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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "This Maple Procedure imple
ments the Euler Approximation to a differential equation of the type \+
" }{XPPEDIT 18 0 "dx/dt = f(t,x);" "6#/*&%#dxG\"\"\"%#dtG!\"\"-%\"fG6$
%\"tG%\"xG" }{TEXT -1 171 ", where f is a vector-valued function and x
 is a vector.  Furthermore, the differential equation is linear in ter
ms of the vector x, that is, it is a First-Order equation." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 664 "with(linalg):\nRKSys:= proc(f,tsta
rt, xstart, tend, n)\nlocal h, i, k, t, x, k1, k2, k3, k4;\nh := evalf
((tend - tstart)/n);\nt:=tstart:   x:=xstart:\nprint('Step','t','x');
\nfor i from 1 to n+1 do\n   print(i-1,t,x);                  # displa
y current values\n   k1 := f(t,x): \011\011\011                # the l
eft-hand slope\n\011  k2 := f(t+h/2,evalm(x+h*k1/2)):\011 # 1st midpoi
nt slope\n\011  k3 := f(t+h/2,evalm(x+h*k2/2)):\011 # 2nd midpoint slo
pe\n\011  k4 := f(t+h,evalm(x+h*k3)): \011\011   # the right-hand slop
e\n   k := evalm((k1+2*k2+2*k3+k4)/6):\011# the average slope\n\011  x
 := evalm(x + h*k):  \011\011         # R-K step to update x\n\011  t \+
:= t + h:     \011\011               # update t\nod:\nprint(f);\nend:
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "The Syntax is as follows: RK
Sys(function, initial t, initial-value vector x, final t, number of st
eps);" }}{PARA 0 "" 0 "" {TEXT -1 16 "Examples follow:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "f:=(t,x) -> evalm(<<2,3,3>|<3,4,3>|
<1,0,0>> &* x + <<sin(t),cos(t),tan(t)>>);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGf*6$%\"tG%\"xG6\"6$%)operatorG%&arrowGF)-%&evalmG
6#,&-%#&*G6$-%$<|gr>G6%-%$<,>G6%\"\"#\"\"$F;-F86%F;\"\"%F;-F86%\"\"\"
\"\"!FB9%FA-F86#-F86%-%$sinG6#9$-%$cosGFJ-%$tanGFJFAF)F)F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "RKSys(f,0,<<1,1,0>>,1,10);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%%%StepG%\"tG%\"xG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"\"!F#-%'RTABLEG6%\")GMo@-%'MATRIXG6#7%7#\"\"\"F,7#F#%
'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"$\"+++++5!#5-%'matri
xG6#7%7#$\"+IO*fv\"!\"*7#$\"+\"4=^4#F.7#$\"+*\\%o6&)F&" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6%\"\"#$\"+++++?!#5-%'matrixG6#7%7#$\"+Hj?>L!\"*7#$
\"+50mGTF.7#$\"+.\"p1]#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$$\"+
++++I!#5-%'matrixG6#7%7#$\"+_$)fEk!\"*7#$\"+x#o+(zF.7#$\"+r(\\no&F." }
}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%$\"+++++S!#5-%'matrixG6#7%7#$\"+
32Z\\7!\")7#$\"+fV4H:F.7#$\"+QKu#=\"F." }}{PARA 11 "" 1 "" {XPPMATH 
20 "6%\"\"&$\"+++++]!#5-%'matrixG6#7%7#$\"+YskCC!\")7#$\"+tr\\IHF.7#$
\"+&[OZO#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"'$\"+++++g!#5-%'mat
rixG6#7%7#$\"+m@i\"p%!\")7#$\"+()H.>cF.7#$\"+lgGQYF." }}{PARA 11 "" 1 
"" {XPPMATH 20 "6%\"\"($\"+++++q!#5-%'matrixG6#7%7#$\"+S)3m0*!\")7#$\"
+')3Cy5!\"(7#$\"+:<%)4!*F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\")$\"
+++++!)!#5-%'matrixG6#7%7#$\"+\"=!RX<!\"(7#$\"+\"\\(Qq?F.7#$\"+Y*)RT<F
." }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"*$\"+++++!*!#5-%'matrixG6#7%7
#$\"+j2>gL!\"(7#$\"+lDFxRF.7#$\"+i8&pN$F." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#5$\"+++++5!\"*-%'matrixG6#7%7#$\"+o]([Y'!\"(7#$\"+]d
oUwF.7#$\"+Gh]ikF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6$%\"tG%\"xG6
\"6$%)operatorG%&arrowGF'-%&evalmG6#,&-%#&*G6$-%$<|gr>G6%-%$<,>G6%\"\"
#\"\"$F9-F66%F9\"\"%F9-F66%\"\"\"\"\"!F@9%F?-F66#-F66%-%$sinG6#9$-%$co
sGFH-%$tanGFHF?F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "g:
= (t,x) -> evalm(<<t,t^2>|<exp(t),sin(t)>> &* x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gGf*6$%\"tG%\"xG6\"6$%)operatorG%&arrowGF)-%&evalmG
6#-%#&*G6$-%$<|gr>G6$-%$<,>G6$9$*$)F9\"\"#\"\"\"-F76$-%$expG6#F9-%$sin
GFB9%F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "RKSys(g,0,<<
1,1>>,1,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%%StepG%\"tG%\"xG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!F#-%'RTABLEG6%\")/7@@-%'MATRIXG6#
7$7#\"\"\"F,%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"$\"++++
+?!#5-%'matrixG6#7$7#$\"+L*[iC\"!\"*7#$\"+vzHB5F." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"\"#$\"+++++S!#5-%'matrixG6#7$7#$\"+e*4+i\"!\"*7#$\"+M
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#7$7#$\"+1\\q5A!\"*7#$\"+,$\\tK\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6
%\"\"%$\"+++++!)!#5-%'matrixG6#7$7#$\"+(Gar?$!\"*7#$\"+xPq#z\"F." }}
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xG6\"6$%)operatorG%&arrowGF'-%&evalmG6#-%#&*G6$-%$<|gr>G6$-%$<,>G6$9$*
$)F7\"\"#\"\"\"-F56$-%$expG6#F7-%$sinGF@9%F'F'F'" }}}{EXCHG {PARA 0 ">
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