《数值分析》课程PPT教学课件（英文版）Chapter 01 The Solution of Nonlinear Equations 1.2 Bracketing Methods for Locating a Root

1.2 Bracketing Methods for Locating a root

1.2 Bracketing Methods for Locating a Root

Definition 1.3 (Root of an Equation, Zero of a Function. Assume that f(c) is a continuous function. Any number r for which f(r)=0 is called a root of the equation f(a)=0. Also, we say r is a zero of the function f(a)

1.2 1 The bisection method of bolzano

1.2.1 The Bisection Method of Bolzano

If f(a) and f(c have opposite signs, a zero lies in [a, c If f(c) and f(b) have opposite signs, a zero lies in c, bl If f(c=0, then the zero is c

Theorem 1.4(Bisection Theorem). Assume that f E Cla, b and that there exists a number r E [a, b such that f(r)=0. If f(a) and f(b) have opposite signs, and icn ingo represents the sequence of midpoints generated by the bisection process of (122)and(1.23),then r-cn|≤ (1.24) and there fore the sequence icn_o converges to the zero =r; that is m Cn=7 (1.25)

Example 1. 7. The function h(a)=. sin()occurs in the study of undamped forced oscillations. Find the value of r that lies in the interval 0, 2, where the function takes on the value h(a)=1(the function sin(a) is evaluated in radians)

k Left end point, ak Midpoint, Ck Right end point, bk Function value, f( 0.158529 11.0 1.5 2.0 0.496242 21.00 1.25 1.50 0.186231 31.000 1.125 1.250 0.015051 41.000 1.0615 1.1250 0.071827 51.06250 1.09375 1.12500 0.028362 61.093750 1.109375 1.125000 0.006643 71.1093750 1.11718751.1250000 0.004208 81.10937500 1113281251.11718750 0.001216

N= int In(b-a)-In(8)

Method of false position (Regula false method .2 0.2

Method of false position (Regula false method)

where the points(a, f(a) and(b, f(b))are used 0-f(