[工学]数值分析英文版课件 2.ppt

Chapter 3 Solution of Nonlinear Equations Lecturer: GUO Tongtong Time: 15th December, 2010 The 11th Lecture Topics of Today 3.0 Introduction 3.1 Bisection( Interval Halving) Method 3.2 Newton’s Method 3.3 Secant Method Topics of Today 3.0 Introduction 3.1 Bisection( Interval Halving) Method 3.2 Newton’s Method 3.3 Secant Method 3.0 Introduction(1) The objective of this chapter is : To solve the roots of equations (or zeros of functions). To solve a system of nonlinear equations: to find x such that f (x) = 0 or finding X = (x1,x2,…,x n) T so that F (X) = 0. 3.0 Introduction(2) Examples of nonlinear equations can be found in many applications. In the theory of diffraction of light, we need the roots of the equation 3.0 Introduction(3) In the calculation of planetary orbits, we need the roots of Kepler’s equation for various values of a and b. 3.0 Introduction(4) In this chapter, we begin with three simple methods that are quite useful: The bisection method, Newton’s method, and the secant method. Also, we discuss special methods for computing the zeros of polynomials. Topics of Today 3.0 Introduction 3.1 Bisection( Interval Halving) Method 3.1.0 Introduction 3.1.1 Bisection Algorithm 3.1.2 Error Analysis 3.2 Newton’s Method 3.3 Secant Method 3.1.0 Introduction (1) If f is a continuous function on the interval [a, b] and if f (a) f (b) 0, then f must have a zero in (a, b). Since f (a) f (b) 0, the function f changes sign on the interval [a, b] and, therefore, it has at least one zero in the interval. 3.1.0 Introduction (2) This is a consequence of the Intermediate-Value Theorem (中值定理). The bisection method exploits this idea in the following way: If f (a) f (b) 0, then we compute c = ?(a+b) and test whether f (a) f (c) 0. 3.1.0 Introduction (3) If this is true, then f has a zero in [a, c]. So we rename c as b and start again with the new interval [a, b], which is half as large as the original interval. If f (a) f (c