Transcript lee-06-chap7

Chapter 7
Simple Data Types
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Objectives
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No programming language can
predefine all the data types that a
programmer may need.
C allows a programmer to create new
data types.
Simple Data Type: a data type used to
store a single value.
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Representation and Conversion
of Numeric Types
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int vs. double: why having more than one
numeric type is necessary?
Can the data type double be used for all
numbers?
Operations involving integers are faster than
those involving numbers of type double.
Operations with integers are always precise,
whereas some loss of accuracy or round-off
error may occur when dealing with double
numbers.
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Internals Formats
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All data are represented in memory as binary strings.
Integers are represented by standard binary numbers.
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For example, 13 = 01101
Doubles are represented by two sections:
mantissa and exponent.
real number = mantissa * 2exponent
e.g. 4.0 = 0010 * 20001 = 2 * 21
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Implementation-Specific Ranges
for Positive Numeric Data
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Run demo (Fig. 7.2)
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Integer Types in C
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Floating-Point Types in C
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Numerical Inaccuracies
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Certain fractions cannot be represented
exactly in the decimal number system
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e.g., 1/3= 0.33333……
The representational error (round-off error)
will depend on the number of binary
numbers in the mantissa.
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Example: Representational
Error
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for (trial = 0.0;
trial != 10.0;
trial = trial + 0.1) {
….
}
Adding 0.1 one hundred times in not exactly 10.0.
The above loop may fail to terminate on some
computers.
trail < 10.0: the loop may execute 100 times on one
computer and 101 times on another.
It is best to use integer variable for loop control
whenever you can predict the exact number of times
a loop body should be repeated.
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Manipulating Very Large and Very
Small Real Numbers
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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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Arithmetic Underflow: an error in which a very small
computational result is represented as zero.
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e.g., 1000.0 + 0.0000001234 is equal to 1000.0
e.g., 0.00000001 * 10-1000000 is equal to 0
Arithmetic Overflow: a computational result is too large.
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e.g., 999999999 * 109999999 may become a negative value on
some machines.
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Automatic Conversion of Data Types
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The data of one numeric type may be
automatically converted to another numeric types.
int
k = 5,
m = 4;
n;
double x = 1.5,
y = 2.1,
z;
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Explicit Conversion of Data Types
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C also provides an explicit type
conversion operation called a cast.
e.g.,
int n = 2, d = 4;
double frac;
frac = n / d;
frac = (double) n / (double) d;
frac = (double) (n / d);
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//frac = 0.0
//frac = 0.5
//frac = 0.0
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Representation and Conversion of char
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Character values can be compared by
the equality operators == and !=, or by
the relational operators <, <=, >, and >=.
e.g., letter = ‘A’;
if (letter < ‘Z’) …
Character values may also be compared,
scanned, printed, and converted to type
int.
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ASCII (American Standard Code for
Information Interchange)
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Each character has its own unique numeric code.
A widely used standard is called American Standard
Code for Information Interchange (ASCII). (See
Appendix A in the textbook)
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The printable characters have codes from 32 to 126, and
others are the control characters.
For example, the digit characters from ‘0’ to ‘9’ have
code values from 48 to 57 in ASCII.
The comparison of characters (e.g., ‘a’<‘c’)
depends on the corresponding code values in ASCII.
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Print Part of the Collating
Sequence (Fig. 7.3)
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Enumerated Types
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Good solutions to many programming
problems require new data types.
Enumerated type: a data type whose
list of values is specified by the
programmer in a type declaration.
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Accumulating Weekday Hours Worked
(Fig 7.5)
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Iterative Approximations
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Numerical Analysis: to develop algorithms for
solving computational problems.
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Finding solutions to sets of equations,
Performing operations on matrices,
Finding roots of equations, and
Performing mathematical integration.
Many real-world problems can be solved by
finding roots of equations.
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Six Roots for the Equation f(x) = 0
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Case Study: Bisection Method for
Finding Roots
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Function Parameters
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The bisection routine would be far more useful if
we could call it to find a root of any function.
Declaring a function parameter is accomplished
by simply including a prototype of the function in
the parameter list.
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Case Study: Bisection Method for
Finding Roots
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First, tabulate function values to find an
appropriate interval in which to search
for a root.
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Bisect this interval
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Three possibilities that wrise when the
Iinterval [xleft, xright] is Bisected
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Epsilon
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A fourth possibility is that the length of
the interval is less than Epsilon.
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Epsilon is a very small constant.
In this case, any point in the interval is
an acceptable root approximation.
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Finding a Function Root Using
the Bisection Method
Run demo
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Figure 7.11 Sample Run of
Bisection Program with Trace
Code Included
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Homework #8
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Due: 2006/12/9
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複數運算
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以長度 2 的一維陣列 ( float [2] ) ，來表示複數，並實作出加
減乘除、次方 ( 根號 ) 的運算，為強化乘除的計算，本題的乘
除、次方 ( 根號 ) 運算需使用極座標系統 ( 複數的 乘法 、 除
法 以及 指數 以及開方運算，在極坐標中會比在直角坐標中容
易得多，請見reference 的複數部份說明 ) 。
作業要求 :
1. 使用者輸入二對 X,Y 代表二複數 a = (X 1 +iY 1 ), b = (X 2
+iY 2 )
2. 計算出 a+b
3. 計算出 a/b
4. 使用者輸入欲計算 a 次方大小 (exp)
5. 計算出 a 的 exp 次方
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Summary
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Representation and Conversion of
Numeric Types
Representation and Conversion of
Type Char
Enumerated Types
Iterative Approximations
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