Question

Newton's Method Derivation (20 pts) Derive Newton's method, also known as Newton- Raphson method, starting from Taylor Series. (a) Write the first order Taylor series expansion for f (x) about xo. We will call this polynomial To(x). Define your step as (xı - xo) (b) What kind of curve is To(x) (line, parabola, cubic, ...)? (c) Solve for the root of To(x). This will give you x1. If you're not sure what to do (d) Repeat the steps above but about to solve for r2. You can call your new Taylor (e) How would your derivation change if you wanted to know when your function was here, first answer the question: What value should To(x1) take? series Ti(r). If f(22) is within your error tolerance, what could say about x2? equal to 3, instead of 0? (There are two methods of doing this, give one of them)

Answer 1

Derivation of NEWTON-RAPHSON method using taylor series: Given that Let f(x) be a polynomial a) First order taylor series expansion of f(x) about x_0 is given by T_0 (x) middot f(x_0) + f^'(x_0) (x_1 - x_0) -(1) b) T_0 (x) is a line with f^'(x_0) is slope and passing through f(x_0) c) Solving for the root of T_0(x) we get 0 = f(x_0) + f^'(x_0) (x_1 - x_0) x_1 - x_0 = -f(x_0)/f^'(x_0) x_1 = x_0 - f(x_0)/f^'(x_0) i.e. T-0 (x) = Delta_at x = x_1 d) Now i.e. T_1 (x) = f(x_1) + f^'(x_1) (x_2 - x_1) solving for root of T_1 (x) we get 0 = f(x_1) + f^'(x_1)(x_2 - x_1) x_2 - x_1 = -f(x_1)/f^'(x_1) \r\n x_2 = x_1 - f(x_1)/f^'(x_1) similarly we can write x_i + 1 = x_i - f(x_i)/f(x_i) which represents Newton-Raphson Root finding equation if f(x_2) is within the error tolerance we can say x_2 is the root of f(x) e) If the function is equal to 3 instead of 0