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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 442 "#Dr. Gustafson,\n#I
 am in your 2250-4 class and am continually struggling with the codes \+
for\n#the problems in section 2.4, 2.5, and 2.6.  Actually 2.5 (Improv
ed Euler)\n#seems to be going pretty well, but I can't get very good  \+
results for 2.4\n#(Euler) or 2.6 (Runge-Kutta Idea).  Is there a websi
te where I may be able\n#to find help and/or codes on these sections, \+
as nothing I type in for my\n#codes will seem to work?  Thanks for you
r time.\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 844 "# =========
================================\n# Can't copy with the mouse? A work-
around:\n# ========================================\n# Run the applica
tion \"xclipboard &\" to capture the mouse copies of this\n# file. Kee
p xclipboard near the xmaple window. Go to the netscape\n# window, cop
y with the mouse. Switch to the xclipboard window. Copy\n# with the mo
use from the xclipboard. Then paste with mouse button 2 or\n# mouse bu
tton 3 into xmaple.\n# =========================================\n# Ma
ple code doesn't work? Read this:\n# =================================
=======\n# To type in a group, hold shift then press return, except\n#
 for the last line of group, in which case use just return.\n# If you \+
copy multiple groups with the mouse, then split\n# them using key F3 w
ith the cursor placed at the front of\n# a line where the split is to \+
happen.\n\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 314 "# Warning:
 These snips of code made for y'=1-x-y, y(0)=3.\n#          Code compu
tes approx values for y(0.1) to y(1.0).\n# 'Dots' is the list of dots \+
for connect-the-dots graphics.\n# ====================================
====\n# Euler. Group 1, initialize.\n  f:=(x,y)->1-x-y:\n  x0:=0:y0:=3
:h:=0.1:Dots:=[x0,y0]:n:=10:\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 127 "# Group 2, repeat n times. Euler's method\n for i from 1 to n
 do\n  Y:=y0+h*f(x0,y0);\n  x0:=x0+h:y0:=Y:Dots:=Dots,[x0,y0];\n od:\n
 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "# Group 3, display dot
s and plot.\n  seq(Dots[j],j=1..n+1);\n  plot([Dots]);\n\n " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "# =========================
===============\n # Heun. Group 1, initialize.\n  f:=(x,y)->1-x-y:\n  \+
x0:=0:y0:=3:h:=0.1:Dots:=[x0,y0]:n:=10:\n " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 161 "# Group 2, repeat n times. Heun method.\n for i fr
om 1 to n do\n  Y1:=y0+h*f(x0,y0);\n  Y:=y0+h*(f(x0,y0)+f(x0+h,Y1))/2:
\n  x0:=x0+h:y0:=Y:Dots:=Dots,[x0,y0];\n od:\n " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 77 "# Group 3, display dots and plot.\n  seq(Dots[
j],j=1..n+1);\n  plot([Dots]);\n\n " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 134 "# ========================================\n # RK4. \+
Group 1, initialize.\n  f:=(x,y)->1-x-y:\n  x0:=0:y0:=3:h:=0.1:Dots:=[
x0,y0]:n:=10:\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "# Grou
p 2, repeat n times. RK4 method.\n for i from 1 to n do\n  k1:=h*f(x0,
y0):\n  k2:=h*f(x0+h/2,y0+k1/2):\n  k3:=h*f(x0+h/2,y0+k2/2):\n  k4:=h*
f(x0+h,y0+k3):\n  Y:=y0+(k1+2*k2+2*k3+k4)/6:\n  x0:=x0+h:y0:=Y:Dots:=D
ots,[x0,y0];\n od:\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "# \+
Group 3, display some dots and plot.\n  Dots[1],Dots[2],Dots[n+1];\n  \+
plot([Dots]);\n\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 315 "# Co
de snips for exact/error reports 2.4-#5\n # ==========================
===============\n # Making multiple curves on one plot\n # ===========
=============================\n   Exact:=x->2+x-exp(x);               \+
# An exact solution for 2.4-5\n   plot(\{Exact(x),[Dots]\},x=0..1/2); \+
  # plot exact and approx solutions\n\n " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 232 "# ========================================\n # How t
o create a Dots table for the exact solution\n # =====================
===================\n Exact:= x -> 2+x-exp(x):n:=10:\n ExactDots:=seq(
[Dots[j][1],Exact(Dots[j][1])],j=1..n+1);\n\n " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 416 "# ========================================\n \+
# How to define and print percentage relative error:\n # =============
===========================\n   P:=unapply(evalf(100*abs(exact-approx)
/abs(exact)),(exact,approx));\n   ExactVal:=ExactDots[11][2]:   # Pick
 off exact y-value for x=0.5\n   ApproxVal:=Dots[11][2]:       # Get E
uler approx y-value for x=0.5\n   P(ExactVal),ApproxVal);       # prin
t percent relative error\n\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 278 "# ========================================\n # How to create a \+
Dots table for percentage error\n # ==================================
======\n   P:=unapply(evalf(100*abs(exact-approx)/abs(exact)),(exact,a
pprox));\n   Pdots:=seq([Dots[j][1],P(ExactDots[j][2],Dots[j][2])],j=1
..11);\n\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 272 "# =========
================================\n # Printing results and tables\n # M
ake tables with a pencil, it saves time.\n # =========================
===============\n # To extract and print items 1,101,201,1001 from a l
ist:\n Dots1:=Dots[1],Dots[101],Dots[201],Dots[1001];\n\n " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "# ================================
=========\n # Loop control\n # =======================================
=\n # To automate the production of a Dots list,\n # enclose the desir
ed code between 1 and 2 below.\n #  1. for k from 1 to 10 do\n #  2. o
d:\n #\n # Keyword \"od:\" is short for \"end do:\"\n # Use \":\" to s
top loop results from printing.\n\n " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 387 "# =========================================\n # Debu
g\n # ========================================\n # To remove loop cont
rol and do it by hand, insert\n # pound (#) signs as follows:\n #  1. \+
# for k from 1 to 10 do\n #  2. # od:\n # The hand-done loop is made b
y placing the mouse cursor\n # in the group, then press return. Repeat
 for each loop step,\n # which is 10 times for the loop above.\n " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "# the end" }}}}{MARK "7 0 0" 
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