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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 807 "Ration
alApprox := proc( r,  # irrational number\n                        n  \+
 # upper bound on the denominator\n                     )\n   local p,
 q, eps0, eps1, s, count;\n\n   p, q := floor(r), 1;         # initial
ize p and q\n   eps0 := abs(evalf(p/q-r));   # initialize epsilon0\n\n
   count := 1;\n   s[count] := p/q;\n   while q < n do\n\n      # the \+
simple algorithm\n      if evalf(p/q) < evalf(r) then \n         p := \+
p + 1;\n      else\n         q := q + 1;\n      end if;\n\n      # upd
ate if a better approximation is obtained\n      eps1 := abs(evalf(p/q
-r));   # epsilon 1\n      if eps1 < eps0 then\n         count := coun
t + 1;       # count it\n         eps0 := eps1;             # update t
he error\n         s[count] := p/q;          # store p/q\n      end if
;\n\n   end do;\n\n   return seq(s[k],k=1..count);\nend proc;" }}
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{MPLTEXT 1 0 23 "RationalApprox(Pi,200);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "60\"\"$#\"#8\"\"%#\"#;\"\"&#\"#>\"\"'#\"#A\"\"(#\"$z\"\"#d#\"$,#\"
#k#\"$B#\"#r#\"$X#\"#y#\"$n#\"#&)#\"$*G\"##*#\"$6$\"#**#\"$L$\"$1\"#\"
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{TEXT -1 15 "Harmonic Series" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }
{TEXT 256 597 "HarmonicSeries := proc(  n   # upper bound of the sum\n
                       )\n   local s, j, m, k;\n\n   s := 0;          \+
       # initialize the sum\n   j := 0;                 # initialize t
he denominator\n   m := 1;                 # initialize grid\n   k := \+
Vector[row](n); \n   while s < n do\n   \n      j := j+1;            #
 update the denominator\n      s := s + 1/j;        # add one term\n  \+
    if s >= m then        # when the sum is bigger than m\n         k[
m] := j;        # record this j\n         m := m + 1;       # next num
ber in the grid\n      end if;\n\n   end do;\n\n   return k;\nend proc
;\n" }{TEXT -1 6 "      " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 
"read(\"e:/HarmonicSeries.txt\"):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "k := HarmonicSeries(8);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"kG-%'RTABLEG6%\"*k#=q9-%'VECTORG6#7*\"\"\"\"\"%\"#6\"#J\"#$)
\"$F#\"$;'\"%u;&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The standard deviation problem" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 436 "StanDe
v := proc(  x   # data vector\n                )\n   local n, i, s, mu
, sigma;\n\n   n := LinearAlgebra[Dimension](x);   # get the dimension
\n\n   # 1. compute mu\n   mu := 0;\n   for i from 1 to n do\n      mu
 := mu + x[i]/n;\n   end do;\n\n   # compute the sum inside the square
 root\n   s := 0;\n   for i from 1 to n do\n      s := s + (x[i]-mu)^2
/n;\n   end do;\n\n   # the square root\n   sigma := evalf(sqrt(s));\n
   \n   return sigma;\nend proc;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "read(\"e:/standev.txt\");" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(StanDevGf*6#%\"xG6'%\"nG%\"iG%\"sG
%#muG%&sigmaG6\"F.C)>8$-&%.LinearAlgebraG6#%*DimensionG6#9$>8'\"\"!?(8
%\"\"\"F>F1%%trueG>F:,&F:F>*&&F86#F=F>F1!\"\"F>>8&F;?(F=F>F>F1F?>FG,&F
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "x := LinearAlgebra[RandomVec
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G-%'RTABLEG6%\"*%yNy9-%'VECTORG6#7,\"\"%\"\"(\"\")\"\"'F/F.F0\"\"$F1F1
&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "StanD
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 19 "Retirement problem:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 511 "RetireFund := proc(   B,  # \+
initial balance\n                      r,  # interest rate percentage
\n                      w\n                   )\n   local mr, k, b;\n
\n   mr := (r/12)/100.;  # converge annual percentage rate\n          \+
             # to actual monthly rate\n\n   b[0] := B;\n   if B*mr > w
 then\n      return \"last\",\"forever\"\n   end if;\n\n   k := 0;\n  \+
 while b[k] > 0 do\n\n      k := k + 1;          # update k, the next \+
month\n      b[k] := b[k-1] + b[k-1]*mr - w;\n\n   end do;\n\n   retur
n k-1, b;  \nend proc;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "read(\"e:/retirefund.txt\");" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+RetireFundGf*6%%\"BG%\"rG%\"wG6%%#m
rG%\"kG%\"bG6\"F.C(>8$*(9%\"\"\"#F4\"#7F4$\"$+\"\"\"!!\"\">&8&6#F99$@$
29&*&F?F4F1F4O6$Q%lastF.Q(foreverF.>8%F9?(F.F4F4F.2F9&F=6#FIC$>FI,&FIF
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "k, b := RetireFund(1000000,5,2000);
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)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "k, b := RetireFund(100
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 33 "The coming exam (Monday, Oct. 20)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "1.  Four \+
regular problems,   1 extra credit problems  " }}{PARA 0 "" 0 "" 
{TEXT -1 75 "2.  Time management is important.  30 min/problem. Do eas
y problems first. " }}{PARA 0 "" 0 "" {TEXT -1 115 "3.  Programs are n
ot supposed to be long.  If you are writing a long program, most likel
y you are on a wrong track." }}{PARA 0 "" 0 "" {TEXT -1 49 "4.  My onl
ine book is allowed (but no copy-paste)" }}{PARA 0 "" 0 "" {TEXT -1 
73 "5.  Old class notes are allowed in paper version, not electronic v
ersion." }}{PARA 0 "" 0 "" {TEXT -1 53 "6. Old Maple worksheet are all
owed in paper version. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 9 "Subjects:" }}{PARA 0 "" 0 "" {TEXT -1 95 "1.  if-bloc
ks  (for example,  the leap year problem, or may be nested in a loop, \+
may be nested)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 35 "2.  Iteration (such as Ikeda map)  " }}{PARA 0 "" 0 "" 
{TEXT -1 33 "     (a) initialize first term(s)" }}{PARA 0 "" 0 "" 
{TEXT -1 16 "     (b) a loop " }}{PARA 0 "" 0 "" {TEXT -1 67 "     (c)
 inside the loop, compute the new term(s) using old term(s)" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "3.  Sum" }}
{PARA 0 "" 0 "" {TEXT -1 40 "     (a)  The basic structure of  a sum:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "     \+
" }{TEXT 257 30 "s := 0;   # initialize the sum" }}{PARA 259 "" 0 "" 
{TEXT -1 76 "  for ... from ... to ... do\n     s := s + ...;    # add
 a new term\n  end do" }}{PARA 0 "" 0 "" {TEXT -1 41 "     (b)  Write \+
a sigma notation of a sum" }}{PARA 0 "" 0 "" {TEXT -1 23 "            \+
           " }{XPPEDIT 18 0 "sum((-1)^(k+1)/sum(j^2,j = 1 .. k),k = 1 \+
.. infinity);" "6#-%$sumG6$*&),$\"\"\"!\"\",&%\"kGF)F)F)F)-F$6$*$)%\"j
G\"\"#F)/F1;F)F,F*/F,;F)%)infinityG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 45 "         The sum can be programmed as follows" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 16 "         s := 0
;" }}{PARA 261 "" 0 "" {TEXT -1 29 "         for k from 1 to n do" }}
{PARA 0 "" 0 "" {TEXT -1 35 "                            t := 0;" }}
{PARA 0 "" 0 "" {TEXT -1 48 "                            for j from 1 \+
to k do" }}{PARA 0 "" 0 "" {TEXT -1 44 "                              \+
  t := t + j^2" }}{PARA 0 "" 0 "" {TEXT -1 35 "                       \+
     end do;" }}{PARA 262 "" 0 "" {TEXT -1 34 "            s := s + (-
1)^(k+1)/t;" }}{PARA 263 "" 0 "" {TEXT -1 16 "         end do;" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 12 "4.  While-do" }}{PARA 0 "" 0 "" {TEXT -1 62 "  \+
     (a)  You may need something like  k := 0;   k := k + 1;" }}{PARA 
0 "" 0 "" {TEXT -1 77 "       (b)  The most important part is the cond
ition for the loop to continue" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 17 "5.  Using vectors" }}{PARA 0 "" 0 "" 
{TEXT -1 45 "      (a)   n := LinearAlgebra[Dimension](x);" }}{PARA 0 
"" 0 "" {TEXT -1 35 "      (b)   x := Vector[row](n);   " }}{PARA 0 "
" 0 "" {TEXT -1 67 "      (c)   a := LinearAlgebra[RandomVector][row](
n,generator=....)" }}{PARA 0 "" 0 "" {TEXT -1 50 "      Standard devia
tion problem is a good example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}
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