Showing papers on "Steffensen's method published in 1986"
01 Jan 1986
TL;DR: In this paper, the authors use the Kantorovich theory as a starting point for finding solutions of non-linear equations and systems, where strong hypotheses on differentiability are made; analyticity is assumed.
Abstract: Newton’s method and its modifications have long played a central role in finding solutions of non-linear equations and systems. The work of Kantorovich has been seminal in extending and codifying Newton’s method. Kantorovich’s approach, which dominates the literature in this area, has these features: (a) weak differentiability hypotheses are made on the system, e.g., the map is C 2 on some domain in a Banach space; (b) derivative bounds are supposed to exist over the whole of this domain. In contrast, here strong hypotheses on differentiability are made; analyticity is assumed. On the other hand, we deduce consequences from data at a single point. This point of view has valuable features for computation and its theory. Theorems similar to ours could probably be deduced with the Kantorovich theory as a starting point; however, we have found it useful to start afresh.
389 citations
01 Jan 1986
TL;DR: In this paper, a discrete derivative of the Newton method for continuous iterations is introduced. But it is based on the usual Newton method (for a mapping F of ℝ n → Ω n ) and not on the concept of a derivative.
Abstract: We will now conclude our sight-seeing tour of the behaviour of discrete iterations by defining and analyzing a discrete Newton method. The usual Newton method (for a mapping F of ℝ n → ℝ n ) is based on the concept of a derivative. Since we introduced a discrete derivative in the previous chapter, it seems very natural to try to carry over the ideas behind Newton’s method in the continuous setting into the discrete context.
1 citations