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Chapter 2 Solutions of Equations in One Variable Hung-Yuan Fan (范洪源) Department of Mathematics, National Taiwan Normal University, Taiwan Spring 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chap . 2, Numerical Analysis (I) 1/108 Section 2.1 The Bisection Method (二分法) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chap . 2, Numerical Analysis (I) 2/108 Solutions of Nonlinear Equations Root-Finding Problem (勘根問題) One of the most basic problems in numerical analysis. Try to find a root (or solution) p of a nonlinear equation of the form f(x) = 0, given a real-valued function f, i.e. f(p) = 0. The root p is also called a zero (零根) of f. Note: Three numerical methods will be discussed here: Bisection method Newton’s (or Newton-Raphson) method Secant and False Position (or Regula Falsi) methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chap . 2, Numerical Analysis (I) 3/108 The Procedure of Bisection Method Assume that f is well-defined on the interval [a,b]. Set a = a and b = b. Find the midpoint p of [a ,b ] by 1 1 1 1 1 b −a a +b p =a + 1 1 = 1 1. 1 1 2 2 If f(p ) = 0, set p = p and we are done. 1 1 If f(p1) ̸= 0, then we have f(p ) · f(a ) > 0 ⇒ p ∈ (p ,b ). Set a = p and b = b . 1 1 1 1 2 1 2 1 f(p ) · f(a ) < 0 ⇒ p ∈ (a ,p ). Set a = a and b = p . 1 1 1 1 2 1 2 1 Continue above process until convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chap . 2, Numerical Analysis (I) 4/108
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