Fall, 2006Introduction to FORTRAN1 (2 + 2 = ???) Numbers, Arithmetic Nathan Friedman Fall, 2006.

Fall, 2006 Introduction to FORTRAN 1

(2 + 2 = ???)Numbers, Arithmetic

Nathan Friedman

Fall, 2006


The Speed of Light

How long does it take light to travel from the sun to earth?

Light travels 9.46 x 1012 km a year A year is 365 days, 5 hours, 48

minutes and 45.9747 seconds The average distance between the

earth and sun is 150,000,000 km


Elapsed Timeprogram light_travel

implicit nonereal :: light_minute, distance, timereal :: light_year = 9.46 * 10.0 ** 12

light_minute = light_year / (365.25 * 24.0 * 60.0)distance = 150.0 * 10.0 ** 6time = distance / light_minute

write (*,*) "Light from the sun takes ", time, & "minutes to reach earth."

end program light_travel


Arithmetic Expressions

An arithmetic expression is formed using the operations:+ (addition),

- (subtraction)

* (multiplication),

/ (division)

** (exponentiation)


Watch out for ambiguity

Let’s look at two expressions from our programlight_minute = light_year / (365.25 * 24.0 * 60.0)

distance = 150.0 * 10.0 ** 6




distance = 150.0 * 10.0 ** 6

What value is assigned to distance?




distance = 150.0 * 10.0 ** 6

What value is assigned to distance?(150.0 * 10.0) ** 6 ?

150.0 * (10.0 ** 6) ?




distance = 150.0 * 10.0 ** 6

What value is assigned to distance?(150.0 * 10.0) ** 6 ?

150.0 * (10.0 ** 6) ?

What if the first expression didn’t have parentheses? light_minute = light_year / 365.25 * 24.0 * 60.0


Precedence Rules

Every language has rules to determine what order to perform operations

For example, in FORTRAN ** comes before *

In an expression, all of the **’s are evaluated before the *’s


Precedence Rules First evaluate operators of higher precedence

3 * 4 – 5 ?

3 + 4 * 5 ?



3 * 4 – 5 7

3 + 4 * 5 23



3 * 4 – 5 7

3 + 4 * 5 23 For operators of the same precedence, use

associativity. Exponentiation is right associative, all others are left associative

5 – 4 – 2 ?2 ** 3 ** 2 ?



3 * 4 – 5 7



5 – 4 – 2 -1 (not 3)

2 ** 3 ** 2 512 (not 64)



3 * 4 – 5 7



5 – 4 – 2 -1 (not 3)

2 ** 3 ** 2 512 (not 64) Expressions within parentheses are evaluated first


Precedence of Operators in FORTRAN

Operators in order of precedence and their associativity:Arithmetic

** right to left *, / left to right +, - left to right

Relational <, <=, >, >=, ==, /= no associativity

Logical .NOT. right to left.AND. left to right.OR. left to right .EQV., .NEQV. left to right


Another Example

Last lecture we looked at the problem of finding the roots of a quadratic equation

We focused on the discriminant Here is a program that computes

the roots


! --------------------------------------------------! Solve Ax^2 + Bx + C = 0! --------------------------------------------------PROGRAM QuadraticEquation IMPLICIT NONE

REAL :: a, b, c REAL :: d REAL :: root1, root2

! read in the coefficients a, b and c WRITE(*,*) 'A, B, C Please : ' READ(*,*) a, b, c

! compute the square root of discriminant d

d = SQRT(b*b - 4.0*a*c)

! solve the equation

root1 = (-b + d)/(2.0*a) ! first root root2 = (-b - d)/(2.0*a) ! second root

! display the results

WRITE(*,*) 'Roots are ', root1, ' and ', root2

END PROGRAM QuadraticEquation


Data Types

In the examples we have declared the variables to be of type REAL


Data Types


That is, each memory cell can hold a real number


Data Types



What is a real number?


Data Types



What is a real number? In Mathematics?


Data Types



What is a real number? In Mathematics? In FORTRAN?


Real Numbers(examples)

3.1415910.010.

-123.45+1.0E-3 (0.001)

150.0E6But not:

1,000.000123E5

12.0E1.5


Real Numbers (representation)

A real value is stored in two parts1. A mantissa determines the precision2. An exponent determines the range

Real numbers are typically stored as either

32 bits (4 bytes): type REAL 64 bits (8 bytes): type DOUBLE

(This varies on some computers)


Accuracy of Real Numbers REAL numbers:

Mantissa represented by 24 bits gives about 7 decimal digits of precision

Exponent represented by 8 bits gives range from 10-38 to 1038

DOUBLE numbers: Mantissa represented by 53 bits gives about

15 decimal digits of precision Exponent represented by 11 bits gives

range from 10-308 to 10308


2 + 2 = ???

Be careful not to expect exact results with real numbers

**** example from laptop


Back to The Speed of Light

How long does it take light to travel from the sun to earth?

The result we got was a decimal number of minutes

We’d rather have the number of minutes and seconds


Minutes and Secondsprogram light_travel

implicit nonereal :: light_minute, distance, timereal :: light_year = 9.46 * 10.0**12integer :: minutes, seconds

light_minute = light_year/(365.25 * 24.0 * 60.0)distance = 150.0 * 10**6time = distance / light_minute

minutes = timeseconds = (time - minutes) * 60

write (*,*) "Light from the sun takes ", minutes, & " minutes and ", seconds, " seconds to reach earth."



Integer Numbers Integers are whole numbers represented

using 32 (or sometimes 16 or even 64 bits)

For 32 bit numbers whole numbers with up to nine digits can be represented

0

-987

+17

123456789


Integer Arithmetic The result of performing an operation on

two integers is an integer This may result in some unexpected

results since the decimal part is truncated

12/4 3

13/4 3

1/2 0

2/3 0


Some Simple Examples

1 + 3 4 1.23 - 0.45 0.78 3 * 8 246.5/1.25 5.2 8.4/4.2 2.0 (not 2)



1 + 3 4 1.23 - 0.45 0.78 3 * 8 246.5/1.25 5.2 8.4/4.2 2.0 (not 2)-5**2 -25 (not 25 -- precedence)



1 + 3 4 1.23 - 0.45 0.78 3 * 8 246.5/1.25 5.2 8.4/4.2 2.0 (not 2)-5**2 -25 (not 25)3/5 0 (not 0.6)3./5. 0.6


Another Example2 * 4 * 5 / 3 ** 2 --> 2 * 4 * 5 / [3 ** 2] --> 2 * 4 * 5 / 9 --> [2 * 4] * 5 / 9 --> 8 * 5 / 9 --> [8 * 5] / 9 --> 40 / 9 --> 4

The result is 4 rather than 4.444444 since the operands are all integers.


Still More Examples1.0 + 2.0 * 3.0 / ( 6.0*6.0 + 5.0*44.0) ** 0.25 --> 1.0 + [2.0 * 3.0] / (6.0*6.0 + 5.0*44.0) ** 0.25 --> 1.0 + 6.0 / (6.0*6.0 + 5.0*55.0) ** 0.25 --> 1.0 + 6.0 / ([6.0*6.0] + 5.0*44.0) ** 0.25 --> 1.0 + 6.0 / (36.0 + 5.0*44.0) ** 0.25 --> 1.0 + 6.0 / (36.0 + [5.0*44.0]) ** 0.25 --> 1.0 + 6.0 / (36.0 + 220.0) ** 0.25 --> 1.0 + 6.0 / ([36.0 + 220.0]) ** 0.25 --> 1.0 + 6.0 / 256.0 ** 0.25 --> 1.0 + 6.0 / [256.0 ** 0.25] --> 1.0 + 6.0 / 4.0 --> 1.0 + [6.0 / 4.0] --> 1.0 + 1.5 --> 2.5


Mixed Mode Assignment

In the speed of light example, we assigned an real value to an integer variable

minutes = time

The value being assigned is converted to an integer by truncating (not rounding)


Mixed Mode Assignment

In the speed of light example, we assigned an real value to an integer variable

minutes = time

The value being assigned is converted to an integer by truncating (not rounding)

When assigning an integer to a real variable, the integer is first converted to a real (the internal representation changes)


Mixed Mode Expressions

In the speed of light example, we subtracted the integer value, minutes, from the real value, timeseconds = (time - minutes) * 60


Mixed Mode Expressions

In the speed of light example, we subtracted the integer value, minutes, from the real value, timeseconds = (time - minutes) * 60

If an operation involves an integer and a real, the integer is first converted to a real and then the operation is done.

The result is real.The example has two arithmetic operations,

an assignment and forces two type conversions


Mixed Mode Examples

1 + 2.5 3.5 1/2.0 0.5 2.0/8 0.25 -3**2.0 -9.04.0**(1/2) 1.0 (since 1/2 0)


Another Example25.0 ** 1 / 2 * 3.5 ** (1 / 3) [25.0 ** 1] / 2 * 3.5 ** (1 / 3) 25.0 / 2 * 3.5 ** (1 / 3) 25.0 / 2 * 3.5 ** ([1 / 3]) 25.0 / 2 * 3.5 ** 0 25.0 / 2 * [3.5 ** 0] 25.0 / 2 * 1.0 [25.0 / 2] * 1.0 12.5 * 1.0 12.5


Something’s Not Right Here

program light_travelimplicit nonereal :: light_minute, distance, timereal :: light_year = 9.46 * 10**12

light_minute = light_year/(365.25 * 24.0 * 60.0)distance = 150.0 * 10**6time = distance / light_minute

write (*,*) "Light from the sun takes ", time, & "minutes to reach earth."



What Happened?

Look at this assignment:light_year = 9.46 * 10**12

Precedent rules tell us that 10**12 is evaluated first

Type rules tell us that the result is an integer

Integers can only have about 9 digits, not 12


How do we fix it?

Let’s changelight_year = 9.46 * 10**12

tolight_year = 9.46 * 10.0**12

That works, but why?


Back to roots of a quadratic

Let’s have another look at our program for finding the roots of a quadratic equation

We used the classic formula that involved finding the square root of the discriminant

What if the disciminant is negative?


! --------------------------------------------------! Solve Ax^2 + Bx + C = 0! --------------------------------------------------PROGRAM QuadraticEquation IMPLICIT NONE

REAL :: a, b, c REAL :: d REAL :: root1, root2

! read in the coefficients a, b and c WRITE(*,*) 'A, B, C Please : ' READ(*,*) a, b, c

! compute the square root of discriminant d

d = SQRT(b*b - 4.0*a*c)

! solve the equation

root1 = (-b + d)/(2.0*a) ! first root root2 = (-b - d)/(2.0*a) ! second root

! display the results

WRITE(*,*) 'Roots are ', root1, ' and ', root2

END PROGRAM QuadraticEquation


A Run Time Error

If the disciminant is negative, attempting to take the square root would cause an error during execution

This is called a run time error and the program would abort


Selection

What can we do? Every programming language must

provide a selection mechanism that allows us to control whether or not a statement should be executed

This will depend on whether or not some condition is satisfied (such as the discriminant being positive)


! ------------------------------------------------------------! Solve Ax^2 + Bx + C = 0 ! ------------------------------------------------------------PROGRAM QuadraticEquation IMPLICIT NONE! **** Same old declarations and set up ****! compute the square root of discriminant d

d = b*b - 4.0*a*c IF (d >= 0.0) THEN ! is it solvable? d = SQRT(d) root1 = (-b + d)/(2.0*a) root2 = (-b - d)/(2.0*a) WRITE(*,*) "Roots are ", root1, " and ", root2 ELSE ! complex roots WRITE(*,*) "There is no real root!" WRITE(*,*) "Discriminant = ", d END IFEND PROGRAM QuadraticEquation


FORTRAN SelectionUsed to select between two alternative sequences of

statements.The keywords delineate the statement blocks.Syntax: IF (logical-expression) THEN

first statement block, s1

ELSE

second statement block, s2

END IF


Semantics of IF…THEN…ELSE…END IF

Evaluate the logical expression. It can have value .TRUE. or value .FALSE.

If the value is .TRUE., evaluate s1, the first block of statements

If the value is .FALSE., evaluate s2, the second block of statements

After finishing whichever of s1 or s2 that was chosen, execute the next statement following the END IF

Fall, 2006Introduction to FORTRAN1 (2 + 2 = ???) Numbers, Arithmetic Nathan Friedman Fall, 2006.

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Transcript of Fall, 2006Introduction to FORTRAN1 (2 + 2 = ???) Numbers, Arithmetic Nathan Friedman Fall, 2006.