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}1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Author" -1 258 1 {CSTYLE "" -1 
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0 1 }{PSTYLE "Title" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 
1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" 
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{SECT 0 {EXCHG {PARA 259 "" 0 "" {TEXT -1 9 "Lecture 3" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 50 "if ... then ... elif ... then ... else ..
. end if;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 257 "" 0 "" 
{TEXT 280 49 "Such statements  take one of the following forms:" }
{TEXT -1 2 "\n\n" }{TEXT 260 2 "if" }{TEXT -1 1 " " }{TEXT 272 18 "boo
lean expression" }{TEXT -1 1 " " }{TEXT 261 4 "then" }{TEXT -1 7 " \n \+
    " }{TEXT 273 22 "sequence of statements" }}{PARA 257 "" 0 "" 
{TEXT 276 7 "end if;" }{TEXT 277 1 "\n" }}{PARA 257 "" 0 "" {TEXT -1 
38 "--------------------------------------" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 2 "if" }{TEXT -1 21 " boolean expr
ession  " }{TEXT 263 4 "then" }{TEXT -1 27 " \n  sequence of statement
s\n" }{TEXT 264 4 "else" }}{PARA 0 "" 0 "" {TEXT -1 24 " sequence of s
tatements\n" }{TEXT 278 7 "end if;" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "
" {TEXT -1 38 "--------------------------------------" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 2 "if" }{TEXT -1 24 "   b
oolean expression1  " }{TEXT 266 4 "then" }{TEXT -1 27 " \n  sequence \+
of statements\n" }{TEXT 267 4 "elif" }{TEXT -1 21 " boolean expression
2 " }{TEXT 268 4 "then" }}{PARA 0 "" 0 "" {TEXT -1 23 " sequence of st
atements" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
4 "    " }{TEXT 281 6 ". . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 269 4 "elif" }{TEXT -1 21 " boolean expression3 \+
" }{TEXT 270 4 "then" }{TEXT -1 25 "\n sequence of statements\n" }
{TEXT 271 4 "else" }{TEXT -1 26 "\n  sequence of statements\n" }{TEXT 
279 7 "end if;" }}{PARA 0 "" 0 "" {TEXT -1 38 "-----------------------
---------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 22 "[In Maple V one used  " }{TEXT 406 2 "fi" }{TEXT -1 14 " \+
in place of  " }{TEXT 405 8 "end if ." }{TEXT -1 30 " In Maple 6 and 7
 one may use " }{TEXT 407 7 "fi, end" }{TEXT -1 4 " or " }{TEXT 408 6 
"end if" }{TEXT -1 42 " interchangeably. So if I slip up and use " }
{TEXT 409 2 "fi" }{TEXT -1 48 " from time to time you will know what i
t means.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
3 "By " }{TEXT 259 18 "boolean expression" }{TEXT -1 36 " we mean a st
atement that is either " }{TEXT 383 4 "true" }{TEXT -1 4 " or " }
{TEXT 384 5 "false" }{TEXT -1 52 ". (Note: Maple's boolean expressions
 have the value " }{TEXT 385 4 "true" }{TEXT -1 4 " or " }{TEXT 386 5 
"false" }{TEXT -1 8 " and not" }{TEXT 388 5 " True" }{TEXT -1 4 " or \+
" }{TEXT 387 5 "False" }{TEXT -1 49 ".) Such statements are usually co
nstructed using " }{TEXT 410 18 "<, <=, >, >=, <>. " }{TEXT -1 3 "or \+
" }{TEXT 411 1 "=" }{TEXT -1 46 " together possibly with the logical o
perators " }{TEXT 258 12 "and, or, not" }{TEXT -1 8 ". Note  " }
{XPPEDIT 18 0 "x <> y;" "6#0%\"xG%\"yG" }{TEXT -1 105 " is denoted by \+
x <> y in Maple. Other types of boolean expressions are formed by proc
edures which return " }{TEXT 399 4 "true" }{TEXT -1 4 " or " }{TEXT 
400 5 "false" }{TEXT -1 17 ". And example is " }{TEXT 401 10 "isprime(
n)" }{TEXT -1 43 ". For specific values of n, this is either " }{TEXT 
402 4 "true" }{TEXT -1 4 " or " }{TEXT 403 5 "false" }{TEXT -1 3 ". \n
" }}{PARA 0 "" 0 "" {TEXT -1 55 "One can test a boolean expression by \+
using the command " }{TEXT 404 5 "evalb" }{TEXT -1 30 " as in the foll
owing examples:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 4 "1<2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "evalb(1<2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalb(2<
1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalb(z<w);\n" }}}
{PARA 0 "" 0 "" {TEXT -1 71 "This indicates that Maple doesn't know wh
ether z < w is true or false. " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 4 " \nA " }{TEXT 274 10 "statement " }
{TEXT -1 80 "is a command followed by a colon or semicolon. The follow
ing is an example of a " }{TEXT 275 22 "sequence of statements" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 
282 37 "  y:=i^2;\n   z:=i^3;\n   print(y,z);\n\n" }{TEXT -1 75 "So fa
r as Maple is concerned they could be all on the same line as follows:
" }{TEXT 394 33 "\n\n  y:=i^2;  z:=i^3;  print(y,z);" }}{PARA 0 "" 0 "
" {TEXT -1 108 "\n\nWe now look at some examples. The first illustrate
s a common error. Execute the commands to see the error." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "if t > 1 then \n  x:=3; \nend if;" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "The expre
ssion t  > 1 is called a " }{TEXT 257 18 "boolean expression" }{TEXT 
-1 221 ".  The problem here is that Maple does not know the value of t
 so it cannot tell if t > 1 is true  or false.  This can be fixed by p
utting in a value of t as we now do. Note that a print statement is no
t needed if we use " }{TEXT 412 7 "end if;" }{TEXT -1 16 "  as opposed
 to " }{TEXT 413 6 "end if" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 " Try to anticipate the \+
output." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "t:=3:\nif t > 1 \+
then \n  x:=2;\n  print(x);\nend if:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "t:=3:\nif t > 1 then \n  x:=2;\nend if;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "t:=0:\nif t > 1 then \n   x:=2; \ne
nd if; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 44 "t:=0:\nif t > 1 then \n  x:=2; \nend if;\nx:=3
; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "The above  can be done with
 the following:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "t:=0:\ni
f t > 1 then \n  x:=2; \nelse\n  x:=3;\nend if;" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 23 "Here we illustrate the " }{TEXT 256 52 "if .. then .
. elif .. then else... fi  construction." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 183 "f:=proc(y)\n if y <= 1 then \n   return 1;\n elif y \+
<= 2 then \n   return 2;\n elif y <= 3 then \n   return 3;\n elif y <=
 4 then \n   return 4;\n else \n   return infinity;\n end if;\nend pro
c:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The same thing can be acc
omplished as follows:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 
"g:=proc(y)\nif y <= 1 then return 1; end if;\nif y <= 2 then return 2
; end if;\nif y <= 3 then return 3; end if;\nif y <= 4 then return 4; \+
end if; \nreturn infinity; \nend:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 56 "We check a few values of x to see that f(x) = g(x): The " }
{TEXT 362 5 "by .5" }{TEXT -1 34 " in the do loop causes increments " 
}{TEXT 414 5 "by .5" }{TEXT -1 24 " instead of the default " }{TEXT 
415 4 "by 1" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
108 "for x from 0 to 5  by .5  do \nprint(`If x= ` ,  x  , ` then,   f
(x) = ` , f(x) , `    g(x) =  `, g(x)); \nod;" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 154 "Let's try to plot the function f (t) for t from 0 to \+
5.  The first plot fails, however, the second method works. You can se
e that we have a step function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 24 "plot(f(t),t=0..5, 0..5);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "plot(f,0..5,0..5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 51 "More examples: We create a boolean valued function." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "is_increasing:=proc(a,b,c)\n if (a
 < b) and (b < c) then \n   return true; \n else\n   return false;\n e
nd if;\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "is_inc
reasing(1,2,3);\nis_increasing(1,2,1);\nis_increasing(1,1,1);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Note that we can get these on the \+
same line as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "is
_increasing(1,2,3),is_increasing(1,2,1),is_increasing(1,1,1);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Here's another example of a boolea
n function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "is_composit
e:=proc(n)\n  if isprime(n) = false then \n    return true;  \n  else
\n    return false; \n  end if;\n  end proc:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 35 "is_composite(10);\nis_composite(11);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 24 "Note that the statement " }{MPLTEXT 1 0 
19 "isprime(n) = false " }{TEXT -1 34 "may be replaced by the statemen
t  " }{MPLTEXT 1 0 15 "not(isprime(n))" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 53 "for  ... while .. do ... by .. end do, ne
xt and break" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "In Maple 6 one may
 replace " }{TEXT 288 8 "end do  " }{TEXT 289 4 "by  " }{TEXT 290 2 "o
d" }{TEXT 291 6 "  or  " }{TEXT 292 3 "end" }{TEXT 293 46 ". In Maple \+
V  one can only use the short form " }{TEXT 294 2 "od" }{TEXT 295 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "We lo
ok at some examples: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "for i to 3 do \n  y:=i^2;\n  z:=i^3
;\n  sin(y*z);\n  evalf(%);\nend do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 29 "Note that this is the same as" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "for i from 1 to 3 do \n  y:=i^2;\n  z:=i^3;\n  sin(y*
z);\n  evalf(%);\nend do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 356 "Not
e that the value of the index i is 4 after the above loop for i from 1
 to 3 is executed. Each time one goes throught the loop the value of t
he index is incremented by 1 and then if the value is less that the up
per limit 3 the loop is executed and control is passed to the next sta
tement (if there is one). We now see what the value of the variable i \+
is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "i;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 83 "If we use by 5.5 then the index is increased by
 5.5 each time, as in the following:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "for q from -10 to 10 by 5
.5 do\n print(q);\nend do:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Of \+
course, 5.5 may be replace by any number.\n\nWe now illustrate the use
 of " }{TEXT 393 5 "break" }{TEXT -1 91 " in a loop. This program will
 find the smallest integer w between 1 and 100 that satisfies " }
{XPPEDIT 18 0 "2^w > 10^6" "6#2*$\"#5\"\"')\"\"#%\"wG" }{TEXT -1 1 ":
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "for w from 1 to 100 do
\n  if 2^w > 10^6 then\n    print(w, 2^w);\n    break;\n  end if;\nend
 do;\nprint(`We are finished!`);\nw;\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 10 "Note that " }{TEXT 283 5 "break" }{TEXT -1 64 " means to \+
get out of the loop and go to the statement following " }{TEXT 309 6 "
end do" }{TEXT -1 2 ". " }{TEXT 396 3 "If " }{TEXT 395 5 "break" }
{TEXT 397 87 " is executed then the value of the index w is the value \+
at the time the loop is exited " }{TEXT -1 58 "Note that the same resu
lt is obtained if we leave out the " }{TEXT 333 6 "to 100" }{TEXT -1 
19 " on the first line." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 
"for w from 1 do\n  if 2^w > 10^6 then\n    print(w, 2^w);\n    break;
\n  end if;\nend do;\nprint(`We are finished!`);\nw;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 95 "Let's write a program to find the product of al
l primes less than 100: Try this program with a " }{TEXT 334 9 "semico
lon" }{TEXT -1 7 " after " }{TEXT 335 6 "end do" }{TEXT -1 15 " rather
 than a " }{TEXT 336 5 "colon" }{TEXT -1 44 ". Now see if you can gues
s what the command " }{TEXT 398 4 "next" }{TEXT -1 6 " does." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "P:=1:\nfor m from 1 to 100 \+
do\n  if isprime(m) = false then \n    next; \n  end if;\n  P:= P * m;
\nend do:\nP; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Note that " }
{TEXT 284 5 "next " }{TEXT -1 38 " means to go to the next value of th
e " }{TEXT 363 5 "index" }{TEXT -1 25 " - in the above case the " }
{TEXT 364 5 "index" }{TEXT -1 4 " is " }{TEXT 337 1 "m" }{TEXT -1 88 "
 - if m is not prime. If m is prime then we replace P by the current v
alue of P times m." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 76 "\nIn the following example we seek the smallest positive \+
integer n such that " }{XPPEDIT 18 0 "1000 <= n!;" "6#1\"%+5-%*factori
alG6#%\"nG" }{TEXT -1 56 ". When we find it we will call it a. Note th
at in maple " }{TEXT 285 12 "factorial(n)" }{TEXT -1 5 " and " }{TEXT 
286 2 "n!" }{TEXT -1 23 "  give the same result." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 88 "for n from 1 to 10 do \n if 1000 <= n! then \+
\n  print(n,n!);\n  break;\n end if;\nend do:\nn;\n" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 35 "Another way to do this is by using " }{TEXT 
287 32 "while  boolean expression  do:  " }{TEXT 416 66 "The loop is e
xecuted untill the boolean expression becomes false. " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "q:=0:\nfor n from 1 while q <= 1000
 do\n  q:= n!;\nend do:\nn,n-1,q;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
338 214 "Note that the value of n at the end of the above loop is 8. W
hen a \"while\" loop is exited the index is increment by 1 as we see a
bove. However, when break causes us to exit from a loop the index is n
ot incremented." }{TEXT -1 69 " \n\nSee what happens if we omit the q:
=0; statement at the beginning:\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 
"for n from 1 while q < 1000 do\n  q:= n!;\nend do:" }}}{PARA 0 "" 0 "
" {TEXT -1 118 "This is due to the fact that Maple does not know the v
alue of q at the first step, so cannot tell if  q < 1000 or not." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 44 "Old notation for end do, end if and end proc" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "In versions of \+
Maple prior to Maple 6 one used \n\n   " }{TEXT 373 2 "od" }{TEXT -1 
12 " instead of " }{TEXT 374 6 "end do" }{TEXT -1 1 "\n" }}{PARA 0 "" 
0 "" {TEXT -1 3 "   " }{TEXT 375 2 "fi" }{TEXT -1 12 " instead of " }
{TEXT 376 6 "end fi" }{TEXT -1 5 "\n\n   " }{TEXT 377 3 "end" }{TEXT 
-1 12 " instead of " }{TEXT 378 8 "end proc" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 18 "One may still use " }{TEXT 
379 6 "od, fi" }{TEXT -1 5 " and " }{TEXT 380 4 "end " }{TEXT -1 141 "
in Maple 6 and 7. You may wish to do so since the old way is a little \+
shorter and so far as I know it does not effect the speed of execution
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Also
 in Maple IV one used \", \"\", and \"\"\" instead of %, %%, %%% and \+
" }{TEXT 381 18 "RETURN(expression)" }{TEXT -1 12 " instead of " }
{TEXT 382 17 "return expression" }{TEXT -1 88 " in procedures. Maple 6
 and 7 still recognizes RETURN(expression) but not \",\"\", and \"\"\"
." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "for x in L do ... end do;" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 
"Here is another way to use do loops. Here we specify exactly which va
lues of the index we want the loop to go through:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "for x in [1,3,X,Y, t^2+t+1] do\n  print(x,x^2);\nend \+
do:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "[1,2,X,Y,t^2 + t + 1] is a
n example of a " }{TEXT 296 5 "list " }{TEXT -1 229 "--- we will say m
ore about lists later. You may also use set brackets \{ \} instead of \+
 [  ] as in the following example. Be warned, however, that the order \+
in sets is sometimes lost. If order is important use [   ] instead of \+
\{ \}." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for x in \{1,3,X,
Y, t^2+t+1\} do\n  print(x,x^2);\nend do:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 15 "Nested do loops" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "You may need to ha
ve " }{TEXT 339 27 "do loops inside of do loops" }{TEXT -1 115 ". Here
 are some examples.  The first allows us to generate all ordered pairs
 [i,j ] where i and j are in \{1,2,3,4\}:" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "for i from 1 to 4 do
\n  for j from 1 to 4 do\n    print([i,j]);\n  end do;\nend do;\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Note that if you use (i,j) instea
d of [i,j] in the above code the parentheses will not appear in the ou
tput." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 72 "The following allows us to generate all 2 element subsets
 of  \{1,2,3,4\}:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 84 "for i from 1 to 4 do\n   for j from i+1 to 4
 do\n     print(\{i,j\});\n   end do;\nend do;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Of cou
rse, we may do other  things other than just " }{TEXT 365 12 "print(\{
i,j\})" }{TEXT -1 284 ". We can take i,j and perform various other ope
rations. For example, here we form the set of all sums of i and j  whe
re i < j and put them in a set called S when i + j  is a prime number \+
and we count the number of times that i + j is a prime. For this we ne
ed a \"counter\". We call it " }{TEXT 297 5 "count" }{TEXT -1 27 ". It
 could be any variable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 55 "We set printlevel to 2 so we can see what is going o
n. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "printlevel:=2;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 349 "S:=\{\}:\ncount:=0:\nfor i \+
from 1 to 4 do\n   for j from i+1 to 4 do\n     print(`---------------
--`);\n     print(i,j);\n     n:=i+j;\n     if isprime(n) then\n      \+
 count:=count+1;\n       print(`count = `, count); \n       S:=S union
 \{n\};\n       print(`S = `, S);     \n     end if;\n   end do;\nend \+
do;\nprint(`----------`);\nprint(`S = `, S, `count = `,count);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "Why is c
ount = 4, yet S only contains three elements?\n\nLater we will show ho
w to get rid of unwanted commas such as the commas in the last output \+
statement S = , \{ 3,5,7\}, count = , 4 by use of the " }{TEXT 417 6 "
printf" }{TEXT -1 18 " comma instead of " }{TEXT 418 5 "print" }{TEXT 
-1 136 ".\n\nWe can have do loops nested three deep as in the followin
g example where we form the set of all binary sequences [lists] of len
gth 3:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 "N
ote that the print statement in the following program has been comment
ed out by use of #. Try removing the # and seeing the difference in ou
tput." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 164 "S:=\{\}:\nfor x1 from 0 to 1 do\n  for x2 from 0 to \+
1 do\n   for x3 from 0 to 1 do\n     #print(x1,x2,x3);\n     S:=S unio
n \{ [x1,x2,x3] \};\n   end do:\n  end do:\nend do:\nS;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 198 "We will learn other ways to define this \+
set later. Clearly you would not want to use this method to produce th
e set of all 1024 binary sequences of length 10. That would require 10
 nested do loops. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "Local and Global Variables" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 298 15 "L
ocal variables" }{TEXT -1 29 " are variables that are used " }{TEXT 
340 6 "inside" }{TEXT -1 99 " a procedure. After the procedure is exec
uted the local variables revert to their original values. " }{TEXT 
299 16 "Global variables" }{TEXT -1 121 " may be changed inside the pr
ocedure and the values will be retained after the procedure is over. H
ere are some examples:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re
start:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "We first set the value \+
of the four variables x1,x2,y1,y2 to 1 by executing the following:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x1:=1:\nx2:=1:\ny1:=1:\ny2:
=1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Now we difine a simple pro
cedure called f: Take a minute to try to guess what it will do:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "f:=proc(n)\n global x1,x2;
\n local y1,y2;\n  x1:=2:\n  x2:=3:\n  y1:=5:\n  y2:=7:\n return n*x1*
x2*y1*y2;\nend proc:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Let's s
ee what f(11) returns:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(
11);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "It will help to factor \+
this integer:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ifactor(%)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Now let's see what happened \+
to the variables " }{TEXT 366 14 "x1, x2, y1, y2" }{TEXT -1 1 ":" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "x1,x2, y1,y2;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 150 "You can see that the global variables re
tain the changes made inside the procedure f, but the local variables \+
are unchanged by what happened inside f." }{TEXT 389 51 " I recommend \+
using global variables very sparingly " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "Here's anothe simple exa
mple. Execute these commands in order. Try to anticipate what the outp
ut will be in each case." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "
restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "Note that a procedur
e may have no input. But to execute it one still needs parentheses.  N
ote that we use NULL as output. NULL is the empty sequence." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f2:=proc()\nglobal x;\nx:=2*
x;\nreturn NULL;\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
51 "f3:=proc()\nglobal x;\nx:=3*x;\nreturn NULL;\nend proc:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x:=2;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 5 "f2();" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 2 "x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f3();" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "x;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "print vs return in procedures" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Co
nsider carefully the following procedures named g and h: Notice that t
he output " }{TEXT 392 5 "seems" }{TEXT -1 60 " to be the same for eac
h. However, see below for differences" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 
"g:=proc(x)\nlocal y;\ny:=2*x;\nprint(y);\nend proc:\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "h:=proc(x)\nlocal y;\ny:=2*x;\nretu
rn y;\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(3);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "h(3);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 52 "But note how g misbehaves in the following example
s:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "y:=g(3);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "2+g(3);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 31 "But h behaves as we would wish:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "y:=h(3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 7 "2+h(3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "The moral of this
 section is: DO NOT USE " }{TEXT 390 5 "print" }{TEXT -1 13 " INSTEAD \+
OF  " }{TEXT 391 6 "return" }{TEXT -1 31 " WHEN CONSTRUCTING A PROCEDU
RE." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 
3 "" 0 "" {TEXT -1 57 "Use of print, debug, undebug and printlevel for
 debugging" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 56 "Consider the following attempt to construct a procedure \+
" }{TEXT 419 6 "binexp" }{TEXT -1 63 " whose input is n and whose outp
ut is the smallest k such that " }{XPPEDIT 18 0 "n <= 2^k;" "6#1%\"nG)
\"\"#%\"kG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "binexp:=pro
c(n)\nlocal k;\nfor k from 1 to 10 do\n  if 2^k >= n then\n    return \+
k;\n  end if;\nend do;\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "binexp(10000);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
163 "Note that we get no output for binexp(10000). In this case you ca
n no doubt see the problem but let's illustrate a few debugging techni
ques to try to find the bug." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 29 "First we set printlevel:=100;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "printlevel:=100;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "binexp(10000);\n" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 48 "This doesn't help in this case. Next we try the
 " }{TEXT 424 5 "debug" }{TEXT -1 69 " command after setting the print
level back to the default level of 1:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "printlevel:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "debug(binexp):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "b
inexp(10000);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Again not much h
elp. Use " }{TEXT 423 7 "undebug" }{TEXT -1 53 " to turn off the debug
ger after you are done with it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "undebug(binexp);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Now l
et's try inserting " }{TEXT 425 5 "print" }{TEXT -1 223 " statements i
n the procedure to see what the value of the variables are as the proc
edure runs. (In practice you don't have to recopy the procedure to do \+
this. Just insert the statements as you would with any wordprocessor.)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "binexp:=proc(n)\nlocal
 k;\nfor k from 1 to 10 do\nprint(`k=`,k);\nprint(`2^k = `,2^k);\nprin
t(`n=`, n);\nprint(`---------------------------`);\n  if 2^k >= n then
\n    return k;\n  end if;\nend do;\nend proc:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "binexp(10000);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 87 "Here we see that k only goes up to 10. So we look back an
d see that we need to increase" }}{PARA 0 "" 0 "" {TEXT -1 172 "the va
lue of 10 in the upper limit of the do loop. When doing this it suffic
es to comment out the print statements by inserting # before them as w
e now do as we change the " }{TEXT 420 2 "10" }{TEXT -1 4 " to " }
{TEXT 421 8 "infinity" }{TEXT -1 37 " -- or we could merely leave out \+
the " }{TEXT 422 5 "to 10" }{TEXT -1 159 " part and we would get equiv
alent results. But note that you must be assured that the procedure wo
uld eventually return something else it could  go on forever." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "binexp:=proc(n)\nlocal k;\n
for k from 1 to infinity do\n#print(`k=`,k);\n#print(`2^k = `,2^k);\n#
print(`n=`, n);\n#print(`---------------------------`);\n  if 2^k >= n
 then\n    return k;\n  end if;\nend do;\nend proc:" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 17 "Now we try again:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "binexp(10000);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
17 "Let's check this." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ev
alb(2^13 >= 10000), evalb(2^14 >=10000);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 153 "So the procedure seems to be okay. In general we might w
ant to check more values of n, but in this case we can see clearly tha
t the procedure is correct." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "nam
es (" }{TEXT 431 3 "aka" }{TEXT -1 31 " variables) and reserved words.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
11 "In Maple a " }{TEXT 35 4 "name" }{TEXT -1 2 " (" }{TEXT 341 3 "aka
" }{TEXT -1 4 ", a " }{TEXT 300 9 "variable " }{TEXT -1 3 "or " }
{TEXT 372 13 "variable name" }{TEXT -1 320 ") is usually a letter foll
owed by zero or more letters, digits, and underscores, with lowercase \+
and uppercase letters distinct. The maximum length of a name is system
 dependent. It may even contain spaces if you enclose it in backquotes
 ` `. [Note that in output the backquotes are not shown.] Here's an ex
treme example." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "`this is \+
the name of a variable`:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
35 "`this is the name of a variable`^2;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 88 "If you use underscores where spaces would be you don't ne
ed the back quotes, for example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "this_is_the_name_of_a_variab
le:=3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "But note:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "this is not the name of a variable:
=2;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 427 8 "
Keywords" }{TEXT -1 141 " are reserved words that are used in forming \+
Maple statements. These words cannot be used as variable names. There \+
are 11 operator keywords: " }}{PARA 15 "" 0 "" {TEXT 35 8 "     and" }
{TEXT -1 2 ", " }{TEXT 35 2 "or" }{TEXT -1 2 ", " }{TEXT 35 3 "xor" }
{TEXT -1 2 ", " }{TEXT 35 3 "not" }{TEXT -1 2 ", " }{TEXT 35 7 "implie
s" }{TEXT -1 2 ", " }{TEXT 35 5 "union" }{TEXT -1 2 ", " }{TEXT 35 9 "
intersect" }{TEXT -1 2 ", " }{TEXT 35 6 "subset" }{TEXT -1 2 ", " }
{TEXT 35 5 "minus" }{TEXT -1 2 ", " }{TEXT 35 3 "mod" }{TEXT -1 2 ", \+
" }{TEXT 35 8 "assuming" }{TEXT -1 2 ", " }}{PARA 15 "" 0 "" {TEXT -1 
81 "plus 35 additional language keywords, for a total of 46 reserved w
ords in Maple: " }}{PARA 0 "" 0 "" {TEXT -1 7 "       " }{TEXT 426 
631 "and             assuming   break    by        catch   \n       de
scription  do             done     elif        else    \n       end   \+
          error         export   fi          finally \n       for     \+
         from          global   if          implies \n       in       \+
        intersect     local     minus   mod     \n       module       \+
 next           not       od        option  \n       options        or
              proc     quit       read    \n       return         save
            stop     subset   then    \n       to               try   \+
          union    use       while   \n       xor                     \+
               " }}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 
"" 0 "" {TEXT -1 28 "Be especially wary of using " }{TEXT 428 4 "stop
" }{TEXT -1 2 ", " }{TEXT 429 4 "quit" }{TEXT -1 4 " or " }{TEXT 430 
4 "done" }{TEXT -1 76 ". This may cause Maple to quit and you may lose
 any thing you haven't saved!" }}}{PARA 0 "" 0 "" {TEXT -1 22 "       \+
               " }}{PARA 0 "" 0 "" {TEXT -1 51 "You may find them at a
ny time by using the command " }{MPLTEXT 1 0 9 "?reserved" }{TEXT -1 
19 " . Note that since " }{TEXT 301 2 "in" }{TEXT -1 73 " is reserved \+
it cannot be used as a variable. But `in` can be a variable." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "in:=2;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "`in`:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 7 "`in`^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "But quit even w
ith backquotes will not work as a variable." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 10 "`quit`:=3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 
"A name cannot begin with a number." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "1x:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Indexed varia
bles" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 68 "We can use indexed variables to create subscripted varibles suc
h as " }{XPPEDIT 18 0 "x[1],x[2],x[3],x[4];" "6&&%\"xG6#\"\"\"&F$6#\"
\"#&F$6#\"\"$&F$6#\"\"%" }{TEXT -1 13 ". Here's how:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "x[1], x[2], x[3], x[4];" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 49 "Such variables can be used as ordinary variables." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x[1]:=2; x[3]:=x[1]*x[2];" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Here's another example:\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "We count the number of times each
 digit appears in an integer. We first generate a large positive integ
er. Execute this assignment." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 11 "N:=2^100-1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Now we set u
p a variable " }{XPPEDIT 18 0 "c[i];" "6#&%\"cG6#%\"iG" }{TEXT -1 18 "
 for each integer " }{XPPEDIT 18 0 "i;" "6#%\"iG" }{TEXT -1 47 ". This
 will hold the number of times the digit " }{XPPEDIT 18 0 "i;" "6#%\"i
G" }{TEXT -1 56 " appears in N. First we initialize these variables to
 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "for i from 0 to 9 do
 c[i]:=0; end do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "Now we coun
t the number of times each digit appears using the following loop. Not
e that the least significant digit of N is give by " }{XPPEDIT 18 0 "`
mod`(N,10);" "6#-%$modG6$%\"NG\"#5" }{TEXT -1 112 ".  Then we can get \+
rid of this digit by subtracting it and then dividing by 10.  Usually \+
we would place : after " }{TEXT 432 6 "end do" }{TEXT -1 54 ", but her
e we place ; so you can see what's happening." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 69 "N;\nwhile N > 0 do\nk:=N mod 10;\nc[k]:=c[k]+1;
\nN:=(N - k)/10; \nend do;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "N
ow let's see what the values of the counters are:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 77 "2^100-1;\nfor i from 0 to 9 do \nprint(`digi
t`,i,`appears`, c[i], `times`);\nod:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "assigned and anames" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 433 
11 "assigned(n)" }{TEXT -1 9 " returns " }{TEXT 434 4 "true" }{TEXT 
-1 55 " if n has a value other than its own name, and returns " }
{TEXT 435 5 "false" }{TEXT -1 65 " otherwise. This will be useful in t
he next section. For example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assigned(x
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x:=1;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assigned(x);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 51 "This also works for indexed variables. For example:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "for i from 1 to 10 do\n
if isprime(i) then P[i]:=i; end if;\nend do:\n" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 70 "for i from 1 to 10 do\nif assigned(P[i]) then \+
print(i); end if;\nend do;" }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 4 "The
 " }{TEXT 436 8 "anames()" }{TEXT -1 103 " returns an expression seque
nce of names that are currently assigned values other than their own n
ame. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "anames();" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 437 10 "`isprime/w" }{TEXT -1 6 "` and " 
}{TEXT 438 7 "isprime" }{TEXT -1 29 " are names assigned by Maple." }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "If we restart, you can see that t
here are no names that are assigned variables." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 9 "anames();" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 28 "Squares and Sums of Squares." }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 14 "The procedure " }{MPLTEXT 1 0 9 "root(x,n)" }{TEXT -1 184 " (wh
en x is real or complex and n is a positive integer) computes the nth \+
root of x. If x is an integer that is a perfect nth power then the nth
 root is returned; otherwise the power  " }{XPPEDIT 18 0 "x^(1/n);" "6
#)%\"xG*&\"\"\"F&%\"nG!\"\"" }{TEXT -1 98 "  is returned unevaluated. \+
We can tell if x is a positive integer by use of the boolean function \+
 " }{MPLTEXT 1 0 14 "type(x,posint)" }{TEXT -1 86 " which is true if x
 is a positive integer and false otherwise. Here are some examples:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 11 "root(8,3); " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "root(2,3); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 11 "root(27,3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Compare this
 with the following:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "27^(
1/3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "type(27^(1/3),posint);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "type(root(27,3),posint);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "root(28,3);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "type(%,posint);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 11 "The command" }{TEXT 440 5 " root" }{TEXT -1 100 "
 also works for floating point numbers and complex numbers. Recall tha
t I is the square root of -1. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "root(28.0,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ro
ot(1+2.0*I,3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 159 "We now define \+
a procedure is_sqr which determines whether or not a positive integer \+
 is a  square, that is, whether or not it is the square of another int
eger:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "is_sqr:=n->type(ro
ot(n,2), posint):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "is_s
qr(16);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "is_sqr(17);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "Now we want to determine whether \+
or not the positive integer n is a sum of two positive squares or not.
 If it is, we want to find " }{TEXT 342 14 "all solutions " }{TEXT -1 
17 "to the equations " }{XPPEDIT 18 0 "n = x^2+y^2;" "6#/%\"nG,&*$%\"x
G\"\"#\"\"\"*$%\"yGF(F)" }{TEXT -1 41 " where x and y are positive int
egers and " }{XPPEDIT 18 0 "x <= y;" "6#1%\"xG%\"yG" }{TEXT -1 28 ". N
ote  we may assume that  " }{XPPEDIT 18 0 "x <= sqrt(n/2);" "6#1%\"xG-
%%sqrtG6#*&%\"nG\"\"\"\"\"#!\"\"" }{TEXT -1 0 "" }{TEXT 359 2 " ." }
{TEXT -1 38 "  [Can you see why?  What happens if  " }{XPPEDIT 18 0 "s
qrt(n/2) < x;" "6#2-%%sqrtG6#*&%\"nG\"\"\"\"\"#!\"\"%\"xG" }{TEXT -1 
46 "  ?] Here's one way to write such a procedure:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 210 "sumsqrs:=proc(n)\n local x, y, Sols;\n Sol
s:=\{\};\n for x from 1 to evalf(sqrt(n/2)) do\n   y:= root(n-x^2,2);
\n   if type(y,posint) then  \n     Sols:=Sols union \{\{x,y\}\}; \n  \+
 end if;\n end do;\n return Sols;\nend proc: " }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 34 "\nWhat happens if we leave off the " }{TEXT 360 5 "eva
lf" }{TEXT -1 21 " in the 4-th line of " }{TEXT 361 7 "sumsqrs" }
{TEXT -1 66 "? Remove it and try. Next we illustrate the use of this p
rocedure:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sumsqrs(50);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "We check these two answers. Not
e that \{5\} = \{5,5\}." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "
1^2+7^2, 5^2 + 5^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "What would
 happen in the definition of sumsqrs if you replaced the command" }
{MPLTEXT 1 0 21 " Sols union \{\{x,y\}\}; " }{TEXT -1 5 " by  " }
{MPLTEXT 1 0 17 "Sols union \{x,y\};" }{TEXT -1 1 "?" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 46 "The number of elements in a set S is given by \+
" }{MPLTEXT 1 0 7 "nops(S)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 9 "nops(\{\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "nops(\{a,b,c,d\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "We us
e this to find the number of different solutions to the equation " }
{XPPEDIT 18 0 "n = x^2+y^2" "6#/%\"nG,&*$%\"xG\"\"#\"\"\"*$%\"yGF(F)" 
}{TEXT -1 28 " where x and y are integers " }{XPPEDIT 18 0 "x <= y;" "
6#1%\"xG%\"yG" }{TEXT -1 14 ". For example:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "nops(sumsqrs(50))
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Let's find the smallest n fo
r which " }{XPPEDIT 18 0 "n = x^2+y^2;" "6#/%\"nG,&*$%\"xG\"\"#\"\"\"*
$%\"yGF(F)" }{TEXT -1 36 " has at least 2 different solutions:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "for n from 1 do \nif nops(s
umsqrs(n)) >= 2  then \n   print(n,sumsqrs(n));\n   break; \n end if;
\nend do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Let's find the small
est n for which " }{XPPEDIT 18 0 "n = x^2+y^2;" "6#/%\"nG,&*$%\"xG\"\"
#\"\"\"*$%\"yGF(F)" }{TEXT -1 36 " has at least 3 different solutions:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "for n from 1  do \n if
 nops(sumsqrs(n)) >= 3  then \n   print(n,sumsqrs(n));\n   break; \n e
nd if;\nend do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Here's a much more e
fficient program due to Robert Israel for the determination of integer
s " }{XPPEDIT 18 0 "n <= 20^2+20^2;" "6#1%\"nG,&*$\"#?\"\"#\"\"\"*$F'F
(F)" }{TEXT -1 135 " that can be expressed as the sum of two squares i
n more than two ways:\n\nNote the use of the subscripted variable T[m]
 and the command " }{TEXT 439 8 "assigned" }{TEXT -1 140 " which tells
 us whether or not  something has already been assigned to the variabl
e T[m].  Note that this will cover all integers from 1 to " }{XPPEDIT 
18 0 "1800 = 30^2+30^2;" "6#/\"%+=,&*$\"#I\"\"#\"\"\"*$F'F(F)" }{TEXT 
-1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "restart:\nfor i
 from 1 to 30 do\n     for j from 1 to i do\n       m:= i^2 + j^2;\n  \+
     if assigned(T[m]) then  \n         T[m]:= T[m],[i,j];\n       els
e T[m]:= [i,j]\n       fi\n     od \n   od:" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 23 "Here are some examples:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "T[1];\nT[13];\nT[50];\nT[325];" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 78 "We can get the number of solutions for, say, T[325],
 by the following command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "nops([T[325]]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Using this
 we see which integers up to " }{XPPEDIT 18 0 "2*30^2;" "6#*&\"\"#\"\"
\"*$\"#IF$F%" }{TEXT -1 68 " can be express as a sum of square in thre
e or more different ways: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
73 "for m from 1 to 2*30^2 do\nif nops([T[m]]) >=3 then print(m,T[m]);
 fi;\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "Assignment \+
3  (Due Monday, February 4)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT 343 429 "Perhaps you have heard  the famous stor
y about Ramanujan and Hardy's taxi number: Once when Ramanujan was ill
 in London, Hardy visited Ramanujan in the hospital. When Hardy remark
ed that he had taken taxi number 1729, a singularly unexceptional numb
er, Ramanujan immediately responded that this number was actually quit
e remarkable: it is the smallest positive integer that can be represen
ted in two ways by the sum of two cubes." }{TEXT -1 2 " [" }{TEXT 311 
55 "Eric Weisstein's Treasure Trove of Scientific Biography" }{TEXT 
-1 1 "]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 310 
10 "Problem 1." }{TEXT -1 379 "  \n\n(a) Verify (using Maple) Ramanuja
n's statement that 1729 is the smallest positive integer that can be e
xpressed in two different ways as a sum of two cubes. You should be ab
le to use the ideas in the section above on perfect squares and sums o
f squares. Only here we are talking of cubes instead of squares.  But \+
there are other ways to do the problem that may occur to you. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "(b) Find \+
the next number after 1729 that is a sum of two cubes in two different
 ways." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 303 11 "Problem 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 323 11 "Definition." }{TEXT 344 25 " A positive integer n
 is " }{TEXT 322 7 "perfect" }{TEXT 345 49 " if the sum of its proper \+
positive divisors is n." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
287 "\nFor example, the proper positive divisors of 6 are 1, 2, and 3.
 Further 6 = 1 + 2 + 3. So 6 is perfect. [If you have had number theor
y you may know about the structure of evenperfect numbers. Do not use \+
that knowledge for  this exercise. Use only the definition to write yo
ur programs.]" }}{PARA 0 "" 0 "" {TEXT 321 6 " \n(a) " }{TEXT -1 108 "
Write a procedure which takes as input a positive integer n and return
s the sum of all proper divisors of n." }{TEXT 308 1 " " }{TEXT -1 81 
"Show how your procedure works for the following values of n: 2, 3, 4,
 6, 30, 900." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
302 4 "(b) " }{TEXT 312 34 "Using  the procedure you wrote in " }
{TEXT 346 3 "(a)" }{TEXT 347 1 " " }{TEXT -1 18 "write a procedure " }
{TEXT 348 9 "isperfect" }{TEXT -1 55 " which takes as input a positive
 integer n and returns " }{TEXT 306 4 "true" }{TEXT -1 21 " if n is pe
rfect and " }{TEXT 307 5 "false" }{TEXT -1 9 " if not. " }{TEXT 304 4 
"Note" }{TEXT -1 53 ": proper divisors of n can  not be greater than n
/2. " }{TEXT 324 46 "Use the procedure to find all perfect numbers " }
{XPPEDIT 18 0 "n <= 1000;" "6#1%\"nG\"%+5" }{TEXT 325 2 ". " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 17 "I
nteresting Fact:" }{TEXT 349 293 " It is an open question whether or n
ot there are any odd perfect numbers. So far no one has found one. In \+
number theory you learn the structure of the even perfect numbers. So \+
far only a finite number of even perfect numbers have been found. But \+
it is suspected that there are infinitely many." }{TEXT -1 1 " " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 313 11 "Problem 3 ." }
{TEXT -1 4 "  \n\n" }{TEXT 350 10 "Definition" }{TEXT -1 6 ". The " }
{TEXT 316 5 "order" }{TEXT -1 1 " " }{TEXT 327 13 "of an integer" }
{TEXT -1 3 " a " }{TEXT 326 17 "modulo an integer" }{TEXT -1 59 " m is
 defined to be the least positive integer e such that " }{XPPEDIT 18 
0 "`mod`(a^e,m) = 1;" "6#/-%$modG6$)%\"aG%\"eG%\"mG\"\"\"" }{TEXT -1 
32 ".  \n\nWrite a procedure, call it " }{TEXT 314 3 "Ord" }{TEXT -1 
12 ", such that " }{TEXT 328 8 "Ord(a,m)" }{TEXT -1 9 " returns " }
{TEXT 367 4 "FAIL" }{TEXT -1 46 " if the order does not exist and the \+
order of " }{TEXT 443 10 "a modulo m" }{TEXT -1 18 " if  it  exists.\n
\n" }{TEXT 351 91 "You may use the number theoretic result that the or
der of a modulo m exists if and only if " }{TEXT 329 13 "igcd(a,m) = 1
" }{TEXT 352 46 ". Also you may use the fact that the order of " }
{TEXT 368 10 "a modulo m" }{TEXT 369 23 " is never greater than " }
{TEXT 370 5 "m - 1" }{TEXT 371 24 ". [Note that Maple uses " }{TEXT 
330 4 "igcd" }{TEXT 353 49 " for the greatest common divisor of intege
rs and " }{TEXT 331 3 "gcd" }{TEXT 354 46 " for greatest common diviso
r of polynomials.] " }{TEXT -1 29 "\n\n(a) Check that you obtain: " }
{TEXT 315 65 "\n\n        Ord(0,11) = FAIL,    Ord(2,11) = 10,     Ord
(4,11) = 5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 12 "(b) Compute " }{TEXT 442 10 "Ord(a, 15)" }{TEXT -1 20 " for a f
rom 0 to 20." }}{PARA 0 "" 0 "" {TEXT 441 1 "\n" }}{PARA 0 "" 0 "" 
{TEXT 332 11 "Problem 4. " }{TEXT -1 15 "For each prime " }{TEXT 357 
1 "p" }{TEXT -1 50 " from 2 to 100 find the smallest positive integer \+
" }{TEXT 356 1 "a" }{TEXT -1 11 " such that " }{TEXT 355 14 "Ord(a,p) \+
= p-1" }{TEXT -1 25 ". Print out a table with " }{TEXT 358 4 "p, a" }
{TEXT -1 39 " on each row. \n\n[Such an a is called a " }{TEXT 317 27 
"primitive elements modulo p" }{TEXT -1 47 " and plays an important ro
le in number theory.]" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT 318 5 "Hint:" }{TEXT -1 82 " You might use a loop for p fro
m 2 to 100. If p is not prime then use the command " }{TEXT 319 5 "nex
t;" }{TEXT -1 195 " to go to the next value of p. If p is prime then g
o into another loop for a from 1 to p - 1 looking for an a which has o
rder 1 modulo p. When you find one, print it out and then use the comm
and " }{TEXT 320 6 "break;" }{TEXT -1 19 " to leave the loop." }}}
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0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }