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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 37 "Midpoint Method for Est
imating Roots\n" }{TEXT 256 28 "Mth 351 Sept 30 2001 Maple 6" }}{PARA 
0 "" 0 "" {TEXT 258 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" {TEXT 259 
31 "Filename: 351f2001_midpoint.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 41 "Bisection method for approximating a roo
t" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "restart; Digits:=16:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "midp:=L->(L[1]+L[2])/2;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%midpGR6#%\"LG6\"6$%)operatorG%&arrowGF(,&&9$6#\"\"\"#F0\"\"#*
&F1F0&F.6#F2F0F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "b
isect:=proc(f,L)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "  local a,b,c;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "  a:=L[1];b:=L[2];" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 15 "  c := (a+b)/2;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "  if f(a) = 0 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "    return [a,c]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "  elif f(c)
 = 0 then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "    return [a,c];" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "  elif f(b) = 0 then" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "    return [c,b];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "  elif not signum(f(a)) = signum(f(b)) then" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 39 "    if signum(f(a)) = signum(f(c)) then" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "      return [c,b];" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 8 "    else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 
"      return [a,c];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "    fi; " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "  else" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "    return FAIL;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 
"  fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Here is a sample f
unction from the text" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=x->x-x^(1/3)-2;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,(9$\"\"\"
*$)F-#F.\"\"$F.!\"\"\"\"#F3F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Let's try to locate the root with \+
an error no bigger than  10^(-8)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "L:=[1,5];" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"LG7$\"\"\"\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "while L[2]-L[1] > 2*10^(-8) do L:=bisect(f,L): od: L;
 rf:=evalf(midp(L));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#\"*\"z:jB\"
)k)3r'#\")P(pZ\"\"(/V>%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rfG$\"1%
>$z,(z8_$!#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 32 "Let's check with Maple's solver:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "rftrue:=fso
lve(f(x)=0,x); err:=rftrue-rf;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'r
ftrueG$\"1oX!oqz8_$!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$errG$\"(u
8,&!#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 36 "Here's another example from the text" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "g:=x->x-12*
x^(1/3)+12;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)op
eratorG%&arrowGF(,(9$\"\"\"*&\"#7F.)F-#F.\"\"$F.!\"\"F0F.F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "L:=[0,5];" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"LG7$\"\"!\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 74 "while L[2]-L[1] > 2*10^(-6) do L:=bisect(g,L): od: L;
 rg1:=evalf(midp(L));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#\"(be#e\"(
/V>%#\"(lkX\"\"(w&[5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rg1G$\"1\"=
)>nE**)Q\"!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "L:=[15,25]
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7$\"#:\"#D" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "while L[2]-L[1] > 2*10^(-6) do L:=b
isect(g,L): od: L; rg2:=evalf(midp(L));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7$#\")b;^W\"(_r4##\"):L-*)\"(/V>%" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$rg2G$\"1PkJn6[A@!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 44 "rg1true:=fsolve(g(x)=0,x); err:=rg1true-rg1;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%(rg1trueG$\"1!fc*fF**)Q\"!#:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$errG$\")4%eF*!#:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "rg2true:=fsolve(g(x)=0,x=10..infinity); err:=rg2true-
rg2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(rg2trueG$\"1S:Lm6[A@!#9" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$errG$!'(*[)*!#9" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Now let's find th
e square root of 2 by bisection with an error no bigger than 10^(-9)" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "h:=x->x^2-2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGR6#%\"xG
6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"#\"\"\"F1F0!\"\"F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "L:=[1,2];" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"LG7$\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "while L[2]-L[1] > 2*10^(-9) do L:=bisect(h,L): od: L;
 rh:=evalf(midp(L));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#\"*JD\")*=
\"*Gx@M\"#\"*D,Df(\"*74(o`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rhG$
\"1DHXhN@99!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "err:=eval
f(sqrt(2)-rh);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$errG$\"'q,#*!#:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "23 1" 0 }
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