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Arithmetic Expressions and Built in Functions

1. ARITHMETIC OPERATORS

There are five basic arithmetic operations with Fortran. These are given in the following Table 1

Table 1 Arithmetic operations with FORTRAN

Algebraic Symbol Fortran Symbol Meaning

+ + Addition

- - Subtraction

* Multiplication

/ Division

Ab ** Exponentiation

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2. ARITHMETIC EXPRESSIONArithmetic expressions are formed either by combining integer

variable names and constants with arithmetic operators or by combining real variable names and constants with arithmetic operators. Thus, in Fortran, two

types of expressions are defined.

i) Integer Expression e.g. I, L, +8, -7, N*M, M*N+I, (I*J) / (I+K)+I etc.

ii) Real Expression e.g. A, -A, 4.2, 1.5, -A+B+C*D, (A+B), (A+4.5), (A*B) /(D*C) etc.

Note: If expression consists of integer/real variables then it is called Mixed Mode expression e.g.

20+I+X**3, (A+B)* (K+J) etc.

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3 (a) Integer arithmetic:If the operands are integers, then integer arithmetic is performed to yield an integer.e.g. 5+3 yields 8

5*3 yields 155-3 yields 2

Integer division in FORTRAN, also yields an integer- the integral part of the quotient only i.e. the fractional part of the quotient is deleted in integer division.e.g. 5/4 yields 1

4/5 yields 0-7/2 yields -3

Exponent of integer number/variable can only be integere.g I2 = I** 2 (Valid)

= I**2.0 (Invalid)3(b) Real arithmetic :

If the operands are real, the real arithmetic is performed to yield a real value.e.g. 5.0+3.0 yields 8.0

5.*3. yields 15. 5.0-3.0 yields 2.0 etc.

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Division by real arithmetic in FORTRAN is similar to ordinary division,

e.g. 5.0/4.0 yields 1.25 4.0/5.0 yields 0.8 etc.

Exponent of real number/variable can be integer/real.

e.g. If we have expression, z = x y = x ** y, then it is solved as

i) If y is realTaking log on both sides, log z = y log xTaking antilog,z = antilog (y log x) = e y log x = exp (y log x)

ii) If y is an integer, then x ** y is evaluated by repeated multiplication.

Now, if we want to solve (3.2)7then if we write, 3.2 ** 7.0,then log and antilog will be used by the computer.

If we write, 3.2 ** 7, then simple multiplication will take place. Since, logs takes more computation time as compared to multiplication,

hence exponent should be kept real only if it contains fractional part too, otherwise keep the exponent integer.

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4. HIERARCHICAL RULE FOR THE OPERATONS IN THE EXPRESSIONS

The order, in which arithmetic operations are executed in an expression, is called the hierarchy or precedence of operations.

Table. 2 shows the precedence priority of operators.

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Table 2 Precedence of Operators

Priority Fortran Symbol Meaning/Operation

1. ( ) Parenthesis

2. ** Exponentiation

3. */ Multiplication, Division

4. +- Addition, Subtraction

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Rule 1. All expressions in parentheses are evaluated first.

Rule 2. Operators in the same expression are evaluated according to the hierarchy : ** followed by *, / followed by +, -.

Rule 3. If priority of two operators are equal, then expression is scanned from left to right whichever operator (of same priority) comes first will be operated first.

Rule 4. Priority of operators can be changed by using parentheses.( Expression within parentheses is to be solved first by applying Rule 1 and Rule 2).

Rule 5. Parentheses can be nested also. The inner most parentheses is to be solved first and then next outer and so on.

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Example 1 Consider an expression X** 6+8.0* Y – Z/ (A-8.7)

Following Hierarchical Rule, the solution will be in the steps shown

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Example 2. Let us now consider the solution of an expression

X * Y + Z/ (I+J) – (I** (J/4))

where X,Y and Z are real variables and I and J are integer variables.

This expression is evaluated in following steps with mixed mode as shown below .

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Example 3. The expression x + y/ (Z*A+B**2) is evaluated as follows,

Example 3. The expression x + y/ (Z*A+B**2) is evaluated as follows,

x + y

/

/ (Z*A B**2)+

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Note: Parenthesis can be used to make the expression more clear.By using the parenthesis we can alter the precedence of operatione.g. in b+4*a / b + a the precedence is :

but in the expression, b+4*a / (b + a) the precedence (of (b+a)) changes as shown below:

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e.g (i) The expression , AB + is written as,

A ** B + (C/D)* G+A/(5*B**3) (ii) The expression, is written as,

A/(B*C) – D**2

35B

AG

D

C

2DBC

A

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5. BUILT IN FUNCTIONSBuilt in functions are also known as LIBRARY FUNCTIONS OR INTRINSIC FUNCTIONS.These are the pre-written programs by the manufacturer of the compiler, for commonly used mathematical functions.These functions can be called, in user’s program by writing the name of the function-program and enclosing the argument in the parentheses

Example SQRT (X) = , ABS (X) =

ALOG (X) = loge X etc.

Points to Remember (while using Built-in-functions in the Program)• The argument should be enclosed in parentheses.

If a function has more than one argument, then arguments should be separated by commas.

• Care should be taken to match- the type- the number and- the order of arguments..

Most common functions used in FORTRAN have been summarized in Table 3.

X X

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Table 3.

Function Mathematical Notation FORTRAN

Square root of x SQRT (X)

Absolute value of x ABS (X)

Exponential of x ex EXP (X)

Sine of x sin x SIN (X)

Cosine of x cos x COS (X)

Tangent of x tan x TAN (X)

Inverse of sine sin-1 x ASIN (X)

Inverse of cosine cos-1 x ACOS (X)

Inverse of tangent tan-1 x ATAN (X)

Log of x log10 x ALOG 10(X)

Natural log of x loge x ALOG (X)

Convert integer Into real FLOAT (I)

Truncate real x into integer I FIX (X)

Give only quotient These two functions are DIV (I,J)

Give only remainder Used only for integers. MOD (I,J)

X

X

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e.g:- (i) an expression is written as

SQRT (SQRT ((A-B) / (C+4*D)))

(ii) The expression is written as

A ** (C-1) / ALOG (B)

(iii) The expression tan-1 is written as ATAN (SQRT (Sin (ABS (A))**2))

iv) The exp e b tan3 a is written as EXP (B) * TAN (A)**3.

dc

ba

4

)(log

1

b

a

e

c

a2sin

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6. EXECUTABLE AND NON-EXECUTABLE STATEMENTSExecutable Statements: These are the statements which result in a

machine code i.e. which result in some kind of machine-language instruction

. For example, READ, WRITE, STOP, GOTO, IF, DO etc. Non-Executable Statement : These are the statements which do not result

in any machine code i.e. which will not result in any machine-language

instructions. For example COMMENT statements, TYPE statement,

FORMAT, TYPE, END, ELSE, COMMON, DIMENSION etc.

Note : STOP is an executable statement. Its machine – language equivalent stops the computer from executing the next machine – language instruction of the program. END is a non-executable statement. It tells the compiler that this is the end of the program and there are no more FORTRAN statements to be transferred into machine – language instructions.