Iterative Methods

Iterative Methods:

The term ``iterative method'' refers to a wide range of techniques that use successive approximations to obtain more accurate solutions to a linear system at each step. In this method . In this types of methods, one or more steps are repeatedly executed more than one times to obtain the require solution. This is the most powerful technique to solve the linear problem which has the same operation to be performed more than one times. In iterative technique  we use looping construct to repeat the same operations.

For example the algorithm to add the natural number up to n without iterative method can be

1. Input n

2. Sum=1+2+3+4+5+6+ ……+n;

3. Display sum

           If the number of terms is large this straight forward method is not suitable for the solution. Which is not appropriate algorithm for programming. We can write the iterative algorithm to add n natural number as:

1.      Input n

2.      count = 1 , sum=0

      while count <=1 do  repeat step 3 to 4

        3.  sum = sum + count

        4.  count = count +1

5.      print the sum

In this case  step 3 and 4 are repeatedly executed  until the condition is not true. If the condition is not true then rest of the step are executed.

  Hence we can see that writing the algorithm using iterative method is much more easier and practical to solve the problem.

Control Loops in C

In C iterative algorithms are written using the Loops. Loops are used to repeat a block of code. C gives us a choice of three types of loop, while loop, do while loop  and for loop

• The while loop keeps repeating an action until an associated test returns false. This is useful where the programmer does not know in advance how many times the loop will be traversed.
• The do while loops is similar, but the test occurs after the loop body is executed. This ensures that the loop body is run at least once.
• The for loop is frequently used, usually where the loop will be traversed a fixed number of times. It is very flexible.

However we can write an iterative algorithm to solve a problem using any one  of these three loops.

The while statement :

            The while statement, also called the while loop, executes a block of statements

as long as a specified condition is true. The while statement has the following form:

while (condition)

          statement;

condition is any C expression usually a relational or logical  expression and statement is a single or compound C statement. When program execution reaches a while statement, the following events occur:

  1. The expression condition is evaluated.

  2. If condition evaluates to false (0), the while statement   terminates, and execution  

       passes to the first statement following statement.

  3. If condition evaluates to true (nonzero), the C statement(s) in statement are executed.

  4. Execution returns to step 1.

Example A simple  while statement.

1:   /* Demonstrates a simple while statement */

2:

3:   #include <stdio.h>

4:

5:   int count;

6:

7:   void main()

8:   {

9:       /* Print the numbers 1 through 20 */

10:

11:      count = 1;

12:

13:      while (count <= 20)

14:      {

15:          printf("%3d", count);

16:          count++;

17:      }

18:  }

A small example example:

/* get numbers until you get one greater than 99 */

int number =0;

while (number  <= 99)

    scanf("%d", &number );

/* An example to read the marks of n students and calculates the average */

#include<stdio.h>

void main()

{

 float marks,total,avg;

 int num_of_student,count;

 printf(“\nEnter the no of students:”);

 scanf(“%d”,&num_of_student);

 total=0.0;

 count=0;

 while(count<num_of_student)

 {

  printf(“\nEnter the marks:”);

  scanf(“%f”,&marks);

  total+=marks;

  count++;

 }

 avg=total/count;

 printf(“\nThe average marks is %f “,avg);

}

Run the program and observe the output .

Nesting while Statements

Just like if else while statements can also be nested i.e. the while statement itself contains other while statements as our requirements:The general syntax of nested while is:

 While(expression1)

 {

   while(expression2)

    {

     statements;

    }

   other statements;   

     }

          

Use of break statement in while loop:

The break statement is used to break the loop even if the expression of while loop evaluates true. If break statement is encountered inside the while loop, the rest of the part of the loop is skipped i. e. loop is broken out.

An example:

 sum = 0;

 i = 1;

 while(i <= 10)

 {

   sum += i;

   i++;

 }

The above loops adds the number 1 to 10.

We can break from while loop using break statement writing lines above using break as :

sum = 0;

i = 1;

while(1)

{

  sum += i;

  i++;   

  if(i>10)

   break;

}

Use of continue statement in loop

Using continue statements inside a loop, we can skipped some statements and continue the update part(increment or decrement ) without executing the other statements in loop. Example for continue which is used to add the even number up to 10 is shown below.

i = 2;

sum = 0;

while(i<=10)

{

  sum += i;

  i +=2;

}

this can be equivalently written as

i = 1;

sum = 0;

while(i <= 10)

{

   if(i%2!=0)

   { i++;

    continue;

    }

    sum+= i++;

   

}

 

For statement:

The for statement is a C programming construct that executes a block of one or more statements a certain number of times. It is  called the for loop because program execution typically loops through the statement more than onceA for statement has the following structure:

for ( initialization statement; condition statement; 

      increment/decrement statement )

{

    statement sequences;

}

e.g. for(i = 0;i<= 10; i++)

         sum += i;

Here , the for loop has to execute only one statement: sum+=i; , so , we have not   used left and right curly brackets.

In the above for loop,

1. i  = 1; initialization statement
2. i <= 10 ; condition statement
3. i++  , increment statement

Rules regarding for loop:

1. A for loop generally contains three types of statement:  initialization, condition and increment/decrement.
2. All the three kind of statements in the for loop are optional. It is permitted to write:

for(; ;){

…..

….

}

But , it is absolutely necessary to put two semicolons in for loop even if there are no initialization and condition statements

3.  There can be more than one initialization statements and increment/decrement statements. In such cases each statements should be separated by comma ‘,’ . for example we can write :

for(sum = 0,i = 1; i <= 10;sum += i, i++)

 ;

4. There may not be single statement following for loop, even that case, a semicolon ‘;’ is must be put after the for loop indicating that the for loop executes a null statement.

How for loop works ?

Step 1:  Initialization statements are executed . these are executed only once.

Step 2:  Condition is tested

Step 3:  If condition is true, the statements associated with for loop are executed. If 

             condition is false, exit from the for loop.

Step 4: Increments are executed

Step 5: Go to step 2

A simple example of for loop:

/* This program adds numbers from 1 to 20 and display the sum in screen */

1: #include<stdio.h>

2:

3: void main()

4: {
5:  int sum,i;

6:  sum = 0;

7:  for(i=1;i<=20;i++)

8:    sum += i;

9:  printf(“The sum is  : %d” , sum);

10:}   /*end of main */

The Alternatives for the for statement in line no 7 of above program can be written as:

1.  for(sum = 0, i= 1; i<=20; sum+=i; i++)

;

2.  sum = 0;

i=1;

for(;i <= 20;sum += i,i++)

;

3.  sum = 0;

i=1;

for( ; i <= 20; i++)

 sum += i;

4.  sum = 0;

i=1;

for( ; i <= 20;) {

 sum += i;

 i++;

}

5.  sum = 0;

i = 1;

for(; ;)

{

  sum += i;

  i++;

  if(i>20)

   break;

}

In fifth case the for(;;) loop is infinite loop, as condition part is missing.Which is controlled by using break statement.

Use of continue and break

/* A simple program that adds the odd numbers from 1 to 99 */

#include<stdio.h>

void main()

{

 int sum = 0, i;

 for(i=1;i++)

 {

  if(i%2 == 0)

    continue;

  sum += i;

  if(i >= 99)

    break;

 }             /*end of for loop */

 printf(“The sum of 1,3,… ,99 is : %d”,sum);

}         /*end of main() */

break: breaks the loop and control goes out from the loop .

continue: goes to execute the increment or decrement part 

          of the for loop without executing rest of 

          statements in loop.

The do while statement:

do while is another type of looping construct in C in which statements are executed at least once even if the condition is not true. It executes the statements in the loop block at once and checks the condition . Until  the condition is not true it executes the loop repeatedly and when condition is false, the loop exits.

The general construct of  ‘do while’  statement is :

  do{

     Statements sequences;

    }while(condition );   

condition is any C expression, and statement is a single or compound C statement. When program execution reaches a do...while statement, the following events occur:

  1. The statements in statements sequences are executed.

  2. condition is evaluated. If it's true, execution returns to step 1. If it's

      false, the loop terminates.

The  statement in above program 

   for(sum = 0;i =1; i <= 20;i++)

        Sum += i;

  Can be written using  ‘do while’ loop as:

  sum = 0;

  i = 1;

do{

   sum += i;

   i++;

  }while(i <= 20 );

In while statement, condition is tested at the beginning and if the condition is false the statements in loop are not executed .of the loop  but in ‘do while’ the condition is tested at the end of the loop after executing the statements at once.

Use of continue in  while, do while, and for loop

1. In case of ‘while’ and ‘do while’ loop when the continue encountered, it skips the remaining of the statements of loop and goes to check condition.
2. In case of ‘for’ loop, when the continue is encountered, the  remaining statement of the loop are skipped and increment statements of for  loop executed and then the condition is tested.
3. The ‘continue’ statement is applied to loop only not in ‘switch’
4. The ‘break’ statement exits the loop skipping the remaining statements in loop in all types of loop.

e.g.

for(……; ……;…….)

{

     ……………

     ……………

     if(condition)

          break;

     ………….

     …………

}    /* end for */

while(condition)

{

   …………

    ……………

    if(condition)

       break;

   ………..

   ………….

}  /* end of while */

………..

……….