Square Root of 17 by Newton Raphson Method:
The Newton-Raphson method is a numerical method for finding the approximate square root of a number (or finding approximate solutions to equations in the form f(x) = 0.
The formula to calculate the square root of a number using Newton-Raphson method is given as: \(x_{i+1}=x_i-\frac{f(x_i)}{f'(x_i)}, \), where f′(x) is the derivative of f(x) with respect to x.
Follow the steps given below to find the square root of 17 by Newton-Raphson method:
Step 1: The equation would be written as f(x) = x2 – 17 and repeatedly apply the formula above starting with an initial guess, x0.
Step 2: Take an initial guess, x0 = 4.
Step 3: Calculate the x1 = x0 – f(x0) / f′(x0)
= 4 – (42 – 17) / (2 × 4)
= 4.125.
Step 4: Take x1 = 4.125 as the new guess and apply the above formula for x2.
Step 5: Repeat steps 3 and 4 for better accuracy.
The solution will be the value of x that makes f(x) = 0. So, after a few more iterations, we get the square root of 17 by Newton-Raphson method is approximately 4.123.
