Newton Raphson Method
This is used to determine an equation’s root. We start by making a guess, then we use a formula to get better each time. If our initial guess is correct, it operates quickly.
Example : Find a root of the equation x2−4=0 using Newton Raphson method.
Steps:
- Let’s choose an initial guess:
Let x0=3 - The function:
f(x)=x2−4 - Its derivative:
f′(x)=2x - Newton Raphson formula:
xn+1=xn−f(xn)f′(xn)
First iteration:
- x1=x0−f(x0)f′(x0)
- x1=3−32−42×3
- x1=3−9−46
- x1=3−56
- x1=3−0.833=2.167
Second iteration:
- x2=2.167−2.1672−42×2.167
- x2=2.167−4.697−44.334
- x2=2.167−0.6974.334
- x2=2.167−0.161=2.006
Third iteration:
- x3=2.006−2.0062−42×2.006
- x3=2.006−4.024−44.012
- x3=2.006−0.0244.012
- x3=2.006−0.006=2.000
So, after a few steps, we get very close to the root x=2.
Final Answer:
Root is approximately x=2.
