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Euler's method extrapolates the next point using the slope field at the current point; the improved Euler's method and Runge-Kutta use information from the slope field at points inbetween. See sections 2.4-2.6 in our text for discussion on these methods.\n \nWe will study the IVP:\n\n" }{XPPEDIT 18 0 "diff(y(x),x) = f(x, y(x) );" "6#/-%%diffG6$-%\"yG6#%\"xGF*-%\"fG6$F*-F(6#F*" }{TEXT 258 1 "\n" }{XPPEDIT 18 0 "y(x0) = y0;" "6#/-%\"yG6#%#x0G%#y0G" }{TEXT 259 3 ".\n \n" }{TEXT 260 14 "Euler's Method" }{TEXT 261 337 "\nWe start with an \+ interval [x0,x1] which is subdivided into n equal length subintervals \+ [x[i],x[i+1]], i.e. x[i] = x0 + i*h for a fixed h, our step size. Now we find the point (x[i+1],y[i+1]) inductively: y[i+1] = y[i] + h*k, w here k = f(x[i],y[i]), i.e. the value of the slope field (thus the slo pe of a solution curve) at (x[i],y[i])." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "restart:\nwith(DEtools):\nwith(plots): #clear the \+ memory" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords \+ has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "f : = (x,y) -> (3*x - 2*y)/x; # enter slope field equation" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowG F)*&,&*&\"\"$\"\"\"9$F1F1*&\"\"#F19%F1!\"\"F1F2F6F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "x0 := 2.0; \+ # initial data and number of steps\ny0 := 3.0;\nx1 := 3.0;\nn := 10.0 ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"#?!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#y0G$\"#I!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"#I!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG$\"$+ \"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "h := (x1 - x0)/n ; # step size" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" hG$\"+++++5!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "y[0] := y 0; # start solution vector" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"!$\"#I!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "eqn := diff(u(t),t) = f(t,u(t)); # enter in DE" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%%diffG6$-%\"uG6#%\"tGF,*&,&* &\"\"$\"\"\"F,F1F1*&\"\"#F1F)F1!\"\"F1F,F4" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 49 "soln := dsolve(\{eqn,u(x0)=y0\}); \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG/-%\"uG6#%\"tG,&F)\"\"\"*& \"\"%F+F)!\"#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "printf( \"EULER METHOD\\n\");\nprintf(\"x[i] y[i] Actual y \+ error\\n\");\nfor i from 0 to n do: # loop" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "x[i] := x0 + h*i: # move x \+ over" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "k := f(x[i],y[i]): \+ # evaluate slope field" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "y[i+ 1] := y[i]+k*h: # move along line with slope k" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "valu := subs(t=x[i],rhs(soln)): # get actua l value" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "erro := y[i] - valu: \+ # compute error" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "printf( \"%6.3f %15.7f %15.7f %15.7f\\n\",x[i],y[i],valu,erro); \n \+ # print command" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "od: # finishes do command" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 6 "" 1 "" {TEXT -1 12 "EULER METHOD " }}{PARA 6 "" 1 "" {TEXT -1 50 "x[i] y[i] Actual y error" }}{PARA 6 "" 1 "" {TEXT -1 54 " 2.000 3.0000000 \+ 3.0000000 0.0000000" }}{PARA 6 "" 1 "" {TEXT -1 54 " 2.100 \+ 3.0000000 3.0070295 -.0070295" }}{PARA 6 "" 1 "" {TEXT -1 54 " 2.200 3.0142857 3.0264463 -.0121606" } }{PARA 6 "" 1 "" {TEXT -1 54 " 2.300 3.0402597 3.0561437 \+ -.0158839" }}{PARA 6 "" 1 "" {TEXT -1 54 " 2.400 3.0758893 \+ 3.0944444 -.0185551" }}{PARA 6 "" 1 "" {TEXT -1 54 " 2.500 3.1195652 3.1400000 -.0204348" }}{PARA 6 "" 1 "" {TEXT -1 54 " 2.600 3.1700000 3.1917160 -.0217160" } }{PARA 6 "" 1 "" {TEXT -1 54 " 2.700 3.2261538 3.2486968 \+ -.0225430" }}{PARA 6 "" 1 "" {TEXT -1 54 " 2.800 3.2871795 \+ 3.3102041 -.0230246" }}{PARA 6 "" 1 "" {TEXT -1 54 " 2.900 3.3523810 3.3756243 -.0232433" }}{PARA 6 "" 1 "" {TEXT -1 54 " 3.000 3.4211823 3.4444444 -.0232622" } }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 31 "Now let's plot these solutions :" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 147 "P1:=plot([seq([x[i],y[i]],i =0..n)],color=magenta):\nP2:=plot([seq([x[i],y[i]],i=0..n)],style=poin t,\n symbol=circle,symbolsize=5,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "P3:=plot(rhs(soln),t=x0..x1,color=blue):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "display(\{P1,P2,P3\},scaling=constr ained,title=\"Euler Polygon Method\"); " }}{PARA 13 "" 1 "" {GLPLOT2D 310 310 310 {PLOTDATA 2 "6)-%'CURVESG6$7-7$$\"\"#\"\"!$\"\"$F*7$$\"33+ ++++++@!#JF07$$\"33+++++++EF0$\"3#*************pJF07$$\"3;+ ++++++FF0$\"3))*****f%Q:EKF07$$\"3#)*************z#F0$\"3&******p[zrG$ F07$$\"3!***************GF0$\"3&)*****>&4Q_LF07$F+$\"3>+++mA=@MF0-%'CO LOURG6&%$RGBG$\"*++++\"!\")$F*F*Fjn-F$6&F&-Fgn6&FinFjnF]oF]o-%'SYMBOLG 6$%'CIRCLEG\"\"&-%&STYLEG6#%&POINTG-F$6$7SF'7$$\"3$ommT:(z@?F0$\"3o()4 \\G7N+IF07$$\"3OL$e9ui2/#F0$\"3s*3'GxJ@,IF07$$\"3]mm\"z_\"4i?F0$\"37]' )y.jx-IF07$$\"3YmmT&phN3#F0$\"3JO_Z)of\\+$F07$$\"3UL$e*=)H\\5#F0$\"3(y 1@Lo:x+$F07$$\"3sm;z/3uC@F0$\"3vs<&e3q2,$F07$$\"3z**\\7LRDX@F0$\"3b)>7 &H'>W,$F07$$\"3\"om\"zR'ok;#F0$\"3#\\wk0U&p=IF07$$\"3-+]i5`h(=#F0$\"3# [Y$pcbWBIF07$$\"3YLL$3En$4AF0$\"3,ih%[Z?)GIF07$$\"3cmmT!RE&GAF0$\"3\"4 H,W8]R.$F07$$\"3)*****\\K]4]AF0$\"3dMj:\\<:SIF07$$\"3))****\\PAvrAF0$ \"3XsgaCr\"o/$F07$$\"3#)****\\nHi#H#F0$\"30nH)**eSO0$F07$$\"3bm;z*ev:J #F0$\"36cL[%>l,1$F07$$\"3[LL$347TL#F0$\"3AiT!4!eJoIF07$$\"3=LLLjM?`BF0 $\"3lwIM8CavIF07$$\"38+]7o7TvBF0$\"3o(4)\\3qI%3$F07$$\"3WLLLQ*o]R#F0$ \"39XN?DeP#4$F07$$\"3C+]7=lj;CF0$\"36iF(o`_:5$F07$$\"3%***\\PaRY2a#F0$\"3Rp^'*3LQgJF07$$\"3$om;zXu9c#F0$ \"3ZaN/^N7rJF07$$\"3u*****\\y))Ge#F0$\"3$GZ9OxqC=$F07$$\"3n***\\i_QQg# F0$\"3)Q,\")p>6Q>$F07$$\"3q**\\7y%3Ti#F0$\"3l'HC>&=+0KF07$$\"3\"****\\ P![hYEF0$\"3(o`>Rcqw@$F07$$\"3DLLLQx$om#F0$\"3%z?=Wdl#HKF07$$\"3')**** \\P+V)o#F0$\"3G?jJjE$F07$$\"37L$e9S8&\\FF0$\"3kIMR_iiyKF07$$\"3;+] i?=bqFF0$\"3CM+Lb%f;H$F07$$\"3GLL$3s?6z#F0$\"3uPE0\\hd/LF07$$\"3&***\\ 7`Wl7GF0$\"3%G'RjHzF=LF07$$\"3emmm'*RRLGF0$\"3c/D/dDkJLF07$$\"3_mmTvJg aGF0$\"3)ze2\"\\aZXLF07$$\"34L$e9tOc(GF0$\"3o%[Z^Ra$fLF07$$\"3u*****\\ Qk\\*GF0$\"3)Q3;+sWAP$F07$$\"3@LL3dg6x_<(Q$F07$$\"3_mmmw(G p$HF0$\"3Y=#4lGn1S$F07$$\"3-+]7oK0eHF0$\"38ltz!*>>:MF07$$\"3-+](=5s#yH F0$\"3xv)3Y1D#HMF07$F+$\"3kWWWWWWWMF0-Fgn6&FinF]oF]oFjn-%&TITLEG6#Q5Eu ler~Polygon~Method6\"-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q!Fb_l Fj_l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$\"#?!\"\"$\"#IFe`lF^`l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" }}}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 22 "Improved Euler Metho d\n" }{TEXT 262 368 "This method uses both the current point and the p oint returned from Euler's method. The way this works is to take the \+ slope k1 returned from Euler's method, shoot out along a line with thi s slope and evaluate the slope field here, say we get k2. Then we let our new point be on the line which has slope the average of these, (k 1+k2)/ 2. Here is the improved method:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "z[0] := y0:\nprintf(\"IMPROVED EUL ER'S METHOD\\n\"):\nprintf(\"x[i] Improv. y[i] Actual y \+ error\\n\");\nfor i from 0 to n do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "x[i] := x0 + h*i: # move x over" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 55 "k1 := f(x[i],z[i]): # evaluate sl ope field" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "u[i+1] := z[i]+k1*h: \+ # move along line with slope k" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "k2 := f(x[i] + h,u[i+1]): # get new slope " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "k3 := (k1 + k2)/2: # \+ average the slopes" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "z[i+1] := z[i ] + k3*h: # move out along average" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "valu:=subs(t=x[i],rhs(soln)): # get the actual val ue" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "erro:=y[i]-valu: \+ # compute error" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "printf(\"%6. 2f%15.7f %15.7f %15.7f\\n\",\n x[i],z[i],valu, erro);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 6 "" 1 "" {TEXT -1 23 "IMPROVED EU LER'S METHOD" }}{PARA 6 "" 1 "" {TEXT -1 49 "x[i] Improv. y[i] \+ Actual y error" }}{PARA 6 "" 1 "" {TEXT -1 53 " 2.00 3 .0000000 3.0000000 0.0000000" }}{PARA 6 "" 1 "" {TEXT -1 53 " 2.10 3.0071429 3.0070295 -.0070295" }}{PARA 6 " " 1 "" {TEXT -1 53 " 2.20 3.0266388 3.0264463 -.0121 606" }}{PARA 6 "" 1 "" {TEXT -1 53 " 2.30 3.0563909 3.0561 437 -.0158839" }}{PARA 6 "" 1 "" {TEXT -1 53 " 2.40 3.0947 286 3.0944444 -.0185551" }}{PARA 6 "" 1 "" {TEXT -1 53 " \+ 2.50 3.1403082 3.1400000 -.0204348" }}{PARA 6 "" 1 " " {TEXT -1 53 " 2.60 3.1920388 3.1917160 -.0217160" }}{PARA 6 "" 1 "" {TEXT -1 53 " 2.70 3.2490274 3.2486968 \+ -.0225430" }}{PARA 6 "" 1 "" {TEXT -1 53 " 2.80 3.3105374 \+ 3.3102041 -.0230246" }}{PARA 6 "" 1 "" {TEXT -1 53 " 2.90 \+ 3.3759567 3.3756243 -.0232433" }}{PARA 6 "" 1 "" {TEXT -1 53 " 3.00 3.4447733 3.4444444 -.0232622" }} }{EXCHG {PARA 262 "" 0 "" {TEXT -1 20 "And let's plot this:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "P1:=plot([seq([x[i],z[i]],i=0..n)],color =magenta):\nP2:=plot([seq([x[i],z[i]],i=0..n)],style=point,\n \+ symbol=circle,symbolsize=5,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "P3:=plot(rhs(soln),t=x0..x1,color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "display(\{P1,P2,P3\},scaling=constrained,title=\"Impr oved Euler Method\"); " }}{PARA 13 "" 1 "" {GLPLOT2D 277 277 277 {PLOTDATA 2 "6)-%'CURVESG6$7S7$$\"\"#\"\"!$\"\"$F*7$$\"3$ommT:(z@?!#<$ \"3o()4\\G7N+IF07$$\"3OL$e9ui2/#F0$\"3s*3'GxJ@,IF07$$\"3]mm\"z_\"4i?F0 $\"37]')y.jx-IF07$$\"3YmmT&phN3#F0$\"3JO_Z)of\\+$F07$$\"3UL$e*=)H\\5#F 0$\"3(y1@Lo:x+$F07$$\"3sm;z/3uC@F0$\"3vs<&e3q2,$F07$$\"3z**\\7LRDX@F0$ \"3b)>7&H'>W,$F07$$\"3\"om\"zR'ok;#F0$\"3#\\wk0U&p=IF07$$\"3-+]i5`h(=# F0$\"3#[Y$pcbWBIF07$$\"3YLL$3En$4AF0$\"3,ih%[Z?)GIF07$$\"3cmmT!RE&GAF0 $\"3\"4H,W8]R.$F07$$\"3)*****\\K]4]AF0$\"3dMj:\\<:SIF07$$\"3))****\\PA vrAF0$\"3XsgaCr\"o/$F07$$\"3#)****\\nHi#H#F0$\"30nH)**eSO0$F07$$\"3bm; z*ev:J#F0$\"36cL[%>l,1$F07$$\"3[LL$347TL#F0$\"3AiT!4!eJoIF07$$\"3=LLLj M?`BF0$\"3lwIM8CavIF07$$\"38+]7o7TvBF0$\"3o(4)\\3qI%3$F07$$\"3WLLLQ*o] R#F0$\"39XN?DeP#4$F07$$\"3C+]7=lj;CF0$\"36iF(o`_:5$F07$$\"3%***\\PaR

Y2a#F0$\"3Rp^'*3LQgJF07$$\"3$om;zXu9c# F0$\"3ZaN/^N7rJF07$$\"3u*****\\y))Ge#F0$\"3$GZ9OxqC=$F07$$\"3n***\\i_Q Qg#F0$\"3)Q,\")p>6Q>$F07$$\"3q**\\7y%3Ti#F0$\"3l'HC>&=+0KF07$$\"3\"*** *\\P![hYEF0$\"3(o`>Rcqw@$F07$$\"3DLLLQx$om#F0$\"3%z?=Wdl#HKF07$$\"3')* ***\\P+V)o#F0$\"3G?jJjE$F07$$\"37L$e9S8&\\FF0$\"3kIMR_iiyKF07$$\"3 ;+]i?=bqFF0$\"3CM+Lb%f;H$F07$$\"3GLL$3s?6z#F0$\"3uPE0\\hd/LF07$$\"3&** *\\7`Wl7GF0$\"3%G'RjHzF=LF07$$\"3emmm'*RRLGF0$\"3c/D/dDkJLF07$$\"3_mmT vJgaGF0$\"3)ze2\"\\aZXLF07$$\"34L$e9tOc(GF0$\"3o%[Z^Ra$fLF07$$\"3u**** *\\Qk\\*GF0$\"3)Q3;+sWAP$F07$$\"3@LL3dg6x_<(Q$F07$$\"3_mmm w(Gp$HF0$\"3Y=#4lGn1S$F07$$\"3-+]7oK0eHF0$\"38ltz!*>>:MF07$$\"3-+](=5s #yHF0$\"3xv)3Y1D#HMF07$F+$\"3kWWWWWWWMF0-%'COLOURG6&%$RGBG$F*F*Fjz$\"* ++++\"!\")-F$6$7-F'7$$\"33+++++++@F0$\"3%)*****p&G92IF07$$\"3;+++++++A F0$\"39+++P)Qm-$F07$$\"3#)*************H#F0$\"3y*****H(3RcIF07$$\"3!** ************R#F0$\"3!******RdGZ4$F07$$\"3++++++++DF0$\"3()******o\"3.9 $F07$$\"33+++++++EF0$\"3=+++%yQ?>$F07$$\"3;+++++++FF0$\"3#*******=u-\\ KF07$$\"3#)*************z#F0$\"3-+++-u`5LF07$$\"3!***************GF0$ \"32+++nm&fP$F07$F+$\"3&*******)HtZW$F0-Fgz6&FizF[[lFjzF[[l-F$6&F`[l-F gz6&FizF[[lFjzFjz-%'SYMBOLG6$%'CIRCLEG\"\"&-%&STYLEG6#%&POINTG-%&TITLE G6#Q6Improved~Euler~Method6\"-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG 6%Q\"tFd_lQ!Fd_l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$\"#?!\"\"$\"#IFh`lFa`l " 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" }}}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 107 "You can u se the little magnifying glass on the tool bar to blow up the graphs t o see the difference easier." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 264 " " 0 "" {TEXT -1 18 "Runge-Kutta Method" }{TEXT 263 573 "\nThis method \+ is a bit more complicated. First we use Simpson's rule to estimate th e difference between successive y-values. In this estimation we need \+ the value of the slope field at: 1) our current point (k1), 2) our new point, 3) half between the two. First we use Euler's method to estim ate the slope at the midpoint (k2), then we use this to improve the es timate of the slope at the midpoint (k3). Then we shoot out along thi s direction to find the slope at the new point (k4), then we finally u se a weighted average to advance our solution. Here is the algorithm: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "rk[0] := y0:\nh2 := h/ 2.0:\nprintf(\"RUNGE-KUTTA METHOD\\n\");\nprintf(\"x[i] Runge-K utta y Actual y error\\n\");\nfor i from 0 to n do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "x[i] = x0 + i*h: # move x over" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "k1 := f(x[i],rk[i]): \+ # slope field at current point" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "k2 := f(x[i]+h2,rk[i]+k1*h2): # Euler estimate of midpoin t slope " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "k3 := f(x[i]+h2,rk[i]+k2*h2): # improved estimate" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "k4 := f(x[i]+h,rk[i]+k3*h): # estima te of slope at new point" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "k := (k 1 + 2*k2 + 2*k3 + k4)/6.0; # average slope value" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 46 "rk[i+1] := rk[i] + h*k: # new point" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "valu := subs(t=x[i],rhs(soln)):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "erro := rk[i]-valu:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 60 "printf(\"%6.3f %15.7f %15.7f %15.7f\\n\",x[i], rk[i],valu,erro);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 6 "" 1 "" {TEXT -1 18 "RUNGE-KUTTA METHOD" }}{PARA 6 "" 1 "" {TEXT -1 47 "x[i] Runge-Kutta y Actual y error" }}{PARA 6 "" 1 " " {TEXT -1 54 " 2.000 3.0000000 3.0000000 0.0000000 " }}{PARA 6 "" 1 "" {TEXT -1 54 " 2.100 3.0070295 3.007029 5 .0000001" }}{PARA 6 "" 1 "" {TEXT -1 54 " 2.200 3.02644 64 3.0264463 .0000001" }}{PARA 6 "" 1 "" {TEXT -1 54 " 2. 300 3.0561438 3.0561437 .0000001" }}{PARA 6 "" 1 " " {TEXT -1 54 " 2.400 3.0944446 3.0944444 .0000002 " }}{PARA 6 "" 1 "" {TEXT -1 54 " 2.500 3.1400002 3.140000 0 .0000002" }}{PARA 6 "" 1 "" {TEXT -1 54 " 2.600 3.19171 61 3.1917160 .0000002" }}{PARA 6 "" 1 "" {TEXT -1 54 " 2. 700 3.2486970 3.2486968 .0000002" }}{PARA 6 "" 1 " " {TEXT -1 54 " 2.800 3.3102042 3.3102041 .0000002 " }}{PARA 6 "" 1 "" {TEXT -1 54 " 2.900 3.3756244 3.375624 3 .0000002" }}{PARA 6 "" 1 "" {TEXT -1 54 " 3.000 3.44444 46 3.4444444 .0000001" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 19 "And let's plot this" }}{PARA 266 "> " 0 "" {MPLTEXT 1 0 257 "P1 :=plot([seq([x[i],rk[i]],i=0..n)],color=magenta):\nP2:=plot([seq([x[i] ,rk[i]],i=0..n)],style=point,\n symbol=circle,symbolsize=5,co lor=red):\nP3:=plot(rhs(soln),t=x0..x1,color=blue):display(\{P1,P2,P3 \},scaling=constrained,title=\"Runge-Kutta Method\");" }}{PARA 13 "" 1 "" {GLPLOT2D 271 271 271 {PLOTDATA 2 "6)-%'CURVESG6$7S7$$\"\"#\"\"!$ \"\"$F*7$$\"3$ommT:(z@?!#<$\"3o()4\\G7N+IF07$$\"3OL$e9ui2/#F0$\"3s*3'G xJ@,IF07$$\"3]mm\"z_\"4i?F0$\"37]')y.jx-IF07$$\"3YmmT&phN3#F0$\"3JO_Z) of\\+$F07$$\"3UL$e*=)H\\5#F0$\"3(y1@Lo:x+$F07$$\"3sm;z/3uC@F0$\"3vs<&e 3q2,$F07$$\"3z**\\7LRDX@F0$\"3b)>7&H'>W,$F07$$\"3\"om\"zR'ok;#F0$\"3# \\wk0U&p=IF07$$\"3-+]i5`h(=#F0$\"3#[Y$pcbWBIF07$$\"3YLL$3En$4AF0$\"3,i h%[Z?)GIF07$$\"3cmmT!RE&GAF0$\"3\"4H,W8]R.$F07$$\"3)*****\\K]4]AF0$\"3 dMj:\\<:SIF07$$\"3))****\\PAvrAF0$\"3XsgaCr\"o/$F07$$\"3#)****\\nHi#H# F0$\"30nH)**eSO0$F07$$\"3bm;z*ev:J#F0$\"36cL[%>l,1$F07$$\"3[LL$347TL#F 0$\"3AiT!4!eJoIF07$$\"3=LLLjM?`BF0$\"3lwIM8CavIF07$$\"38+]7o7TvBF0$\"3 o(4)\\3qI%3$F07$$\"3WLLLQ*o]R#F0$\"39XN?DeP#4$F07$$\"3C+]7=lj;CF0$\"36 iF(o`_:5$F07$$\"3%***\\PaRY2a#F0$\"3 Rp^'*3LQgJF07$$\"3$om;zXu9c#F0$\"3ZaN/^N7rJF07$$\"3u*****\\y))Ge#F0$\" 3$GZ9OxqC=$F07$$\"3n***\\i_QQg#F0$\"3)Q,\")p>6Q>$F07$$\"3q**\\7y%3Ti#F 0$\"3l'HC>&=+0KF07$$\"3\"****\\P![hYEF0$\"3(o`>Rcqw@$F07$$\"3DLLLQx$om #F0$\"3%z?=Wdl#HKF07$$\"3')****\\P+V)o#F0$\"3G?jJjE$F07$$\"37L$e9S 8&\\FF0$\"3kIMR_iiyKF07$$\"3;+]i?=bqFF0$\"3CM+Lb%f;H$F07$$\"3GLL$3s?6z #F0$\"3uPE0\\hd/LF07$$\"3&***\\7`Wl7GF0$\"3%G'RjHzF=LF07$$\"3emmm'*RRL GF0$\"3c/D/dDkJLF07$$\"3_mmTvJgaGF0$\"3)ze2\"\\aZXLF07$$\"34L$e9tOc(GF 0$\"3o%[Z^Ra$fLF07$$\"3u*****\\Qk\\*GF0$\"3)Q3;+sWAP$F07$$\"3@LL3dg6x_<(Q$F07$$\"3_mmmw(Gp$HF0$\"3Y=#4lGn1S$F07$$\"3-+]7oK0eHF0$ \"38ltz!*>>:MF07$$\"3-+](=5s#yHF0$\"3xv)3Y1D#HMF07$F+$\"3kWWWWWWWMF0-% 'COLOURG6&%$RGBG$F*F*Fjz$\"*++++\"!\")-F$6$7-F'7$$\"33+++++++@F0$\"30+ ++Y&Hq+$F07$$\"3;+++++++AF0$\"3!******4RYk-$F07$$\"3#)*************H#F 0$\"3))*****H!Q9cIF07$$\"3!**************R#F0$\"37+++&fWW4$F07$$\"3+++ +++++DF0$\"3))*****z:++9$F07$$\"33+++++++EF0$\"3A+++Phr\">$F07$$\"3;++ +++++FF0$\"3\"******\\+(p[KF07$$\"3#)*************z#F0$\"3%*******QU?5 LF07$$\"3!***************GF0$\"3!*******4WivLF07$F+$\"30+++$fWWW$F0-Fg z6&FizF[[lFjzF[[l-F$6&F`[l-Fgz6&FizF[[lFjzFjz-%'SYMBOLG6$%'CIRCLEG\"\" &-%&STYLEG6#%&POINTG-%&TITLEG6#Q3Runge-Kutta~Method6\"-%(SCALINGG6#%,C ONSTRAINEDG-%+AXESLABELSG6%Q\"tFd_lQ!Fd_l-%%FONTG6#%(DEFAULTG-%%VIEWG6 $;$\"#?!\"\"$\"#IFh`lFa`l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 58 "Let's look at all of error from the three methods at o nce:" }}{PARA 268 "> " 0 "" {MPLTEXT 1 0 394 "printf(\"x[i] Actual y Euler E-error Impr.Euler IE-error Runge-Kutta RK-error\");\nfor j from 1 to n do;\nvalu:=subs(t=x[j],rhs(soln)): \+ \ner1:=valu-y[j]: # Euler Method\ner2:=valu-z[j]: # \+ Improved Euler\ner3:=valu-rk[j]: # Runge-Kutta\nprintf(\"%6.3f %12.8f %12.8f %12.8f %12.8f %12.8f %12.8f %12.8f\\n\", \n x[j],valu, y[j],er1,z[j],er2,rk[j],er3);\nod:" }}{PARA 6 "" 1 "" {TEXT -1 95 "x[i ] Actual y Euler E-error Impr.Euler IE-error \+ Runge-Kutta RK-error" }}{PARA 6 "" 1 "" {TEXT -1 97 " 2.100 3.0070 2948 3.00000000 .00702948 3.00714286 -.00011338 3.00702955 \+ -.00000007" }}{PARA 6 "" 1 "" {TEXT -1 97 " 2.200 3.02644628 3.0 1428571 .01216057 3.02663884 -.00019256 3.02644639 -.000000 11" }}{PARA 6 "" 1 "" {TEXT -1 97 " 2.300 3.05614367 3.04025974 \+ .01588393 3.05639087 -.00024721 3.05614380 -.00000014" }} {PARA 6 "" 1 "" {TEXT -1 97 " 2.400 3.09444444 3.07588933 .0185 5512 3.09472857 -.00028413 3.09444460 -.00000015" }}{PARA 6 " " 1 "" {TEXT -1 97 " 2.500 3.14000000 3.11956522 .02043478 3. 14030817 -.00030817 3.14000016 -.00000016" }}{PARA 6 "" 1 "" {TEXT -1 97 " 2.600 3.19171598 3.17000000 .02171598 3.1920387 8 -.00032281 3.19171614 -.00000016" }}{PARA 6 "" 1 "" {TEXT -1 97 " 2.700 3.24869685 3.22615385 .02254300 3.24902742 -.000 33057 3.24869701 -.00000016" }}{PARA 6 "" 1 "" {TEXT -1 97 " 2.800 3.31020408 3.28717949 .02302460 3.31053740 -.00033332 3. 31020424 -.00000016" }}{PARA 6 "" 1 "" {TEXT -1 97 " 2.900 3.37562 426 3.35238095 .02324331 3.37595667 -.00033241 3.37562441 \+ -.00000015" }}{PARA 6 "" 1 "" {TEXT -1 97 " 3.000 3.44444444 3.42 118227 .02326218 3.44477330 -.00032886 3.44444459 -.0000001 5" }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 65 "Now let's study the effect of decreasing the step size parameter " }{TEXT 264 1 "h" }{TEXT -1 28 ". Let's restart everything:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:\nwith(DEtools):\nwith(plots):" } }{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been \+ redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "f := (x,y) - > cos(3*x) - 2*y; # new slope function" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&-%$cosG 6#,$*&\"\"$\"\"\"9$F4F4F4*&\"\"#F49%F4!\"\"F)F)F)" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 74 "x0 := 0.0; # initial \+ data\ny0 := -2/13.0;\nx1 := 2.0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #x0G$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G$!+Q:YQ:!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"#?!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "eqn := diff(w(t),t) = f(t,w(t)); # enter in \+ DE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%%diffG6$-%\"wG6#%\"tGF ,,&-%$cosG6#,$*&\"\"$\"\"\"F,F4F4F4*&\"\"#F4F)F4!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "soln := dsolve(\{eqn, w(x0)=y0\}); \+ # solve DE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG/-%\"wG6#%\"tG, (*&#\"\"#\"#8\"\"\"-%$cosG6#,$*&\"\"$F/F)F/F/F/F/*&#F5F.F/-%$sinGF2F/F /*&#\",(********>\",++++]'F/-%$expG6#,$*&F-F/F)F/!\"\"F/FC" }}}{EXCHG {PARA 270 "" 0 "" {TEXT -1 482 "We will uses Euler's method with n = \+ 2^nsteps, nsteps = 3,4,5,...,10. We'll just have the program spit out values at eight equally spaced points in the interval [x0,x1]. After this we'll plot the solution to see the accuracy of the method. We p rint out error/h^1, note that this is roughly constant, as tyhe theore m we stated in class predicted. For the improved Euler method, we wou ld expect error/h^2 to be constant and for Runge-Kutta, error/h^4 shou ld be roughly constant. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 453 "printf(\"\\n Comparison of different step sizes i n the EULER METHOD\\n\");\nprintf(\"---------------------------------- -----------------------------------------------------------------\\n\" );\nprintf(\" n h | x =%5.3f %7.3f %7.3f %7.3f % 7.3f %7.3f %7.3f %7.3f\\n\",\nseq(x0+j*(x1-x0)/8.0,j=1..8));\npr intf(\"--------------------------------------------------------------- ------------------------------------\\n\");\nfor nsteps from 3 to 10 d o:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "n := 2^nsteps: \+ # number of steps" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "h := (x1-x0)/n : # step size" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "y[0] \+ := y0: # initialize solution vector" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 " for i from 0 to n do: # start Euler Metho d (change in problem 3)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " x[i ] := x0 + i*h: # move x over" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " k := f(x[i],y[i]): # evaluate slope" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 54 " y[i+1] := y[i] + k*h # shoot out along slop e k" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " od: \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "points[nsteps] := [seq([x[j], y[j]],j=0..n)]:\n # we'll use this to plot the graphs" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "jskip := 2^(nsteps-3 ): # selects equally spaced points" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 397 "printf(\"%5d %10.6f | %9.6f %9.6f %9.6f %9.6f %9.6f %9.6f %9.6f %9.6f \\n\",n,h,seq(y[j*jskip],j=1..8));\nprintf(\" \+ D/h = | %9.4f %9.4f %9.4f %9.4f %9.4f %9.4f %9.4f %9.4f\\n\",\ns eq((y[j*jskip]-(subs(t=x[j*jskip],rhs(soln))))/h ,j=1..8)); # error/ h (change in problem 3)\nod:\nprintf(\"------------------------------- --------------------------------------------------------------------\\ n\");\n " }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 55 " Comparison of different step sizes in the EULER METHOD" }}{PARA 6 "" 1 "" {TEXT -1 99 "----------------------------------------------- ----------------------------------------------------" }}{PARA 6 "" 1 " " {TEXT -1 99 " n h | x = .250 .500 .750 1 .000 1.250 1.500 1.750 2.000" }}{PARA 6 "" 1 "" {TEXT -1 99 "--------------------------------------------------------------- ------------------------------------" }}{PARA 6 "" 1 "" {TEXT -1 100 " 8 .250000 | .173077 .269461 .152415 -.080836 -.287916 \+ -.349098 -.227248 .014397 " }}{PARA 6 "" 1 "" {TEXT -1 99 " \+ D/h = | .3593 .5663 .5526 .3222 -.0181 -.30 31 -.3941 -.2528" }}{PARA 6 "" 1 "" {TEXT -1 100 " 16 .1250 00 | .123525 .191976 .077176 -.124168 -.284267 -.306101 -.1 71707 .050311 " }}{PARA 6 "" 1 "" {TEXT -1 99 " D/h = | \+ .3222 .5128 .5033 .2977 -.0070 -.2621 -.3439 \+ -.2183" }}{PARA 6 "" 1 "" {TEXT -1 100 " 32 .062500 | .10245 9 .158559 .044445 -.143418 -.283571 -.288699 -.148956 .06482 9 " }}{PARA 6 "" 1 "" {TEXT -1 99 " D/h = | .3074 . 4909 .4830 .2874 -.0028 -.2459 -.3237 -.2044" }} {PARA 6 "" 1 "" {TEXT -1 100 " 64 .031250 | .092642 .142908 \+ .029061 -.152548 -.283426 -.280787 -.138556 .071413 " }}{PARA 6 "" 1 "" {TEXT -1 99 " D/h = | .3007 .4809 .47 37 .2827 -.0010 -.2385 -.3146 -.1981" }}{PARA 6 "" 1 " " {TEXT -1 100 " 128 .015625 | .087893 .135319 .021590 -.1 57001 -.283398 -.277006 -.133573 .074555 " }}{PARA 6 "" 1 "" {TEXT -1 99 " D/h = | .2975 .4761 .4693 .28 04 -.0001 -.2351 -.3103 -.1950" }}{PARA 6 "" 1 "" {TEXT -1 100 " 256 .007813 | .085556 .131582 .017907 -.159200 - .283393 -.275156 -.131132 .076090 " }}{PARA 6 "" 1 "" {TEXT -1 99 " D/h = | .2960 .4738 .4671 .2793 .0003 -.2334 -.3082 -.1936" }}{PARA 6 "" 1 "" {TEXT -1 100 " 512 \+ .003906 | .084397 .129726 .016078 -.160293 -.283394 -.274 241 -.129924 .076849 " }}{PARA 6 "" 1 "" {TEXT -1 99 " D/h = | .2952 .4726 .4660 .2787 .0005 -.2325 \+ -.3071 -.1929" }}{PARA 6 "" 1 "" {TEXT -1 100 " 1024 .001953 | \+ .083820 .128802 .015167 -.160838 -.283394 -.273786 -.129324 \+ .077226 " }}{PARA 6 "" 1 "" {TEXT -1 99 " D/h = | .29 48 .4721 .4655 .2785 .0006 -.2321 -.3066 -.19 25" }}{PARA 6 "" 1 "" {TEXT -1 99 "----------------------------------- ----------------------------------------------------------------" }}} {EXCHG {PARA 271 "" 0 "" {TEXT -1 55 "Let's plot the solutions now, th is might take a while. 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1 SECOND M APLE LAB QUESTIONS" }}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }{TEXT 266 2 "1 ." }{TEXT 269 259 " Choose some function f(x,y) and initial condition s (x0, y0), moderately interesting, and perform the above analysis on \+ the IVP. (You might need to rechoose f(x,y) if the solution is too bo ring or too complicated). These are the thing I expect to see:\n\n \+ " }{TEXT 276 4 " i.)" }{TEXT 277 148 " A table of values for Euler's \+ method, comparing the computed values to the actual. Graph the numeri cal and actual solution on the same graph.\n " }{TEXT 278 4 "ii.)" } {TEXT 279 154 " A table of values for Improved Euler's method, compar ing the computed values to the actual. Graph the numerical and actual solution on the same graph.\n" }{TEXT 280 6 " iii.)" }{TEXT 281 216 " A table of values for the Runge-Kutta method, comparing the compute d values to the actual. Graph the numerical and actual solution on th e same graph.\n\nYou should use about 10 steps in each of the above me thods.\n\n" }{TEXT 270 3 "2. " }{TEXT 271 19 " What do you see?\n\n" } {TEXT 272 3 "3. " }{TEXT 273 461 " Choose another f(x,y) and initial c onditions(x0,y0) and for either the improved Euler method or Runge-Kut ta, perform the analysis of decreasing step-size as in the second half of the above lesson. The only part of the code you should have to ch ange is the inner loop. Output a table as above for eight equally spa ced values and compute error/h^2 for Improved Euler and error/h^4 for \+ Runge-Kutta. (You'll have to change the power of h in the code as wel l.)\n\n" }{TEXT 274 2 "4." }{TEXT 275 222 " In the above code analyiz ing step-size, change f(x,y) to f(x,y) = cos(3*x) + 2*y and leave the \+ rest the same. What happened? (hint: look at the slope fields for th e different f's, page 138 in your text might help too.)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "30" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }