Natural Science Department Duy Tan University Newtons Method

Natural Science Department – Duy Tan University Newton’s Method In this section, we will learn: How to solve high-degree equations using Newton’s method. Lecturer: Ho Xuan Binh Da Nang-11/2014

Natural Science Department – Duy Tan University 1 INTRODUCTION Suppose that a car dealer offers to sell you a car for $18, 000 or for payments of $375 per month for five years. You would like to know what monthly interest rate the dealer is, in effect, charging you. To find the answer, you have to solve the equation 48 x(1 + x)60 - (1 + x)60 + 1 = 0 How would you solve such an equation? Newton’s method

Natural Science Department – Duy Tan University 2 NEWTON’S METHOD The geometry behind Newton’s method is shown here. We start with a first approximation x 1, which is obtained by one of the following methods: Newton’s method

Natural Science Department – Duy Tan University 2 NEWTON’S METHOD Consider the tangent line L to the curve y = f(x) at the point (x 1, f(x 1)) and look at the x-intercept of L, labeled x 2. Newton’s method

Natural Science Department – Duy Tan University 3 SECOND APPROXIMATION As the x-intercept of L is x 2, we set y = 0 and obtain: 0 - f(x 1) = f’(x 1)(x 2 - x 1) If f’(x 1) ≠ 0, we can solve this equation for x 2: Newton’s method

Natural Science Department – Duy Tan University 4 SUCCESSIVE APPROXIMATIONS If we keep repeating this process, we obtain a sequence of approximations x 1, x 2, x 3, x 4, . . . Newton’s method

Natural Science Department – Duy Tan University 5 SUBSEQUENT APPROXIMATION In general, if the nth approximation is xn and f’(xn) ≠ 0, then the next approximation is given by: Newton’s method

Natural Science Department – Duy Tan University 6 NOTE Consider the situation shown here. You can see that x 2 is a worse approximation than x 1. It might even happen that an approximation falls outside the domain of f, such as x 3. Newton’s method

Natural Science Department – Duy Tan University 7 Example Starting with x 1 = 2, find the third approximation x 3 to the root of the equation x 3 – 2 x – 5 = 0. Newton’s method

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