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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 66 "Derivation of 8 stage, combined o
rder 5 and 6 Runge-Kutta schemes\n" }{TEXT 280 24 "(stage-order 2 exam
ples)" }}{PARA 0 "" 0 "" {TEXT -1 46 "by Peter Stone, Gabriola Island,
B.C., Canada " }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 14.11.2011" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT
1 {PARA 4 "" 0 "" {TEXT -1 58 "load procedures for constructing Runge-
Kutta schemes etc. " }}{PARA 0 "" 0 "" {TEXT -1 18 "The Maple m-files \+
" }{TEXT 262 9 "butcher.m" }{TEXT -1 2 ", " }{TEXT 262 7 "roots.m" }
{TEXT -1 6 " and " }{TEXT 262 6 "intg.m" }{TEXT -1 33 " are required \+
by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 134 "They can be read \+
into a Maple session by commands similar to those that follow, where e
ach file path gives the location of the m-file." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 102 "read \"C:\\\\Maple/procdrs/butcher.m\";\nread
\"C:\\\\Maple/procdrs/roots.m\";\nread \"C:\\\\Maple/procdrs/intg.m\"
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 51 "#==============================================
====" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 55 "Step by step construction
of the Prince-Dormand scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 5 "See: " }}{PARA 0 "" 0 "" {TEXT -1 83 "(1) \+
High order embedded Runge\226Kutta formulae, by P.J. Prince and J. \+
R. Dormand, " }}{PARA 0 "" 0 "" {TEXT -1 88 " Journal of Comput
ational and Applied Mathematics, Vol.7, (1981), pages 67 to 75." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "(2) On
Runge-Kutta Processes of High Order, by J. C. Butcher," }}{PARA 0 ""
0 "" {TEXT -1 90 " Journal of the Australian Mathematical Soci
ety, Vol. 4, (1964), pages 179 to 194." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 155 "As suggested by Prince and Dormand i
n (1) their scheme can be constructed in a step-by-step manner similar
to that used by Butcher for his order 6 schemes." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "-------------------------
-------------------------------------------------------------------" }
}{PARA 0 "" 0 "" {TEXT -1 17 "We specify that " }{XPPEDIT 18 0 "c[2] \+
= 1/10;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#5!\"\"" }{TEXT -1 3 ", " }
{XPPEDIT 18 0 "c[3] = 2/9;" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"\"\"*!\"\""
}{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 3/5;" "6#/&%\"cG6#\"\"&*&\"\"
$\"\"\"F'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 4/5;" "6#/&%
\"cG6#\"\"'*&\"\"%\"\"\"\"\"&!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "
c[7]=1" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8
]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[2]=0
" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[8]=1/10
" "6#/&%\"bG6#\"\")*&\"\"\"F)\"#5!\"\"" }{TEXT -1 7 " and " }
{XPPEDIT 18 0 "a[8,7] = 0;" "6#/&%\"aG6$\"\")\"\"(\"\"!" }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 281 6 "Ste
p 1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 21 "We use the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 "
" }{XPPEDIT 18 0 "c[4]=c[3]/(15*c[3]^2-10*c[3]+2)" "6#/&%\"cG6#\"\"%*
&&F%6#\"\"$\"\"\",(*&\"#:F,*$&F%6#F+\"\"#F,F,*&\"#5F,&F%6#F+F,!\"\"F3F
,F8" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 14 "to determine " }
{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 36 " and use the qua
drature conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18
0 "Sum(b[i],i = 1 .. 7) = 1;" "6#/-%$SumG6$&%\"bG6#%\"iG/F*;\"\"\"\"\"
(F-" }{TEXT -1 15 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^(k-1
),i = 2 .. 7) = 1/k;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\")&%\"cG6#F+,&%
\"kGF,F,!\"\"F,/F+;\"\"#\"\"(*&F,F,F2F3" }{TEXT -1 7 ", " }
{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" }{TEXT -1 8 " . . 6, " }}{PARA
0 "" 0 "" {TEXT -1 21 "to find the weights " }{XPPEDIT 18 0 "b[1]" "6
#&%\"bG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[3]" "6#&%\"bG6#\"
\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[4];" "6#&%\"bG6#\"\"%" }
{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[5];" "6#&%\"bG6#\"\"&" }{TEXT -1 3
", " }{XPPEDIT 18 0 "b[6]" "6#&%\"bG6#\"\"'" }{TEXT -1 3 ", " }
{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 356 "e0 := \{
c[2]=1/10,c[3]=2/9,c[5]=3/5,c[6]=4/5,c[7]=1,c[8]=1,b[2]=0,b[8]=1/10,a[
8,7]=0\}:\ne1 := `union`(e0,\{subs(e0,c[4]=c[3]/(15*c[3]^2-10*c[3]+2))
\}):\nc[4]=subs(e1,c[4]);\n``;\nquad_cdns := [add(b[i],i=1..8)=1,seq(a
dd(b[i]*c[i]^(k-1),i=2..8)=1/k,k=2..6)]:\nquad_eqns := subs(e1,quad_cd
ns):\nmatrix(ListTools[Enumerate](quad_eqns));\n``;\nindets(quad_eqns)
;\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%#\"\"$\"
\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#K%'matrixG6#7(7$\"\"\"/,0&%\"bG6#F(F(#F(\"#5F(&F,6#\"\"
$F(&F,6#\"\"%F(&F,6#\"\"&F(&F,6#\"\"'F(&F,6#\"\"(F(F(7$\"\"#/,.F.F(*
F@\"\"*F(F0F(F(*F2F>F(F3F(F(*F2F8F(F6F(F(*F5F8F(F9F(F(F
\"\")O+jNF(&F%6$\"\"(\"\"'F(!\"\""
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"##!%F:\"%OJ" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$,\"(4\"\\D\"(7tA'
\"\"\"*\"*vGcP#\"+\"R4kl\"F-&F%6$\"\"(\"\"'F-F-" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,\")`X'[&\")?*H0*\"\"\"*\")qtyK
\"*\\]2c\"F-&F%6$\"\"(\"\"'F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/&%\"aG6$\"\")\"\"&,\"-xo@Mj7\"-%yye&e@!\"\"*\"+]FGhK\",81CyH\"\"
\"\"&F%6$\"\"(\"\"'F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"
\")\"\"',&*\"#b\"$k$\"\"\"&F%6$\"\"(F(F.!\"\"#\"'@N#*\"(g\"46F." }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(\"\"!" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*/&%#b*G
6#\"\"\"#\"(VkB\"\")?Jf;/&F&6#\"\"#\"\"!/&F&6#\"\"$#\")^4oV\"*!)p8d\"/
&F&6#\"\"%#\"*`_[z$\"++q7<=/&F&6#\"\"\"+Z,1H;\",?\\U%o&F&6#\"\"'#
\",$zgs8I\"-+!))Hr4\"/&F&6#\"\"(#\"#7\"$n\"/&F&6#\"\")F0" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_
8eqs := [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8,'ex
panded'))]:\n`RK5_8eqs*` := subs(b=`b*`,OrderConditions(5,8,'expanded'
)):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Ch
eck: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "simplify(subs(e7,R
K6_8eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsimplify(subs(e7,`RK5_
8eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$
F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH
20 "6#\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F
$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 ""
}}{PARA 0 "" 0 "" {TEXT -1 21 "We find a value for " }{XPPEDIT 18 0 "
a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 39 " by applying the simplif
ying condition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum
(`b*`[i]*a[i,j],i = j+1 .. s) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b
*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,%\"sG*&&F)6#F0F,,&F,F,&%
\"cG6#F0!\"\"F," }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 2" "6#/%\"jG
\"\"#" }{TEXT -1 7 " . . . " }{TEXT 271 1 "s" }{TEXT -1 1 "," }}{PARA
0 "" 0 "" {TEXT -1 6 "with " }{XPPEDIT 18 0 "s=8" "6#/%\"sG\"\")" }
{TEXT -1 23 " for the eighth stage." }}{PARA 0 "" 0 "" {TEXT -1 8 "Ta
king " }{XPPEDIT 18 0 "s = 8;" "6#/%\"sG\"\")" }{TEXT -1 7 " and "
}{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 9 ", we have" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`b*`[7]*a[7,6]+`b*`[8
]*a[8,6] = `b*`[6]*(1-c[6]);" "6#/,&*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F)\"
\"'F*F**&&F'6#\"\")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*"
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 93 "Applying this equation to embedded order 5 scheme gives a
n equation which can be solved for " }{XPPEDIT 18 0 "a[7,6];" "6#&%\"
aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(j=6,'add'(`b*`[i]*a[i,j
],i=j+1..8)=`b*`[j]*(1-c[j]));\nvalue(%);\nsubs(e7,%);\na[7,6]=solve(%
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$addG6$*&&%#b*G6#%\"iG\"\"\"&
%\"aG6$F+\"\"'F,/F+;\"\"(\"\")*&&F)6#F0F,,&F,F,&%\"cGF7!\"\"F," }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F)\"\"
'F*F**&&F'6#\"\")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*&%\"cGF7!\"\"F*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*\"#7\"$n\"\"\"\"&%\"aG6$\"\"(\"
\"'F)F)#\"*.(p@(*\",+SGtc\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG
6$\"\"(\"\"'#\"*.(p@(*\"++SAE6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 39 "We now complete the construction with "
}{XPPEDIT 18 0 "a[7,6]=972169703/1126224000" "6#/&%\"aG6$\"\"(\"\"'*&
\"*.(p@(*\"\"\"\"++SAE6!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 ""
{TEXT 260 4 "Note" }{TEXT -1 88 ": See the following subsection for fu
rther remarks concerning this choice of value for " }{XPPEDIT 18 0 "a
[7,6];" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "e8 := `union
`(subs(a[7,6]=972169703/1126224000,e7),\{a[7,6]=972169703/1126224000\}
):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e8" }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1169 "
e8 := \{`b*`[3] = 43680951/157136980, a[8,5] = -6116292391/16604298368
, a[8,3] = 26560509/49818496, a[5,4] = 3284078/5138991, a[6,4] = -2600
4300/372178963, c[8] = 1, a[5,2] = -18239/102487, `b*`[1] = 1236443/16
593120, b[4] = 8252237/43524000, `b*`[8] = 0, `b*`[7] = 12/167, a[5,3]
= 3243/761332, c[3] = 2/9, b[2] = 0, `b*`[2] = 0, a[6,5] = 9368775900
/30948112231, a[4,2] = -1053/2401, b[6] = 28629151/119448000, b[5] = 1
43496441/1058947600, b[7] = 11/1120, a[6,3] = 98740944/564271331, a[8,
7] = 0, c[7] = 1, a[6,2] = 92664/208537, a[4,1] = 957/9604, c[2] = 7/3
9, c[5] = 23/33, c[6] = 24/31, b[8] = 13/200, a[6,1] = -11597952/14868
6881, a[8,6] = 5233891417/7453555200, a[7,4] = 1592286101/1436292000, \+
a[7,5] = -2279466607/2965053280, a[3,1] = 16/189, c[4] = 3/7, a[4,3] =
1053/1372, a[7,6] = 972169703/1126224000, b[3] = 19683/68432, a[7,3] \+
= 156399/1505504, a[8,4] = 576719677/1357948800, `b*`[6] = 30137260793
/109712988000, b[1] = 14459/198720, `b*`[5] = 1629060147/17684424920, \+
a[2,1] = 7/39, a[8,1] = 118627013/607606272, a[7,1] = 38665819/9180864
0, `b*`[4] = 379485253/1817127000, a[5,1] = 2563741/11068596, a[7,2] =
-897/1232, a[8,2] = -1527/3136, a[3,2] = 26/189\}:" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
119 "subs(e8,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),
\n[`b`,seq(b[i],i=1..8)],[`b*`,seq(`b*`[i],i=1..8)]]));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+#\"\"(\"#RF(%!GF+F+F+F+F+F+7+#\"
\"#\"\"*#\"#;\"$*=#\"#EF2F+F+F+F+F+F+7+#\"\"$F)#\"$d*\"%/'*#!%`5\"%,C#
\"%`5\"%s8F+F+F+F+F+7+#\"#B\"#L#\"(TPc#\")'fo5\"#!&R#=\"'([-\"#\"%VK\"
'K8w#\"(ySG$\"(\"**Q^F+F+F+F+7+#\"#C\"#J#!)_zf6\"*\")oo[\"#\"&kE*\"'P&
3##\")W4u)*\"*J8Fk!)+V+E\"*j*y@P#\"++fxo$*\",JA6[4$F+F+F+7+\"\"\"#\"
)>emQ\")S'3=*#!$(*)\"%K7#\"'*Rc\"\"(/b]\"#\"+,hG#f\"\"++?HO9#!+2mYzA\"
+!G`]'H#\"*.(p@(*\"++SAE6F+F+7+F_o#\"*8qi=\"\"*sig2'#!%F:\"%OJ#\")40cE
\")'\\=)\\#\"*x'>nd\"++)[zN\"#!+\"R#H;h\",o$)H/m\"#\"+<9*QB&\"++_b`u\"
\"!F+7+%\"bG#\"&fW\"\"'?()>Feq#\"&$o>\"&K%o#\"(PAD)\")+S_V#\"*Tk\\V\"
\"++w%*e5#\")^\"H'G\"*+![%>\"#\"#6\"%?6#\"#8\"$+#7+%#b*G#\"(VkB\"\")?J
f;Feq#\")^4oV\"*!)p8d\"#\"*`_[z$\"++q7<=#\"+Z,1H;\",?\\U%o<#\",$zgs8I
\"-+!))Hr4\"#\"#7\"$n\"FeqQ(pprint76\"" }}}{PARA 0 "" 0 "" {TEXT -1 0
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 ""
{TEXT -1 52 "#---------------------------------------------------" }}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "note concerning the value for "
}{XPPEDIT 18 0 "a[7, 6];" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "e7: coefficients of the \+
combined scheme with " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'
" }{TEXT -1 16 " as a parameter" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1331 "e7 := \{`b*`[3] = 43680951
/157136980, a[8,1] = 5675675/25316928-1194875/35630036*a[7,6], a[7,1] \+
= 97577/425040+152075/685193*a[7,6], a[5,4] = 3284078/5138991, a[6,4] \+
= -26004300/372178963, c[8] = 1, a[5,2] = -18239/102487, `b*`[1] = 123
6443/16593120, b[4] = 8252237/43524000, `b*`[8] = 0, `b*`[7] = 12/167,
a[5,3] = 3243/761332, c[3] = 2/9, b[2] = 0, `b*`[2] = 0, a[6,5] = 936
8775900/30948112231, a[4,2] = -1053/2401, b[6] = 28629151/119448000, b
[5] = 143496441/1058947600, b[7] = 11/1120, a[6,3] = 98740944/56427133
1, a[8,7] = 0, c[7] = 1, a[8,5] = -126334216877/215855878784+326128275
0/12978240613*a[7,6], a[6,2] = 92664/208537, a[4,1] = 957/9604, c[2] =
7/39, c[5] = 23/33, c[6] = 24/31, b[8] = 13/200, a[8,6] = -55/364*a[7
,6]+923521/1109160, a[6,1] = -11597952/148686881, a[3,1] = 16/189, c[4
] = 3/7, a[4,3] = 1053/1372, a[7,4] = -194747/2127840+16691752/1200577
3*a[7,6], b[3] = 19683/68432, a[8,3] = 2549109/6227312+237562875/16564
09391*a[7,6], a[7,5] = 25026793/37532320-1660289400/998326201*a[7,6], \+
a[7,3] = 217176/235235-17277300/18202301*a[7,6], `b*`[6] = 30137260793
/109712988000, b[1] = 14459/198720, `b*`[5] = 1629060147/17684424920, \+
a[2,1] = 7/39, `b*`[4] = 379485253/1817127000, a[5,1] = 2563741/110685
96, a[7,2] = -897/1232, a[8,2] = -1527/3136, a[8,4] = 54864553/9052992
0-32787370/156075049*a[7,6], a[3,2] = 26/189\}:" }}}{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 5 "Let " }
{XPPEDIT 18 0 "T[6, 8];" "6#&%\"TG6$\"\"'\"\")" }{TEXT -1 127 " denot
e the vector whose components are the principal error terms of the 8 s
tage, order 6 scheme (the error terms of order 7)." }}{PARA 0 "" 0 ""
{TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5,8];" "6#&%#T*G6$\"\"&\"\")"
}{TEXT -1 145 " denote the vector whose components are the principal \+
error terms of the embedded 8 stage, order 5 scheme (the error terms o
f order 6) and let " }{XPPEDIT 18 0 "`T*`[6,8];" "6#&%#T*G6$\"\"'\"\"
)" }{TEXT -1 99 " denote the vector whose components are the error te
rms of order 7 of the embedded order 5 scheme." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of thes
e vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6#-%$absG6#-F$6#&%
\"TG6$\"\"'\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[5,8])
);" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\")" }{TEXT -1 7 " and " }
{XPPEDIT 18 0 "abs(abs(`T*`[6,8]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\"
\")" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 "
" }{XPPEDIT 18 0 "A[7] = abs(abs(T[6, 8]));" "6#/&%\"AG6#\"\"(-%$absG
6#-F)6#&%\"TG6$\"\"'\"\")" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] =
abs(abs(`T*`[6,8]))/abs(abs(`T*`[5,8]));" "6#/&%\"BG6#\"\"(*&-%$absG6
#-F*6#&%#T*G6$\"\"'\"\")\"\"\"-F*6#-F*6#&F/6$\"\"&F2!\"\"" }{TEXT -1
6 ", " }{XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,8]-T[6,8]))/abs(abs(`
T*`[5,8]));" "6#/&%\"CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\")\"\"
\"&%\"TG6$F2F3!\"\"F4-F*6#-F*6#&F06$\"\"&F3F8" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand a
nd Prince have suggested that as well as attempting to ensure that "
}{XPPEDIT 18 0 "A[7]" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, \+
if the embedded scheme is to be used for error control, " }{XPPEDIT
18 0 "B[7]" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[
7]" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should be chosen so that " }}
{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and als
o not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "errterms6_8 := PrincipalErr
orTerms(6,8,'expanded'):\n`errterms5_8*` := subs(b=`b*`,PrincipalError
Terms(5,8,'expanded')):\n`errterms6_8*` := subs(b=`b*`,PrincipalErrorT
erms(6,8,'expanded')):\nsnmB := sqrt(add(subs(e7,`errterms6_8*`[i])^2,
i=1..nops(`errterms6_8*`))):\nsdnB := sqrt(add(subs(e7,`errterms5_8*`[
i])^2,i=1..nops(`errterms5_8*`))):\nsnmC := sqrt(add((subs(e7,`errterm
s6_8*`[i])-subs(e7,errterms6_8[i]))^2,i=1..nops(errterms6_8))):\nB_7 :
= snmB/sdnB:\nC_7 := snmC/sdnB:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "'B[7]' = evalf[8](eval(B_7,
a[7,6]=972169703/1126224000));\n'C[7]' = evalf[8](eval(C_7,a[7,6]=9721
69703/1126224000));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($
\")EKc?!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\")C#zX
\"!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6
"When " }{XPPEDIT 18 0 "a[7,6]=972169703/1126224000" "6#/&%\"aG6$\"\"
(\"\"'*&\"*.(p@(*\"\"\"\"++SAE6!\"\"" }{TEXT -1 11 ", we have " }
{XPPEDIT 18 0 "B[7]" "6#&%\"BG6#\"\"(" }{TEXT -1 1 " " }{TEXT 272 1 "~
" }{TEXT -1 14 " 2.056 and " }{XPPEDIT 18 0 "C[7]" "6#&%\"CG6#\"\"(
" }{TEXT -1 1 " " }{TEXT 273 1 "~" }{TEXT -1 8 " 1.458. " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The following pict
ure shows the graphs of " }{XPPEDIT 18 0 "B[7]" "6#&%\"BG6#\"\"(" }
{TEXT -1 5 " in " }{TEXT 259 5 "brown" }{TEXT -1 6 " and " }
{XPPEDIT 18 0 "C[7]" "6#&%\"CG6#\"\"(" }{TEXT -1 5 " in " }{TEXT 261
3 "red" }{TEXT -1 3 ". \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
529 "aa := 972169703/1126224000:\nbb := evalf(eval(B_7,a[7,6]=aa)):\nc
c := evalf(eval(C_7,a[7,6]=aa)):\np1 := plot([B_7,C_7,1],a[7,6]=-.12..
1.85,0..2.8,color=[brown,red,blue]):\np2 := plot([[aa,0],[aa,2.9]],col
or=COLOR(RGB,0,.7,0)):\np3 := plot([[[aa,bb],[aa,cc]]$3],style=point,s
ymbol=[circle,diamond,cross],color=[black,coral$2]):\np4 := plot([[[aa
,0]]$3],style=point,symbol=[circle,diamond,cross],color=black):\np5 :=
plot([[[aa,bb],[0,bb]],[[aa,cc],[0,cc]]],linestyle=3,color=[brown,red
]):\nplots[display]([p||(1..5)],font=[HELVETICA,9]);" }}{PARA 13 "" 1
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