**Problem 1:**Write programs to find the real root of the following equations by using Newton Raphson Method.

a) f(x)
= x

^{3}– 3x – 1 = 0; correct to 5 decimal point near x = 0, 2, -2
b) x
sin x + cos x = 0; correct to 5 decimal point near x = 3

**(a)**

__Solve:__
function y = f1(x);

y
= x^3-3*x-1;

function y = f2(x);

y
= 3*x^2-3;

**When x = 0,**

clc

clear
all

close
all

e
= 10^(-5);

x0=0;

x1=0;

for i=1:100

x1=x0-(f1(x0)/f2(x0));

y=f1(x1);

if abs(y)<=e

break;

else x0=x1;

end

end

fprintf('The value of root is %.5f \n',x1);

fprintf('Number of iteration required %.f \n',i);

**When x = 2,**

clc

clear
all

close
all

e
= 10^(-5);

x0=2;

x1=0;

for i=1:100

x1=x0-(f1(x0)/f2(x0));

y=f1(x1);

if abs(y)<=e

break;

else x0=x1;

end

end

fprintf('The value of root is %.5f \n',x1);

fprintf('Number of iteration required %.f \n',i);

**When x =**-

**2,**

clc

clear
all

close
all

e
= 10^(-5);

x0=3;

x1=0;

for i=1:100

x1=x0-(f1(x0)/f2(x0));

y=f1(x1);

if abs(y)<=e

break;

else x0=x1;

end

end

fprintf('The value of root is %.5f \n',x1);

fprintf('Number of iteration required %.f \n',i);

**(b)**

__Solve:__
function y=g1(x);

y=x*sin(x)+cos(x);

function y=g2(x);

y=x*cos(x);

clc

clear
all

close
all

e
= 10^(-5);

x0=3;

x1=0;

for i=1:100

x1=x0-(g1(x0)/g2(x0));

y=g1(x1);

if abs(y)<=e

break;

else x0=x1;

end

end

fprintf('The value of root is %.5f \n',x1);

fprintf('Number of iteration required %.f \n',i);

**(c)**

__Solve:__
function y = h1(x);

y
= x-exp(-x);

function y = h2(x);

y
= 1 + exp(-x);

clc

clear
all

close
all

e
= 10^(-5);

x0=2;

x1=0;

for i=1:100

x1=x0-(h1(x0)/h2(x0));

y=h1(x1);

if abs(y)<=e

break;

else x0=x1;

end

end

fprintf('The value of root is %.5f \n',x1);

fprintf('Number of iteration required %.f \n',i);

**Problem 2:**Solve 1 (a) using

**roots, fzero, fsolve**Matlab function

__Solve:__

__ROOTS:__
clc

clear
all

close
all

A
= [1 0 -3 -1]

X
= roots(A)

__FZERO:__

clc

clear
all

close
all

a
= input('Enter the nearest guess value:');

x
=fzero(@(x) x^3-3*x-1,a)

__FSOLVE:__
clc

clear
all

close
all

g
= input('Enter the nearest guess value:');

X
= fsolve(@(x) x^3-3*x-1, g)

**Problem 3:**Solve 1(b) and 1(c) using

**fzero, fsolve**Matlab function.

**1 (b)**

__Solve:__

__FZERO:__
clc

clear
all

close
all

x
= fzero(@(x) x*sin(x)+cos(x),3)

__FSOLVE:__
clc

clear
all

close
all

root=fsolve(@(x)
x*sin(x)+cos(x),3)

**1 (c)**

__Solve:__

__FZERO:__
clc

clear
all

close
all

root
= fzero(@(x) x - exp(-x), 2)

__FSOLVE:__
clc

clear
all

close
all

root
= fsolve(@(x) x - exp(-x), 2)

**Problem 4:**What is the number of iterations required to find the root of problem 1(a), 1 (b), 1(c).

**Solve:**

Using
Newton Raphson Method,

No
of iteration required for 1(a) = 3

No
of iteration required for 1(b) = 3

No
of iteration required for 1(c) = 4

**Problem 5:**Write the advantage and disadvantage of Newton Raphson Method

**Solve:**

- The method is very expensive - It
needs the function evaluation and then the derivative evaluation.
- If the tangent is parallel or
nearly parallel to the x-axis, then the method does not converge.
- Usually Newton method is expected
to converge only near the solution.
- The advantage of the method is its
order of convergence is quadratic.
- Convergence rate is one of the
fastest when it does converge
- If the solution met convergence,
the required number of iteration decreases.

**Discussion:**

In
this experiment, we have used MATLAB Functions to find the value of root.

¢ The first method we used is
Newton Raphson Method. Which is an open iterative method. The required
equations and functionalities were studied at first. Then the procedure has
been coded and implemented on MATLAB. The root was obtained by providing an
initial guess value to the software.

¢ During the experiment a
minimum amount of error level was maintained. All the values of roots are 5
decimal place accurate.

¢ During conducting the
experiment none of the equations met divergence. So, the output was very fast
and found after a few numbers of iteration.

¢ We have compared the no of
iterations required in N/R Method with False Position Method. It seems N/R is a
very fast method and required no of iteration is less than False Position
Method.

¢ We have also learned
alternative ways to find out the root. The alternative functions include roots,
fzero, fsolve method.

¢ On roots no initialization
is required. By inputting the value of co-efficient the roots are obtained.
This function is only applicable for polynomial equations.

¢ fzero and fsolve function
can solve any type of equation. But a initial guess value is required to find
the nearby value of root.