Numerical Methods Part 1

Hello!
I am Prijimol V B
Asst Prof.
Christ College Of Engineering
Irinjalakuda

, Newton Raphson Method


To solve equation f(x)=0

Newton Raphson Iteration Formula is

𝒇(𝒙𝒏 )
𝒙𝒏+𝟏 = 𝒙𝒏 −
𝒇′ (𝒙𝒏 )

, 1. Using Newton’s method find a root of the equation 𝑥 3 + 𝑥 − 1 = 0

Ans: 𝑓 𝑥 = 𝑥 3 + 𝑥 − 1

𝑇𝑜 𝑓𝑖𝑛𝑑 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑟𝑜𝑜𝑡 𝑤𝑒 𝑢𝑠𝑒 𝑡𝑟𝑖𝑎𝑙 𝑎𝑛𝑑 𝑒𝑟𝑟𝑜𝑟 𝑚𝑒𝑡ℎ𝑜𝑑
𝑓 0 = −1 < 0

𝑓 1 =1>0

Since there is a sign change of function at 0 and 1, therefore root lies between 0

Location of roots is (0,1)
0+1
Let 𝑥0 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑟𝑜𝑜𝑡 = = 0.5 (midpoint)
2

, 𝑓 𝑥0
𝑥1 = 𝑥0 −
𝑓′ 𝑥0
𝑓 𝑥 = 𝑥 3 + 𝑥 − 1 and 𝑓 ′ 𝑥 = 3(𝑥)2
(0.5)3 + 0.5 −1 𝑓 𝑥0 = 𝑓 0.5 = (0.5)3 + 0.5 − 1 =
= 0.5 −
3(0.5)2 +1
𝑓 ′ 𝑥0 = 3(0.5)2 +1 = 1.75
= 0.7142857

𝑓 𝑥1
𝑥2 = 𝑥1 −
𝑓′ 𝑥1


(0.7142857 )3 + 0.7142857 −1
= 0.7142857 −
3(0.7142857 )2 +1

= 0.6831797