The Faà Di Bruno Formula With Applications

This dissertation mainly study some problems in algebraic combinatorics and numerical approximation. The study aims at the divided difference form of Faàdi Bruno's formula and its applications on determinant identities. Besides, a new explicit representation of Hermite interpolation also is given by using cycle index of symmetric group. Also, the numerical differentiation of the Hermite interpolation and the asymptotic representations for the remainder are presented.In the approximation theory of parametric curves, a general framework for high-accuracy, parametric, polynomial interpolation methods has been developed . Usually, we choose a system of nodes t₀,t₁,…, t_n, construct a polynomial p_n by interpolation at these nodes to approximate the curve f and guarantees the approximation order is as high as possible. Hereand g is a strictly increasing function.Suppose that the composite function h := f o g is smooth enough, then a bound of the higher derivative of the composite function h is often considered in the error analysis. In fact, what is more required is a bound on certain divided difference of h if h is not smooth enough. It is the first motivation of this paper to derive an explicit formula that expresses the divided difference of a composite function h = f o g on the given nodes t₀, t₁,…, t_n in terms of divided differences of f on the nodes g(t₀),g(t₁),…, g(t_n) and divided differences of g on the nodes t₀,t₁,…,t_n.Faàdi Bruno's formula has always played a very important role in combinatorial analysis. It is applied to solve the problem on the higher derivative of a composite function. It also has wide applications in statistics. The expression of Faa di Bruno's formula is where the exponential partial Bell polynomialandIn the third chapter of this paper, by mean of the chain rule for divided differenceand the Steffensen formula, four different expressions of Faa di Bruno's formula are derived. In particular, when the given system of nodes are equal to each other completely, it will lead to Faa di Bruno's formula of higher derivatives of function h. Various approaches to proving this formula have been proposed since Faàdi Bruno first found the general formula in 1857. Our proof can be regard as a new approach to proving this formula. In order to understand the formula well, we also provide some examples.Mina's determinant identity is a very beautiful identity:It is so interesting that many mathematicians gave various proofs and generalizations . By the properties of divided differences and the divided difference form of Faa di Bruno's formula, a discrete Mina's determinant identity is given. As a special case of the discrete Mina's determinant identity, when all nodes are equal to each other completely, we derive the Mina determinant identity again. Furthermore, we proposed a generalization of the discrete Mina's determinant identity with composite functions. Meanwhile, we study the determinant identities with weighed functions. All of these results generalize those of Chu[28] which include a lot of classical identities in combinatorial analysis. As applications we present some examples. Besides, we discuss Mina's determinant identity with q-derivative. By the relationship between divided differences and q-derivatives, we propose an analogous of Mina's determinant identity. Recently, Several generalizations of Faa di Bruno's formula for multivariate composite functions have appeared in the literature [33, 36, 71]. However, it seems that the details of the proofs and expressions are so cumbersome that they are difficult to be used for computing in practice. One of our aims is to find a simple expression which is fairly easy to remember and to generalize the divided difference form of Faa di Bruno's formula. The functions f and g satisfy f: V→F, g : U→V, and the function h : U→F is the composite function f o g, which is denoted by t→f(g(t)). Here F denotes the set of real numbers R or the set of complex numbers C. Moreover, U (?)F, V (?) F⁸ and s is a positive integer. We follow the definition of the divided difference of g from [27]. So we can obtain the following generalized Faa di Bruno's formula,whereis the ordinary partial Bell polynomial with several vector variables x₁,x₂,…,x_n.It is well known that interpolation and divided differences are very important tools in numerical analysis. They have wide applications on solving nonlinear equations and quadratures. However, few numerical analysis textbooks discuss the explicit expression of the general Hermite interpolation in detail. They often discuss the special Hermite interpolation which is only given the information of the values of the function and the first derivative. In this paper, we use a useful tool, i.e., the cycle index of symmetric group in combinatorics,Z_n(t_k) := Z_n(t_k|k∈[n]) := Z_n(t₁,t₂,…, t_n). In fact, it has the following representation: whereÏ€_n := {a := (a₁, a₂,…, a_n)∈Nⁿ|1a₁ + 2a₂ +…+na_n = n}is a set whose any element a := (a₁, a₂,…, a_n) corresponds to a way of partitioning n∈N into a₁ integers 1, a₂ integers 2,…, a_n integers n. Moreover, for a∈π_n,is the number of permutations of n symbols composed of a₁ cycles of length 1, a₂ cycles of length 2, ... , a_n cycles of length n. We derive an explicit representations for Hermite interpolation by using the cycle index of symmetric group. Furthermore, we exploit its numerical differential method and the asymptotic representations for the remainder. If the interpolated function f is sufficiently smooth on a neighborhood of a given point x, we use the derivatives of the Hermite interpolation polynomial to approximate the corresponding derivatives of the function f(x).