1. Lesson 12 Arrays 2007.11.7

2. 2 Objectives: About the Fibonacci sequence One-Dimensional Arrays Array Initialization Arrays as Function Arguments Case Study: Computing Averages and Standard Deviations Two-Dimensional Arrays

3. 3 Fibonacci Numbers One pair rabbit give birth to a new pair baby rabbits each year The baby rabbits need to wait one year to bear new baby f n+1 = f n + f n-1 f 0 = 1 f 1 = 1

4. 4 Fibonacci Numbers in Nature 1 2 3 5 8 13 21

5. 5 Compute the Fibonacci numbers recursively int fib(int n) { if (n <= 2) return 1; else return fib(n-1) + fib(n-2); }

6. 6 Time complexity of the recursive algorithm fib(5) fib(3) fib(4) + The growth rate of the computing time with the problem size fib(1)fib(2) + fib(0)fib(1) + fib(2)fib(3) + fib(1)fib(2) + … exponential

7. 7 Use iterative algorithm Liner complexity O(n)

8. 8 Use the Golden Ratio: (1+ sqr(5))/2 F(n) = round ( pow (1+sqr(5)) / 2, n) / sqr(5) )

9. 9 By recursive powering formula for multiplying two symmetric 2x2 matrices: [ F(n+1) + F(n) F(n) + F(n-1) ] [ F(n +1 ) F(n) ] [ F(n+2) F(n+1)] [ F(n+1) F(n) ]

10. 10 By recursive powering (cont.) [ 1 1] [ 1 0] n [ 1 1] [ 1 0] n/2 [ 1 1] [ 1 0] n/2 * + [ 1 1] [ 1 0] logn

11. 11 Arrays: Introduction Atomic variable: variable whose value cannot be further subdivided into a built-in data type Aggregate data type: data type with two main characteristics  Its values can be decomposed into individual data elements, each of which is either atomic or another Aggregate data type  It provides an access scheme for locating individual data elements within the data structure

12. 12 Name + space + access scheme 00611001 00611002 00611003 00611004 00611005 00611006 00611007 00611008 00611009 00611010 00611011 刘栩佳 樊怡 张昊 严霄 李伟强 叶翔 蔡浩 李鑫 李彬彬 安马太 宫照恒 uid list name list Arrays: Introduction

13. 13 Arrays A set of data items with same type Stored in consecutive memory  Fixed in size throughout program —— not dynamic index Arrays: Introduction

14. 14 Create a one-dimensional array #define NUMELS 5 int grades[NUMELS]; Const expression

15. 15 To access an element Format: arrayname [ index value ]  First element at position 0 position offset deviation grades[3] One-Dimensional Arrays

16. 16 Use a subscripted variable Subscripted variables can be used anywhere scalar variables are valid  grades[0] = 98;  grades[1] = grades[0] - 11; Any expression that evaluates an integer may be used as a subscript #define NUMELS 5 total = 0; for (i = 0; i < NUMELS; i++) total = total + grades[i];

17. 17 Input and Output of Array Values Array elements can be assigned values by:  using assignment statements  interactively, using the scanf() function #define NUMELS 5 for(i = 0; i < NUMELS; i++) { printf("Enter a grade: "); scanf("%d", &grades[i]); } Be careful: C does not check the value of the index being used

18. 18 Sample output: Enter a grade: 85 Enter a grade: 90 Enter a grade: 78 Enter a grade: 75 Enter a grade: 92 grades 0 is 85 grades 1 is 90 grades 2 is 78 grades 3 is 75 grades 4 is 92 Input and Output of Array Values From temporal sequence To spatial sequence Next round Next element

19. 19 Input and Output of Array Values

20. 20 Array Initialization Examples of initializations:  int grades[5] = {98, 87, 92, 79, 85};  double length[7] = {8.8, 6.4, 4.9};  char codes[5]= {‘h', ‘e', ‘l', ‘l', ‘o'};  char codes[] = {‘h', ‘e', ‘l', ‘l', ‘o'};  char codes[] = "sample"; /* size is 7 */

21. 21 Passing an array to a function By reference Via pointer (left behind)

22. 22 Pass array by reference

23. 23 Pass array by reference

24. 24 Find MAX: write proof inside the function

25. 25 Case Study: Computing Averages and Standard Deviations n-1 Average = ∑ a[ i ] n i =0 Standard deviation = √ ∑(a[ i ] – Ave) 2 ／ n

26. 26 Requirements Specification Need two functions  Determine the average…  Determine the standard deviation… …of a list of integer numbers Each function must accept the numbers as an array and return the calculated values to the calling function We now apply the top-down development procedure to developing the required functions

27. 27 Analyze the Problem Array of data Main control function Find the average ave Find the average ave std double findAvg( int [] ); double stdDev( int [], double) ?

28. 28 Select an Overall Solution Problem-Solver Algorithm is adapted:  Initialize an array of integers  Call the average function  Call the standard deviation function  Display the returned value of the average function  Display the returned value of the standard deviation function

29. 29 Computing Averages and Standard Deviations

30. 30 Computing Averages and Standard Deviations

31. 31 Two-Dimensional Arrays A two-dimensional array, or table, consists of both rows and columns of elements Can be treated as array of arrays int ar[3][4];

32. 32 Two-Dimensional Arrays (continued) Initialization: # define NUMROWS 3 #define NUMCOLS 4 int val[NUMROWS][NUMCOLS] = { {8,16,9,52}, {3,15,27,6}, {14,25,2,10} }; The inner braces can be omitted: int val[NUMROWS][NUMCOLS] = {8,16,9,52,3,15,27, 6,14,25,2,10}; Initialization is done in row order

33. 33 Two-Dimensional Arrays (continued)

34. 34 Two-Dimensional Arrays (exp)

35. 35 Common Programming Errors Forgetting to declare the array Using a subscript that references a nonexistent array element Not using a large enough conditional value in a for loop counter to cycle through all the array elements Forgetting to initialize the array

36. 36 Summary A single-dimensional array is a data structure that can store a list of values of the same data type Elements are stored in contiguous locations  Referenced using the array name and a subscript Single-dimensional arrays may be initialized when they are declared Single-dimensional arrays are passed to a function by passing the name of the array as an argument

37. 37 homework ： Program exercise: 8.2 4; 8.3 5; 8.4 11a Due time: 12/Nov.