DIVIDED DIFFERENCES AND POLYNOMIAL CONVERGENCES
نویسندگان
چکیده
منابع مشابه
Divided Differences
Starting with a novel definition of divided differences, this essay derives and discusses the basic properties of, and facts about, (univariate) divided differences. “It belongs to the most beautiful I have been able to do.”[Newton 1676]
متن کاملInterpolating polynomials and divided differences
We shall take it as known that a polynomial of degree n has at most n distinct zeros (a proof is given in Lemma 1 below). Given n+1 distinct real numbers xj and any numbers αj (0 ≤ j ≤ n), there is a unique polynomial p of degree at most n satisfying p(xj) = αj (0 ≤ j ≤ n). The polynomial is unique, since if p1 and p2 were two such polynomials, then p1−p2 would be zero at each xj: since it has ...
متن کاملOn Modified Divided Differences II
From (7) and (9) it is clear that the only essential information contained in Ei is the set of values i\u and the index rt. Note further that in (8), the successive Ei are used with increasing I and it follows from (9) that it is necessary to know ri before using the r¡n. On the other hand, in (6), the E¡ are used in decreasing sequence of / but from (7) it is not necessary to know rt until aft...
متن کاملOn Modified Divided Differences I
Although divided differences of a function are of basic importance in the theory of numerical analysis, they are not nearly as useful as ordinary differences in application. In the first place, they are much harder to generate ; not only because they involve unequal intervals of the argument, but because divisions are involved, often by small numbers. Furthermore, they are harder to interpret t...
متن کاملDivided differences of implicit functions
Divided differences can be viewed as a discrete analogue of derivatives and are commonly used in approximation theory, see [1] for a survey. Recently, the second author and Lyche established two univariate chain rules for divided differences [2], both of which can be viewed as analogous to Faà di Bruno’s formula for differentiating composite functions [6,7]. One of these formulas was simultaneo...
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ژورنال
عنوان ژورنال: Journal of the Korea Society for Industrial and Applied Mathematics
سال: 2016
ISSN: 1226-9433
DOI: 10.12941/jksiam.2016.20.001