Newton-Raphson's Method

↪ The Newton-Raphson’s Method or Newton’s Method is a powerful technique for solving

 equations numerically.

↪ Let f(x) be a well-behaved function and let x be a root of the equation f(x) = 0. We start

 with an estimate x₀ of x. From x₀, we produce an improved estimate x₁. From x₁ we

 produced a new estimate x₂. From x₂, we produce a new estimate x₃. We go on until we

 are close enough to r or until it becomes clear that we are getting nowhere. The above

 general style of proceeding is called iterative. Of the many iterative root finding

 procedures, the Newton Raphson’s Method, with its combination of simplicity and power,   
 is most widely used.

⁕ NEWTON RAPHSON’S ITERATION

↪ Let f(x) = 0 be an equation and x be its root. Let x₀ be a good estimate of x then for a

 point (x, y) sufficiently close to it the function can be approximated by its tangent line.

↪ This tangent line crosses the x-axis when y = 0. Denote this new value of x by x1

↪ This x intercept gives a better approximation to the function's root than the original

 guess.

     The next estimate x₂ is obtained from x₁ in exactly the same way as x₁ was obtained from        x₀

↪ Continue in this way, if xn is the current estimate, then the next estimate x_n + 1

is given by