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 Bracketing Methods for Locating a Root
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, b If f(c=0, then the zero is c
1.2.1 The Bisection Method of Bolzano
Theorem 1.4(Bisection Theorem). Assume that f e C[a, 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 icninso represents the sequence of midpoints generated by the bisection process of (1.22)and(1.23),then 2n+1 7=0,1, (1.24) and there fore the sequence (cn ino converges to the zero a=r; that is lim c=r 1.25 n→
Example 1.7. The function h(c)=a sin(z) occurs in the study of undamped forced oscillations. Find the value of r that lies in the interval 0, 2, where the function takes h on the value nl r )=1(the function sin(a)is evaluated in radians)
k Left end point, ak Midpoint, Ck Right end point, bk Function value, f(ck) 0.158529 11.0 2.0 0.496242 21.00 1.25 1.50 0.186231 31.000 1.125 1.250 0.015051 410000 1.0615 1.1250 0.071827 51.06250 1.09375 1.12500 0.028362 61.093750 1.109375 1.125000 0.006643 71.1093750 1.1171875 1.1250000 0.004208 81.10937500 1.113281251.11718750 0.001216 N= int In(b-a)-In(8) ln(2)
Method of false position (Regula false method) .3
Method of false position (Regula false method)
where the points(a, f(a) and(6, f(b)are used, and f(6) m
Equating the slopes in(1.30) and (1.31), we have f(b)-f(a)0-f(b) which is easily solved for c to get c=b f(b)(b f(b)-f(a 32) The three possibilities are the same as before
If f(a) and f(c) have opposite signs, a zero lies in a, c].(1 If f(c) and f(b) have opposite signs, a zero lies in [c, b (1.34) If f(c)=0, then the zero is c (1.35)
1.2.2 Convergence of the false Position method bn f(bn )(6n-an)
1.2.2 Convergence of the False Position Method