Iteration. The final line is simply ending the do word in the first line. You can put as many lines as you like in between the do and the end do

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Herbert Pierce
6 years ago
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1 Intruction Iteration Iteration: the for loop Computers are superb at repeating a set of instructions over and over again - very quickly and with complete reliability. Here is a short maple program ce that follows shows how to do that with a very simple example. restart : for i from 1 to 8 by 2 do print "here's ", i, "squared - in various forms: ", i $i, i $ i $ 1.0, evalf i $i, i 2, i 2.0 ; end do "here's ", 1, "squared - in various forms: ", 1, 1.0, 1., 1, 1. "here's ", 3, "squared - in various forms: ", 9, 9.0, 9., 9, 9. "here's ", 5, "squared - in various forms: ", 25, 25.0, 25., 25, 25. "here's ", 7, "squared - in various forms: ", 49, 49.0, 49., 49, 49. (2.1) The first line of the ce is saying:run the next lines again and again for i going up from 1 to 8 in steps of 2 The second line is printing i squared using various meths. See if you can figure out why the answers are the way they are; some with decimal points - some without. If you don't know what evalf means then type?evalf and press the return key - and you'll get some help; it means evaluate using floating point arithmetic [ not exact arithmetic ]. The quotation marks in the output are a bit ugly; they can be got rid of but that's not important here. The final line is simply ending the do word in the first line. You can put as many lines as you like in between the do and the end do You should now try your own for loop by mifying this one. Note that at the end of the first and second lines in the ce above you should press shift return rather than just return. Return forces a calculation, shift return gives you a new line. Note there is no semi-colon at the end of the for line. Iteration : the while loop i d 1 : while i 2 % 64 do print "here's ", i, "squared - in various ways: ", i $i, i $ i $ 1.0, evalf i $ i, i 2, i 2.0 ; i d i C 2 : end do: "here's ", 1, "squared - in various ways: ", 1, 1.0, 1., 1, 1. 1

2 "here's ", 3, "squared - in various ways: ", 9, 9.0, 9., 9, 9. "here's ", 5, "squared - in various ways: ", 25, 25.0, 25., 25, 25. "here's ", 7, "squared - in various ways: ", 49, 49.0, 49., 49, 49. (3.1) This while loop does the same thing as the for loop above. It seems a little bit clumsier, but it is very powerful - especially when you don't know in advance just how many times you need to run through a loop: Note in this example the while condition is that i 2 % 64 - not the simpler previous i % 8. The while loop above can be read as follows: Take my counting variable i and set it equal to 1. Start a loop that will run until 64! i 2. Print out i squared. Increment my counting variable by 2. End the loop. Note the use of the colon and the semicolon. Using the colon at the end of the do loop I tried to suppress computer output - otherwise I'd keep getting the i:= i + 2 line giving me the latest value of i. Note also that in this latest version of Maple [ 13 ], the semi-colon is often unnecessary. Space is too short to give examples here of when it is and is not necessary but you'll pick it up very quickly. Iteration: the do loop i d 1 : do if i = 250 then print "i should be 250, check here:", i ; break: end if: i d i C 1 : end do: "i should be 250, check here:", 250 (4.1) This do loop would run forever unless the loop at some stage reaches the break command. This type of loop can be very powerful indeed, although some purists might frown on its use! predicates: the if word... here is an example of the if word in Maple a d 7 a := 7 (5.1) if a! 7 then print " a is less than seven: ", a ; 2

3 if a O 7 then print " a is greater than seven: ", a ; if a = 7 then print " a equals seven: ", a ; print " a equals seven: ", 7 (5.2) if a R 7 then print " a is O= ", a ; print " a is O= ", 7 (5.3) if a % 7 then print " a is!= ", a ; print " a is!= ", 7 (5.4) The if statement can be enhanced a bit using the else word as follows: if a! 7 then print " a is! seven: ", a else print " a is O= 7 ", a ; print " a is O= 7 ", 7 (5.5) note that we can use the "not equal to" construction in maple as follows: if not a = 8 then print " a is not equal to eight: ", a else print " a is equal to 8 ", a ; print " a is not equal to eight: ", 7 (5.6) Note the kind-of reverse logic on the first line above. Using the palettes on the left you can find slightly nicer ways to write this: if a s 8 then print " a is not equal to eight: ", a else print " a is equal to 8 ", a ; print " a is not equal to eight: ", 7 (5.7) 3

4 You can think of else as meaning the same thing as "otherwise". Let's try something slightly more ambitious using for loops and while loops and if. Using a for loop let's find the smallest positive integer n such that 1000! n C 1 3 K n 3 restart : for n from 1 to 20 do if n C 1 3 K n 3 O 1000 then print n end if end do (5.8) Remember that you use the cursor keys to, for example, get down from the power of 3 to the base line before you type the subtraction symbol. Note also the complete departure from old versions of Maple; in this example there are no semi colons or colons. Remember to get to the "next lines" in this example you press shift-enter. Note that the ce can also be written all on one line, although its hard to see any benefit from doing so apart from saving paper maybe. for n to 20 do if n C 1 ^3Kn^3 O 1000 then print n end if end do Note in the next example the cryptic but cute shorthand where "fi" is like a bookend to the intructory "if"; the suggestion being "fi" is shorthand for "end if". Similarly with "" - its the mirror-opposite of "do" so to speak, and so ends a "do" statement, replacing "end do". (5.9) for n from 1 to 20 do if n C 1 3 K n 3 O 1000 then print n fi (5.10) Note above the attempt at indentation of ce. Not important in such a short piece but for long complex ce you will come to bless the facility to indent ce. Its makes it easier to read. The "if" 4

5 section is kind-of contained within the "for" section. And the the "print" section contained within the "if" section. for n from 1 to 20 do if n C 1 3 K n 3 O 1000 then print n fi (5.11) 17 C 1 3 K C 1 3 K (5.12) (5.13) Our ce above works okay but isn't very go. First: how was I to know that the answer was about 18 or 19 and so only needed to do my loop that many times? The while loop will do something similar: n d 1 n := 1 (5.14) Note that we need to "start n off" with a value here. We call this "initialising". while n C 1 3 K n 3! 1000 do n d n C 1; end do; n := 2 n := 3 n := 4 n := 5 n := 6 n := 7 n := 8 n := 9 n := 10 n := 11 n := 12 5

6 n := 13 n := 14 n := 15 n := 16 n := 17 n := C 1 3 K (5.15) (5.16) You can use the sometimes invaluable break word in Maple to tidy up the for loop above: restart : for n from 1 to 20 do if n C 1 3 K n 3 O 1000 then print n, n C 1 3 K n 3 ; break; fi; ; 18, 1027 (5.17) note again the use of the shorthand above for end if [ fi ] and end do [ ] The while loop can be tidied by suppressing output until the end: n d 1 : while n C 1 3 K n 3! 1000 do n d n C 1; : n; 18 (5.18) Summing using a loop total d 0 : Note the variable above is initialised. for i from 1 to 10 do total d total C i total := 1 total := 3 total := 6 total := 10 total := 15 6

7 total := 21 total := 28 total := 36 total := 45 total := 55 (6.1) Make sure you can understand the above program and output. Next, examine this ce and output: total d 1 : Note it wouldn't make sense to initialise this total variable to zero for i from 1 to 10 do total d total i total := 1 total := 2 total := 6 total := 24 total := 120 total := 720 total := 5040 total := total := total := total d 1 : Note: in the ce above can also be written:- for i from 1 to 10 do total d total $i # where the "$" is the obtained from typing the asterisk symbol. total := 1 total := 2 total := 6 total := 24 total := 120 total := 720 total := 5040 total := total := (6.2) 7

8 total := (6.3) Can you see what the above program is doing? If you need a clue - see the program below now: for i from 1 to 10 do i! (6.4) You can run a loop that stops when two values get sufficiently close [ converge ] to each other. Here is an example: restart : m d 1 : i d 2 : do n d 1 / i; if abs mkn! then print "okay - stopping now, n=", n, "and m = ", m, "; m-n=", evalf nkm, "and number of iterations =", i ; break: fi: m d n : i d i C 1 : : "okay - stopping now, n=", "and number of iterations =", , "and m = ", 1, "; m-n=", K , 100 (6.5) The above loop demonstrates several things: 1. Convergence: an inverse is being calculated and compared in size to the previous inverse calculation. When the two are sufficiently close the values are printed out and the calculation ends. 2. The open do loop and the use of break. 3. The use of abs to give the absolute value. Was abs really necessary? 4. Just before the end of the loop, m is updated to have the most recent value of n, just before n is recalculated at the start of the next loop. 5. Can you see why we initialised i as 2 rather than 1? Note that i is incremented near the end of the 8

9 loop. That's needed because we're not using a for loop. If we'd written "for i from 1 to 100 do", we would not have needed to manually increment i. What is the possible disadvantage in this example of using a simple for loop? Newton-Raphson iteration: find the root of a function First we demonstrate a special Maple meth for illustrating Newton-Raphson: restart : with Student Calculus1 : NewtonsMeth x 3 K 3 x 2 C x K 17, x = 20, output = sequence, iterations = 10 20, , , , , , , , , (7.1) Note: the successive values of x are printed above. NewtonsMeth x 3 K 3 x 2 C x K 17, x = 20, output = plot, showroot = true, iterations = 10 9

10 Newton's Meth x f x Tangent lines From the initial point x = 20, at most 9 iteration(s) of Newton's meth for f x = x 3 K 3 x 2 C x K 17 The above demonstrates Maple using its inbuilt Newton-Raphson meth. You can see the successive x values and the associated tangents to the curve - very rapidly converging on the root. 10

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Section 1: Numerical Calculations

Section 1: Numerical Calculations In this section you will use Maple to do some standard numerical calculations. Maple's ability to produce exact answers in addition to numerical approximations gives you