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20061129 chap7 2
Objectives 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.
20061129 chap7 3
Representation and Conversion of Numeric Types 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.
20061129 chap7 4
Internals Formats All data are represented in memory as binary
strings. Integers are represented by standard binary
numbers. 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
20061129 chap7 8
Numerical Inaccuracies Certain fractions cannot be
represented exactly in the decimal number system e.g., 1/3= 0.33333……
The representational error (round-off error) will depend on the number of binary numbers in the mantissa.
20061129 chap7 9
Example: Representational Error 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.
20061129 chap7 10
Manipulating Very Large and Very Small Real Numbers
Cancellation Error: an error resulting from applying an arithmetic operation to operands of vastly different magnitudes; effect of smaller operand is lost. e.g., 1000.0 + 0.0000001234 is equal to 1000.0
Arithmetic Underflow: an error in which a very small computational result is represented as zero. e.g., 0.00000001 * 10-1000000 is equal to 0
Arithmetic Overflow: a computational result is too large. e.g., 999999999 * 109999999 may become a negative value
on some machines.
20061129 chap7 11
Automatic Conversion of Data Types 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;
20061129 chap7 12
Explicit Conversion of Data Types C also provides an explicit type
conversion operation called a cast.
e.g., int n = 2, d = 4;
double frac;
frac = n / d; //frac = 0.0
frac = (double) n / (double) d; //frac = 0.5
frac = (double) (n / d); //frac = 0.0
20061129 chap7 13
Representation and Conversion of char
Character values can be compared by the equality operatorsequality operators == and !=, or by the relational operatorsrelational operators <, <=, >, and >=.e.g., letter = ‘A’;
if (letter < ‘Z’) … Character values may also be
compared, scanned, printed, and converted to type int.
20061129 chap7 14
ASCII (American Standard Code for Information Interchange) Each character has its own unique
numeric code. A widely used standard is called American American
Standard Code for Information Interchange Standard Code for Information Interchange (ASCII)(ASCII). (See Appendix A in the textbook) The printable charactersprintable characters have codes from 32 to
126, and others are the control characterscontrol 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.
20061129 chap7 16
Enumerated Types 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.
20061129 chap7 18
Iterative Approximations Numerical Analysis: to develop
algorithms for solving computational problems. 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.
20061129 chap7 19
Six Roots for the Equation f(x) = 0
Case Study: Bisection Method for Finding Roots
20061129 chap7 20
Function Parameters 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.
20061129 chap7 21
Case Study: Bisection Method for Finding Roots First, tabulate function values
to find an appropriate interval in which to search for a root.
20061129 chap7 22
Bisect this interval Three possibilities that wrise
when the Iinterval [xleft, xright] is Bisected
20061129 chap7 23
Epsilon A fourth possibility is that the
length of the interval is less than Epsilon. Epsilon is a very small constant.
In this case, any point in the interval is an acceptable root approximation.
20061129 chap7 26
Homework #8 Due: 2006/12/9 複數運算
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