This document presents an overview of Newton Raphson's method for finding the roots of a function. It describes the theoretical background of using linearization and iterative improvement to approximate roots. The algorithm involves choosing an initial guess, defining the function, calculating derivatives, and iteratively applying the Newton Raphson formula until reaching the desired accuracy. An example finds the real root of a cubic equation. Advantages include fast convergence and efficiency, while limitations are sensitivity to initial guesses and the need to calculate derivatives. Applications include solving nonlinear equations, optimization, differential equations, and calculating option implied volatility.

1. Exploring Numerical
Analysis & An Overview of
Newton Raphson's Method
Presented by Raihan Hossain

2. Newton
Raphson's
Method
Newton-Raphson's Method is an
iterative technique for finding the roots
of a given function.
The formula for the Newton-Raphson
method is given as:
Here, Xn+1 is the next approximation of
the root, Xn is the current
approximation, f(Xn) is the value of the
function at the current approximation,
and f'(Xn) is the derivative of the
function at the current approximation.

3. Theoretical Background
The Newton-Raphson Method is
based on the idea of linearizing a
nonlinear equation by using its first
derivative. The method requires an
initial guess and involves iteratively
improving the guess until a desired
level of accuracy is achieved. The
method is widely used in fields such
as physics, engineering, and finance.

4. Algorithm for
Newton Raphson method

5. Step - 1
Choose an initial guess for the root, x0.

6. Step - 2
Define the function f(x) whose root is to be
found.

7. Step - 3
Calculate the derivative f'(x).

8. Step - 4
Calculate the next approximation to the
root, x1 using Newton Raphson formula

9. Step - 5
Check the difference between x1 and x0,if
both value are same for the given decimal
place, stop the iteration and output x1 as
the root. Otherwise, calculate x2 and repeat
steps 3 to 5 until desired outcome is
achieved.

10. Example
Finding a real root of the equation X3-2X2-4 = 0
Choice initial
value X0 = 2.5
Define the function
f(x) = X3-2X2-4
The derivative
f'(x) = 3X2-4X
Repeat 3 to 5
1 4 5
2 3

11. Advantages
Overall, the Newton-Raphson method is a powerful and efficient tool for
finding the roots of a function.
Fast convergence Flexibility Efficiency
Easy to implement Local convergence

12. Limitations and Drawbacks
04
Sensitivity to
the initial guess
Derivative
requirement
Multiple roots Not global
convergence
02
01
Divergence in
some cases
03
04 05

13. The Newton-Raphson Method has
various applications such as
● finding roots of nonlinear
equations
● optimization problems
● solving differential equations.
● In finance, the method is used to
calculate the implied volatility of
an option.
● The method can also be used to
solve complex engineering
problems such as heat transfer
and fluid dynamics.
Applications

14. Conclusion
The Newton-Raphson method is
powerful for finding roots but sensitive
to initial guess and derivative
requirement.

15. THANKS
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