This document discusses the Newton-Raphson method, an iterative method for finding the roots of a function. It provides the algorithm for the Newton-Raphson method, beginning with an initial guess that is improved upon in each step by using the tangent line of the function to get closer to the root. The document also includes source code in C that implements the secant method, another root-finding algorithm, to find the root of the function f(x) = xlog10(x) - 1.2.

1. S.N.PATEL INSTITUTE OF
TECHNOLOGY AND RESEARCH
CENTRE,UMRAKH
Secant Method
Prepared By: 150490105035
Subject : CEM(2140505)
1. Mr. Tejas Patel
2. Mrs. Jagruti Patel
Guided by:

2. Features of Newton’s Method:
 Type – open bracket
 No. of initial guesses – 1
 Convergence – quadratic
 Rate of convergence – faster
 Accuracy – good
 Programming effort – easy
 Approach – Taylor’s series

3. Newton Raphson Method:-
 The most used ittration method is Newton-raphson method, also
called newton`s methodand,
 When an apporimation value of a root of an equation is given, a
better and closer approximation to the root can be found using
this method.
 This method is a particular form of the ittration method dicussed
earlier and divided as follow,
 Suppose is a root of and is an estimate of . Then
by Taylor series expansion we have,

4.  By newton raphson method

5. Newton Raphson Method Algorithm:
 Start
 Read x, e, n, d
*x is the initial guess
e is the absolute error i.e the desired degree of accuracy
n is for operating loop
d is for checking slope*
 Do for i =1 to n in step of 2
 f = f(x)
 f1 = f'(x)
 If ( [f1] < d), then display too small slope and goto 11.
*[ ] is used as modulus sign*
 x1 = x – f/f1If ( [(x1 – x)/x1] < e ), the display the root as x1 and goto 11.
*[ ] is used as modulus sign*
 x = x1 and end loop
 Display method does not converge due to oscillation.
 Stop

6. Source code for secant method in C:
 #include<stdio.h>
 #include<math.h>
 float f(float x)
 {
 return x*log10(x) - 1.2;
 }
 float df (float x)
 {
 return log10(x) + 0.43429;
 }
 void main()
 {
 int itr, maxmitr;
 float h, x0, x1, allerr;
 printf("nEnter x0, allowed error and maximum iterationsn");

7.  scanf("%f %f %d", &x0, &allerr, &maxmitr);
 for (itr=1; itr<=maxmitr; itr++)
 {
 h=f(x0)/df(x0);
 x1=x0-h;
 printf(" At Iteration no. %3d, x = %9.6fn", itr, x1);
 if (fabs(h) < allerr)
 {
 printf("After %3d iterations, root = %8.6fn", itr, x1);
 return 0;
 }
 x0=x1;
 }
 printf(" The required solution does not converge or iterations are
insufficientn");
 return 1;
 }

8. References
 http://www.codewithc.com/c-program-for-newton-raphson-method/
 http://www.codewithc.com/newton-raphson-method-algorithm-flowchart/

9. Thank You