Trees(12):\nno ps(T12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%mZ" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ST12 := Simp leTrees(12):\nnops(ST12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%C5" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "nn := 461:\nST12[nn];\nDrawTree(ST12[nn],show_ordercondition=tru e,scale=1.3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7\"7#7%F$F$7%F$F$7# 7#F$" }}{PARA 13 "" 1 "" {GLPLOT2D 532 285 285 {PLOTDATA 2 "69-%%TEXTG 6&7$$\"\"!F($!#l!\"#QGb~c~(a~(a~c~~(a~c~~(a~(a~c)))))~=~~~~~6\"-%&COLO RG6&%$RGBGF'F'$\"\"\"F(-%%FONTG6$%(COURIERG\"#5-F$6&7$F'$!$U%!\"$QI~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1F-F.F4-F$6&7$F'$!$2&F>QM~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_____F-F.F4-F$6&7$F'$!$B*F>QM~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~38880F-F.F4-F$6&7$F'$!$;%F>QI~~~~~~~~~~~~2 ~~~~~2~~~~~~~~~~~~~~~~~~~~~F-F.F4-%'CURVESG6&7.7$F'F'7$$!3++++++++]!#= F27$$\"3++++++++]FZF27$Ffn$\"\"#F(7$$\"3emmmmmmm;FZ$\"\"$F(7$FfnF^o7$$ \"3qLLLLLLL$)FZF^o7$$\"35AAAAAAAsFZ$\"\"%F(7$FboFgo7$$\"3?WWWWWWW%*FZF go7$F[p$\"\"&F(7$F[p$\"\"'F(-%'SYMBOLG6#%'CIRCLEG-F/6&F1F'F'F'-%&STYLE G6#%&POINTG-FS6&FU-Fdp6#%(DIAMONDGFgpFip-FS6&FU-Fdp6#%&CROSSGFgpFip-FS 6%7$FVFW-F/6&F1F2F'F'-%*THICKNESSG6#Fjn-FS6%7$FVFenFjqF\\r-FS6%7$FenFh nFjqF\\r-FS6%7$FhnF[oFjqF\\r-FS6%7$FhnF`oFjqF\\r-FS6%7$FhnFaoFjqF\\r-F S6%7$FaoFdoFjqF\\r-FS6%7$FaoFioFjqF\\r-FS6%7$FaoFjoFjqF\\r-FS6%7$FjoF] pFjqF\\r-FS6%7$F]pF`pFjqF\\r-FS6%7#7$F'Fap-%'COLOURG6&F1$F8!\"\"F'F'-F jp6#%%LINEG-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!F-F`u-F56#%(DEFAULT G-%%VIEWG6$FcuFcu" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "We can find this simple tree among the list of all trees \+ of order 12." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "for ii to n ops(T12) do\n if lhs(TreeOrderCondition0(T12[ii]))=lhs(TreeOrderCond ition0(ST12[461])) then break end if;\nend do: ii;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"%CD" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "nn := 2524:\nT12[nn];\nDrawTree(T12 [nn],show_ordercondition=true,scale=1.3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7\"7#7%F$F$7%F$F$7#7#F$" }}{PARA 13 "" 1 "" {GLPLOT2D 510 287 287 {PLOTDATA 2 "69-%%TEXTG6&7$$\"\"!F($!#l!\"#QGb~c ~(a~(a~c~~(a~c~~(a~(a~c)))))~=~~~~~6\"-%&COLORG6&%$RGBGF'F'$\"\"\"F(-% %FONTG6$%(COURIERG\"#5-F$6&7$F'$!$U%!\"$QI~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~1F-F.F4-F$6&7$F'$!$2&F>QM~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~_____F-F.F4-F$6&7$F'$!$B*F>QM~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~38880F-F.F4-F$6&7$F'$!$;%F>QI~~~~~~~~~~~~2~~~~~2~~~~~~~~~~~~~~~~~~~ ~~F-F.F4-%'CURVESG6&7.7$F'F'7$$!3++++++++]!#=F27$$\"3++++++++]FZF27$Ff n$\"\"#F(7$$\"3emmmmmmm;FZ$\"\"$F(7$FfnF^o7$$\"3qLLLLLLL$)FZF^o7$$\"35 AAAAAAAsFZ$\"\"%F(7$FboFgo7$$\"3?WWWWWWW%*FZFgo7$F[p$\"\"&F(7$F[p$\"\" 'F(-%'SYMBOLG6#%'CIRCLEG-F/6&F1F'F'F'-%&STYLEG6#%&POINTG-FS6&FU-Fdp6#% (DIAMONDGFgpFip-FS6&FU-Fdp6#%&CROSSGFgpFip-FS6%7$FVFW-F/6&F1F2F'F'-%*T HICKNESSG6#Fjn-FS6%7$FVFenFjqF\\r-FS6%7$FenFhnFjqF\\r-FS6%7$FhnF[oFjqF \\r-FS6%7$FhnF`oFjqF\\r-FS6%7$FhnFaoFjqF\\r-FS6%7$FaoFdoFjqF\\r-FS6%7$ FaoFioFjqF\\r-FS6%7$FaoFjoFjqF\\r-FS6%7$FjoF]pFjqF\\r-FS6%7$F]pF`pFjqF \\r-FS6%7#7$F'Fap-%'COLOURG6&F1$F8!\"\"F'F'-Fjp6#%%LINEG-%*AXESSTYLEG6 #%%NONEG-%+AXESLABELSG6%Q!F-F`u-F56#%(DEFAULTG-%%VIEWG6$FcuFcu" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Cur ve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We can find a tree whos e order condtion involves the powers 2, 3 and 4." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 204 "for ii to nops(T12) do\n ee := lhs(TreeOrde rCondition0(T12[ii]));\n if has(ee,2) and has(ee,3) and has(ee,4) th en break end if;\nend do: ii;\nT12[ii];\nDrawTree(T12[ii],show_orderco ndition=true,scale=0.6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%$p%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%7\"F$7&F$F$F$7&F$F$F$F$" }}{PARA 13 "" 1 "" {GLPLOT2D 623 297 297 {PLOTDATA 2 "69-%%TEXTG6&7$$\"\"!F($!#I! \"#Q8b~c~~(a~c~~(a~c~))~=~~~6\"-%&COLORG6&%$RGBGF'F'$\"\"\"F(-%%FONTG6 $%(COURIERG\"#5-F$6&7$F'$!$/#!\"$Q:~~~~~~~~~~~~~~~~~~~~~~~~1F-F.F4-F$6 &7$F'$!$M#F>Q<~~~~~~~~~~~~~~~~~~~~~~~~___F-F.F4-F$6&7$F'$!$E%F>Q<~~~~~ ~~~~~~~~~~~~~~~~~~~540F-F.F4-F$6&7$F'$!$#>F>Q:~~~~2~~~~~3~~~~~4~~~~~~~ ~F-F.F4-%'CURVESG6&7.7$F'F'7$$!3++++++++]!#=F27$F'F27$$\"3++++++++]FZF 27$$\"3:LLLLLLLLFZ$\"\"#F(7$$\"3?WWWWWWWWFZF\\o7$$\"3!ebbbbbbb&FZF\\o7 $$\"3ImmmmmmmmFZF\\o7$$\"3['H'H'H'H'H'FZ$\"\"$F(7$$\"325Kaw)4Ka'FZFjo7 $$\"3iB,zcM7!z'FZFjo7$$\"3?Pq.Pq.PqFZFjo-%'SYMBOLG6#%'CIRCLEG-F/6&F1F' F'F'-%&STYLEG6#%&POINTG-FS6&FU-Ffp6#%(DIAMONDGFipF[q-FS6&FU-Ffp6#%&CRO SSGFipF[q-FS6%7$FVFW-F/6&F1F2F'F'-%*THICKNESSG6#F]o-FS6%7$FVFenF\\rF^r -FS6%7$FVFfnF\\rF^r-FS6%7$FfnFinF\\rF^r-FS6%7$FfnF^oF\\rF^r-FS6%7$FfnF aoF\\rF^r-FS6%7$FfnFdoF\\rF^r-FS6%7$FdoFgoF\\rF^r-FS6%7$FdoF\\pF\\rF^r -FS6%7$FdoF_pF\\rF^r-FS6%7$FdoFbpF\\rF^r-FS6%7#7$F'Fjo-%'COLOURG6&F1$F 8!\"\"F'F'-F\\q6#%%LINEG-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!F-Fbu- F56#%(DEFAULTG-%%VIEWG6$FeuFeu" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 22 "This is a simple tree." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 131 "for ii to nops(ST12) do\n if lhs(TreeOrderCondit ion0(ST12[ii]))=lhs(TreeOrderCondition0(T12[4693])) then break end if; \nend do: ii;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$r*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The first tree of or der 12: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "nn := 1;\nT12[nn ];\nDrawTree(T12[nn],show_ordercondition=true,scale=1.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nnG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7#7#7#7#7#7#7#7#7#7#7#7\"" }}{PARA 13 "" 1 "" {GLPLOT2D 596 437 437 {PLOTDATA 2 "68-%%TEXTG6&7$$\"\"!F($!#v!\"#QWb~(a~(a~(a~(a~(a~(a~(a~(a ~(a~(a~c))))))))))~=~~~~~~~~~6\"-%&COLORG6&%$RGBGF'F'$\"\"\"F(-%%FONTG 6$%(COURIERG\"#5-F$6&7$F'$!$5&!\"$QY~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~1F-F.F4-F$6&7$F'$!$&eF>Q[o~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_________F-F.F4-F$6&7$F'$!%l5F>Q[o~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~479001600F-F.F4-%'C URVESG6&7.7$F'F'7$F'F27$F'$\"\"#F(7$F'$\"\"$F(7$F'$\"\"%F(7$F'$\"\"&F( 7$F'$\"\"'F(7$F'$\"\"(F(7$F'$\"\")F(7$F'$\"\"*F(7$F'$F8F(7$F'$\"#6F(-% 'SYMBOLG6#%'CIRCLEG-F/6&F1F'F'F'-%&STYLEG6#%&POINTG-FM6&FO-Fjo6#%(DIAM ONDGF]pF_p-FM6&FO-Fjo6#%&CROSSGF]pF_p-FM6%7$FPFQ-F/6&F1F2F'F'-%*THICKN ESSG6#FT-FM6%7$FQFRF`qFbq-FM6%7$FRFUF`qFbq-FM6%7$FUFXF`qFbq-FM6%7$FXFe nF`qFbq-FM6%7$FenFhnF`qFbq-FM6%7$FhnF[oF`qFbq-FM6%7$F[oF^oF`qFbq-FM6%7 $F^oFaoF`qFbq-FM6%7$FaoFdoF`qFbq-FM6%7$FdoFfoF`qFbq-FM6%7#Ffo-%'COLOUR G6&F1$F8!\"\"F'F'-F`p6#%%LINEG-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q! F-Fet-F56#%(DEFAULTG-%%VIEWG6$FhtFht" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The last tree of order 12: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "nn := 4766;\nT12[nn];\nDrawTree(T12[nn],show_ordercon dition=true,scale=0.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nnG\"%mZ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-7\"F$F$F$F$F$F$F$F$F$F$" }} {PARA 13 "" 1 "" {GLPLOT2D 836 189 189 {PLOTDATA 2 "69-%%TEXTG6&7$$\" \"!F($!#?!\"#Q*b~c~~~=~~6\"-%&COLORG6&%$RGBGF'F'$\"\"\"F(-%%FONTG6$%(C OURIERG\"#5-F$6&7$F'$!$O\"!\"$Q,~~~~~~~~~~1F-F.F4-F$6&7$F'$!$c\"F>Q-~~ ~~~~~~~~__F-F.F4-F$6&7$F'$!$%GF>Q-~~~~~~~~~~12F-F.F4-F$6&7$F'$!$G\"F>Q ,~~~~11~~~~~F-F.F4-%'CURVESG6&7.7$F'F'7$$!3++++++++]!#=F27$$!3A+++++++ SFZF27$$!3))**************HFZF27$$!35+++++++?FZF27$$!3/+++++++5FZF27$F 'F27$$\"3/+++++++5FZF27$$\"35+++++++?FZF27$$\"3))**************HFZF27$ $\"3A+++++++SFZF27$$\"3++++++++]FZF2-%'SYMBOLG6#%'CIRCLEG-F/6&F1F'F'F' -%&STYLEG6#%&POINTG-FS6&FU-Fbp6#%(DIAMONDGFepFgp-FS6&FU-Fbp6#%&CROSSGF epFgp-FS6%7$FVFW-F/6&F1F2F'F'-%*THICKNESSG6#\"\"#-FS6%7$FVFenFhqFjq-FS 6%7$FVFhnFhqFjq-FS6%7$FVF[oFhqFjq-FS6%7$FVF^oFhqFjq-FS6%7$FVFaoFhqFjq- FS6%7$FVFboFhqFjq-FS6%7$FVFeoFhqFjq-FS6%7$FVFhoFhqFjq-FS6%7$FVF[pFhqFj q-FS6%7$FVF^pFhqFjq-FS6%7#Fao-%'COLOURG6&F1$F8!\"\"F'F'-Fhp6#%%LINEG-% *AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!F-F^u-F56#%(DEFAULTG-%%VIEWG6$Fa uFau" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8 " "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Cu rve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Number of trees of a gi ven order" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[order,`|`,1,2,3,4,5,6,7,8,9,10,11,12 ,13,14,15,16],\n[`number of trees`,`|`,1,1,2,4,9,20,48,115,286,719,184 2,4766,12486,32973,87811,235381]])" "6#-%'matrixG6#7$74%&orderG%\"|grG \"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#: \"#;74%0number~of~treesGF)F*F*F+F-F2\"#?\"#[\"$:\"\"$'G\"$>(\"%U=\"%mZ \"&'[7\"&tH$\"&6y)\"'\"QN#" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[or der, `|`, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [`no . of trees with order` <= n, `|`, 1, 2, 4, 8, 17, 37, 85, 200, 486, 12 05, 3047, 7813, 20299, 53272, 141083, 376464]]);" "6#-%'matrixG6#7$74% &orderG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\" #7\"#8\"#9\"#:\"#;741%8no.~of~trees~with~orderG%\"nGF)F*F+F-F1\"#<\"#P \"#&)\"$+#\"$'[\"%07\"%ZI\"%8y\"&*H?\"&sK&\"'$3T\"\"'kkP" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "Th e sequence of rooted trees of a given order is A000081(M1180) in Neil J. Sloane's database of integer sequences. " }}{PARA 0 "" 0 "" {TEXT -1 5 "See: " }{URLLINK 17 "http://www.research.att.com/~njas/sequences /A000081" 4 "http://www.research.att.com/~njas/sequences/A000081" "" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "and : " }{URLLINK 17 "http://www.research.att.com/~njas/se quences/b000081.txt" 4 "http://www.research.att.com/~njas/sequences/b0 00081.txt" "" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "This sequence can be computed using Maple code \+ due to Joe Riel June 23, 2008." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "A := proc(n) \n local k;\n A(n) := add(k*A(k)*S(n-1,k),k= 1..n-1)/(n-1);\nend proc:\nA(0) := 0: A(1) := 1:\nS := proc(n,k)\n l ocal j;\n S(n,k) := add(A(n+1-j*k),j=1..iquo(n,k));\nend proc:\nB := proc(n)\n add(A(j),j=1..n);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "convert([[`n`,`no. of trees of order n`],seq([n,A(n)],n=1..20)],matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#777$%\"nG%8no.~of~trees~of~order~nG7$ \"\"\"F+7$\"\"#F+7$\"\"$F-7$\"\"%F17$\"\"&\"\"*7$\"\"'\"#?7$\"\"(\"#[7 $\"\")\"$:\"7$F4\"$'G7$\"#5\"$>(7$\"#6\"%U=7$\"#7\"%mZ7$\"#8\"&'[77$\" #9\"&tH$7$\"#:\"&6y)7$\"#;\"'\"QN#7$\"#<\"'Z[j7$\"#=\"(f6s\"7$\"#>\"(w ')o%7$F7\")Gi#G\"Q)pprint226\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "convert([[`n`,`no. of trees \+ with order`<=n],seq([n,B(n)],n=1..20)],matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#777$%\"nG1%8no.~of~trees~with~orderGF(7$\" \"\"F,7$\"\"#F.7$\"\"$\"\"%7$F1\"\")7$\"\"&\"#<7$\"\"'\"#P7$\"\"(\"#&) 7$F3\"$+#7$\"\"*\"$'[7$\"#5\"%077$\"#6\"%ZI7$\"#7\"%8y7$\"#8\"&*H?7$\" #9\"&sK&7$\"#:\"'$3T\"7$\"#;\"'kkP7$F6\"(68,\"7$\"#=\"(qCt#7$\"#>\"(Y6 U(7$\"#?\")utC?Q)pprint236\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 63 "Some trees of order 16 and their corresponding order co nditions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "read \"C:\\\\Maple/data/trees.m\";" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "trs16[1]; \ntrs16[235381];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7#7#7#7#7#7#7#7# 7#7#7#7#7#7#7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#717\"F$F$F$F$F$F$F$ F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 62 "trs16[117691];\nDrawTree(%,show_ordercondition =true,width=1.7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(7\"F$F$F$7#7#F$ 7$F%7#F%" }}{PARA 13 "" 1 "" {GLPLOT2D 561 282 282 {PLOTDATA 2 "6?-%%T EXTG6&7$$\"\"!F($!\"&!\"\"QSb~c~~(a~(a~c))~(a~(a~(a~c))~(a~(a~(a~c)))) ~=~~~~~~6\"-%&COLORG6&%$RGBGF'F'$\"\"\"F(-%%FONTG6$%(COURIERG\"#5-F$6& 7$F'$!#M!\"#QU~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1F-F. F4-F$6&7$F'$!#RF>QZ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ______F-F.F4-F$6&7$F'$!#rF>QZ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~110592F-F.F4-F$6&7$F'$!#KF>QU~~~~4~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~F-F.F4-%'CURVESG6&727$F'F'7$$!3++++++++]!#=F27$$! 3))**************HFZF27$$!3/+++++++5FZF27$$\"3/+++++++5FZF27$$\"3))*** ***********HFZF27$F_o$\"\"#F(7$F_o$\"\"$F(7$$\"3++++++++]FZF27$$\"3[LL LLLLLVFZFbo7$F[pFeo7$F[p$\"\"%F(7$$\"3_mmmmmmmcFZFbo7$FbpFeo7$FbpF_p7$ Fbp$\"\"&F(-%'SYMBOLG6#%'CIRCLEG-F/6&F1F'F'F'-%&STYLEG6#%&POINTG-FS6&F U-Fjp6#%(DIAMONDGF]qF_q-FS6&FU-Fjp6#%&CROSSGF]qF_q-FS6%7$FVFW-F/6&F1F2 F'F'-%*THICKNESSG6#Fco-FS6%7$FVFenF`rFbr-FS6%7$FVFhnF`rFbr-FS6%7$FVF[o F`rFbr-FS6%7$FVF^oF`rFbr-FS6%7$F^oFaoF`rFbr-FS6%7$FaoFdoF`rFbr-FS6%7$F VFgoF`rFbr-FS6%7$FgoFjoF`rFbr-FS6%7$FjoF]pF`rFbr-FS6%7$F]pF^pF`rFbr-FS 6%7$FgoFapF`rFbr-FS6%7$FapFdpF`rFbr-FS6%7$FdpFepF`rFbr-FS6%7$FepFfpF`r Fbr-FS6%7#7$F'Fgp-%'COLOURG6&F1$F8F+F'F'-F`q6#%%LINEG-FS6%7#7$$!3w**** *********\\)FZF'FcuFgu-FS6%7#7$$\"3w*************\\)FZF'FcuFgu-%*AXESS TYLEG6#%%NONEG-%+AXESLABELSG6%Q!F-F]w-F56#%(DEFAULTG-%%VIEWG6$F`wF`w" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Cur ve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15 " "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "C urve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "t1 := [[ [],[],[[]],[[],[],[]]],[[],[]],[[],[]]];\nt2 := [[[]],[[],[],[[],[]],[ [],[]]],[[],[],[]]];\nDrawTree(t1,show_ordercondition=true);\nDrawTree (t2,show_ordercondition=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t 1G7%7&7\"F'7#F'7%F'F'F'7$F'F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# t2G7%7#7\"7&F'F'7$F'F'F)7%F'F'F'" }}{PARA 13 "" 1 "" {GLPLOT2D 444 229 229 {PLOTDATA 2 "6=-%%TEXTG6&7$$\"\"!F($!\"&!\"\"QEb~(a~c~~(a~c)~( a~c~))~(a~c~)~~=~~~~~6\"-%&COLORG6&%$RGBGF'F'$\"\"\"F(-%%FONTG6$%(COUR IERG\"#5-F$6&7$F'$!#M!\"#QG~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1F-F.F 4-F$6&7$F'$!#RF>QK~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_____F-F.F4-F$6 &7$F'$!#rF>QK~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~10368F-F.F4-F$6&7$F' $!#KF>QG~~~~~~~2~~~~~~~~~~~3~~~~~~~2~2~~~~~~~~F-F.F4-%'CURVESG6&727$F' F'7$$!3++++++++]!#=F27$$!3ImmmmmmmmFZ$\"\"#F(7$$!3!ebbbbbbb&FZFhn7$$!3 ?WWWWWWWWFZFhn7$F^o$\"\"$F(7$$!3:LLLLLLLLFZFhn7$$!3].Pq.Pq.PFZFao7$Fdo Fao7$$!3!G'H'H'H'H'HFZFao7$F'F27$$!3emmmmmmm;FZFhn7$$\"3emmmmmmm;FZFhn 7$$\"3++++++++]FZF27$$\"3:LLLLLLLLFZFhn7$$\"3ImmmmmmmmFZFhn-%'SYMBOLG6 #%'CIRCLEG-F/6&F1F'F'F'-%&STYLEG6#%&POINTG-FS6&FU-F^q6#%(DIAMONDGFaqFc q-FS6&FU-F^q6#%&CROSSGFaqFcq-FS6%7$FVFW-F/6&F1F2F'F'-%*THICKNESSG6#Fin -FS6%7$FWFenFdrFfr-FS6%7$FWFjnFdrFfr-FS6%7$FWF]oFdrFfr-FS6%7$F]oF`oFdr Ffr-FS6%7$FWFcoFdrFfr-FS6%7$FcoFfoFdrFfr-FS6%7$FcoFioFdrFfr-FS6%7$FcoF joFdrFfr-FS6%7$FVF]pFdrFfr-FS6%7$F]pF^pFdrFfr-FS6%7$F]pFapFdrFfr-FS6%7 $FVFdpFdrFfr-FS6%7$FdpFgpFdrFfr-FS6%7$FdpFjpFdrFfr-FS6%7#7$F'Fao-%'COL OURG6&F1$F8F+F'F'-Fdq6#%%LINEG-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q! 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Pq.PFZF[o7$$!3!G'H'H'H'H'HFZF[o7$$!35AAAAAAAAFZF[o7$F'F27$$!3emmmmmmm; FZFhn7$$\"3emmmmmmm;FZFhn7$$\"3++++++++]FZF27$$\"3:LLLLLLLLFZFhn7$FdpF hn7$$\"3ImmmmmmmmFZFhn-%'SYMBOLG6#%'CIRCLEG-F/6&F1F'F'F'-%&STYLEG6#%&P OINTG-FS6&FU-F^q6#%(DIAMONDGFaqFcq-FS6&FU-F^q6#%&CROSSGFaqFcq-FS6%7$FV FW-F/6&F1F2F'F'-%*THICKNESSG6#Fin-FS6%7$FWFenFdrFfr-FS6%7$FenFjnFdrFfr -FS6%7$FWF]oFdrFfr-FS6%7$F]oF`oFdrFfr-FS6%7$F]oFcoFdrFfr-FS6%7$F]oFfoF drFfr-FS6%7$F]oFioFdrFfr-FS6%7$FVF\\pFdrFfr-FS6%7$F\\pF]pFdrFfr-FS6%7$ F\\pF`pFdrFfr-FS6%7$FVFcpFdrFfr-FS6%7$FcpFfpFdrFfr-FS6%7$FcpFipFdrFfr- FS6%7$FcpFjpFdrFfr-FS6%7#7$F'F[o-%'COLOURG6&F1$F8F+F'F'-Fdq6#%%LINEG-% *AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!F-Fev-F56#%(DEFAULTG-%%VIEWG6$Fh vFhv" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8 " "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Cu rve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve \+ 21" "Curve 22" "Curve 23" "Curve 24" }}}{PARA 13 "" 1 "" {GLPLOT2D 490 235 235 {PLOTDATA 2 "6=-%%TEXTG6&7$$\"\"!F($!\"&!\"\"QHb~(a~c)~(a~ (a~c~)~(a~c~))~(a~c~)~=~~~~~6\"-%&COLORG6&%$RGBGF'F'$\"\"\"F(-%%FONTG6 $%(COURIERG\"#5-F$6&7$F'$!#M!\"#QJ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~1F-F.F4-F$6&7$F'$!#RF>QN~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~__ ___F-F.F4-F$6&7$F'$!#rF>QN~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1536 0F-F.F4-F$6&7$F'$!#KF>QJ~~~~~~~~~~~~~~~~2~~~~~~3~~~~~~~4~~~~~~~~~F-F.F 4-%'CURVESG6&727$F'F'7$$!3++++++++]!#=F27$FX$\"\"#F(7$F'F27$$!3emmmmmm m;FZFfn7$$!3!zxxxxxxx#FZ$\"\"$F(7$$!3Ebbbbbbbb!#>F_o7$$\"3emmmmmmm;FZF fn7$$\"3EbbbbbbbbFdoF_o7$FfoF_o7$$\"3!zxxxxxxx#FZF_o7$$\"3++++++++]FZF 27$$\"3:LLLLLLLLFZFfn7$$\"3?WWWWWWWWFZFfn7$$\"3!ebbbbbbb&FZFfn7$$\"3Im mmmmmmmFZFfn-%'SYMBOLG6#%'CIRCLEG-F/6&F1F'F'F'-%&STYLEG6#%&POINTG-FS6& FU-F_q6#%(DIAMONDGFbqFdq-FS6&FU-F_q6#%&CROSSGFbqFdq-FS6%7$FVFW-F/6&F1F 2F'F'-%*THICKNESSG6#Fgn-FS6%7$FWFenFerFgr-FS6%7$FVFhnFerFgr-FS6%7$FhnF inFerFgr-FS6%7$FinF\\oFerFgr-FS6%7$FinFaoFerFgr-FS6%7$FhnFeoFerFgr-FS6 %7$FeoFhoFerFgr-FS6%7$FeoF[pFerFgr-FS6%7$FeoF\\pFerFgr-FS6%7$FVF_pFerF gr-FS6%7$F_pFbpFerFgr-FS6%7$F_pFepFerFgr-FS6%7$F_pFhpFerFgr-FS6%7$F_pF [qFerFgr-FS6%7#7$F'F_o-%'COLOURG6&F1$F8F+F'F'-Feq6#%%LINEG-%*AXESSTYLE G6#%%NONEG-%+AXESLABELSG6%Q!F-Ffv-F56#%(DEFAULTG-%%VIEWG6$FivFiv" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9 " "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "C urve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Some equivalent trees of order 17" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 563 "beta := [[[ ],[[]]],[[],[[[]]]],[[],[[]],[[[]]]]];\nPM := map(combinat[permute],be ta):\nP := [seq(seq(seq(subsop(seq(j=op(i||j,op(j,PM)),j=1..3),beta), \n i1=1..nops(op(1,PM))),i2=1..nops(op(2,PM))),i3=1..nops(op(3,PM)) )]:\ntrs := ListTools[FlattenOnce](map(combinat[permute],P)):\nm := 4: ## number of trees per row\nnn := nops(trs): q := iquo(nn,m,'r'):\nfo r i to nn do \n p||i := DrawTree(trs[i]);\nend do:\npp := plot([[1,1] ],style=line,axes=none):\nplots[display](convert([seq([p||((k-1)*m+1.. m*k)],k=1..q),\n `if`(r>0,[p||(m*q+1..nn),pp$(m*(q+1)-nn)],NULL)],arr ay));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG7%7$7\"7#F'7$F'7#F(7% F'F(F*" }}{PARA 13 "" 1 "" {GLPLOT2D 949 3387 3387 {PLOTDATA 2 "6^_v-% 'CURVESG6&737$$\"#!*\"\"!$!+++++Y!\"(7$$\"++++]#)!\")$!++++]XF-7$$\"++ +++!)F1$!+++++XF-7$F5$!++++]WF-7$F5$!+++++WF-7$$\"+++++&)F1F77$F(F27$$ \"++++]()F1F77$FDF:7$F(F77$$\"++++]#*F1F77$FIF:7$FIF=7$$\"++++](*F1F27 $$\"+++++&*F1F77$$\"+++++5F-F77$FTF:-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&%$ RGBG$F*F*FinFin-%&STYLEG6#%&POINTG-F$6&F&-FX6#%(DIAMONDGFenFjn-F$6&F&- FX6#%&CROSSGFenFjn-F$6%7$F'F.-Ffn6&Fhn$\"\"\"F*FinFin-%*THICKNESSG6#\" \"#-F$6%7$F.F4F[pF_p-F$6%7$F4F9F[pF_p-F$6%7$F9FF-7$Fft$!++++]=F-7$F[u$!+++++=F- 7$FcuFd`t7$Fcu$!++++] " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 280 27 "____________ _______________" }{TEXT 260 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 99 "Procedures for the construction o f order conditions and related concepts for Runge-Kutta schemes: \n" } {TEXT 0 15 "OrderConditions" }{TEXT -1 2 ", " }{TEXT 0 16 "RowSumCondi tions" }{TEXT -1 2 ", " }{TEXT 0 20 "StageOrderConditions" }{TEXT -1 2 ", " }{TEXT 0 24 "PrincipalErrorConditions" }{TEXT -1 2 ", " }{TEXT 0 19 "PrincipalErrorTerms" }{TEXT -1 2 ", " }{TEXT 0 20 "QuadratureCon ditions" }{TEXT -1 2 ", " }{TEXT 0 23 "SubquadratureConditions" } {TEXT -1 3 ", \n" }{TEXT 0 21 "SimpleOrderConditions" }{TEXT -1 2 ", \+ " }{TEXT 0 29 "SimpleSubquadratureConditions" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "Acknowledgemen t: These procedures are based on Maple code from \"Runge-Kutta Methods , Trees, and Maple\", " }}{PARA 0 "" 0 "" {TEXT -1 92 "by Folkmar Born emann, Selcuk Journal of Applied Mathematics, Vol. 2, No. 1, pp. 3-15, 2001. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{URLLINK 17 "http://wwwzeng er.informatik.tu-muenchen.de/selcuk/sjam012101.html" 4 "http://wwwzeng er.informatik.tu-muenchen.de/selcuk/sjam012101.html" "" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 210 "OrderConditions, Ro wSumConditions, StageOrderConditions, PrincipalErrorConditions, \nPrin cipalErrorTerms,QuadratureConditions, SubquadratureConditions, SimpleO rderConditions,\nSimpleSubquadratureConditions: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 18 "Callin g Sequence:\n" }}{PARA 0 "" 0 "" {TEXT -1 33 " OrderConditions(orde r,stages)" }{TEXT 266 1 "\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 " RowSumConditions(stages)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 37 " PrincipalErrorTerms(order,stages)" } {TEXT 267 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 42 " PrincipalErrorCond itions(order,stages)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 44 " StageOrderConditions(stage_order,stages)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 " QuadratureC onditions(order,stages)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 " SubquadratureConditions(order,stages)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " SimpleOrder Conditions(order,stages)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 47 " SimpleSubquadratureConditions(order,stages)" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 11 "Parame ters:" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 23 17 " order - " }{TEXT -1 69 " a positive integer givi ng the order of the Runge-Kutta scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 18 " stage_order - " }{TEXT -1 89 " \+ a positive integer giving the stage-order of various stages of the R unge-Kutta scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 19 " stages - " }{TEXT -1 74 " a positive integer g iving the number of stages of the Runge-Kutta scheme" }}{PARA 0 "" 0 " " {TEXT -1 28 " " }{TEXT 260 14 " \+ OR" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 122 " \+ (in the case of the procedure \"StageOrderCond itions\") a range of positive integers " }}{PARA 0 "" 0 "" {TEXT -1 99 " giving the stages to which \+ the stage-order condition applies" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 125 " If the \"stages\" argument is omit ted from any appropriate procedure, then abreviated forms for the equa tions are given. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 " " {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 15 "OrderConditions" } {TEXT -1 105 " constructs a list of order conditions to be satisfied b y an explicit Runge-Kutta scheme of given order. " }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 16 "RowSumConditions" }{TEXT -1 113 " constructs a list of the simplifying row sum conditions to be sa tisfied by a Runge-Kutta scheme of given order. " }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 19 "PrincipalErrorTerms" }{TEXT -1 101 " constructs a list of the principal error terms associated wit h a Runge-Kutta scheme of given order. " }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 24 "PrincipalErrorConditions" }{TEXT -1 120 " constructs a list of the conditions or equations obtained by equ ating the principal error terms to zero.\nThe procedure " }{TEXT 0 20 "StageOrderConditions" }{TEXT -1 163 " constructs a list of all the si mplifying stage-order conditions associated with a given stage-order o r a specific consecutive sub-list of stage-order conditions. " }} {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 20 "QuadratureCo nditions" }{TEXT -1 103 " constructs a list of the quadrature conditio ns to be satisfied by a Runge-Kutta scheme of given order." }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 23 "SubquadratureConditi ons" }{TEXT -1 107 " constructs a list of the sub-quadrature condition s to be satisfied by a Runge-Kutta scheme of given order." }}{PARA 0 " " 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 21 "SimpleOrderConditions " }{TEXT -1 102 " constructs a sub-list of the order conditions to be \+ satisfied by a Runge-Kutta scheme of given order." }}{PARA 0 "" 0 "" {TEXT -1 79 "The list of simpler conditions is sometimes sufficient to determine the scheme." }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure \+ " }{TEXT 0 29 "SimpleSubquadratureConditions" }{TEXT -1 72 " construct s the corresponding sub-list of the sub-quadrature conditions." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 7 "Option:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "The op tional argument \"expanded\" given to any of the procedures causes all the summations in the output equation or expression to be expanded. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 260 16 "How to activate:" }}{PARA 0 "" 0 "" {TEXT -1 158 "To make the procedures active, open the subsections, pla ce the cursor anywhere after the prompt [ > and press [Enter].\nYou c an then close up the subsections." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 221 "OrderConditions, RowSumConditions, StageOrderConditions, Principa lErrorConditions, \nPrincipalErrorTerms, QuadratureConditions, Subquad ratureConditions, SimpleOrderConditions,\nSimpleSubquadratureCondition s: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 7924 "OrderConditions := proc(order::posint,stag es::\{posint,name\})\n local s1,s2,trs;\n if order>16 then\n \+ error \"the 1st argument for the order must be no greater than 16\"\n \+ end if;\n s1 := \{a,b,c,e\};\n unassign('a','b','c','e');\n s2 := \{a,b,c,e\} minus s1;\n if s2<>\{\} then \n WARNING(\"the v ariables %1 have been unassigned for use as coefficients\",s2);\n en d if;\n trs := AllTrees(order);\n if nargs=1 then\n map(TreeO rderCondition0,trs)\n else\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v, w,d,h\};\n unassign('i','j','k','l','m','n','p','q','r','s','t',' u','v','w','d','h');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h\} \+ minus s1;\n if s2<>\{\} then \n WARNING(\"the variables % 1 have been unassigned for possible use as indices\",s2);\n end i f;\n if nargs=3 and args[3]='expanded' then\n if not type (stages,posint) or stages15 then\n error \"the 1st argument for the \+ order must be no greater than 15\"\n end if;\n s1 := \{a,b,c\};\n \+ unassign('a','b','c');\n s2 := \{a,b,c\} minus s1;\n if s2<>\{\} then \n WARNING(\"the variables %1 have been unassigned for use \+ as coefficients\",s2);\n end if;\n trs := Trees(order+1);\n if n args=1 then\n map(TreeOrderCondition0,trs)\n else\n s1 := \+ \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h\};\n unassign('i','j','k','l',' m','n','p','q','r','s','t','u','v','w','d','h');\n s2 := \{i,j,k, l,m,n,p,q,r,s,t,u,v,w,d,h\} minus s1;\n if s2<>\{\} then \n \+ WARNING(\"the variables %1 have been unassigned for possible use as indices\",s2);\n end if;\n if nargs=3 and args[3]='expanded ' then\n if not type(stages,posint) or stages15 then\n e rror \"the 1st argument for the order must be no greater than 15\"\n \+ end if;\n s1 := \{a,b,c\};\n unassign('a','b','c');\n s2 := \{a ,b,c\} minus s1;\n if s2<>\{\} then \n WARNING(\"the variables \+ %1 have been unassigned for use as coefficients\",s2);\n end if;\n \+ trs := Trees(order+1);\n if nargs=1 then\n map(TreeErrorTerm0 ,trs)\n else\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h\};\n \+ unassign('i','j','k','l','m','n','p','q','r','s','t','u','v','w','d' ,'h');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h\} minus s1;\n \+ if s2<>\{\} then \n WARNING(\"the variables %1 have been un assigned for possible use as indices\",s2);\n end if;\n if n args=3 and args[3]='expanded' then\n if not type(stages,posint ) or stages\{\} then \n WARNING(\"the va riables %1 have been unassigned for use as coefficients\",s2);\n end if;\n if nargs=2 and args[2]='expanded' then\n [seq(add(a[i,j] ,j=1..i-1)=c[i],i=2..stages)]\n else\n s1 := \{j\};\n unas sign('j');\n s2 := \{j\} minus s1;\n if s2<>\{\} then \n \+ WARNING(\"the variable %1 has been unassigned for use as an index \",j);\n end if;\n [seq(Sum(a[i,j],j=1..i-1)=c[i],i=2..stage s)]\n end if;\nend proc: # RowSumConditions\n\nStageOrderConditions \+ := proc(stage_order::posint,stages::\{posint,posint..posint\})\n loc al s1,s2,aa,bb;\n\n if stage_order<2 then\n error \"the 1st arg ument for the stage order must be greater than or equal to 2\"\n end if;\n s1 := \{a,c\};\n unassign('a','c');\n s2 := \{a,c\} minus s1;\n if s2<>\{\} then \n WARNING(\"the variable(s): %1 have b een unassigned for use as coefficients\",s2);\n end if;\n\n if typ e(stages,posint) then\n aa := 3;\n bb := stages;\n else\n \+ aa := max(3,op(1,stages));\n bb := op(2,stages);\n end if; \n\n if nargs=3 and args[3]='expanded' then\n [seq(add(a[i,j]*c [j]^(stage_order-1),j=2..i-1)=\n c[i]^stage_orde r/stage_order,i=aa..bb)]\n else\n s1 := \{j\};\n unassign( 'j');\n s2 := \{j\} minus s1;\n if s2<>\{\} then \n \+ WARNING(\"the variable %1 has been unassigned for use as an index\",j) ;\n end if;\n [seq(Sum(a[i,j]*c[j]^(stage_order-1),j=2..i-1) =\n c[i]^stage_order/stage_order,i=aa..bb)]\n \+ end if;\nend proc: # StageOrderConditions\n\nQuadratureConditions := p roc(order::posint,stages::\{posint,name\})\n local s1,s2;\n s1 := \+ \{b,c,e\};\n unassign('b','c','e');\n s2 := \{b,c,e\} minus s1;\n \+ if s2<>\{\} then \n WARNING(\"the variables %1 have been unassi gned for use as coefficients\",s2);\n end if;\n if nargs=1 then\n \+ [b*e=1,seq(b*c^(j-1)=1/j,j=2..order)] \n elif nargs=3 and arg s[3]='expanded' then\n if not type(stages,posint) or stages\{\} then \n WARNING(\"the variable %1 has been unassigned for use as an index\",j);\n end if;\n [seq(Sum(b[i]*c[i]^ (j-1),i=1..stages)=1/j,j=1..order)];\n end if;\nend proc: # Quadratu reConditions\n\nSubquadratureConditions := proc()\n local L1,L2;\n \+ L1 := OrderConditions(args[1..nargs]);\n L2 := QuadratureConditions (args[1..nargs]);\n remove(member,L1,L2);\nend:\n\nSimpleOrderCondit ions := proc(order::posint,stages::\{posint,name\})\n local s1,s2,tr s;\n if order>16 then\n error \"the 1st argument for the order \+ must be no greater than 16\"\n end if;\n s1 := \{a,b,c,e\};\n un assign('a','b','c','e');\n s2 := \{a,b,c,e\} minus s1;\n if s2<>\{ \} then \n WARNING(\"the variables %1 have been unassigned for us e as coefficients\",s2);\n end if;\n trs := AllSimpleTrees(order); \n if nargs=1 then\n map(TreeOrderCondition0,trs)\n else\n \+ s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h\};\n unassign('i','j', 'k','l','m','n','p','q','r','s','t','u','v','w','d','h');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h\} minus s1;\n if s2<>\{\} then \n WARNING(\"the variables %1 have been unassigned for possib le use as indices\",s2);\n end if;\n if nargs=3 and args[3]= 'expanded' then\n if not type(stages,posint) or stages " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 40 "Examples are given in the next section. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Order co nditions for an order 4 Runge-Kutta scheme" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 17 "Abbreviat ed form " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "OrderConditions(4):\nconvert(ListTools[Enumerate](%), matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7$\"\"\"/*&% \"bGF(%\"eGF(F(7$\"\"#/*&F+F(%\"cGF(#F(F.7$\"\"$/*&F+F(-%!G6#*&%\"aGF( F1F(F(#F(\"\"'7$\"\"%/*&F+F()F1F.F(#F(F47$\"\"&/*&F+F(-F86#*&F;F(F7F(F (#F(\"#C7$F=/*(F+F(F1F(F7F(#F(\"\")7$\"\"(/*&F+F(-F86#*&FBF(F;F(F(#F( \"#77$FQ/*&F+F()F1F4F(#F(F?Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 23 "Long (summation ) form " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 271 1 "s" } {TEXT -1 25 " is the number of stages " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "OrderConditions(4,s):\nc onvert(ListTools[Enumerate](%),matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7$\"\"\"/-%$SumG6$&%\"bG6#%\"iG/F0;F(%\"sGF(7$\"\" #/-F+6$*&F-F(&%\"cGF/F(F1#F(F57$\"\"$/-F+6$*&F-F(-F+6$*&&%\"aG6$F0%\"j GF(&F;6#FIF(/FI;F(,&F0F(F(!\"\"F(F1#F(\"\"'7$\"\"%/-F+6$*&F-F()F:F5F(F 1#F(F>7$\"\"&/-F+6$*&F-F(-F+6$*&FFF(-F+6$*&&FG6$FI%\"kGF(&F;6#FboF(/Fb o;F(,&FIF(F(FOF(FLF(F1#F(\"#C7$FQ/-F+6$*(F-F(F:F(FCF(F1#F(\"\")7$\"\"( /-F+6$*&F-F(-F+6$*&FFF()FJF5F(FLF(F1#F(\"#77$F`p/-F+6$*&F-F()F:F>F(F1# F(FSQ(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 14 "Ex panded form " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 102 " The following \"expanded\" forms of the order condition s are for a 4 stage order 4 Runge-Kutta method. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "OrderConditi ons(4,4,'expanded'):\nconvert(ListTools[Enumerate](%),matrix);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7$\"\"\"/,*&%\"bG6#F(F(& F,6#\"\"#F(&F,6#\"\"$F(&F,6#\"\"%F(F(7$F0/,(*&F.F(&%\"cGF/F(F(*&F1F(&F F(F(F(F(#F(\"\"'7$F6/,(*&F.F()F;F0F(F( *&F1F()F>F0F(F(*&F4F()F@F0F(F(#F(F37$\"\"&/**F4F(FOF(FFF(F;F(#F(\"#C7$ FR/,&**F1F(F>F(FFF(F;F(F(*(F4F(F@F(FJF(F(#F(\"\")7$\"\"(/,&*(F1F(FFF(F WF(F(*&F4F(,&*&FLF(FWF(F(*&FOF(FYF(F(F(F(#F(\"#77$Fco/,(*&F.F()F;F3F(F (*&F1F()F>F3F(F(*&F4F()F@F3F(F(#F(F6Q(pprint46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 50 "Order conditions for an order 5 Runge-Kutta scheme" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 17 "Abbreviated form " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "OrderConditions(5):\nconvert (ListTools[Enumerate](%),matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#737$\"\"\"/*&%\"bGF(%\"eGF(F(7$\"\"#/*&F+F(%\"cGF(#F(F.7$\" \"$/*&F+F(-%!G6#*&%\"aGF(F1F(F(#F(\"\"'7$\"\"%/*&F+F()F1F.F(#F(F47$\" \"&/*&F+F(-F86#*&F;F(F7F(F(#F(\"#C7$F=/*(F+F(F1F(F7F(#F(\"\")7$\"\"(/* &F+F(-F86#*&FBF(F;F(F(#F(\"#77$FQ/*&F+F()F1F4F(#F(F?7$\"\"*/*&F+F(-F86 #*&F;F(FHF(F(#F(\"$?\"7$\"#5/*&F+F(-F86#*(F;F(F1F(F7F(F(#F(\"#S7$\"#6/ *(F+F(FHF(F1F(#F(\"#I7$FZ/*&F+F(-F86#*&F;F(FVF(F(#F(\"#g7$\"#8/*&F+F() F7F.F(#F(\"#?7$\"#9/*(F+F(F1F(FVF(#F(\"#:7$Ffq/*(F+F(FBF(F7F(#F(Fdo7$ \"#;/*&F+F(-F86#*&FhnF(F;F(F(F_q7$\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 23 "Long (summ ation ) form " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 270 1 "s" }{TEXT -1 25 " is the number of stages " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "OrderConditions(5,s):\nconvert(ListTools[Enumerate](% ),matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#737$\"\"\"/- %$SumG6$&%\"bG6#%\"iG/F0;F(%\"sGF(7$\"\"#/-F+6$*&F-F(&%\"cGF/F(F1#F(F5 7$\"\"$/-F+6$*&F-F(-F+6$*&&%\"aG6$F0%\"jGF(&F;6#FIF(/FI;F(,&F0F(F(!\" \"F(F1#F(\"\"'7$\"\"%/-F+6$*&F-F()F:F5F(F1#F(F>7$\"\"&/-F+6$*&F-F(-F+6 $*&FFF(-F+6$*&&FG6$FI%\"kGF(&F;6#FboF(/Fbo;F(,&FIF(F(FOF(FLF(F1#F(\"#C 7$FQ/-F+6$*(F-F(F:F(FCF(F1#F(\"\")7$\"\"(/-F+6$*&F-F(-F+6$*&FFF()FJF5F (FLF(F1#F(\"#77$F`p/-F+6$*&F-F()F:F>F(F1#F(FS7$\"\"*/-F+6$*&F-F(-F+6$* &FFF(-F+6$*&F`oF(-F+6$*&&FG6$Fbo%\"lGF(&F;6#FerF(/Fer;F(,&FboF(F(FOF(F eoF(FLF(F1#F(\"$?\"7$\"#5/-F+6$*&F-F(-F+6$*(FFF(FJF(F]oF(FLF(F1#F(\"#S 7$\"#6/-F+6$*(F-F(FjnF(F:F(F1#F(\"#I7$F\\q/-F+6$*&F-F(-F+6$*&FFF(-F+6$ *&F`oF()FcoF5F(FeoF(FLF(F1#F(\"#g7$\"#8/-F+6$*&F-F()FCF5F(F1#F(\"#?7$ \"#9/-F+6$*(F-F(F:F(FgpF(F1#F(\"#:7$F^v/-F+6$*(F-F(FXF(FCF(F1#F(F^s7$ \"#;/-F+6$*&F-F(-F+6$*&FFF()FJF>F(FLF(F1Feu7$\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 14 " Expanded form " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 102 " The following \"expanded\" forms of the order condition s are for a 6 stage order 5 Runge-Kutta method. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "L1 := Order Conditions(5,6,'expanded'):\nfor ct to nops(L1) do\n print(`equation `||ct);print(L1[ct]);print(``);\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+equation~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,.&% \"bG6#\"\"\"F(&F&6#\"\"#F(&F&6#\"\"$F(&F&6#\"\"%F(&F&6#\"\"&F(&F&6#\" \"'F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+equation~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&& %\"bG6#\"\"#\"\"\"&%\"cGF(F*F**&&F'6#\"\"$F*&F,F/F*F**&&F'6#\"\"%F*&F, F4F*F**&&F'6#\"\"&F*&F,F9F*F**&&F'6#\"\"'F*&F,F>F*F*#F*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+equati on~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**(&%\"bG6#\"\"$\"\"\"&%\"a G6$F)\"\"#F*&%\"cG6#F.F*F**&&F'6#\"\"%F*,&*&&F,6$F5F.F*F/F*F**&&F,6$F5 F)F*&F0F(F*F*F*F**&&F'6#\"\"&F*,(*&&F,6$FAF.F*F/F*F**&&F,6$FAF)F*F=F*F **&&F,6$FAF5F*&F0F4F*F*F*F**&&F'6#\"\"'F*,**&&F,6$FPF.F*F/F*F**&&F,6$F PF)F*F=F*F**&&F,6$FPF5F*FLF*F**&&F,6$FPFAF*&F0F@F*F*F*F*#F*FP" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%+equation~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&&%\"bG6#\"\"#\" \"\")&%\"cGF(F)F*F**&&F'6#\"\"$F*)&F-F0F)F*F**&&F'6#\"\"%F*)&F-F6F)F*F **&&F'6#\"\"&F*)&F-FF*&F3F(F*F*F*F*F*F*#F*\"#C" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%+equation~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,***&%\"bG6#\"\"$\" \"\"&%\"cGF(F*&%\"aG6$F)\"\"#F*&F,6#F0F*F**(&F'6#\"\"%F*&F,F5F*,&*&&F. 6$F6F0F*F1F*F**&&F.6$F6F)F*F+F*F*F*F**(&F'6#\"\"&F*&F,FAF*,(*&&F.6$FBF 0F*F1F*F**&&F.6$FBF)F*F+F*F**&&F.6$FBF6F*F7F*F*F*F**(&F'6#\"\"'F*&F,FP F*,**&&F.6$FQF0F*F1F*F**&&F.6$FQF)F*F+F*F**&&F.6$FQF6F*F7F*F**&&F.6$FQ FBF*FCF*F*F*F*#F*\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%+equation~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**(&%\"bG6#\"\"$\"\"\"&%\"aG6$F)\"\"#F*)&%\"cG6#F.F.F *F**&&F'6#\"\"%F*,&*&&F,6$F6F.F*F/F*F**&&F,6$F6F)F*)&F1F(F.F*F*F*F**&& F'6#\"\"&F*,(*&&F,6$FCF.F*F/F*F**&&F,6$FCF)F*F>F*F**&&F,6$FCF6F*)&F1F5 F.F*F*F*F**&&F'6#\"\"'F*,**&&F,6$FSF.F*F/F*F**&&F,6$FSF)F*F>F*F**&&F,6 $FSF6F*FNF*F**&&F,6$FSFCF*)&F1FBF.F*F*F*F*#F*\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+equation~8G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&&%\"bG6#\"\"#\"\"\")&%\"cGF(\"\" $F*F**&&F'6#F.F*)&F-F1F.F*F**&&F'6#\"\"%F*)&F-F6F.F*F**&&F'6#\"\"&F*)& F-FF*F**&&F,6$FCF6F*)&F1F5 F)F*F*F*F**&&F'6#\"\"'F*,**&&F,6$FSF.F*F/F*F**&&F,6$FSF)F*F>F*F**&&F,6 $FSF6F*FNF*F**&&F,6$FSFCF*)&F1FBF)F*F*F*F*#F*\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~17G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&&%\"bG6#\"\"#\"\"\")&%\"cGF(\" \"%F*F**&&F'6#\"\"$F*)&F-F1F.F*F**&&F'6#F.F*)&F-F7F.F*F**&&F'6#\"\"&F* )&F-F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 55 "Principal error terms for an order 5 Runge-Kutta scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 17 "Abbreviated \+ form " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "PrincipalErrorTerms(5):\nconvert(ListTools[Enumerate] (%),matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#767$\"\"\" ,&*&%\"bGF(-%!G6#*&%\"aGF(-F-6#*&F0F(-F-6#*&F0F(-F-6#*&F0F(%\"cGF(F(F( F(F(F(#F(\"$?(!\"\"7$\"\"#,&*&F+F(-F-6#*&F0F(-F-6#*(F0F(F:F(F7F(F(F(F( #F(\"$S#F=7$\"\"$,&*&F+F(-F-6#*(F0F(F:F(F4F(F(F(#F(\"$!=F=7$\"\"%,&*(F +F(F:F(F1F(F(#F(\"$W\"F=7$\"\"&,&*(F+F(F4F(F7F(F(#F(\"#sF=7$\"\"',$*&# F(F?F(-F-6#,&*&F+F(-F-6#*&F0F(-F-6#*&F0F(-F-6#*&)F:F?F(F0F(F(F(F(F(#F( \"$g$F=F(F(7$\"\"(,$*&F]oF(-F-6#,&*&F+F(-F-6#*&F0F()F7F?F(F(F(#F(\"$? 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tF(F1#F(\"$o\"7$F]u/-F+6$*(F-F(FXF(FjqF(F1Fdcm7$\"#h/-F+6$*(F-F(FjnF(F gpF(F1#F(\"$E\"7$\"#i/-F+6$*&F-F(-F+6$*&FFF(-F+6$*&F`oF(-F+6$*&FcrF()F frF>F(FhrF(FeoF(FLF(F1Fihl7$\"#j/-F+6$*&F-F(-F+6$*(FFF(FJF(Fe]lF(FLF(F 1Fdcm7$\"#k/-F+6$*(F-F(Fb]lF(F:F(F1#F(\"$S\"7$\"#l/-F+6$*&F-F()FCF>F(F 1#F(Fgam7$\"#m/-F+6$*&F-F(-F+6$*(FFF(FjpF(FhtF(FLF(F1Fadm7$\"#n/-F+6$* (F-F(Fd^lF(F:F(F1#F(\"$0\"7$\"#o/-F+6$*(F-F(FXF(FetF(F1F__m7$\"#p/-F+6 $*(F-F(Fe_lF(F:F(F1#F(\"#q7$Fjhm/-F+6$*(F-F(FcsF(FXF(F1F[gm7$\"#r/-F+6 $**F-F(F:F(FCF(FgpF(F1#F(Fdhl7$Fj[l/-F+6$*&F-F(-F+6$*(FFF(F^alF(FJF(FL F(F1Fdcm7$\"#t/-F+6$*(F-F(F:F(F[alF(F1Fbfm7$\"#u/-F+6$*&F-F(-F+6$*(FFF (F^wF(F]oF(FLF(F1F__m7$\"#v/-F+6$*(F-F(F[wF(FCF(F1F[gm7$\"#w/-F+6$*(F- F(FjnF(FbqF(F1Ffim7$\"#x/-F+6$*&F-F(-F+6$*&FFF(-F+6$*&F`oF()FcoFSF(Feo F(FLF(F1F_^m7$\"#y/-F+6$*&F-F()FgpF5F(F1#F(Fdem7$\"#z/-F+6$*(F-F(FXF(F duF(F1#F(F`_l7$\"#!)/-F+6$*(F-F(F[wF(FXF(F1Fe]n7$\"#\")/-F+6$*(F-F(Fbq F(FgpF(F1#F(F\\[l7$\"##)/-F+6$*(F-F(F`clF(F:F(F1#F(Ffbl7$\"#$)/-F+6$*( F-F(FewF(FCF(F1#F(Fhu7$F`_m/-F+6$*&F-F(-F+6$*&FFF()FJFenF(FLF(F1Ffim7$ \"#&)/-F+6$*&F-F()F:FQF(F1#F(FbpQ)pprint106\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 52 "Row sum conditions for an 8 stage Runge-Kutta scheme" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "RowSumConditions(8,'expanded'):\nconvert(ListTools[Enumerate](%) ,matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$\"\"\"/&% \"aG6$\"\"#F(&%\"cG6#F-7$F-/,&&F+6$\"\"$F(F(&F+6$F6F-F(&F/6#F67$F6/,(& F+6$\"\"%F(F(&F+6$F@F-F(&F+6$F@F6F(&F/6#F@7$F@/,*&F+6$\"\"&F(F(&F+6$FL F-F(&F+6$FLF6F(&F+6$FLF@F(&F/6#FL7$FL/,,&F+6$\"\"'F(F(&F+6$FZF-F(&F+6$ FZF6F(&F+6$FZF@F(&F+6$FZFLF(&F/6#FZ7$FZ/,.&F+6$\"\"(F(F(&F+6$FdoF-F(&F +6$FdoF6F(&F+6$FdoF@F(&F+6$FdoFLF(&F+6$FdoFZF(&F/6#Fdo7$Fdo/,0&F+6$\" \")F(F(&F+6$FfpF-F(&F+6$FfpF6F(&F+6$FfpF@F(&F+6$FfpFLF(&F+6$FfpFZF(&F+ 6$FfpFdoF(&F/6#FfpQ)pprint116\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 58 "Stage-order 2 conditions for a 10 stage Runge-Kutta sch eme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "StageOrderConditions(2,10,'expanded'):\nconvert(ListT ools[Enumerate](%),matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7*7$\"\"\"/*&&%\"aG6$\"\"$\"\"#F(&%\"cG6#F/F(,$*&#F(F/F(*$)&F16#F .F/F(F(F(7$F//,&*&&F,6$\"\"%F/F(F0F(F(*&&F,6$F@F.F(F8F(F(,$*&F5F(*$)&F 16#F@F/F(F(F(7$F./,(*&&F,6$\"\"&F/F(F0F(F(*&&F,6$FPF.F(F8F(F(*&&F,6$FP F@F(FHF(F(,$*&F5F(*$)&F16#FPF/F(F(F(7$F@/,**&&F,6$\"\"'F/F(F0F(F(*&&F, 6$F]oF.F(F8F(F(*&&F,6$F]oF@F(FHF(F(*&&F,6$F]oFPF(FenF(F(,$*&F5F(*$)&F1 6#F]oF/F(F(F(7$FP/,,*&&F,6$\"\"(F/F(F0F(F(*&&F,6$FcpF.F(F8F(F(*&&F,6$F cpF@F(FHF(F(*&&F,6$FcpFPF(FenF(F(*&&F,6$FcpF]oF(F[pF(F(,$*&F5F(*$)&F16 #FcpF/F(F(F(7$F]o/,.*&&F,6$\"\")F/F(F0F(F(*&&F,6$F\\rF.F(F8F(F(*&&F,6$ F\\rF@F(FHF(F(*&&F,6$F\\rFPF(FenF(F(*&&F,6$F\\rF]oF(F[pF(F(*&&F,6$F\\r FcpF(FdqF(F(,$*&F5F(*$)&F16#F\\rF/F(F(F(7$Fcp/,0*&&F,6$\"\"*F/F(F0F(F( *&&F,6$FhsF.F(F8F(F(*&&F,6$FhsF@F(FHF(F(*&&F,6$FhsFPF(FenF(F(*&&F,6$Fh sF]oF(F[pF(F(*&&F,6$FhsFcpF(FdqF(F(*&&F,6$FhsF\\rF(F`sF(F(,$*&F5F(*$)& F16#FhsF/F(F(F(7$F\\r/,2*&&F,6$\"#5F/F(F0F(F(*&&F,6$FguF.F(F8F(F(*&&F, 6$FguF@F(FHF(F(*&&F,6$FguFPF(FenF(F(*&&F,6$FguF]oF(F[pF(F(*&&F,6$FguFc pF(FdqF(F(*&&F,6$FguF\\rF(F`sF(F(*&&F,6$FguFhsF(F_uF(F(,$*&F5F(*$)&F16 #FguF/F(F(F(Q)pprint126\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "A 6 stage, order 5 Runge-Kutta scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "RK5_6eqs := [op(RowSumConditions(6,'expanded')),op(OrderConditions(5,6,'expanded' ))]:\nSOeqs := [op(StageOrderConditions(2,6,'expanded')),\n \+ op(StageOrderConditions(3,6,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Prince-Dormand metho d: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 348 "e1 := \{c[2]=1/5,c[4 ]=4/5,c[5]=8/9\}:\ne2 := solve(subs(e1,\{op(RK5_6eqs),op(SOeqs)\})):\n e3 := `union`(e1,e2):\nsubs(e3,matrix([[c[2],a[2,1],``,``,``,``,``],\n [c[3],a[3,1],a[3,2],``,``,``,``],[c[4],a[4,1],a[4,2],a[4,3],``,``,``], \n[c[5],a[5,1],a[5,2],a[5,3],a[5,4],``,``],[c[6],a[6,1],a[6,2],a[6,3], a[6,4],a[6,5],``],\n[``,b[1],b[2],b[3],b[4],b[5],b[6]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7)#\"\"\"\"\"&F(%!GF+F+F+F+7)# \"\"$\"#5#F.\"#S#\"\"*F1F+F+F+F+7)#\"\"%F*#\"#W\"#X#!#c\"#:#\"#KF3F+F+ F+7)#\"\")F3#\"&s$>\"%hl#!&g`#\"%(=##\"&[W'FD#!$7#\"$H(F+F+7)F)#\"%#!%(=#\"%%y'#\"#6\"#%)Q)pprint106\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 281 27 "___________________________" }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 77 "A procedur e for constructing the stability function of a Runge-Kutta scheme: " } {TEXT 0 16 "StabiltyFunction" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Stabilit y Function: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 283 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT -1 35 " StabilityFunction(order,stages)" }{TEXT 284 4 "\nor \+ " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " StabilityFunction( stages)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 " " {TEXT 23 17 " order - " }{TEXT -1 69 " a positive inte ger giving the order of the Runge-Kutta scheme " }{TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 17 " stages - " }{TEXT -1 79 " a positive integer giving the number of stages of the Runge-Kutta scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 17 "St abilityFunction" }{TEXT -1 85 " constructs the stability function of a n explicit Runge-Kutta scheme of given order. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 285 7 "Option:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "The optional argument \+ \"expanded\" causes all the summations in the expression for the stabi lity function to be expanded. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 16 "How to a ctivate:" }}{PARA 0 "" 0 "" {TEXT -1 155 "To make the procedure active , open the subsection, place the cursor anywhere after the prompt [ > \+ and press [Enter].\nYou can then close up the subsection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Stability Function: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8014 "StabilityFunction := proc()\n l ocal order,stages,s1,s2,tt,phi,ct,u_,trs,i_,cffs,stbfcn;\n\n if narg s=0 then\n error \"%1 uses a 1st argument, (of type posint), whic h is missing\",'procname';\n end if;\n if nargs>0 then\n if n ot type(args[1],posint) then \n error \"the 1st argument, %1, \+ is invalid .. it should be a positive integer\",args[1];\n end if ;\n end if;\n if nargs=1 or nargs=2 and args[2]='expanded' then\n \+ stages := args[1];\n order := -1;\n else\n order := a rgs[1];\n end if; \n if nargs>1 then\n if args[2]<>'expande d' then\n if not type(args[2],posint) then \n error \"the 2nd argument, %1, is invalid .. it should be a positive integer or the option 'expanded'\",args[2];\n end if;\n if ar gs[2]>27 then\n error \"the 2nd argument, %1, is invalid .. it should be no greater than 27\",args[2];\n end if;\n \+ stages :=args[2];\n end if;\n end if;\n if nargs>2 and (arg s[3]<>'expanded' or nargs>3) then\n error \"the only valid option al argument for %1 is 'expanded'\",procname;\n end if; \n\n s1 := \+ \{a,b,c\};\n unassign('a','b','c');\n s2 := \{a,b,c\} minus s1;\n \+ if s2<>\{\} then \n WARNING(\"the variables, %1 have been unass igned for use as coefficients\",s2);\n end if;\n s1 := \{z\};\n \+ unassign('z');\n s2 := \{z\} minus s1;\n if s2<>\{\} then \n \+ WARNING(\"the variable, %1 has been unassigned for use as a parameter \",op(s2));\n end if;\n if stages=1 or stages=2 then\n s1 := \+ \{i\}; unassign('i'); s2 := \{i\} minus s1;\n elif stages=3 then\n \+ s1 := \{i,j\}; unassign('i','j'); s2 := \{i,j\} minus s1;\n el if stages=4 then\n s1 := \{i,j,k\}; unassign('i','j','k'); s2 := \{i,j,k\} minus s1;\n elif stages=5 then\n s1 := \{i,j,k,l\}; \+ unassign('i','j','k','l'); s2 := \{i,j,k,l\} minus s1;\n elif stage s=6 then\n s1 := \{i,j,k,l,m\}; unassign('i','j','k','l','m'); s 2 := \{i,j,k,l,m\} minus s1;\n elif stages=7 then\n s1 := \{i,j ,k,l,m,n\}; unassign('i','j','k','l','m','n'); s2 := \{i,j,k,l,m,n\} m inus s1;\n elif stages=8 then\n s1 := \{i,j,k,l,m,n,p\}; unassi gn('i','j','k','l','m','p');\n s2 := \{i,j,k,l,m,n,p\} minus s1; \n elif stages=9 then\n s1 := \{i,j,k,l,m,n,p,q\}; unassign('i' ,'j','k','l','m','p','q');\n s2 := \{i,j,k,l,m,n,p,q\} minus s1; \n elif stages=10 then\n s1 := \{i,j,k,l,m,n,p,q,r\}; unassign( 'i','j','k','l','m','p','q','r');\n s2 := \{i,j,k,l,m,n,p,q,r\} m inus s1;\n elif stages=11 then\n s1 := \{i,j,k,l,m,n,p,q,r,s\}; unassign('i','j','k','l','m','p','q','r','s');\n s2 := \{i,j,k,l ,m,n,p,q,r,s\} minus s1;\n elif stages=12 then\n s1 := \{i,j,k, l,m,n,p,q,r,s,t\};\n unassign('i','j','k','l','m','p','q','r','s' ,'t');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t\} minus s1;\n elif stage s=13 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u\};\n unassign('i ','j','k','l','m','p','q','r','s','t','u');\n s2 := \{i,j,k,l,m,n ,p,q,r,s,t,u\} minus s1;\n elif stages=14 then\n s1 := \{i,j,k, l,m,n,p,q,r,s,t,u,v\};\n unassign('i','j','k','l','m','p','q','r' ,'s','t','u','v');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v\} minus s1 ;\n elif stages=15 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w\} ;\n unassign('i','j','k','l','m','p','q','r','s','t','u','v','w') ;\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w\} minus s1;\n elif stag es=16 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d\};\n unas sign('i','j','k','l','m','p','q','r','s','t','u','v','w','d');\n \+ s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d\} minus s1;\n elif stages=17 t hen\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h\};\n unassign( 'i','j','k','l','m','p','q','r','s','t','u','v','w','d','h');\n s 2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h\} minus s1;\n elif stages=18 \+ then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota\};\n una ssign('i','j','k','l','m','p','q','r','s','t','u','v','w','d','h','iot a');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota\} minus s1;\n elif stages=19 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h, iota,kappa\};\n unassign('i','j','k','l','m','p','q','r','s','t', 'u','v','w','d','h','iota','kappa');\n s2 := \{i,j,k,l,m,n,p,q,r, s,t,u,v,w,d,h,iota,kappa\} minus s1;\n elif stages=20 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda\};\n unas sign('i','j','k','l','m','p','q','r','s','t','u','v','w','d','h','iota ',\n 'kappa','lambda');\n s2 := \{i,j,k,l,m,n ,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda\} minus s1;\n elif stages=21 \+ then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda, mu\};\n unassign('i','j','k','l','m','p','q','r','s','t','u','v', 'w','d','h','iota',\n 'kappa','lambda','mu');\n \+ s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda,mu\} minu s s1;\n elif stages=22 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v ,w,d,h,iota,kappa,lambda,mu,nu\};\n unassign('i','j','k','l','m', 'p','q','r','s','t','u','v','w','d','h','iota',\n \+ 'kappa','lambda','mu','nu');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v, w,d,h,iota,kappa,lambda,mu,nu\} minus s1;\n elif stages=23 then\n \+ s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda,mu,nu,rho \};\n unassign('i','j','k','l','m','p','q','r','s','t','u','v','w ','d','h','iota',\n 'kappa','lambda','mu','nu','rh o');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda, mu,nu,rho\} minus s1;\n elif stages=24 then\n s1 := \{i,j,k,l,m ,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda,mu,nu,rho,sigma\};\n una ssign('i','j','k','l','m','p','q','r','s','t','u','v','w','d','h','iot a',\n 'kappa','lambda','mu','nu','rho','sigma');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda,mu,nu, rho,sigma\} minus s1;\n elif stages=25 then\n s1 := \{i,j,k,l,m ,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda,mu,nu,rho,sigma,tau\};\n \+ unassign('i','j','k','l','m','p','q','r','s','t','u','v','w','d','h', 'iota',\n 'kappa','lambda','mu','nu','rho','sigma' ,'tau');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lam bda,mu,nu,rho,sigma,tau\} minus s1;\n elif stages=26 then\n s1 \+ := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda,mu,nu,rho,sigma ,tau,upsilon\};\n unassign('i','j','k','l','m','p','q','r','s','t ','u','v','w','d','h','iota',\n 'kappa','lambda',' mu','nu','rho','sigma','tau','upsilon');\n s2 := \{i,j,k,l,m,n,p, q,r,s,t,u,v,w,d,h,iota,kappa,lambda,mu,nu,rho,sigma,tau,upsilon\} minu s s1;\n elif stages=27 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v ,w,d,h,iota,kappa,lambda,mu,nu,rho,sigma,tau,upsilon,omega\};\n u nassign('i','j','k','l','m','p','q','r','s','t','u','v','w','d','h','i ota',\n 'kappa','lambda','mu','nu','rho','sigma',' tau','upsilon','omega');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d, h,iota,kappa,lambda,mu,nu,rho,sigma,tau,upsilon,omega\} minus s1;\n \+ end if;\n if s2<>\{\} then \n WARNING(\"the variables, %1 have \+ been unassigned for possible use as indices\",s2);\n end if;\n if \+ order<0 then\n tt[1] :=[];\n phi := u_->[u_];\n for ct \+ to stages do tt[ct+1] := phi(tt[ct]) end do;\n cffs := map(lhs,[s eq(subs(_S=stages,TreeOrderCondition(tt[i_])),i_=1..stages)]);\n \+ stbfcn := 1+add(cffs[i_]*z^i_,i_=1..stages);\n if member('expande d',[args]) then\n subs(c[1]=0,eval(subs('Sum'='add',stbfcn))); \n else\n stbfcn;\n end if;\n else\n if stage s[ u_];\n for ct to stages do tt[ct+1] := phi(tt[ct]) end do;\n \+ cffs := [seq(subs(_S=stages,TreeOrderCondition(tt[i_])),i_=1..stages) ];\n cffs := map(u_->`if`(rhs(u_)>=1/order!,rhs(u_),lhs(u_)),cffs );\n stbfcn := 1+add(cffs[i_]*z^i_,i_=1..stages);\n if nargs =3 and args[3]='expanded' then\n subs(c[1]=0,eval(subs('Sum'=' add',stbfcn)));\n else\n stbfcn;\n end if;\n end i f;\nend proc: # StabilityFunction" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are given in the next \+ section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "StabilityFunction(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.\"\" \"F$*&-%$SumG6$&%\"bG6#%\"iG/F,;F$\"\"&F$%\"zGF$F$*&-F'6$*&F)F$&%\"cGF +F$F-F$)F0\"\"#F$F$*&-F'6$*&F)F$-F'6$*&&%\"aG6$F,%\"jGF$&F66#FCF$/FC;F $,&F,F$F$!\"\"F$F-F$)F0\"\"$F$F$*&-F'6$*&F)F$-F'6$*&F@F$-F'6$*&&FA6$FC %\"kGF$&F66#FXF$/FX;F$,&FCF$F$FIF$FFF$F-F$)F0\"\"%F$F$*&-F'6$*&F)F$-F' 6$*&F@F$-F'6$*&FVF$-F'6$*&&FA6$FX%\"lGF$&F66#FioF$/Fio;F$,&FXF$F$FIF$F enF$FFF$F-F$)F0F/F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "StabilityFunction(5,'expanded');" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#,.\"\"\"F$*&,,&%\"bG6#F$F$&F(6#\"\"#F $&F(6#\"\"$F$&F(6#\"\"%F$&F(6#\"\"&F$F$%\"zGF$F$*&,**&F*F$&%\"cGF+F$F$ *&F-F$&F;F.F$F$*&F0F$&F;F1F$F$*&F3F$&F;F4F$F$F$)F6F,F$F$*&,(*(F-F$&%\" aG6$F/F,F$F:F$F$*&F0F$,&*&&FG6$F2F,F$F:F$F$*&&FG6$F2F/F$F=F$F$F$F$*&F3 F$,(*&&FG6$F5F,F$F:F$F$*&&FG6$F5F/F$F=F$F$*&&FG6$F5F2F$F?F$F$F$F$F$)F6 F/F$F$*&,&**F0F$FOF$FFF$F:F$F$*&F3F$,&*(FWF$FFF$F:F$F$*&FZF$FJF$F$F$F$ F$)F6F2F$F$*.F3F$FZF$FOF$FFF$F:F$)F6F5F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "StabilityFunction( 5,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.\"\"\"F$%\"zGF$*&#F$\"\"#F$ *$)F%F(F$F$F$*&#F$\"\"'F$*$)F%\"\"$F$F$F$*&#F$\"#CF$*$)F%\"\"%F$F$F$*& #F$\"$?\"F$*$)F%\"\"&F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "StabilityFunction(5,6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,0\"\"\"F$%\"zGF$*&\"\"#!\"\"F%F'F$*& \"\"'F(F%\"\"$F$*&\"#CF(F%\"\"%F$*&\"$?\"F(F%\"\"&F$*&-%$SumG6$*&&%\"b G6#%\"iGF$-F46$*&&%\"aG6$F:%\"jGF$-F46$*&&F?6$FA%\"kGF$-F46$*&&F?6$FG% \"lGF$-F46$*&&F?6$FM%\"mGF$&%\"cG6#FSF$/FS;F$,&FMF$F$F(F$/FM;F$,&FGF$F $F(F$/FG;F$,&FAF$F$F(F$/FA;F$,&F:F$F$F(F$/F:;F$F*F$)F%F*F$F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "StabilityFunction(5,6,'expanded');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0\"\"\"F$%\"zGF$*&\"\"#!\"\"F%F'F$*&\"\"'F(F%\"\"$F$*&\"#CF(F% \"\"%F$*&\"$?\"F(F%\"\"&F$*0&%\"bG6#F*F$&%\"aG6$F*F1F$&F76$F1F.F$&F76$ F.F+F$&F76$F+F'F$&%\"cG6#F'F$)F%F*F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "StabilityFunction(2,'ex pand');" }}{PARA 8 "" 1 "" {TEXT -1 127 "Error, (in StabilityFunction) the 2nd argument, expand, is invalid .. it should be a positive integ er or the option 'expanded'\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 282 27 "___________________________" }{TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }