259 Lecture 25: Simple Programming

Transcription

1 259 Lecture 25: Simple Programming In[1]:= In[2]:= Off General::spell Off General::spell1 Note: To type a command in a Mathematica notebook, use the mouse to move the cursor until it is horizontal, click the left mouse button, and type the command. To enter a command, use the mouse to move the cursor until it is vertical and over the command line, then click the left mouse button, and finally press and hold the Shift key followed by Enter key or use the Enter key on the smaller number keypad (lower right-hand corner of keyboard). To expand cells, double click on the cell bracket (blue vertical bar to the right) with the downward pointing arrow. Def: A computer program or program is a sequence of instructions that a computer can interpret and execute. (from dictionary.com) Def: A computer program is one or more instructions that are intended for execution by a computer. Without programs, computers would not run. Moreover, a computer program does nothing unless its instructions are executed by a central processor. Computer programs are the result of the compilation or interpretation of programming languages, are embedded into hardware, or may be manually inputted to the central processor of a computer. (from Wikipedia) Mathematica can be used to write computer programs when more than single step or an iterative process is required for some application or problem that needs to be solved. In addition to the logical operators AND, OR, NOT, etc, there are several built-in functions similar to the IF command in Excel that can be used to write computer programs! We will look at these logical operators, as well as WHICH, SWITCH, NEST, PRINT, DO, FOR, and WHILE, and finish up with a simple program for Newton's Method. Logical Operators (And, Or, Not) And e 1 && e 2 && is the logical AND function. It evaluates its arguments in order, giving False immediately if any of them are False, and True, if they are all True. And[e 1, e 2, ] can be input in StandardForm and InputForm as e 1 e 2.

2 2 259Lecture25sp17.nb The character can be entered as Ç&&, Çand or [And]. Example 1 In[3]:= 2 3 4&&4 2 Out[3]= False In[4]:= Out[4]= False In[5]:= Out[5]= And 2 3 5, 4 4 True In[6]:= Out[6]= a True, False True, False In[7]:= TableForm Table And a i, a j, i, 1, 2, j, 1, 2, TableHeadings a, a Out[7]//TableForm= True False True True False False False False Or e 1 e 2 is the logical OR function. It evaluates its arguments in order, giving True immediately if any of them are True, and False if they are all False. Or[e 1, e 2, ] can be input in StandardForm and InputForm as e 1 e 2. The character can be entered as Ç, Çor or [Or]. Example 2 In[8]:= Out[8]= True In[9]:= Out[9]= False In[10]:= Out[10]= Or 1 Sin 2, 3 True In[11]:= Out[11]= a True, False True, False

3 259Lecture25sp17.nb 3 In[12]:= Out[12]//TableForm= TableForm Table Or a i, a j, i, 1, 2, j, 1, 2, TableHeadings a, a True False True True True False True False Not!expr is the logical NOT function. It gives False if expr is True, and True if it is False. Not[expr] can be input in StandardForm and InputForm as Ÿexpr. The character Ÿ can be entered as Ç!, Çnot or [Not]. In[13]:= 3 3 Example 3 Out[13]= True In[14]:= Out[14]= 3 3 False In[15]:= Out[15]= Not 3 3 False In[16]:= Out[16]= Not 2 3 True In[17]:= Out[17]= True False In[18]:= Out[18]= 7 4 False In[19]:= Out[19]= Not Not True True In[20]:= Out[20]= True False False In[21]:= Out[21]= True In[22]:= Out[22]= True False False

4 4 259Lecture25sp17.nb In[23]:= Out[23]= a True, False True, False In[24]:= Out[24]= Table a i, i, 1, 2 False, True More Logical Operators (Xor, Nor, Nand) Xor Xor[e 1, e 2, ] is the logical XOR (exclusive OR) function. It gives True if an odd number of the e i are True, and the rest are False. It gives False if an even number of the e i are True, and the rest are False. Xor[e 1, e 2, ] can be input in StandardForm and InputForm as e 1 ƒ e 2 ƒ. The character ƒ can be entered as Çxor or [Xor]. Example 4 In[25]:= Out[25]= Xor True, 2 4, False True In[26]:= Out[26]= Xor True, 2 4, False, True False In[27]:= Out[27]= a True, False True, False In[28]:= TableForm Table Xor a i, a j, i, 1, 2, j, 1, 2, TableHeadings a, a Out[28]//TableForm= True False True False True False True False Nor Nor[e 1, e 2, ] is the logical NOR function. It evaluates its arguments in order, giving False immediately if any of them are True, and True if they are all False. Nor[e 1, e 2, ] can be input in StandardForm and InputForm as e 1 e 2. The character can be entered as Çnor or [Nor].

5 259Lecture25sp17.nb 5 Nor[e 1, e 2, ] is equivalent to Not[Or[e 1, e 2, ]]. Example 5 In[29]:= Out[29]= a True, False True, False In[30]:= TableForm Table Nor a i, a j, i, 1, 2, j, 1, 2, TableHeadings a, a Out[30]//TableForm= True False True False False False False True In[31]:= TableForm Table Or a i, a j, i, 1, 2, j, 1, 2, TableHeadings a, a Out[31]//TableForm= True False True False False False False True Nand Nand[e 1, e 2, ] is the logical NAND function. It evaluates its arguments in order, giving True immediately if any of them are False, and False if they are all True. Nand[e 1, e 2, ] can be input in StandardForm and InputForm as e 1 e 2. The character can be entered as Çnand or [Nand]. Nand[e 1, e 2, ] is equivalent to Not[And[e 1, e 2, ]]. Example 6 In[32]:= Out[32]= a True, False True, False In[33]:= TableForm Table Nand a i, a j, i, 1, 2, j, 1, 2, TableHeadings a, a Out[33]//TableForm= True False True False True False True True In[34]:= TableForm Table And a i, a j, i, 1, 2, j, 1, 2, TableHeadings a, a Out[34]//TableForm= True False True False True False True True

6 6 259Lecture25sp17.nb If If[condition, t, f] gives t if condition evaluates to True, and f if it evaluates to False. If[condition, t, f, u] gives u if condition evaluates to neither True nor False. If evaluates only the argument determined by the value of the condition. If[condition, t, f] is left unevaluated if condition evaluates to neither True nor False. If[condition, t] gives Null if condition evaluates to False. Example 7 If Syntax: In[35]:= If 3 3,, 0,7 Out[35]= 0 In[36]:= If 3 3 True,, 0,7 Out[36]= In[37]:= If x,, 0,7 Out[37]= 7 A piecewise-defined function that can be differentiated! In[38]:= Clear f In[39]:= f x_ : x ; x 0 f x_ : x ; x 0 In[41]:= Plot f x, x, 2, Out[41]=

7 259Lecture25sp17.nb 7 In[42]:= Plot f x, x, 2, Out[42]= In[43]:= Out[43]= f' x f x In[44]:= In[45]:= In[46]:= Clear f f x_ : If x 0, x, x Plot f x, x, 2, Out[46]=

8 8 259Lecture25sp17.nb In[47]:= Plot f x, x, 2, Out[47]= In[48]:= Out[48]= f' x If x 0, 1, 1 In[49]:= Out[49]= f'' x If x 0, 0, 0 Which Which[test 1, value 1, test 2, value 2, ] evaluates each of the test i in turn, returning the value of the value i corresponding to the first one that yields True. If any of the test i evaluated by Which give neither True nor False, then a Which object containing these remaining elements is returned unevaluated. You can make Which return a default value by taking the last test i to be True. Example 8 Which can be used to construct piecewise-defined functions! In[50]:= g x_ : Which x 2, 2 x, 2 x 0, x 2,x 0, Cos x

9 259Lecture25sp17.nb 9 In[51]:= Plot g x, x, 3, Out[51]= In[52]:= Plot g x, x, 3, Out[52]= In[53]:= g' x Out[53]= Which x 2, 1, 2 x 0, Here is the Heaviside Function: 1 2 x 2,x 0, Sin x In[54]:= H x_ : Which x 0, 0, x 0, 1

10 10 259Lecture25sp17.nb In[55]:= Plot H x, x, 1, Out[55]= In[56]:= Out[56]= Which True, 45, y, 23 In[57]:= Out[57]= Which False, 45, y, 23 Which y, 23 In[58]:= Out[58]= Which y, 23, False, 45, True, 87 Which y, 23, False, 45, True, 87 In[59]:= Out[59]= 87 Which True, 87, y, 23, False, 45 Switch Switch[expr, form 1, value 1, form 2, value 2, ] evaluates expr, then compares it with each of the form i in turn, evaluating and returning the value i corresponding to the first match found. Only the value i corresponding to the first form i that matches expr is evaluated. Each form i is evaluated only when the match is tried. If the last form i is the pattern _, then the corresponding value i is always returned if this case is reached. If none of the form i match expr, the Switch is returned unevaluated. Example 9 This use of Switch defines a function that depends on the values of its argument modulo 3. In[60]:= r x_ : Switch Mod x, 3, 0, a, 1, b, 2, c, _, "Input is not an integer "

11 259Lecture25sp17.nb 11 In[61]:= Out[61]= r 1 b In[62]:= Out[62]= r 11 c In[63]:= Out[63]= r 234 True, False In[64]:=?a Global`a a True, False In[65]:= In[66]:= Out[66]= Clear a r 234 a In[67]:= Out[67]= r 2.3 Input is not an integer In[68]:= Out[68]= r à Input is not an integer In[69]:= Out[69]= r à à a Nest Nest[f, expr, n] gives an expression with f applied n times to expr. Example 10 The Nest command can be used to find closed-form solutions of recurrence relations! a. x n = 1 + r x n - 1 In[70]:= In[71]:= Clear f f x_ : 1 r x

12 12 259Lecture25sp17.nb In[72]:= Out[72]//TableForm= TableForm Table n, Simplify Nest f, x 0,n, n, 0, 10, TableHeadings None, "n", "x n " n x n 0 x r x r 2 x r 3 x r 4 x r 5 x r 6 x r 7 x r 8 x r 9 x r 10 x 0 In[73]:= Clear x b. x n =ax n - 1 +b In[74]:= In[75]:= g x_ : x TableForm Table n, Simplify Nest g, x 0,n, n, 0, 10, TableHeadings None, "n", "x n " Out[75]//TableForm= n x n 0 x 0 1 x 0 2 x 0 3 x x x x x x x x 0 Print Print[expr 1, expr 2, ] prints the expr i concatenated together. Example 11 Try each command!

13 259Lecture25sp17.nb 13 In[76]:= Print "Hello World " Hello World In[77]:= p 4 Out[77]= 4 In[78]:= Print "p is equal to ", p p is equal to 4 In[79]:= Print "p 2 is equal to ", p 2, "." p 2 is equal to 16. In[80]:= Print "Part of this sentence \n", "is printed on a new line " Part of this sentence is printed on a new line In[81]:= Print "The graph of y sin x on the interval 0, looks like this:\n", Plot Sin x, x, 0,, "\nwhat would the graph look like on,?" The graph of y sin x on the interval 0, looks like this: What would the graph look like on,? Do Do[expr, {i max }] evaluates expr i max times. Do[expr, {i, i max }] evaluates expr with the variable i successively taking on the values 1 through i max (in steps of 1). Do[expr, {i, i min, i max }] starts with i = i min Do[expr, {i, i min, i max, di}] uses steps di. Do[expr, {i, i min, i max }, {j, j min, j max }, ] evaluates expr looping over different values of j, etc. for each i. (As with Table, the outermost iterator comes first...) Unless an explicit Return is used, the value returned by Do is Null. Null is a symbol used to indicate the absence of an expression or a result. When it appears as an output expression, no output is printed.

14 14 259Lecture25sp17.nb Example 12 Here is a way to add the sum of the squares of the first 10 positive integers! In[82]:= sum 0; In[83]:= Do sum sum i^2, i, 10 In[84]:= sum Out[84]= 385 In[85]:= In[86]:=?i Clear sum Note that in a Do loop, the counter i is a local variable. The variable "sum" was defined externally, so it is global. In[87]:=? sum Global`sum Check with the known formula : In[88]:= Out[88]= Sum i 2, i, 1, n 1 n 1 n 1 2n 6 In[89]:= Out[89]= This loop can be simplified by replacing the =sum+ with +=: In[90]:= sum 0; In[91]:= Do sum i^2, i, 1000 In[92]:= sum Out[92]= In[93]:= Clear sum We can print out the intermediate sums with Print: In[94]:= sum 0; In[95]:= Do Print "The sum of the squares of the first ", i, " positive integers is ", sum i^2, i, 10

15 259Lecture25sp17.nb 15 The sum of the squares of the first 1 positive integers is 1 The sum of the squares of the first 2 positive integers is 5 The sum of the squares of the first 3 positive integers is 14 The sum of the squares of the first 4 positive integers is 30 The sum of the squares of the first 5 positive integers is 55 The sum of the squares of the first 6 positive integers is 91 The sum of the squares of the first 7 positive integers is 140 The sum of the squares of the first 8 positive integers is 204 The sum of the squares of the first 9 positive integers is 285 The sum of the squares of the first 10 positive integers is 385 How could we "fix" this Print command so that it uses "correct" grammar? One possible solution In[96]:= sum 0; In[97]:= Do If i 1, Print "The sum of the squares of the first ", i, " positive integer is ", sum i^2, Print "The sum of the squares of the first ", i, " positive integers is ", sum i^2, i, 10 The sum of the squares of the first 1 positive integer is 1 The sum of the squares of the first 2 positive integers is 5 The sum of the squares of the first 3 positive integers is 14 The sum of the squares of the first 4 positive integers is 30 The sum of the squares of the first 5 positive integers is 55 The sum of the squares of the first 6 positive integers is 91 The sum of the squares of the first 7 positive integers is 140 The sum of the squares of the first 8 positive integers is 204 The sum of the squares of the first 9 positive integers is 285 The sum of the squares of the first 10 positive integers is 385 In[98]:= Clear sum For For[start, test, incr, body] executes start, then repeatedly evaluates body and incr until test fails to give True. For[start, test, incr] does the loop with a null body.

16 16 259Lecture25sp17.nb The sequence of evaluation is test, body, incr. The For loop exits as soon as test fails. If Break[ ] is generated in the evaluation of body, the For loop exits. Continue[ ] exits the evaluation of body, and continues the loop by evaluating incr. Unless an explicit Return is used, the value returned by For is Null. Example 13 Try each command! In[99]:= In[100]:= For i 0, i 5, i, i^2 Print Null Unless otherwise specified, For returns Null. Null is a symbol used to indicate the absence of an expression or a result. When it appears as an output expression, no output is printed. One way to get output from this command is to use the Print function inside the For loop! In[101]:= For i 0, i 5, i, Print i^ In[102]:=?i Note that unlike the Do command, the counter variables are global so they actually change and are stored in memory! i 6 One way to get rid of this "problem" is to use the Module command, which specifies which variables are local, i.e. not saved in memory as a global variable! The syntax for Module is: Module[{x, y, }, expr]. In this case, occurrences of the symbols x, y, in expr are treated as local. Module[{x = x 0, }, expr] defines initial values for local variables x,. In[103]:= Module i, For i 0, i 15, i, Print "When i ", i, ", i 2 ", i^2, "."

17 259Lecture25sp17.nb 17 In[104]:=?i When i 0, i 2 0. When i 1, i 2 1. When i 2, i 2 4. When i 3, i 2 9. When i 4, i When i 5, i When i 6, i When i 7, i When i 8, i When i 9, i When i 10, i When i 11, i When i 12, i When i 13, i When i 14, i When i 15, i i 6 In[105]:= Clear i In[106]:= L 4; For i 0, i L, i, Print "Hello World, number ", i Hello World, number 0 Hello World, number 1 Hello World, number 2 Hello World, number 3 Hello World, number 4 In[108]:=?i i 5 In[109]:=?L Global`L L 4

18 18 259Lecture25sp17.nb In[110]:= Clear L, i In[111]:= Module L, L 4; For i 0, i L, i, Print "Hello World, number ", i Hello World, number 0 Hello World, number 1 Hello World, number 2 Hello World, number 3 Hello World, number 4 In[112]:=?L Global`L In[113]:=?i i 5 In[114]:= Clear L, i In[115]:= Module L, i, L 4; For i 0, i L, i, Print "Hello World, number ", i Hello World, number 0 Hello World, number 1 Hello World, number 2 Hello World, number 3 Hello World, number 4 In[116]:=?L Global`L In[117]:=?i Anything that results in a numerical value can be used in the counter! Also note that placement of the iterator can affect the For loop's output! In[118]:= For i 0, i 5, i, Print "Hello World, number ", i

19 259Lecture25sp17.nb 19 Hello World, number 0 Hello World, number 1 Hello World, number 2 Hello World, number 3 Hello World, number 4 Hello World, number 5 In[119]:=?i i 6 In[120]:= For i 2, i 5, i, Print "Hello World, number ", i Hello World, number 2 Hello World, number 1 Hello World, number 0 Hello World, number 1 Hello World, number 2 Hello World, number 3 Hello World, number 4 Hello World, number 5 In[121]:=?i i 6 In[122]:= For i 0.1, i 5, i, Print "Hello World, number ", i Hello World, number 0.1 Hello World, number 1.1 Hello World, number 2.1 Hello World, number 3.1 Hello World, number 4.1 In[123]:=?i i 5.1 In[124]:= For i,i 5, i, Print "Hello World, number ", i Hello World, number Hello World, number 1

20 20 259Lecture25sp17.nb In[125]:=?i i 2 In[126]:= For i 2, i 5, Print "Hello World, number ", i, i Hello World, number 1 Hello World, number 0 Hello World, number 1 Hello World, number 2 Hello World, number 3 Hello World, number 4 Hello World, number 5 Hello World, number 6 In[127]:=?i i 6 The Continue command can be used to bypass calculation for certain values of the counter in a For loop! In[128]:= For i 2, i 5, i, If i 3 i 5, Continue, Print "Hello World, number ", i Hello World, number 2 Hello World, number 1 Hello World, number 0 Hello World, number 1 Hello World, number 2 Hello World, number 4 In[129]:=?i i 6 In[130]:= Clear i While While[test, body] evaluates test, then body, repetitively, until test first fails to give True.

21 259Lecture25sp17.nb 21 While[test] does the loop with a null body. If Break[ ] is generated in the evaluation of body, the While loop exits. Continue[ ] exits the evaluation of body, and continues the loop. Unless an explicit Return is used, the value returned by While is Null. Example 14 Use While to simulate flipping a pair of coins and counting the number of heads in each toss. In[131]:= tosses 20; i 0; While i i 1 tosses, Print "Number of Heads in Toss ", i, " is ", RandomInteger 0, 2 Number of Heads in Toss 1 is 1 Number of Heads in Toss 2 is 1 Number of Heads in Toss 3 is 0 Number of Heads in Toss 4 is 1 Number of Heads in Toss 5 is 2 Number of Heads in Toss 6 is 2 Number of Heads in Toss 7 is 2 Number of Heads in Toss 8 is 2 Number of Heads in Toss 9 is 0 Number of Heads in Toss 10 is 0 Number of Heads in Toss 11 is 0 Number of Heads in Toss 12 is 2 Number of Heads in Toss 13 is 0 Number of Heads in Toss 14 is 0 Number of Heads in Toss 15 is 2 Number of Heads in Toss 16 is 1 Number of Heads in Toss 17 is 2 Number of Heads in Toss 18 is 0 Number of Heads in Toss 19 is 0 Number of Heads in Toss 20 is 1 In[134]:=?i

22 22 259Lecture25sp17.nb i 21 In[135]:= In[137]:= Example 15 Use While to randomly choose an integer from 0 to 10 until the number 2 is chosen. 0; While 2, RandomInteger 0, 10 ; Print Note that the loop can run for a very long time, as there is no preset bound on how many times the loop will execute. We can set an upper limit on the number of iterations of the For loop by including an And within the statement! 0; i 0; While 2&&i 20, RandomInteger 0, 10 ; Print ; i

23 259Lecture25sp17.nb In[140]:=?i i 17 Newton's Method Example In[141]:= Clear f, g, x In[142]:= f x_ : x 3 2x 5

24 24 259Lecture25sp17.nb In[143]:= Plot f x, x, 5, 5, PlotRange All Out[143]= In[144]:= g x_ : x f x f' x In[145]:= Out[145]= g x 5 2x x3 x 2 3x 2 In[146]:= Newton's Method "by hand"! Choose x = 0 as the initial guess for the root of g(x) = 0. Run ten iterations by putting the output back into g(x) to get the next guess! N g 0 Out[146]= 2.5 In[147]:= N g Out[147]= In[148]:= N g Out[148]= In[149]:= N g Out[149]= In[150]:= N g Out[150]= In[151]:= N g Out[151]= In[152]:= N g Out[152]= In[153]:= N g Out[153]=

25 259Lecture25sp17.nb 25 In[154]:= N g Out[154]= In[155]:= N g Out[155]= Version 1 In[156]:= x 0; Do x N g x, n, 1, 10 ; x Out[157]= Version 2 In[158]:= x 0; Do Print "n ", n, " x ", x ; x N g x, n, 0, 10 n 0 x 0 n 1 x 2.5 n 2 x n 3 x n 4 x n 5 x n 6 x n 7 x n 8 x n 9 x n 10 x Version 3 In[160]:= Clear x, y In[161]:= tolerence 0.001; In[162]:= For x 0; y 0; n 0, n 50, n, Print "n ", n, " x ", x, " and change from last guess is ", x y ; y x; x N g x ; If Abs x y tolerence, Print "n ", n 1, " x ", x, " change from last guess is ", x y ; Break

26 26 259Lecture25sp17.nb n 0 x 0 and change from last guess is 0 n 1 x 2.5 and change from last guess is 2.5 n 2 x and change from last guess is n 3 x and change from last guess is n 4 x and change from last guess is n 5 x and change from last guess is n 6 x and change from last guess is n 7 x and change from last guess is n 8 x and change from last guess is n 9 x and change from last guess is n 10 x and change from last guess is n 11 x and change from last guess is n 12 x and change from last guess is n 13 x and change from last guess is n 14 x and change from last guess is n 15 x and change from last guess is n 16 x and change from last guess is n 17 x and change from last guess is n 18 x change from last guess is Notice that within the command are a way to stop calculation after at most 50 steps (the "n<50") or stop if the tolerence is achieved (the Break[] within the If statement). Here's a version of the previous command that doesn't print out the intermediate steps! In[163]:= For x 0; n 0, n 50, n, y x; x N g x ; If Abs x y tolerence, Print "After ", n 1, " steps, x ", x, ", with absolute error ", Abs x y, "." ; Break ; After 18 steps, x , with absolute error Note: Portions of this lecture are based on the book Mathematica by S. Wolfram. The rest was created by M. A. Karls in Fall 2007 and revised in Fall 2008.