1. Chapter 7 Simple Date Types Dr. Jiung-yao Huang Dept. Comm. Eng.
Nat. Chung Cheng Univ.
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
Range in Typical Microprocessor Implementation
short
~ 32767
unsigned short
0 ~ 65535
int
unsigned
long ~
unsigned long
0 ~

8. 7.1 (cont) Floating-Point Types in C
Approximate Range*
Significant
Digits*
float
10-37 ~1038 6
double
~10308 15
long double
~104932 19

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
Context of Conversion
Example
Explanation
Expression with binary operator and operands of different numeric types
k + x
value is 6.5
Value of int k is converted to type double
Assignment statement with type double target variable and type int expression
z = k / m;
expression value is 1; value assigned to z is 1.0
Expression is evaluated first. The result is converted to type double
Assignment statement with type int target variable and type double expression
n = x * y;
expression value is 3.15; value assigned to n is 3
Expression is evaluated first. The result is converted to type int

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 Enumeration constant
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：
{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
Call to evaluate
Output Produced
evaluate(sqrt, 0.25, 25.0, 100.0)
f( )=
f( )=
f( )=
evaluate(sin, 0.0, , 0.5* )
f( )=
f( )=
f( )=

29. 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
xmid =
xleft
+ xright
2.0

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 7.9 Three Possibilities That 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
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.