Avram Sidi

Avram Sidi

In a series of recent publications of the author, three interpolation procedures, denoted IMPE, IMMPE, and ITEA, were proposed for vector-valued functions $F(z)$, where $F : \C \to\C^N$, and their algebraic properties were studied. The convergence studies of two of the methods, namely, IMPE and IMMPE, were also carried out as these methods are being applied to meromorphic functions with simple poles, and de Montessus and König type theorems for them were proved. In the present work, we concentrate on ITEA. We study its convergence properties as it is applied to meromorphic functions with simple poles, and prove de Montessus and König type theorems analogous to those obtained for IMPE and IMMPE.

Avram Sidi

Let the scalars $A^{(j)}_n$ be defined via the linear equations $$A_l=A^{(j)}_n+\sum^n_{k=1}\barα_ku_{k+l-1},\ \ l=j,j+1,\ldots,j+n\ .$$ Here the $A_i$ and $u_i$ are known and the $\barα_k$ are additional unknowns, and the quantities of interest are the $A^{(j)}_n$. This problem arises, for example, when one computes infinite-range integrals by the higher-order $G$-transformation of Gray, Atchison, and McWilliams. One efficient procedure for computing the $A^{(j)}_n$ is the rs-algorithm of Pye and Atchison. In the present work, we develop yet another procedure that combines the FS-algorithm of Ford and Sidi and the qd-algorithm of Rutishauser, and we denote it the FS/qd-algorithm. We show that the FS/qd-algorithm has a smaller operation count than the rs-algorithm. We also show that the FS/qd algorithm can also be used to implement the transformation of Shanks, and compares very favorably with the $\varepsilon$-algorithm of Wynn that is normally used for this purpose.

Avram Sidi

We say that a function $α(x)$ belongs to the set ${\bf A}^{(γ)}$ if it has an asymptotic expansion of the form $α(x)\sim \sum^\infty_{i=0}α_ix^{γ-i}$ as $x\to\infty$, which can be differentiated term by term infinitely many times. A function $f(x)$ is in the class ${\bf B}^{(m)}$ if it satisfies a linear homogeneous differential equation of the form $f(x)=\sum^m_{k=1}p_k(x)f^{(k)}(x)$, with $p_k\in {\bf A}^{(i_k)}$, $i_k$ being integers satisfying $i_k\leq k$. These functions have been shown to have many interesting properties, and their integrals $\int^\infty_0 f(x)\,dx$, whether convergent or divergent, can be evaluated very efficiently via the Levin--Sidi $D^{(m)}$-transformation. (In case of divergence, they are defined in some summability sense, such as Abel summability or Hadamard finite part or a mixture of these two.) In this note, we show that if $f(x)$ is in ${\bf B}^{(m)}$, then so is $(f\circ g)(x)=f(g(x))$, where $g(x)>0$ for all large $x$ and $g\in {\bf A}^{(s)}$, $s$ being a positive integer. This enlarges the scope of the $D^{(m)}$-transformation considerably to include functions of complicated arguments. We demonstrate the validity of our result with an application of the $D^{(3)}$ transformation to two integrals $I[f]$ and $I[f\circ g]$, for some $f\in{\bf B}^{(3)}$ and $g\in{\bf A}^{(2)}$.

Avram Sidi

An important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors $\{\xx_m\}$, where $\xx_m\in \C^N$, $N$ being very large. Such sequences arise, for example, in the solution of systems of linear or nonlinear equations by fixed-point iterative methods, and $\lim_{m\to\infty}\xx_m$ are simply the required solutions. In most cases of interest, however, these sequences converge to their limits extremely slowly. One practical way to make the sequences $\{\xx_m\}$ converge more quickly is to apply to them vector extrapolation methods. Two types of methods exist in the literature: polynomial type methods and epsilon algorithms. In most applications, the polynomial type methods have proved to be superior convergence accelerators. Three polynomial type methods are known, and these are the {minimal polynomial extrapolation} (MPE), the {reduced rank extrapolation} (RRE), and the {modified minimal polynomial extrapolation} (MMPE). In this work, we develop yet another polynomial type method, which is based on the singular value decomposition, as well as the ideas that lead to MPE. We denote this new method by SVD-MPE. We also design a numerically stable algorithm for its implementation, whose computational cost and storage requirements are minimal. Finally, we illustrate the use of {SVD-MPE} with numerical examples.