A de Montessus Type Convergence Study for a Vector-Valued Rational Interpolation Procedure of Epsilon Class

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 K\"{o}nig 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 K\"{o}nig type theorems analogous to those obtained for IMPE and IMMPE

SVD-MPE: An SVD-Based Vector Extrapolation Method of Polynomial Type

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

A Further Property of Functions in Class B(m){\bf B}^{\boldsymbol(m)}

We say that a function $\alpha(x)$ belongs to the set ${\bf A}^{(\gamma)}$ if it has an asymptotic expansion of the form $\alpha(x)\sim \sum^\infty_{i=0}\alpha_ix^{\gamma-i}$ 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 $g\in{\bf A}^{(2)}$

Acceleration of Convergence of Some Infinite Sequences {An}\boldsymbol{\{A_n\}} Whose Asymptotic Expansions Involve Fractional Powers of n\boldsymbol{n}

In this paper, we deal with the acceleration of the convergence of infinite series $\sum^\infty_{n=1}a_n$, when the terms $a_n$ are in general complex and have asymptotic expansions that can be expressed in the form $$a_n\sim[\Gamma(n)]^{s/m}\exp\left[Q(n)\right]\sum^\infty_{i=0}w_i n^{\gamma-i/m}\quad\text{as $n\to\infty$},$$ where $\Gamma(z)$ is the gamma function, $m\geq1$ is an arbitrary integer, $Q(n)=\sum^{m}_{i=0}q_in^{i/m}$ is a polynomial of degree at most $m$ in $n^{1/m}$, $s$ is an arbitrary integer, and $\gamma$ is an arbitrary complex number. This can be achieved effectively by applying the $\tilde{d}^{(m)}$ transformation of the author to the sequence $\{A_n\}$ of the partial sums $A_n=\sum^n_{k=1}a_k$, $n=1,2,\dots\ .$ We give a detailed review of the properties of such series and of the $\tilde{d}^{(m)}$ transformation and the recursive W-algorithm that implements it. We illustrate with several numerical examples of varying nature the remarkable performance of this transformation on both convergent and divergent series. We also show that the $\tilde{d}^{(m)}$ transformation can be used efficiently to accelerate the convergence of some infinite products of the form $\prod^\infty_{n=1}(1+v_n)$, where $$v_n\sim \sum^\infty_{i=0}e_in^{-t/m-i/m}\quad \text{as $n\to\infty$,\ \ $t\geq m+1$ an integer,}$$ and illustrate this with numerical examples. We put special emphasis on the issue of numerical stability, we show how to monitor stability, or lack thereof, numerically, and discuss how it can be achieved/improved in suitable ways

A New Algorithm for the Higher-Order GG-Transformation

Let the scalars $A^{(j)}_n$ be defined via the linear equations $$A_l=A^{(j)}_n+\sum^n_{k=1}\bar{\alpha}_ku_{k+l-1},\ \ l=j,j+1,\ldots,j+n\ .$$ Here the $A_i$ and $u_i$ are known and the $\bar{\alpha}_k$ 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

Minimal Polynomial and Reduced Rank Extrapolation Methods Are Related

Minimal Polynomial Extrapolation (MPE) and Reduced Rank Extrapolation (RRE) are two polynomial methods used for accelerating the convergence of sequences of vectors $\{{x}_m\}$. They are applied successfully in conjunction with fixed-point iterative schemes in the solution of large and sparse systems of linear and nonlinear equations in different disciplines of science and engineering. Both methods produce approximations $s_k$ to the limit or antilimit of $\{{x}_m\}$ that are of the form ${s}_k=\sum^k_{i=0}\gamma_i{x}_i$ with $\sum^k_{i=0}\gamma_i=1$, for some scalars $\gamma_i$. The way the two methods are derived suggests that they might, somehow, be related to each other; this has not been explored so far, however. In this work, we tackle this issue and show that the vectors $s_{k}^\text{MPE}$ and $s_k^\text{RRE}$ produced by the two methods are related in more than one way, and independently of the way the $x_m$ are generated. One of our results states that RRE stagnates, in the sense that ${s}_k^\text{RRE}={s}_{k-1}^\text{RRE}$, if and only if ${s}_{k}^\text{MPE}$ does not exist. Another result states that, when ${s}_{k}^\text{MPE}$ exists, there holds $$\mu_k{s}_k^\text{RRE} = \mu_{k-1}{s}_{k-1}^\text{RRE} + \nu_k{s}_{k}^\text{MPE} \quad \text{with} \quad \mu_k = \mu_{k-1} + \nu_k,$$ for some positive scalars $\mu_k$, $\mu_{k-1}$, and $\nu_k$ that depend only on ${s}_k^\text{RRE}$, ${s}_{k-1}^\text{RRE}$, and ${s}_{k}^\text{MPE}$, respectively. Our results are valid when MPE and RRE are defined in any weighted inner product and the norm induced by it. They also contain as special cases the known results pertaining to the connection between the method of Arnoldi and the method of generalized minimal residuals, two important Krylov subspace methods for solving nonsingular linear systems

Screened Potential of a Moving Meson In a Quark Gluon plasma

We consider the quark antiquark potential of a quarkonium moving with velocity $v$ through a quark-gluon plasma at temperature T. An explicit, configuration-space expression is found for the screened interaction between the quarks constituting the meson. This potential is non-spherical, but axially symmetric about the direction of $v$.Comment: 3 page

On a Vectorized Version of a Generalized Richardson Extrapolation Process

Let \{\xx_m\} be a vector sequence that satisfies \xx_m\sim \sss+\sum^\infty_{i=1}\alpha_i \gg_i(m)\quad\text{as $m\to\infty$}, \sss being the limit or antilimit of \{\xx_m\} and $\{\gg_i(m)\}^\infty_{i=1}$ being an asymptotic scale as $m\to\infty$, in the sense that $$\lim_{m\to\infty}\frac{\|\gg_{i+1}(m)\|}{\|\gg_{i}(m)\|}=0,\quad i=1,2,\ldots.$$ The vector sequences $\{\gg_i(m)\}^\infty_{m=0}$, $i=1,2,\ldots,$ are known, as well as \{\xx_m\}. In this work, we analyze the convergence and convergence acceleration properties of a vectorized version of the generalized Richardson extrapolation process that is defined via the equations \sum^k_{i=1}\braket{\yy,\Delta\gg_{i}(m)}\widetilde{\alpha}_i=\braket{\yy,\Delta\xx_m},\quad n\leq m\leq n+k-1;\quad \sss_{n,k}=\xx_n+\sum^k_{i=1}\widetilde{\alpha}_i\gg_{i}(n), \sss_{n,k} being the approximation to \sss. Here \yy is some nonzero vector, $\braket{\cdot\,,\cdot}$ is an inner product, such that \braket{\alpha\aaa,\beta\bb}=\bar{\alpha}\beta\braket{\aaa,\bb}, and \Delta\xx_m=\xx_{m+1}-~\xx_m and $\Delta\gg_i(m)=\gg_i(m+1)-\gg_i(m)$. By imposing a minimal number of reasonable additional conditions on the $\gg_i(m)$, we show that the error \sss_{n,k}-\sss has a full asymptotic expansion as $n\to\infty$. We also show that actual convergence acceleration takes place and we provide a complete classification of it

Dissociation of a Boosted Quarkonium in Quark Gluon Plasma

I consider the dissociation of a boosted quarkonium in a quark-gluon plasma. This dissociation is due to absorption of a thermal gluon. I discuss the dissociation in terms of the velocity of the quarkonium and the temperature of the quark-gluon plasma. I compare this dissociation rate to the one calculated without including the velocity of the quarkonium.Comment: 4 page

From moment explosion to the asymptotic behavior of the cumulative distribution for a random variable

We study the Tauberian relations between the moment generating function (MGF) and the complementary cumulative distribution function of a random variable whose MGF is finite only on part of the real line. We relate the right tail behavior of the cumulative distribution function of such a random variable to the behavior of its MGF near the critical moment. We apply our results to an arbitrary superposition of a CIR process and the time-integral of this process