This document provides an overview of the Newton-Raphson method for finding the roots of numerical equations. It discusses that the method uses tangent lines to locate approximate roots, and that the intersection points of successive tangents converge to the true root. It also outlines that the method is derived by taking the Taylor series expansion of the function and neglecting higher order terms, resulting in the Newton-Raphson formula to iteratively calculate better approximations. The document notes that the method converges quickly for small problems but can have difficulties with large systems due to computational challenges in calculating and inverting functions.

1. Numerical and Statical Methods
Government
Engineering Collage
Modasa

2. NEWTON RAPHSON
METHOD
BhargavTank

3. Introduction
■ Named after Isaac Newton and Joseph Raphson. This method is one of the best
method to solve the complex numerical equation. this method is based on linear
approximation.This method has fast rate of convergence than other methods.
■ This method uses tangent line of curve to locate the approximate range of root of the
equation.
■ Both tangents are always intersect with axis except when both are parallel in that case
there is no root possible.The intersection points of tangent are converge to the true
root.

4. Features of Newton-Raphson
method
■ Efficient for small molecules, converges quickly
■ Calculation and inversion is computationally difficult for large molecules
■ Approximation of quadratic surface poor, particularly far from minimum

5. Derivation of Newton-Raphson method
■ Let 𝑥0 be an approximate root of the equation 𝑓 𝑥 = 0,
■ If 𝑥1 = 𝑥0 + 𝑥0 be an exact root then 𝑓 𝑥1 = 0
■ Expanding 𝑓(𝑥0 + ℎ) byTaylor's series
𝑓 𝑥0 + 𝑓ℎ′
𝑥0 +
h2
2!
f′′
𝑥0 + … = 0
■ Since h is small, neglecting h2 and higher powers of h,
𝑓 𝑥0 + ℎ𝑓′ 𝑥0 = 0
h = -
𝑓 𝑥0
𝑓′ 𝑥0
𝑥1 = 𝑥0 + ℎ = 𝑥0 -
𝑓 𝑥0
𝑓′ 𝑥0

6. Conti…
■ Similarly, starting with 𝑥1, a still better approximation 𝑥2 is obtained
𝑥2 = 𝑥1 -
𝑓 𝑥1
𝑓′ 𝑥1
𝑥 𝑛+1 = 𝑥 𝑛 -
𝑓 𝑥 𝑛
𝑓′ 𝑥 𝑛
■ This equation is known as the newton-Raphson formula or newton's iteration formula.

7. ThankYou