1 class Lecture4 { 2 3 "Loops" / References 8 [1] Ch. 5 in YDL 9 / Zheng-Liang Lu Java Programming 125 / 207

Transcription

1 1 class Lecture4 { 2 3 "Loops" 4 5 } 6 7 / References 8 [1] Ch. 5 in YDL 9 / Zheng-Liang Lu Java Programming 125 / 207

2 Loops A loop can be used to make a program execute statements repeatedly without having to code the same statements. For example, a program prints Hello, Java. for 100 times. 1 System.out.println("Hello, Java."); 2 System.out.println("Hello, Java."); // Copy and paste for 100 times 5. 6 System.out.println("Hello, Java."); Zheng-Liang Lu Java Programming 126 / 207

3 1 int cnt = 0; 2 while (cnt < 100) { 3 System.out.println("Hello, Java."); 4 cnt++; 5 } This is a simple example to show the power of loops. In practice, any routine which repeats couples of times 1 can be done by folding them into a loop. 1 I d like to call them patterns. Zheng-Liang Lu Java Programming 127 / 207

4 Flow Controls (Recap) The concept of looping, which is one of elements in algorithms, is fundamental to programming. Loops provide significant computation power. Loops bring an efficient way of programming. Loops could consume a lot of time. 2 2 We will visit the analysis of algorithm. Zheng-Liang Lu Java Programming 128 / 207

5 while Loops A while loop executes statements repeatedly while the condition is true. 1 while (loop continuation condition) { 2 statements; // loop body 3 } The loop-continuation-condition is a Boolean expression which controls the execution of the body. It is evaluated each time to determine if the loop body is executed. If true, the loop body is executed. Otherwise, the entire loop terminates. Zheng-Liang Lu Java Programming 129 / 207

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7 Example Write a program which sums up all integers from 1 to 100. You can choose to write a program like this: 1 int sum=0; 2 sum=sum+1; 3 sum=sum+2; sum=sum+100; Pros: Not efficient, not general (what if sum up to 10 10?) Zheng-Liang Lu Java Programming 131 / 207

8 Using a while loop, the program looks like this: int sum = 0; 3 int i = 0; 4 while (++i <= 100) 5 sum = sum + i; 6... Make sure that the loop-continuation-condition eventually becomes false so that the loop will terminate. It is really easy to make an infinite loop while(true); 3... Zheng-Liang Lu Java Programming 132 / 207

9 Besides, replacing 100 by n determined by the user makes this program more general Scanner input = new Scanner(System.in); 3 int n = input.nextint(); 4 int sum = 0; 5 int i = 0; 6 while (++i <= n) 7 sum = sum + i; 8... In practice, the number of loop steps is unknown until the input data is given. For example, the bisection algorithm provides a numerical solution to root-finding problems. Zheng-Liang Lu Java Programming 133 / 207

10 Example Write a program which sums two random integers and lets the user repeatedly enter a new answer until it is correct public static void main (String[] args) { 3 // random integer generation 4 int number1 = (int)(math.random() % 10); 5 int number2 = (int)(math.random() % 10); 6 7 Scanner input = new Scanner(Sysmte.in); 8 System.out.println(number1 + " + " + number2 + " =?"); 9 int ans = input.nextint(); while (number1 + number2!= ans) { 12 System.out.println("Wrong answer. Try again?"); 13 ans = input.nextint(); 14 } 15 System.out.println("Congrats! You are smart!"); 16 } Zheng-Liang Lu Java Programming 134 / 207

11 Loop Design Strategies Writing a correct loop is not an easy task for novice programmers. Consider 4 steps when writing a loop: Identify the statements that need to be repeated. Wrap these statements in a loop. Find an appropriate loop continuation condition 3. Add, if necessary, extra statements for controlling the loop 4. 3 Not unique. 4 Make sure that the condition can turn to false. Zheng-Liang Lu Java Programming 135 / 207

12 Sentinel-Controlled Loop Another common technique for controlling a loop is to designate a special value when reading and processing a set of values. This special input value, known as a sentinel value, signifies the end of the loop. Zheng-Liang Lu Java Programming 136 / 207

13 Example Write a program which sums real numbers from the input except for -1 to exit, and displays the sum Scanner in = new Scanner(System.in); 3 double sum = 0; 4 System.out.println("Please enter a real number ( 1 to exit): "); 5 double x = in.nextdouble(); 6 while(x!= 1) { 7 sum += x; 8 System.out.println("Please enter a real number ( 1 to exit): "); 9 x = in.nextdouble(); 10 } 11 System.out.println("Sum = " + sum); 12 in.close(); Try more features? Zheng-Liang Lu Java Programming 137 / 207

14 do-while Loops A do-while loop is the same as a while loop except that it does execute the loop body first and then checks the loop continuation condition do { 3 statements; 4 } while (loop continuation condition); 5 // Do not miss the semicolon! 6... Note that there is a semicolon at the end the do-while loop. do-while loops are also called posttest loop, in contrast to while loops, which are pretest loops. Zheng-Liang Lu Java Programming 138 / 207

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16 Example Write a program which allows the user to enter positive integers except for -1 to exit, and displays the maximum. Zheng-Liang Lu Java Programming 140 / 207

17 Scanner in = new Scanner(System.in); 3 int max = 0, x; 4 do{ 5 System.out.println("Please enter a positive integer ( 1 to exit): "); 6 x = in.nextint(); 7 if(max < x) { 8 max = x; 9 } 10 System.out.println("Max = " + max); 11 } while(x!= 1); 12 in.close(); Zheng-Liang Lu Java Programming 141 / 207

18 for Loops A for loop generally uses a variable to control how many times the loop body is executed and when the loop terminates for(init; condition; increment) { 3 statements; // loop body 4 } 5... init: initialize a variable condition: a criteria that a loop continues increment: how the variable changes after each iteration Note that the three terms are separated by a semicolon. Zheng-Liang Lu Java Programming 142 / 207

19 Example Sum from 1 to 100 Write a program which sums from 1 to int sum = 0; 3 for (int i = 1; i <= 100; ++i) 4 sum = sum + i; 5... Compared to the while version, int sum = 0; 3 int i = 0; 4 while (++i <= 100) 5 sum = sum + i; 6... Zheng-Liang Lu Java Programming 143 / 207

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21 Example: Selection Resided in Loop Display all even numbers Write a program which displays all even numbers smaller than 100. An even number is an integer of the form x = 2k, where k is an integer. You may use modular operator (%) for (int i = 1; i <= 100; i++) { 3 if (i % 2 == 0) 4 System.out.println(i); 5 } 6... How about odd numbers? Zheng-Liang Lu Java Programming 145 / 207

22 for Loops: Multiple Loop Variables The init in a for loop can be a list of zero or more commaseparated variable declaration statements or assignment expressions for (int i = 0, j = 0; (i + j < 10); i++, j+=2) { 3 System.out.println("(i,j) = " + "(" + i + ", " + j + ")" ); 4 } 5... The last line on display is? Zheng-Liang Lu Java Programming 146 / 207

23 Jump Statements Two keywords, break and continue, can be used in loop statements to provide additional controls. break: The loop is terminated right after a break statement is executed. continue: The loop skips this iteration right after a continue statement is executed. Normally, jump statements are used within selection structures in loops. Zheng-Liang Lu Java Programming 147 / 207

24 Example Check if the input number is a prime) Write a program which determine if the input integer is a prime. Recall that a prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Let x be any natural number. The most naive approach is to divide x by all natural numbers smaller than x. A better approach is to to divide x by all natural numbers smaller than x. (Why?) Zheng-Liang Lu Java Programming 148 / 207

25 Scanner in = new Scanner(System.in); 3 System.out.println("Please enter an integer: "); 4 int x = in.nextint(); 5 in.close(); 6 for (int i = 2; i <= (int) Math.sqrt(x); i++) { 7 if ((x % i) == 0) { 8 System.out.println(x + " is not a prime."); 9 break; 10 } 11 if (i == (int) Math.sqrt(x)) 12 System.out.println(x + " is a prime."); 13 } Zheng-Liang Lu Java Programming 149 / 207

26 Equivalence: while and for Loops Compounding problem Write a program which determines the number of years n if the initial amount of saving is 10, 000 NTD, then the compounding amount of saving exceeds 15, 000 NTD. Zheng-Liang Lu Java Programming 150 / 207

27 1 import java.util.scanner; 2 3 public class compoudingmain { 4 public static void main(string[] args) { 5 Scanner in = new Scanner(System.in); 6 System.out.println("r =?"); 7 double r = in.nextdouble(); 8 System.out.println("x =?"); 9 double x = in.nextdouble(); 10 System.out.println("y =?"); 11 double y = in.nextdouble(); 12 int n = 0; 13 while (x = (1 + r)) <= y) 14 n = n + 1; 15 System.out.println("n = " + n); 16 System.out.println("x = " + x); 17 in.close(); 18 } 19 } Note that the listing is a generalized version. Try to extend your work as general as possible. It could be hard, but worth. Zheng-Liang Lu Java Programming 151 / 207

28 int n; 3 for (n = 1; true; n++) { 4 x = (1 + r); 5 if (x >= y) 6 break; 7 } 8... A for loop can be an infinite loop by setting true or simply leaving empty in the condition in for. An infinite for loop with a break statement is equivalent to a normal while loop. Zheng-Liang Lu Java Programming 152 / 207

29 Equivalence: while and for Loops (Concluded) You can use a for loop, a while loop, or a do-while loop, whichever is convenient. In general, a for loop may be used if the number of repetitions is known in advance. If not, then a while loop is preferred. Zheng-Liang Lu Java Programming 153 / 207

30 Nested Loops A loop can be nested inside another loop. Nested loops consist of an outer loop and one or more inner loops. Each time the outer loop is repeated, the inner loops are reentered, and started anew. Zheng-Liang Lu Java Programming 154 / 207

31 Example Multiplication table Write a program which displays a multiplication table. Zheng-Liang Lu Java Programming 155 / 207

32 for (int j = 0; j <= 10; ++j) { 3 if (j == 0) 4 System.out.printf(" "); 5 else if (j!= 10) 6 System.out.printf("%3d", j); 7 else { 8 System.out.printf("\n"); 9 for (int k = 0; k <= j; ++k){ 10 System.out.printf(" "); 11 if (k == j) 12 System.out.printf("\n"); 13 } 14 } 15 } 16 for (int i = 1; i < 10; ++i){ 17 System.out.printf("%3d ", i); 18 for (int j = 1; j < 10; ++j) { 19 System.out.printf("%3d", i j); 20 if (j==9) 21 System.out.printf("\n"); 22 } 23 } Zheng-Liang Lu Java Programming 156 / 207

33 Exercise Zheng-Liang Lu Java Programming 157 / 207

34 1 // Case (a) 2 public class printstars { 3 public static void main(string[] args){ 4 for(int i = 1; i <= 5; i++) { 5 for(int j = 1; j <= i; j++){ 6 System.out.printf(" "); 7 if (j == i) 8 System.out.printf("\n"); 9 } 10 } 11 } 12 } Try on your own for (b), (c), and (d). Allow the user to decide the number of lines? Zheng-Liang Lu Java Programming 158 / 207

35 Problem Set Exercise 4.1 (Greatest common divisor) Write a program which receives two positive integers and returns the greatest common divisor which is the largest positive integer that divides the numbers without a remainder. Exercise 4.2 (Find all prime numbers smaller than 1000) Write a program which displays all prime numbers smaller than Exercise 4.3 (π estimated by Monte Carlo) Write a program which estimates π by Monte Carlo Simulation. Zheng-Liang Lu Java Programming 159 / 207

36 Exercise 4.4 (Find the two highest scores) Write a program that prompts the user to enter the number of students and each student s name and score, and finally displays the student with the highest score and the student with the second-highest score. Exercise 4.5 (Find numbers divisible by 5 and 6) Write a program that displays all the numbers from 100 to 1,000, ten per line, that are divisible by 5 and 6. Numbers are separated by exactly one space. Zheng-Liang Lu Java Programming 160 / 207

37 Exercise 4.6 (Continued from 4.6) Write a program that displays all the numbers from 100 to 200, ten per line, that are divisible by 5 or 6, but not both. Numbers are separated by exactly one space. Exercise 4.7 Write a program that finds the smallest n such that n 2 > using a while loop. Exercise 4.8 Write a program which finds the largest n such that n 3 > using a while loop. Zheng-Liang Lu Java Programming 161 / 207

38 Exercise 4.9 (Display pyramid) Write a program that prompts the user to enter an integer from 1 to 15 and displays a pyramid, as shown in the following sample run: Zheng-Liang Lu Java Programming 162 / 207

39 Exercise 4.10 (Display numbers in a pyramid pattern) Write a nested for loop that prints the following output: Zheng-Liang Lu Java Programming 163 / 207

40 Exercise 4.11 (Display four patterns using loops) Use nested loops that display the following patterns in four separate programs: Zheng-Liang Lu Java Programming 164 / 207

41 Exercise 4.12 (Display leap years) Write a program that displays all the leap years, ten per line, in the twenty-first century (from 2001 to 2100), separated by exactly one space. Exercise 4.13 (Compute π) You can approximate by using the following series: π = 4( ( 1)i+1 2i 1 ) Write a program that displays the value for i = 10000, 20000,..., and Zheng-Liang Lu Java Programming 165 / 207

42 Exercise 4.14 (Perfect number) A positive integer is called a perfect number if it is equal to the sum of all of its positive divisors, excluding itself. For example, 6 is the first perfect number because 6 = The next is 28 = There are four perfect numbers less than 10,000. Write a program to find all these four numbers. Exercise 4.15 (Game: scissor, rock, paper) Exercise 3.17 gives a program that plays the scissor-rockpaper game. Revise the program to let the user continuously play until either the user or the computer wins more than two times. Zheng-Liang Lu Java Programming 166 / 207

43 Exercise 4.16 Histogram for math grades Write a program which allows the user to enter the math grades in order (-1 to exit), and makes a histogram. Zheng-Liang Lu Java Programming 167 / 207

44 1 class Lecture5 { 2 3 "Methods" 4 5 } 6 7 / References 8 [1] Ch. 5 in YDL 9 [1] Ch. 20 in YDL 10 / Zheng-Liang Lu Java Programming 168 / 207

45 Methods 5 Methods can be used to define reusable code, and organize and simplify code. 5 Some programming languages refer to methods as procedures and functions. Zheng-Liang Lu Java Programming 169 / 207

46 1 modifier returntype methodname(list of Parameter) { 2 // method body 3 } So far, the modifier could be static, public, and final. returntype could be primitive data types, referential data types, and void. 6 List of Parameter refers to the list of parameters as input of the method. Parameters are optional; that is, a method doesn t have to contain any parameters. The method name and the parameter list together constitute the method signature. 6 A void method does not return a value. Zheng-Liang Lu Java Programming 170 / 207

47 Example Write a program which allows the user to enter the math grades in order (-1 to exit), and makes a histogram. How to count? Zheng-Liang Lu Java Programming 171 / 207

48 Scanner in = new Scanner(System.in); 3 int a, b, c, d, e, x; 4 a = b = c = d = e = 0; 5 do { 6 System.out.printf("Enter ( 1 to exit): "); 7 x = in.nextint(); 8 if (x >= 90) ++a; 9 else if (x >= 80) ++b; 10 else if (x >= 70) ++c; 11 else if (x >= 60) ++d; 12 else if (x >= 0) ++e; 13 else if (x == 1) System.out.println("End of Input..."); 14 else System.out.println("Invalid input."); 15 } while (x!= 1); 16 in.close(); System.out.printf("Total: %d\n", a + b + c + d + e); System.out.printf("%3d %3d: ", 100, 90); 21 for (int i = 0; i < a ; i++) 22 System.out.printf(" "); 23 System.out.printf("\n"); 24 Zheng-Liang Lu Java Programming 172 / 207

49 25 System.out.printf("%3d %3d: ", 89, 80); 26 for (int i = 1; i <= b; i++) 27 System.out.printf(" "); 28 System.out.printf("\n"); System.out.printf("%3d %3d: ", 79, 70); 31 for (int i = 1; i <= c; i++) 32 System.out.printf(" "); 33 System.out.printf("\n"); System.out.printf("%3d %3d: ", 69, 60); 36 for (int i = 1; i <= d; i++) 37 System.out.printf(" "); 38 System.out.printf("\n"); System.out.printf("%3d %3d: ", 59, 0); 41 for (int i = 1; i <= e; i++) 42 System.out.printf(" "); 43 System.out.printf("\n"); Line repeat the same pattern as Line Zheng-Liang Lu Java Programming 173 / 207

50 Recall that we may use a loop when a bundle of instructions are applied to different data. Organize the print() method which displays the stars given the interval (a, b] and the number of stars cnt static void print(int a, int b, int cnt) { 3 System.out.printf("%3d %3d: ", b, a); 4 for (int i = 1; i <= cnt; i++) 5 System.out.printf(" "); 6 System.out.printf("\n"); 7 } 8... In this case, print() does not return a value. Zheng-Liang Lu Java Programming 174 / 207

51 Then one can reorganize main() like this: public static void main(string[] args) { System.out.printf("Total: %d\n", a + b + c + d + e); 5 print(100, 90, a); 6 print(89, 80, b); 7 print(79, 70, c); 8 print(69, 60, d); 9 print(59, 0, e); 10 } Be aware that this is not unique. Zheng-Liang Lu Java Programming 175 / 207

52 Exercise Write a program which prints a certain number of star lines determined by the user repeatedly until the input is negative. Input: n 0 A while loop which makes the program alive until n < 0 Allow the user to choose a symbol in addition to *? Zheng-Liang Lu Java Programming 176 / 207

53 static void printstar(int x, String marker) { 3 for (int i = 1; i <= x; ++i){ 4 for (int j = 1; j <= i; ++j) 5 System.out.printf("%s", marker); 6 System.out.println(); 7 } 8 } Scanner in = new Scanner(System.in); 3 int n; 4 String marker; 5 while (true) { 6 System.out.println("Enter a positive integer? "); 7 n = in.nextint(); 8 if (n < 0) break; 9 System.out.println("Choose a symbol? "); 10 marker = in.next(); 11 printstar(n, marker); 12 } 13 in.close(); 14 System.out.println("Bye."); Zheng-Liang Lu Java Programming 177 / 207

54 Invoking A Method Invoking a method executes the code in the method. When a program 7 calls a method, program control is transferred to the called method. Each time a method is invoked, the system creates an activation frame that stores parameters and variables for the method and places it in the call stack. A called method returns control to the caller when its return 8 statement is executed or when its end brace is reached. 7 Aka a callee. 8 In addition to break and continue is used as the exit point of methods and programs. Zheng-Liang Lu Java Programming 178 / 207

55 Note that the parameters are new declared variables within the method. When calling a method, you need to provide arguments, which must match the parameters in order, number, and compatible type, as defined in the method signature. This mechanism is so-called pass-by-value. Zheng-Liang Lu Java Programming 179 / 207

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57 Scope of Variables The scope of a variable is the part of the program where the variable can be referenced. A variable defined within a method is referred to as a local variable. For example, a parameter is actually a local variable. The scope of a method parameter covers the entire method. Moreover, you can declare a local variable with the same name in different blocks in a method Yet, you cannot declare a local variable twice in the same block or in nested blocks. Zheng-Liang Lu Java Programming 181 / 207

58 Example 1 public class incrementmain { 2 3 static int i = 1; // a local variable within this class 4 5 public static void main(string[] args) { 6 System.out.printf("%d\n", i); 7 int i = 2; 8 i++; 9 System.out.printf("%d\n", i); 10 p(); 11 System.out.printf("%d\n", i); 12 } static void p() { 15 i = i + 1; 16 System.out.printf("%d\n", i); 17 } 18 } Zheng-Liang Lu Java Programming 182 / 207

59 Overloading Methods Overloading methods enables you to define the methods with the same name as long as their signatures are different. Overloading methods can make programs clearer and more readable. Note that overloaded methods must have different parameter lists. You cannot overload methods based on different modifiers or return types. Zheng-Liang Lu Java Programming 183 / 207

60 Exercise Upgraded star printing Write a program which prints a certain number of lines with some symbol determined by the user repeatedly until the number of lines is negative. If the user enter -1, use * as the default symbol. Zheng-Liang Lu Java Programming 184 / 207

61 static void print(int x, String marker) { 3 for (int i = 1; i <= x; ++i){ 4 for (int j = 1; j <= i; ++j) 5 System.out.printf("%s", marker); 6 System.out.println(); 7 } 8 } 9 10 static void print(int x) { 11 for (int i = 1; i <= x; ++i){ 12 for (int j = 1; j <= i; ++j) 13 System.out.printf(" "); 14 System.out.println(); 15 } 16 } print()s are overloading. Zheng-Liang Lu Java Programming 185 / 207

62 public static void main (String[] args) { while (true) { 5 System.out.println("Enter a positive integer? "); 6 n = in.nextint(); 7 if (n < 0) break; 8 System.out.println("Choose a symbol? ( 1 to default symbol )"); 9 marker = in.next(); 10 if (marker.equals(" 1")) 11 print(n); 12 else 13 print(n, marker); 14 } } in.next() returns a string from keyboard. Be aware that in.next() skips the blanks in the beginning of line. marker.equals() is one method of String object. Zheng-Liang Lu Java Programming 186 / 207

63 Math Class The Math class contains the methods needed to perform basic mathematical functions. Note that the Math class is public and all methods within it are public and static. Two constants defined in Math class: Math.PI 9 and Math.E 10 Full document of Math class can be found here. Common methods are like: max, min, round, ceil, floor, abs, pow, exp, sqrt, cbrt, log, log10, sin, cos, asin, acos, random 9 π is a mathematical constant, the ratio of a circle s circumference to its diameter, commonly approximated as e is the base of the natural logarithm. It is approximately equal to Zheng-Liang Lu Java Programming 187 / 207

64 Example Password generator Write a program which generate a random string as a password of ten characters. Note that there may be lower-case letters, upper-case letters, digital characters in the password. Recall that a character is coded using an integer. How to generate the feasible characters randomly? Zheng-Liang Lu Java Programming 188 / 207

65 static char getrandomuppercaseletter() { 3 return (char)(math.random() ( Z A ) + A ); 4 } 5 6 static char getrandomlowercaseletter() { 7 return (char)(math.random() ( z a ) + a ); 8 } 9 10 static char getrandomdigitalcharacter() { 11 return (char)(math.random() ( 9 0 ) + 0 ); 12 } Zheng-Liang Lu Java Programming 189 / 207

66 public static void main (String[] args) { 3 System.out.println("Generating a password..."); 4 int key = 0; 5 for (int i = 1; i <= 10; ++i){ 6 key = (int)((math.random() 10) % 3); 7 switch (key) { 8 case 0: 9 System.out.printf("%s", getrandomuppercaseletter()); 10 break; 11 case 1: 12 System.out.printf("%s", getrandomlowercaseletter()); 13 break; 14 case 2: 15 System.out.printf("%s", getrandomdigitalcharacter()); 16 break; 17 } 18 } 19 } Zheng-Liang Lu Java Programming 190 / 207

67 Encapsulation The key to developing software is to apply the concept of abstraction. The client can use a method without knowing how it is implemented. The details of the implementation are encapsulated 11 in the method and hidden from the client who invokes the method. If you decide to change the implementation, the client program will not be affected, provided that you do not change the method signature. For example, do you really know how System.out.println() works? 11 Encapsulation, inheritance, and polymorphism. Zheng-Liang Lu Java Programming 191 / 207

68 A Black Box Zheng-Liang Lu Java Programming 192 / 207

69 Divide and Conquer When developing a large program, you can use the divide-andconquer strategy, aka stepwise refinement, to decompose it into subproblems. The subproblems can be further decomposed into smaller, more manageable problems. Pros: easier to write, reuse, debug, test, modify, maintain, and better facilitating teamwork Do not write the entire program at once. Sitting in front of the computer is the last thing when programming. Zheng-Liang Lu Java Programming 193 / 207

70 Recursion 12 Recursion is the process of defining something in terms of itself. A method that calls itself is said to be recursive. Recursion is an alternative form of program control. It is essentially repetition without a loop. 12 Recursion is a commom pattern in nature. Zheng-Liang Lu Java Programming 194 / 207

71 Try Fractal. Zheng-Liang Lu Java Programming 195 / 207

72 In general, to solve a problem using recursion, you break it into subproblems. Each subproblem is the same as the original problem but smaller in size. You can apply the same approach to each subproblem to solve it recursively. Zheng-Liang Lu Java Programming 196 / 207

73 Example The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. 13 For example, 5! = 5 4! = 5 (4 3!) = 5 (4 (3 2!)) = 5 (4 (3 (2 1))) = 5 (4 (3 2)) = 5 (4 6) = 5 24 = 120. Can you find the pattern? 13 Note that 0! = 1. Zheng-Liang Lu Java Programming 197 / 207

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75 Write a program which determines the first 10 factorials static int factorial(int n) { 3 if (n == 1) 4 return 1; 5 else 6 return n factorial(n 1); 7 } 8... Note that there must be a base condition in recursion. So, can you replace a recursive method by a loop? Zheng-Liang Lu Java Programming 199 / 207

76 Equivalence: Recursion Replaced by Loops for (int i = 1, j = 1; i <= 10; ++i) 3 System.out.println(i + "! = " + (j = i)); 4... In this case, the loop approach seems more compact than the recursive approach. One more intriguing question is, Can we always turn a recursive method into an iterative one? Yes, theoretically The Church-Turing thesis proves it if the memory serves. Zheng-Liang Lu Java Programming 200 / 207

77 In Practice Recursion bears substantial overhead. So, the recursive algorithm may execute a bit more slowly than the iterative equivalent. Besides, a deeply recursive method depletes the call stack, which is limited, and causes a exception fast. 15 The decision whether to use recursion or iteration should be based on the nature of, and your understanding of, the problem you are trying to solve. 15 Stack overflow. Zheng-Liang Lu Java Programming 201 / 207

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79 Exercise Fibonacci numbers Write a program which determines the first 10 Fibonacci numbers. A sequence F n of Fibonacci numbers is defined by the recurrence relation F n = F n 1 + F n 2, where F 0 = 0, F 1 = 1, and n 2. The first 10 Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34. Zheng-Liang Lu Java Programming 203 / 207

80 The recursive implementation of fib is very simple and straightforward. Yet, it isn t efficient since it requires more time and memory fib is an exponential-time algorithm while factorial is a linear-time algorithm. (Why?) Zheng-Liang Lu Java Programming 204 / 207

81 static int fib(int n) { 3 if (n == 1 n == 0) 4 return (n == 1)? 1 : 0; 5 else 6 return fib(n 1) + fib(n 2); 7 } int f0 = 0, f1 = 1; 3 int tmp = 0; 4 for (int i = 0; i < 10; ++i){ 5 if (i == 0 i == 1) 6 System.out.println("F " + i + " = " + ((i == 1)? f1 : f0)); 7 else { 8 tmp = f0 + f1; 9 f0 = f1; 10 f1 = tmp; 11 System.out.println("F " + i + " = " + f1); 12 } 13 } Zheng-Liang Lu Java Programming 205 / 207

82 Problem Set Exercise 6.1 (Greatest common divisor) Write a program which receives two positive integers and returns the greatest common divisor using a recursive approach. Exercise 5.x (Towers of Hanoi) Write a program which performs the classical problem, Towers of Hanoi, and shows the movements of disks. Zheng-Liang Lu Java Programming 206 / 207

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