Chapter 7
Simple Date
Types
Dr. Jiung-yao Huang
Dept. Comm. Eng.
Nat. Chung Cheng Univ.
E-mail : [email protected]
TA: 鄭筱親 陳昱豪
本章重點
Enumerated type
Declaring a function parameter
Bisection method
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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
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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
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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
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Figure 7.2 Program to Print ImplementationSpecific Ranges for Positive Numeric Data
p.805, limits.h, float.h
%e : print DBL_MIN, DBL_MAX in scientific notation
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7.1 (cont) Integer Types in C
Type
short
Range in Typical Microprocessor
Implementation
-32767 ~ 32767
unsigned short 0 ~ 65535
int
-32767 ~ 32767
unsigned
0 ~ 65535
long
-2147483647 ~ 2147483647
unsigned long
0 ~ 4294967295
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7.1 (cont) Floating-Point Types in C
Type
float
double
Approximate
Range*
10-37 ~1038
10-307 ~10308
long double 10-4931 ~104932
Significant
Digits*
6
15
19
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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
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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
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7.1 (cont) Automatic Conversion of Data Types
variable initialized
int
k = 5, m = 4, n;
double x = 1.5, y = 2.1, z;
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7.1 (cont) Automatic Conversion of Data Types
Context of Conversion
Example
Expression with binary
operator and operands of
different numeric types
k+x
value is 6.5
Assignment statement
with type double target
variable and type int
expression
Assignment statement
with type int target
variable and type double
expression
Explanation
Value of int k is
converted to
type double
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
n = x * y;
Expression is
expression value is evaluated first.
The result is
3.15; value
assigned to n is 3 converted to
type int
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7.1 (cont) Explicit Conversion of Data Types
cast
p.63 Table 2.9
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)
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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’) …
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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’
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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
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Figure 7.3 Program to Print Part of the
Collating Sequence
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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.
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Figure 7.4 Enumerated Type for Budget
Expenses
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Figure 7.4
Enumerated
Type for Budget
Expenses (cont’d)
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Figure 7.4 Enumerated Type for Budget
Expenses (cont’d)
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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;
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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.
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Figure 7.5 Accumulating Weekday Hours
Worked
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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.
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Figure 7.6 Six Roots for the Equation
f(x) = 0
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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.
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7.4 (cont) Calls to Function evaluate and the
Output Produced
Call to evaluate
Output Produced
evaluate(sqrt, 0.25, 25.0, f(0.25000)=0.50000
100.0)
f(25.00000)=5.00000
f(100.00000)=10.00000
evaluate(sin, 0.0,
f(0.00000)=0.00000
3.14156, 0.5*3.14156)
f(3.14159)=0.00000
f(1.57079)=1.00000
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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
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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
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Figure 7.8 Change of Sign Implies an Odd
Number of Roots
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Figure 7.9
Three Possibilities
That Arise When
the Interval [xleft,
xright] Is Bisected
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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
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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
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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)
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Figure 7.10
Finding a
Function Root
Using the
Bisection
Method
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Figure 7.10
Finding a
Function Root
Using the
Bisection
Method (cont’d)
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Figure 7.10
Finding a
Function Root
Using the
Bisection
Method (cont’d)
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Figure 7.11 Sample Run of Bisection
Program with Trace Code Included
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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
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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.
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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.
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