ABSTRACT
Broyden’s method is only one (though the one most widely known and used) representative of the broad class of iterative methods collectively called secant update methods. The methods of this class, given an initial pair (x0,A0) ∈ D × L(H) with invertible A0, generate the sequence (xn,An) ∈ D × L(H) as follows:
x+ := x−Af(x) , A+ := A+B , (5.1) where the update B is a linear operator of a finite rank (most often 1 or 2) such that A+ is invertible and satisfies the secant equation
A−1 + (x+ − x) = f(x+)− f(x) . (5.2)
As we have seen in Chapter 3, there is a great variety of updates B satisfying this condition. This fact raises the question: which of them is most preferable? The answer to this question depends on a criterion enabling us to compare any two given updates and to decide which one is better than the other. The criteria that have been used for justifying Broyden’s update hardly can serve in that capacity because they are unrelated to the purpose of the iterative method being designed: to locate a solution of the equation
f(x) = 0 . (5.3)
In this chapter, we use as such a criterion one that directly reflects this purpose: the entropy of the solution’s position within a set of its guaranteed existence and uniqueness (see Section 1.4).
