Bisection Method Lecture Notes

the bisection method is a Root finding method that applies to any continuous for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval defined by these values and then selecting the subinterval in which the function changes sign, and therefore must contain a root. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods. The method is also called the interval halving method f(x) 3 First, two numbers a and b have to be found such that above function, a=1 and b=2 satisfy this criterion, as f(a)= 13 − 1 − 2 = −2 ; f(a) and f(b) have opposite signs. For the f(b)= 23 − 2 − 2 = 4 Lets C1= (1+2)/2 = 1.5 f(c1)= 1.5 3 − 1.5 − 2 = −0.125 Since f(c1 ) is positive, replace a=1.5 to ensure that for the next Iteration f(a) and f(b) have opposite signs Iteration an bn cn f(cn) 1 1 2 1.5 −0.125 2 1.5 2 1.75 1.6093750 3 1.5 1.75 1.625 0.6660156 4 1.5 1.625 1.5625 0.2521973 5 1.5 1.5625 1.5312500 0.0591125 6 1.5 1.5312500 1.5156250 −0.0340538 7 1.5156250 1.5312500 1.5234375 0.0122504 8 1.5156250 1.5234375 1.5195313 −0.0109712 9 1.5195313 1.5234375 1.5214844 0.0006222 10 1.5195313 1.5214844 1.5205078 −0.0051789 11 1.5205078 1.5214844 1.5209961 −0.0022794 12 1.5209961 1.5214844 1.5212402 −0.0008289 13 1.5212402 1.5214844 1.5213623 −0.0001034 14 1.5213623 1.5214844 1.5214233 0.0002594 15 1.5213623 1.5214233 1.5213928 0.0000780