Newton-Raphson Method: Finding Roots of Equations, Lecture notes of Engineering Mathematics

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NewtonMethod

f (x )

f(x )

x = x -

i i i i

• 1 Newton Method Finding the root of an equation Advantages

• Converges fast (quadratic convergence), if it converges.
• Requires only one guess

Compare the absolute relative approximate error with the pre-specified relative error

tolerance.

Also, check if the number of iterations has exceeded the maximum number of iterations

allowed. If so, one needs to terminate the algorithm and notify the user.

s 

Is?

Yes

No

Go to Step 2 using new

estimate of the root.

Stop the algorithm

 a  s

Step 4:

Example 1: 𝑓 𝑥 = 2 𝑥 2 − 3 , 𝑓 𝑥 = 0 Use Newton Raphson method to approximate the real zero close to x=1 until successive approximations differ by less than 0.. 𝑓′ 𝑥 = 4𝑥

2 − 3

f (x )

f(x )

x = x -

i i i i

• 1 N 𝒙𝒏 𝒇(𝒙𝒏) 𝒇′(𝒙𝒏)

1.225 – 1.2247 =0.003 < 0.005 (STOP)