Chapter 2
                                                       Solutions of Equations in One
                                                                                                                                Variable
                                                                                                      Hung-Yuan Fan (范洪源)
                                                                                                              Department of Mathematics,
                                                                                        National Taiwan Normal University, Taiwan
                                                                                                                                  Spring 2016
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              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)
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              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
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              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.
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              Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan                                                                                                    Chap . 2, Numerical Analysis (I)                                                        4/108