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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 67 "The method of steps for a
 linear scalar delay differential equation" }}{PARA 19 "" 0 "" {TEXT 
-1 39 "Robert M. Corless and Jane M. Heffernan" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
173 "The DDE is entered in operator format, supposed to be equated ide
ntically to zero.  The inputs are\nt -- the variable used for time\ny \+
-- the operator t->y(t) for the solution" }}{PARA 0 "" 0 "" {TEXT -1 
143 "z -- the operator t->y(t-1) for the delay term\nj -- the start of
 the time interval (often not used)\n\nThe first example is dde = 0, w
here dde is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "lambda := -1
/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lambdaG#!\"\"\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "dde := (t,y,z,j) -> D(y)(t) \+
- lambda*z(t) + y(t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ddeGf*6&%
\"tG%\"yG%\"zG%\"jG6\"6$%)operatorG%&arrowGF+,(--%\"DG6#9%6#9$\"\"\"*&
%'lambdaGF7-9&F5F7!\"\"-F4F5F7F+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 74 "This means that the DDE is in mathematical notation y'(t) = lam
bda*y(t-1)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 69 "Change the next s
tatement to reflect where you put the file March.mpl" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 39 "currentdir(\"C:/data/local/grads/jane\");
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q9C:\\data\\local\\grads\\jane6\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read \"March.mpl\";" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&MarchGf*6%'%\"NG%'posintG%\"fG%(His
toryG6+%\"jG%#dyG%\"yG%\"YG%&thetaG%$solG%%dsolG%%strtG%$tmeG6\"F6C+>8
&-%&arrayG6#;\"\"!9$>8%F:>8,-F;6#;\"\"\"F?>&F96#F>9&>&FAFJ-%\"DG6#FK?(
8$FGFGF?%%trueGC&>8+-%%timeGF6>&F96#FR-%(unapplyG6$-%%subsG6$-%'dsolve
G6%<$-9%6&8(8'&F96#,&FRFGFG!\"\"FR/-FdoFJ-Feo6#FG-Fdo6#Fco%'useintGF]p
Fco>&FAFen-FO6#FZ>&FCFen,&FWFGFVFho>8)f*6#%\"tGF66$%)operatorG%&arrowG
F6-&T#6#-%&floorG6#,&F?FGFGFG6#-%%fracGFeqF6F66$F/F9>8*f*FjpF6F\\qF6F_
qF6F66$F.FAO6&FhpF\\rFCF9F6F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
52 "Now we loop over the integrations.  We loop N times." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N := 4;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"NG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 89 "We integrate this delay differential equa
tion with a piecewise constant initial function." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "History := \+
t -> piecewise( t>0.2 and t <=0.4, -1,  t>0.6 and t<=0.8, -1, 1) ;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(HistoryGf*6#%\"tG6\"6$%)operatorG%&
arrowGF(-%*piecewiseG6'32$\"\"#!\"\"9$1F4$\"\"%F3F332$\"\"'F3F41F4$\"
\")F3F3\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot
( History, 0..1, discont=true, colour=BLACK );" }}{PARA 13 "" 1 "" 
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$$\"3)\\Q%*[j^PL*FcpF*7$$\"3%y'3S(HJpP*FcpF*7$$\"3hl9R*[egT*FcpF*7$$\"
3G1H3X[$)e%*FcpF*7$$\"33+#\\@1%3*\\*FcpF*7$$\"3=>#o74c6a*FcpF*7$$\"3O
\\))z*=*G#e*FcpF*7$$\"3L^K%>t^`i*FcpF*7$$\"3WZG8yg#om*FcpF*7$$\"3_eUQ(
eR#4(*FcpF*7$$\"3i\\\\O\"*=I^(*FcpF*7$$\"3A/8I!y_**y*FcpF*7$$\"33YHxg5
DM)*FcpF*7$$\"3C</\"4(>(Q()*FcpF*7$$\"33Lt$\\7;h\"**FcpF*7$$\"3](ot0<
\\l&**FcpF*7$F*F*-%'COLOURG6&%$RGBGF(F(F(-%+AXESLABELSG6$Q!6\"F`jm-%%V
IEWG6$;F(F*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "We mar
ch N steps into the future, using this dde starting from this history:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "sol,dsol, tme, YY := Ma
rch( N, dde, History );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6&%$solG%%
dsolG%$tmeG%#YYG6&%$solG%%dsolG%$tmeG%\"yG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 45 "Now plot the computing times (on a log scale)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "plot( [seq([k,log[10](tme[k])], k=
1..N )], style=POINT, symbolsize=30, symbol=DIAMOND, colour=BLACK );" 
}}{PARA 13 "" 1 "" {GLPLOT2D 682 682 682 {PLOTDATA 2 "6(-%'CURVESG6#7&
7$$\"\"\"\"\"!$!3D+++W]L9_!#=7$$\"\"#F*$!3$)*******)RY7sF-7$$\"\"$F*$!
31+++I3@&o%F-7$$\"\"%F*$!3++++9,>k=F--%&STYLEG6#%&POINTG-%'SYMBOLG6$%(
DIAMONDG\"#I-%'COLOURG6&%$RGBG$F*F*FJFJ-%+AXESLABELSG6$Q!6\"FN-%%VIEWG
6$%(DEFAULTGFS" 1 5 2 1 30 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 
0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Total number of mi
nutes of cpu taken:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "tme[
N]/60;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++&3\"!#6" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 96 "Now look at the solution...to do this, de
fine a variable theta taking on values between 0 and 1." }}}{EXCHG }
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "for k to N do frac(k+theta) \+
:= theta; od: floor(theta) := 0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
-%&floorG6#%&thetaG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"sol( theta );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6'7$,&#
\"\"\"\"\"#!\"\"*&#\"\"$F*F)-%$expG6#,$%&thetaGF+F)F)2F3#F)\"\"&7$,(#F
)F*F)*&F-F)F/F)F)-F06#,&F5F)F3F+F+2F3#F*F67$,*#F)F*F+*&F-F)F/F)F)F;F+-
F06#,&F?F)F3F+F)2F3#F.F67$,,F9F)*&F-F)F/F)F)F;F+FDF)-F06#,&FHF)F3F+F+2
F3#\"\"%F67$,.#F)F*F+*&F-F)F/F)F)F;F+FDF)FLF+-F06#,&FPF)F3F+F)1FPF3" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "sol( 1+theta );" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6'7$,(#\"\"\"\"\"%F)*&#\"\"$F*F
)*&-%$expG6#,$%&thetaG!\"\"F)F3F)F)F4*&F/F),.#F-F*F4*&#F-\"\"#F)-F06#F
4F)F)-F06##!\"%\"\"&F4-F06##!\"$FAF)-F06##!\"#FAF4-F06##F4FAF)F)F)2F3#
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)*&F3F)FVF)F)F)2F3FU7$,0F(F)*&#F-F*F)F.F)F4F5F)*&#F-\"#5F)-F06#,&FUF)F
3F4F)F4*&FUF)FVF)F)*&FZF)FenF)F)*&#F)F:F)*&F3F)F^oF)F)F42F3#F-FA7$,4#F
)F*F4*&#F-F*F)F.F)F4F5F)*&#F-F]oF)F^oF)F4*&FUF)FVF)F)*&FZF)*&F3F)-F06#
,&FgoF)F3F4F)F)F)*&FNF)FbpF)F)*&FZF)FenF)F)*&#F)F:F)FeoF)F42F3#F*FA7$,
8F(F)*&#F-F*F)F.F)F4F5F)*&#F-F]oF)F^oF)F4*&#F)F]oF)-F06#,&FjpF)F3F4F)F
4*&FUF)FVF)F)*&FZF)FapF)F)*&FNF)FbpF)F)*&FZF)FenF)F)*&#F)F:F)FeoF)F4*&
#F)F:F)*&F3F)FcqF)F)F41FjpF3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 10 "eval(dde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6&%\"tG%\"yG%\"z
G%\"jG6\"6$%)operatorG%&arrowGF),(--%\"DG6#9%6#9$\"\"\"*&%'lambdaGF5-9
&F3F5!\"\"-F2F3F5F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Now c
heck that the computed solution satisfies the delay differential equat
ion (this guards against bugs)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 91 "zer := unapply( subs( D(Y)(t)='dsol'(t), Z(t)='sol'(t-1), Y(t)
='sol'(t), dde(t,Y,Z) ), t );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ze
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