The Method
The method is applicable when we wish to solve the equation f(x) = 0 for the real variable x, where f is a continuous function defined on an interval and f(a) and f(b) have opposite signs. In this case a and b are said to bracket a root since, by the intermediate value theorem, the f must have at least one root in the interval (a, b).
At each step the method divides the interval in two by computing the midpoint c = (a+b) / 2 of the interval and the value of the function f(c) at that point. Unless c is itself a root (which is very unlikely, but possible) there are now two possibilities: either f(a) and f(c) have opposite signs and bracket a root, or f(c) and f(b) have opposite signs and bracket a root. The method selects the subinterval that is a bracket as a new interval to be used in the next step. In this way the interval that contains a zero of f is reduced in width by 50% at each step. The process is continued until the interval is sufficiently small.
Explicitly, if f(a) and f(c) are opposite signs, then the method sets c as the new value for b, and if f(b) and f(c) are opposite signs then the method sets c as the new a. (If f(c)=0 then c may be taken as the solution and the process stops.) In both cases, the new f(a) and f(b) have opposite signs, so the method is applicable to this smaller interval.
Read more about this topic: Bisection Method
Famous quotes containing the word method:
“I have a new method of poetry. All you got to do is look over your notebooks ... or lay down on a couch, and think of anything that comes into your head, especially the miseries.... Then arrange in lines of two, three or four words each, don’t bother about sentences, in sections of two, three or four lines each.”
—Allen Ginsberg (b. 1926)
“The insidiousness of science lies in its claim to be not a subject, but a method. You could ignore a subject; no subject is all-inclusive. But a method can plausibly be applied to anything within the field of consciousness.”
—Katharine Fullerton Gerould (1879–1944)