Chapter 7

1. Chapter 7 Simple Date Types Dr. Jiung-yao Huang Dept. Comm. Eng. Nat. Chung Cheng Univ. E-mail : comjyh@ccu.edu.tw TA: 鄭筱親 陳昱豪

2. 本章重點 • Enumerated type • Declaring a function parameter • Bisection method

3. outline • 7.1 REPRESENTATION AND CONVERSION OF NUMERIC TYPES • 7.2 REPRESENTATION AND CONVERSION OF TYPE CHAR • 7.3 ENUMERATED TYPES • 7.4 ITERATIVE APPROXIMATIONS • CASE STUDY: BISECTION METHOD FOR FINDING ROOTS • 7.5 COMMON PROGRAMMING ERRORS

4. 7.1 Representation and Conversion of Numeric Types • Simple data type • A data type used to store a single value • Uses a single memory cell to store a variable • Different numeric types has different binary strings representation in memory

5. Figure 7.1 Internal Formats of Type int and Type double mantissa: binary fraction between 0.5~1.0 for positive numbers -0.5~-1.0 for negative numbers exponent: integer real number: mantissa x 2exponent

6. Figure 7.2 Program to Print Implementation-Specific Ranges for Positive Numeric Data p.857, limits.h, float.h %e : print DBL_MIN, DBL_MAX in scientific notation

7. 7.1 (cont) Integer Types in C

8. 7.1 (cont) Floating-Point Types in C

9. 7.1 (cont) Numerical Inaccuracies • Representational error (round-off error) • An error due to coding a real number as a finite number of binary digits • Cancellation error • An error resulting from applying an arithmetic operation to operands of vastly different magnitudes; effect of smaller operand is lost

10. 7.1 (cont) Numerical Inaccuracies • Arithmetic underflow • An error in which a very small computational result is represented as zero • Arithmetic overflow • An error that is an attempt to represent a computational result that is too large

11. 7.1 (cont) Automatic Conversion of Data Types • variable initialized • int k = 5, m = 4, n; • double x = 1.5, y = 2.1, z;

12. 7.1 (cont) Automatic Conversion of Data Types

13. 7.1 (cont) Explicit Conversion of Data Types p.63 Table 2.9 • cast • an explicit type conversion operation • not change what is stored in the variable • Ex. • frac = (double) n1 / (double) d1; • Average = (double) total_score / num_students(p.63)

14. 7.2 Representation and Conversion of Type char • A single character variable or value may appear on the right-hand side of a character assignment statement. • Character values may also be compared, printed, and converted to type int. #define star ‘*’ char next_letter = ‘A’; if (next_letter < ‘Z’) …

15. 7.2 (cont) Three Common character codes(Appendix A) • Digit character • ASCII ‘0’ ~’9’ have code value 48~57 • ‘0’ < ‘1’ < ‘2’…….< ‘9’ • Uppercase letters • ASCII ‘A’~’Z’ have code values 65~90 • ‘A’ < ‘B’ < ‘C’……< ‘Z’ • Lowercase letters • ASCII ‘a’~’z’ have code values 97~122 • ‘a’ < ‘b’ < ‘c’…….< ‘z’

16. 7.2 (cont) Example 7.1 • collating sequence • A sequence of characters arranged by character code number • Fig. 7.3 uses explicit conversion of type int to type char to print part of C collating sequence

17. Figure 7.3 Program to Print Part of the Collating Sequence

18. 7.3 Enumerated Types • Enumerated type • A data type whose list of values is specified by the programmer in a type declaration • Enumeration constant • An identifier that is one of the values of an enumerated type • Fig. 7.4 shows a program that scans an integer representing an expense code and calls a function that uses a switch statement to display the code meaning.

19. Figure 7.4 Enumerated Type for Budget Expenses

20. Figure 7.4 Enumerated Type for Budget Expenses (cont’d)

21. Figure 7.4 Enumerated Type for Budget Expenses (cont’d)

22. 7.3 (cont) Enumerated Type Definition • Syntax： typedef enum {identifier_list} enum_type; • Example： typedef enum {sunday, monday, tuesday, wednesday, thursday, friday, saturday} day_t;

23. 7.3 (cont) Example 7.3 • The for loop in Fig. 7.5 scans the hours worked each weekday for an employee and accumulates the sum of these hours in week_hours.

24. Figure 7.5 Accumulating Weekday Hours Worked

25. 7.4 Iterative Approximations • root (zero of a function) • A function argument value that causes the function result to be zero • Bisection method • Repeatedly generates approximate roots until a true root is discovered.

26. Figure 7.6 Six Roots for the Equation f(x) = 0

27. Figure 7.7 Using a Function Parameter • Declaring a function parameter is accomplished by simply including a prototype of the function in the parameter list.

28. 7.4 (cont) Calls to Function evaluate and the Output Produced

29. xleft + xright xmid = 2.0 7.4 (cont) Case Study: Bisection Method for Finding Roots • Problem • Develop a function bisect that approximates a root of a function f on an interval that contains an odd number of roots. • Analysis

30. 7.4 (cont) Case Study: Bisection Method for Finding Roots • Analysis • Problem Inputs • double x_left • double x_right • double epsilon • double f(double farg) • Problem Outputs • double root • int *errp

31. Figure 7.8 Change of Sign Implies an Odd Number of Roots

32. Figure 7.9Three PossibilitiesThat Arise When the Interval [xleft, xright] Is Bisected

33. 7.4 (cont) Case Study: Bisection Method for Finding Roots • Design • Initial Algorithm • 1.if the interval contains an even number of roots • 2.Set error flag • 3.Display error message • else • 4.Clear error flag • 5.repeat as long as interval size is greater than • epsilon and root is not found • 6.Compute the function value at the midpoint of the interval • 7.if the function value is zero, the midpoint is a root • else • 8.Choose the left or right half of the interval • in which to continue the search • 9.Return the midpoint of the final interval as the root

34. 7.4 (cont) Case Study: Bisection Method for Finding Roots • Design • Program variables • int root_found • double x_mid • double f_left, f_mid, f_right • Refinement • 1.1 f_left = f(x_left) • 1.2 f_right = f(x_right) • 1.3 if signs of f_left and f_right are the same

35. 7.4 (cont) Case Study: Bisection Method for Finding Roots • Design • Refinement • 5.1 while x_right – x_left > epsilon and !root_found • 8.1 if root is in left half of interval (f_left*f_mid<0.0) 8.2 Change right end to midpoint else 8.3 Change left end to midpoint • Implementation (Figure 7.10)

36. Figure 7.10 Finding a Function Root Using the Bisection Method

37. Figure 7.10 Finding a Function Root Using the Bisection Method (cont’d)

38. Figure 7.10 Finding a Function Root Using the Bisection Method (cont’d)

39. Figure 7.11 Sample Run of Bisection Program with Trace Code Included

40. 7.5 Common Programming Errors • Arithmetic underflow and overflow resulting from a poor choice of variable type are causes of erroneous results. • Programs that approximate solutions to numerical problems by repeated calculations often magnify small errors. • Not reuse the enumerated identifiers in another type or as a variable name • C does not verify the value validity in enum variables

41. Chapter Review(1) • Type int and double have different internal representations. • Arithmetic with floating-point data may not be precise, because not all real numbers can be represented exactly. • Type char data are represented by storing a binary code value for each symbol. • Defining an enumerated type requires listing the identifier that are the values of the type.

42. Chapter Review(2) • A variable or expression can be explicitly converted to another type by writing the new type’s name in parentheses before the value to convert. • A function can take another function as a parameter. • The bisection method is a technique for iterative approximation of a root of a function.