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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 -1 170 ", where f is a vector-valued function and x
 is a vector. Furthermore, the differential equation is linear in term
s of the vector x, that is, it is a First-Order equation." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 447 "with(linalg):\nSysEuler:= proc(f,t
start, xstart, tend, n)\nlocal h, i, k, t, x;\nh := evalf((tend - tsta
rt)/n);\nt:=tstart:   x:=xstart:\nprint('step', 't', 'x');\nfor i from
 1 to n+1 do\n    print(i-1,t,x);                    # display current
 values    \n    k := f(t,x): \011\011                    # the left-h
and slope \n\011   x := evalm(x + h*k):  \011            # Euler step \+
to update x\n\011   t := t + h:     \011                  # update t\n
\011   od:\nprint(f);\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "T
he Syntax is as follows: SysEuler(function, initial t, initial-value v
ector x, final t, number of steps);" }}{PARA 0 "" 0 "" {TEXT -1 16 "Ex
amples 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)-%&evalmG6#,&-%#&*G6$-%$<|gr>G6%-%$<,>G6%\"\"#\"\"$F;-F86%F;
\"\"%F;-F86%\"\"\"\"\"!FB9%FA-F86#-F86%-%$sinG6#9$-%$cosGFJ-%$tanGFJFA
F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "SysEuler(f,0,<<1,
1,0>>,1,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%%stepG%\"tG%\"xG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!F#-%'RTABLEG6%\")Wl\\>-%'MATRIXG6
#7%7#\"\"\"F,7#F#%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"$
\"+++++5!#5-%'matrixG6#7%7#$\"+++++:!\"*7#$\"+++++=F.7#$\"+++++gF&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#$\"+++++?!#5-%'matrixG6#7%7#$\"+U
L)*4C!\"*7#$\"+</]pIF.7#$\"+nM.+;F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6
%\"\"$$\"+++++I!#5-%'matrixG6#7%7#$\"+;/q#*R!\"*7#$\"+XAI=^F.7#$\"+*f
\\TE$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%$\"+++++S!#5-%'matrixG
6#7%7#$\"+a\")p#o'!\"*7#$\"+<p'*e%)F.7#$\"+BSQGgF." }}{PARA 11 "" 1 "
" {XPPMATH 20 "6%\"\"&$\"+++++]!#5-%'matrixG6#7%7#$\"+53()>6!\")7#$\"+
Bp%RR\"F.7#$\"+(G;81\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"'$\"++
+++g!#5-%'matrixG6#7%7#$\"+L\\&H(=!\")7#$\"+hFE'H#F.7#$\"+KY#4#=F." }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"($\"+++++q!#5-%'matrixG6#7%7#$\"+O
O;CJ!\")7#$\"+,x!\\y$F.7#$\"+38`yIF." }}{PARA 11 "" 1 "" {XPPMATH 20 "
6%\"\")$\"+++++!)!#5-%'matrixG6#7%7#$\"+\")Rw)>&!\")7#$\"+%HoPC'F.7#$
\"+$fv'f^F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"*$\"+++++!*!#5-%'ma
trixG6#7%7#$\"+&Q)yM')!\")7#$\"+(=(yI5!\"(7#$\"+h;t-')F." }}{PARA 11 "
" 1 "" {XPPMATH 20 "6%\"#5$\"+++++5!\"*-%'matrixG6#7%7#$\"+19AK9!\"(7#
$\"+Btw-<F.7#$\"+KJ\")H9F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6$%\"t
G%\"xG6\"6$%)operatorG%&arrowGF'-%&evalmG6#,&-%#&*G6$-%$<|gr>G6%-%$<,>
G6%\"\"#\"\"$F9-F66%F9\"\"%F9-F66%\"\"\"\"\"!F@9%F?-F66#-F66%-%$sinG6#
9$-%$cosGFH-%$tanGFHF?F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 51 "g:= (t,x) -> evalm(<<t,t^2>|<exp(t),sin(t)>> &* x);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"tG%\"xG6\"6$%)operatorG%&arrowGF)-
%&evalmG6#-%#&*G6$-%$<|gr>G6$-%$<,>G6$9$*$)F9\"\"#\"\"\"-F76$-%$expG6#
F9-%$sinGFB9%F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "SysE
uler(g,0,<<1,1>>,1,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%%stepG%\"tG
%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!F#-%'RTABLEG6%\")_r[>-%'
MATRIXG6#7$7#\"\"\"F,%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"
\"$\"+++++?!#5-%'matrixG6#7$7#$\"+++++7!\"*7#$F#\"\"!" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6%\"\"#$\"+++++S!#5-%'matrixG6#7$7#$\"+_0G#\\\"!\"*7
#$\"+mQL\\5F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$$\"+++++g!#5-%'m
atrixG6#7$7#$\"+KuuC>!\"*7#$\"+:G\")y6F." }}{PARA 11 "" 1 "" {XPPMATH 
20 "6%\"\"%$\"+++++!)!#5-%'matrixG6#7$7#$\"+AXI&e#!\"*7#$\"+(=;0X\"F.
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"&$\"+++++5!\"*-%'matrixG6#7$7#
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\"tG%\"xG6\"6$%)operatorG%&arrowGF'-%&evalmG6#-%#&*G6$-%$<|gr>G6$-%$<,
>G6$9$*$)F7\"\"#\"\"\"-F56$-%$expG6#F7-%$sinGF@9%F'F'F'" }}}{EXCHG 
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