Solved Problems of Newton Raphson Method
Problem 1: For the initial value x0 = 1, approximate the root of f(x)=x2β5x+1.
Solution:
Given, x0 = 1 and f(x) = x2-5x+1
f'(x) = 2x-5
f'(x0) = 2 β 5 = -3
f(x0) = f(1) = 1 β 5 + 1 = -3
Using Newton Raphson method:
x1 = x0 β f(x0)/f'(x0)
β x1 = 1 β (-3)/-3
β x1 = 1 -1
β x1 = 0
Problem 2: For the initial value x0 = 2, approximate the root of f(x)=x3β6x+1.
Solution:
Given, x0 = 2 and f(x) = x3-6x+1
f'(x) = 3x2 β 6
f'(x0) = 3(4) β 6 = 6
f(x0) = f(2) = 8 β 12 + 1 = -3
Using Newton Raphson method:
x1 = x0 β f(x0)/f'(x0)
β x1 = 2 β (-3)/6
β x1 = 2 + 1/2
β x1 = 5/2 = 2.5
Problem 3: For the initial value x0 = 3, approximate the root of f(x)=x2β3.
Solution:
Given, x0 = 3 and f(x) = x2-3
f'(x) = 2x
f'(x0) = 6
f(x0) = f(3) = 9 β 3 = 6
Using Newton Raphson method:
x1 = x0 β f(x0)/f'(x0)
β x1 = 3 β 6/6
β x1 = 2
Problem 4: Find the root of the equation f(x) = x3 β 3 = 0, if the initial value is 2.
Solution:
Given x0 = 2 and f(x) = x3 β 3
f'(x) = 3x2
f'(x0 = 2) = 3 Γ 4 = 12
f(x0) = 8 β 3 = 5
Using Newton Raphson method:
x1 = x0 β f(x0)/f'(x0)
β x1 = 2 β 5/12
β x1 = 1.583
Using Newton Raphson method again:
x2 = 1.4544
x3 = 1.4424
x4 = 1.4422
Therefore, the root of the equation is approximately x = 1.442.
Problem 5: Find the root of the equation f(x) = x3 β 5x + 3 = 0, if the initial value is 3.
Solution:
Given x0 = 3 and f(x) = x3 β 5x + 3 = 0
f'(x) = 3x2 β 5
f'(x0 = 3) = 3 Γ 9 β 5 = 22
f(x0 = 3) = 27 β 15 + 3 = 15
Using Newton Raphson method:
x1 = x0 β f(x0)/f'(x0)
β x1 = 3 β 15/22
β x1 = 2.3181
Using Newton Raphson method again:
x2 = 1.9705
x3 = 1.8504
x4 = 1.8345
x5 = 1.8342
Therefore, the root of the equation is approximately x = 1.834.
Newton Raphson Method
Newton Raphson Method or Newton Method is a powerful technique for solving equations numerically. It is most commonly used for approximation of the roots of the real-valued functions. Newton Rapson Method was developed by Isaac Newton and Joseph Raphson, hence the name Newton Rapson Method.
Newton Raphson Method involves iteratively refining an initial guess to converge it toward the desired root. However, the method is not efficient to calculate the roots of the polynomials or equations with higher degrees but in the case of small-degree equations, this method yields very quick results. In this article, we will learn about Newton Raphson Method and the steps to calculate the roots using this method as well.
Table of Content
- What is Newton Raphson Method?
- Newton Raphson Method Formula
- Newton Raphson Method Calculation
- Convergence of Newton Raphson Method
- Articles related to Newton Raphson Method:
- Newton Raphson Method Example
- Solved Problems of Newton Raphson Method
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