What is the formula for xn1 when fx X2 A If we take A 2 a

What is the formula for xn+1 when f(x) = X^2 - A? If we take A = 2 and x1 = 1.4 we get x1 = 1.4 , x2 = 1.4142857, x3 = 1.4142136, x4 = 1.4142136, which is already correct to 7 decimals! Notice that the number of correct decimals at least doubled each time. This is essentially guaranteed by the inequality (*) when M/m

Solution

F[X]=X^2 - A A = 2 USING NEWTONS FORMULA F\'[X]= 2X X[N+1] = X[N] - [F{X(N)} / F\'{X(N)}] X[N+1] = X[N] - [{X(N)^2 -A ] / [ 2*X(N)]