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:
“Traditional scientific method has always been at the very best 20-20 hindsight. It’s good for seeing where you’ve been. It’s good for testing the truth of what you think you know, but it can’t tell you where you ought to go.”
—Robert M. Pirsig (b. 1928)
“In child rearing it would unquestionably be easier if a child were to do something because we say so. The authoritarian method does expedite things, but it does not produce independent functioning. If a child has not mastered the underlying principles of human interactions and merely conforms out of coercion or conditioning, he has no tools to use, no resources to apply in the next situation that confronts him.”
—Elaine Heffner (20th century)