OPEN METHODS

For the bracketing methods in the previous chapter, the root is located within an interval prescribed by a lower and an upper bound. Repeated application of these methods always results in doser estimates of the true value of the root.

Such methods are said to be convergent because they move closer to the truth as the computation progresses. For the contrast, the open methods described in this chapter are based on formulas that require only a single starting value of x or two starting values that do not necessarily bracket the root. As such, they sometimes diverge or move away from the true root as the computation progresses.

 However, when the open methods converge, they usually do so much more quickly lhan the bracketing methods. We will begin ours discussion of open techniques with a simple version that is useful for illustrating their general form and also for demonstrating the concept of convergence.