introduction numerical analysis lecture .ppt

Lectures on Numerical Methods 1
An Example
 Compute Squares
 Prints a table of squares as shown
below
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
10 100
/* Compute Squares */
#include <stdio.h>
main() {
int n, n_square;
int lower, upper, step;
lower = 1; /* Lower imit */
upper = 10; /* Upper limit */
step = 1;
n = lower;
while ( n <= upper ) {
n_square = n * n;
printf(“%3dt%6dn”,n,n_square);
n = n + 1;
}
}

An Example
 Functions
 C programs contain functions
and variables
 Function names are followed by
a list of arguments enclosed in
parenthesis.
 Function definition consists of a
group of statements enclosed in
curly brackets.
 printf is a library function.
The information about this
function is contained in a file
named stdio.h.
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}

An Example
 Variables
 Variables refer the objects that can
be stored in computer’s memory
locations. Variables are named
(identifiers)
 Variables can store data of various
types. Some basic datatypes are
char characters (1/2 bytes)
int integers (2/4 bytes)
float rationals (4 bytes)
double rationals(8 bytes)
 Variables must be declared and
initialized.
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}

An Example
 Declarations
 Declarations have form
type var1, var2, var3;
 Merely announces the data type to
be associated with a variable.
 Initial value of variable is not known.
 Examples
char name;
int numStudents;
float applePrice;
double veryAccurateVar;
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}

An Example
 Assignments
Statement
 The assignment statements have
the following form
someVar = expression ;
 Expressions consist of variables,
functions and operators.
 Examples
c = 5 * ( f – 32 ) / 9
s = sin(x) / x
y = log(x) + v0
I = P * R * T / 100
Tax = 0.3 * ( I – 60000 )
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}

An Example
 Loops
 The form of the while statement is
While ( condition )
statement ;
While ( condition ) {
statements }
 If the condition is true the
statements in the body of the while
loop are executed. At the end of the
loop the the condition is tested
again and executes the loop if the
condition is true. This procedure
continues till the condition fails to be
true.
 Conditions are usually logical
expressions using operators like ==,
<, <=, >, >= etc. These have
boolean value
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}

An Example
 Execution /* Compute Squares */
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n ?? ??
n_square ?? ??
lower ?? ??
upper ?? ??
step ?? ??

An Example
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n ?? ??
n_square ?? ??
lower ?? ??
upper ?? ??
step ?? ??

An Example
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n ?? ??
n_square ?? ??
lower ?? 1
upper ?? ??
step ?? ??

An Example
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n ?? ??
n_square ?? ??
lower 1 1
upper ?? 10
step ?? ??

An Example
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n ?? ??
n_square ?? ??
lower 1 1
upper 10 10
step ?? 1

An Example
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n ?? 1
n_square ?? ??
lower 1 1
upper 10 10
step 1 1

An Example
 Execution
 n <= upper is true.
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n 1 1
n_square ?? ??
lower 1 1
upper 10 10
step 1 1

An Example
 Execution
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n 1 1
n_square ?? 1
lower 1 1
upper 10 10
step 1 1

An Example
 Execution
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n 1 1
n_square 1 1
lower 1 1
upper 10 10
step 1 1

An Example
 Execution
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n 1 2
n_square 1 1
lower 1 1
upper 10 10
step 1 1

An Example
 Execution
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n 2 2
n_square 1 1
lower 1 1
upper 10 10
step 1 1

An Example
 Execution
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n 2 2
n_square 1 4
lower 1 1
upper 10 10
step 1 1

An Example
 Execution
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n 2 2
n_square 4 4
lower 1 1
upper 10 10
step 1 1

An Example
 Execution
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n 2 3
n_square 4 4
lower 1 1
upper 10 10
step 1 1

An Example
 Execution
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n 10 11
n_square 100 100
lower 1 1
upper 10 10
step 1 1

An Example
 Execution
 n <= upper is false.
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n 11 11
n_square 100 100
lower 1 1
upper 10 10
step 1 1

An Example
 Execution
 n <= upper is false.
 Program ends
#include <stdio.h>
main() {
int n, n_square;
step = 1;
n = lower;
n_square = n * n;
n = n + 1;
}
}
Variable Before
exeution
After
executio
n
n 11 11
n_square 100 100
lower 1 1
upper 10 10
step 1 1

Printf Function
 The first argument of printf is a format specifier.
 Each % sign in the format specifier is a formatting instruction.
 %d print a decimal integer
 %f print a floating number
 %wd prints a decimal number with min width of w chars
 %w.pf prints a floating number with precision p and min width w
 Other characters in the format specifier are just printed.

Printf Function
 Examples
 Printf( “I am Charu” ); I am Charu
 Printf( “|%3d|” , 8 ); | 8|
 Printf( “|%03d|” , 8 ); |8 |
 Printf( “|%3d|” , 8000 ); |8000|
 Printf( “|%f|” , 123.456 ); |
123.456000|
 Printf( “|%f|” , 123.4567890123 ); |123.456789|
 Printf( “|%8.3d|” , 123.4567890123 ); | 123.456|
 Printf( “ Age = %d years” , 22 ); Age = 22 years
 Printf( “%d and %d” , 1, 2 ); 1 and 2

An Example
 Area of a circle
 The program produces the following
output when 5.0 is input
Enter the radius 5.0
Area = 78.539749
Peri = 31.415899
 And produces the following when –
3.0 is input.
Enter the radius -3.0
Negative radius
/* Compute Area and Perimeter of a
circle */
#include <stdio.h>
main() {
float rad;
float area, peri;
printf( “Enter the radius “ );
scanf(“%f” , &rad);
if ( rad > 0.0 ) {
area = 3.14159 * rad * rad;
peri = 6.28318 * rad;
printf( “Area = %fn” , area );
printf( “Peri = %fn” , peri );
}
else
printf( “Negative radiusn”);
}

An Example
 Decisions
 The form of if-else statement is
as follows:
if ( condition ) statements
if ( condtion ) statements
else statements
 If condition is true then if-block of
statements is executed otherwise
else-block of statements is
executed.
circle */
#include <stdio.h>
main() {
float rad;
float area, peri;
if ( rad > 0.0 ) {
area = 3.14159 * rad * rad;
peri = 6.28318 * rad;
}
else
}

An Example
 Execution /* Compute Area and Perimeter of a
circle */
#include <stdio.h>
main() {
float rad;
float area, peri;
if ( rad > 0.0 ) {
area = 3.14159 * rad * rad;
peri = 6.28318 * rad;
}
else
}
Variable Before
exeution
After
executio
n
rad ?? ??
area ?? ??
peri ?? ??

An Example
circle */
#include <stdio.h>
main() {
float rad;
float area, peri;
if ( rad > 0.0 ) {
area = 3.14159 * rad * rad;
peri = 6.28318 * rad;
}
else
}
Variable Before
exeution
After
executio
n
rad ?? 5.000000
area ?? ??
peri ?? ??

An Example
 Execution
 Condition rad > 0.0 is true
circle */
#include <stdio.h>
main() {
float rad;
float area, peri;
if ( rad > 0.0 ) {
area = 3.14159 * rad * rad;
peri = 6.28318 * rad;
}
else
}
Variable Before
exeution
After
executio
n
rad 5.000000 5.000000
area ?? ??
peri ?? ??

An Example
circle */
#include <stdio.h>
main() {
float rad;
float area, peri;
if ( rad > 0.0 ) {
area = 3.14159 * rad * rad;
peri = 6.28318 * rad;
}
else
}
Variable Before
exeution
After
executio
n
rad 5.00000 5.00000
area ?? 78.53975
peri ?? ??

An Example
circle */
#include <stdio.h>
main() {
float rad;
float area, peri;
if ( rad > 0.0 ) {
area = 3.14159 * rad * rad;
peri = 6.28318 * rad;
}
else
}
Variable Before
exeution
After
executio
n
rad 5.00000 5.00000
area 78.53975 78.53975
peri ?? 31.41590

An Example
circle */
#include <stdio.h>
main() {
float rad;
float area, peri;
if ( rad > 0.0 ) {
area = 3.14159 * rad * rad;
peri = 6.28318 * rad;
}
else
}
Variable Before
exeution
After
executio
n
rad 5.00000 5.00000
area 78.53975 78.53975
peri 31.41590 31.41590

An Example
 Execution
 Program ends
circle */
#include <stdio.h>
main() {
float rad;
float area, peri;
if ( rad > 0.0 ) {
area = 3.14159 * rad * rad;
peri = 6.28318 * rad;
}
else
}
Variable Before
exeution
After
executio
n
rad 5.00000 5.00000
area 78.53975 78.53975
peri 31.41590 31.41590

An Example
circle */
#include <stdio.h>
main() {
float rad;
float area, peri;
if ( rad > 0.0 ) {
area = 3.14159 * rad * rad;
peri = 6.28318 * rad;
}
else
}
Variable Before
exeution
After
executio
n
rad ?? -3.00000
area ?? ??
peri ?? ??

An Example
 Execution
 The condition rad > 0.0 is false
circle */
#include <stdio.h>
main() {
float rad;
float area, peri;
if ( rad > 0.0 ) {
area = 3.14159 * rad * rad;
peri = 6.28318 * rad;
}
else
}
Variable Before
exeution
After
executio
n
rad -3.00000 -3.00000
area ?? ??
peri ?? ??

An Example
 Execution
 The condition rad > 0.0 is false
 Prints “Negative radius” and
 Program ends
circle */
#include <stdio.h>
main() {
float rad;
float area, peri;
if ( rad > 0.0 ) {
area = 3.14159 * rad * rad;
peri = 6.28318 * rad;
}
else
}
Variable Before
exeution
After
executio
n
rad -3.00000 -3.00000
area ?? ??
peri ?? ??

Scanf Function
 The first argument of scanf is a format specifier.
 Each % sign in the format specifier is a formatting instruction.
 %d scans a decimal integer and stores it in int variable
 %f scans a floating number and stores it in float variable
 %wd maximum width of w chars are scanned
 %wf maximum width of w chars are scanned
 Other characters in the format specifier are expected to be input as is.
Best to avoid any other chars in the input.

Scanf Function
Format “%d%f” “%3d%6f”
Input Intvar Fltvar Intvar Fltvar
3 4.5 3 4.5 3 4.5
3t4.5 (tab) 3 4.5 3 4.5
3n4.5(newline) 3 4.5 3 4.5
34.5 34 0.5 34 0.5
003 4.5 3 4.5 3 4.5
0034.5 34 0.5 3 4.5
0003 4.5 3 4.5 0 3.0
00034.5 34 0.5 0 34.5
3 000004.5 3 4.5 3 4.0
3 4.5678912 3 4.567891 3 4.567800
A3a4.5 ?? ?? ?? ??
3a4.5 3 ?? 3 ??
 scanf ( format, &intvar, &fltvar );

Algorithms
 Problem
 Let S be a finite sequence of
positive nonzero integers ak, except
the last number which is always 0.
Find the largest number in the
sequence.
 Example
 7, 9, 4, 6, 2, 5, 8, 0
 The largest is 9.
 If Sk is a subsequence of first k
numbers of S and mk is the largest
term of Sk then
 mk+1 = max { mk , ak+1 }
Algorithm
1. The m1 be the largest in
S1. Clearly m1 = a1.
2. Let I = 1.
3. If aI+1 is zero, go to step
7.
4. mI+1 = max { mI, aI+1}.
5. Increment I.
6. Go to step 3.
7. Print mI.

An Example
 Find Largest
 The program produces the following
output
Enter a number 7
Enter a number 9
Enter a number 4
Enter a number 6
Enter a number 2
Enter a number 0
Largest is 9
/* Find the largest number */
#include <stdio.h>
main() {
int number, largest;
printf( "Enter a number " );
scanf( "%d" , &largest );
scanf( "%d" , &number );
while ( number > 0 ) {
if ( number > largest )
largest = number;
scanf( "%d" , &number );
}
printf( "Largest is %dn" ,
largest );
}

Lab Assignment 1
 Print a table of sine values of angles starting from 0o
upto 180o
in steps of
10o
. ( There is a standard library function sin(x) declared in math.h )
 Read a sequence of positive integers and count the number of input
integers and calculate their sum. Calculate the average of the integers.
 Let ax2
+bx+c=0. Read a, b and c. Print the solutions to the quadratic
equation. (The solutions may be complex. )
 Print a sequence of Fibonacci numbers less than 100.
 Input a positive integer and determine if it is a prime.

introduction numerical analysis lecture .ppt

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