arXiv subjects

Avram Sidi

Publications and source records attributed to Avram Sidi.

14 recordsLinked to original sources

arXiv2024

Analogue of the Cauchy-Schwarz inequality for determinants: a simple proof

In this note, we present a simple proof of an analogue of the Cauchy-Schwarz inequality relevant to products of determinants. Specifically, we show that $$ |\det(A^*MB)|^2\leq \det(A^*MA)\cdot \det(B^*MB),\quad A,B\in \mathbb{C}^{m\times n},$$ where $M\in\mathbb{C}^{m\times m}$ is hermitian positive definite. Here $m$ and $n$ are arbitrary. In case $m\leq n$, equality holds trivially. Equality holds when $m>n$ and $\text{rank}(A)=\text{rank}(B)=n$ if and only if the columns of $A$ and the columns of $B$ span the same subspace of $\mathbb{C}^m$.

math.GM↗

arXiv2021

Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form $f(x)=0$. In a recent work [A. Sidi, Generalization of the secant method for nonlinear equations. {\em Appl. Math. E-Notes}, 8:115--123, 2008] we presented a generalization of the secant method that uses only one evaluation of $f(x)$ per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer $k$, this method generates a sequence $\{x_n\}$ of approximations to a real root of $f(x)$, where, for $n\geq k$, $x_{n+1}=x_n-f(x_n)/p'_{n,k}(x_n)$, $p_{n,k}(x)$ being the polynomial of degree $k$ that interpolates $f(x)$ at $x_n,x_{n-1},\ldots,x_{n-k}$, the order $s_k$ of this method satisfying $1<s_k<2$. Clearly, when $k=1$, this method reduces to the secant method with $s_1=(1+\sqrt{5})/2$. In addition, $s_1<s_2<s_3<\cdots,$ such that and $\lim_{k\to\infty}s_k=2$. In this note, we study the application of this method to simple complex roots of a real or complex function $f(z)$. We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.

math.NA↗

arXiv2021

Unified Compact Numerical Quadrature Formulas for Hadamard Finite Parts of Singular Integrals of Periodic Functions

We consider the numerical computation of finite-range singular integrals $$I[f]=\intBar^b_a f(x)\,dx,\quad f(x)=\frac{g(x)}{(x-t)^m},\quad m=1,2,\ldots,\quad a<t<b,$$ that are defined in the sense of Hadamard Finite Part, assuming that $g\in C^\infty[a,b]$ and $f(x)\in C^\infty(\mathbb{R}_t)$ is $T$-periodic with $\mathbb{R}_t=\mathbb{R}\setminus\{t+ kT\}^\infty_{k=-\infty}$, $T=b-a$. Using a generalization of the Euler--Maclaurin expansion developed in [A. Sidi, {Euler--Maclaurin} expansions for integrals with arbitrary algebraic endpoint singularities. {\em Math. Comp.}, 81:2159--2173, 2012], we unify the treatment of these integrals. For each $m$, we develop a number of numerical quadrature formulas $\widehat{T}^{(s)}_{m,n}[f]$ of trapezoidal type for $I[f]$. For example, three numerical quadrature formulas of trapezoidal type result from this approach for the case $m=3$, and these are \begin{align*} \widehat{T}^{(0)}_{3,n}[f]&=h\sum^{n-1}_{j=1}f(t+jh)-\frac{π^2}{3}\,g'(t)\,h^{-1} +\frac{1}{6}\,g'''(t)\,h, \quad h=\frac{T}{n}, \widehat{T}^{(1)}_{3,n}[f]&=h\sum^n_{j=1}f(t+jh-h/2)-π^2\,g'(t)\,h^{-1},\quad h=\frac{T}{n}, \widehat{T}^{(2)}_{3,n}[f]&=2h\sum^n_{j=1}f(t+jh-h/2)- \frac{h}{2}\sum^{2n}_{j=1}f(t+jh/2-h/4),\quad h=\frac{T}{n}.\end{align*}

math.NA↗

arXiv2021

Exactness and Convergence Properties of Some Recent Numerical Quadrature Formulas for Supersingular Integrals of Periodic Functions

In a recent work, we developed three new compact numerical quadrature formulas for finite-range periodic supersingular integrals $I[f]=\intBar^b_a f(x)\,dx$, where $f(x)=g(x)/(x-t)^3,$ assuming that $g\in C^\infty[a,b]$ and $f(x)$ is $T$-periodic, $T=b-a$. With $h=T/n$, these numerical quadrature formulas read \begin{align*} \widehat{T}{}^{(0)}_n[f]&=h\sum^{n-1}_{j=1}f(t+jh) -\frac{π^2}{3}\,g'(t)\,h^{-1}+\frac{1}{6}\,g'''(t)\,h, \widehat{T}{}^{(1)}_n[f]&=h\sum^n_{j=1}f(t+jh-h/2) -π^2\,g'(t)\,h^{-1}, \widehat{T}{}^{(2)}_n[f]&=2h\sum^n_{j=1}f(t+jh-h/2)- \frac{h}{2}\sum^{2n}_{j=1}f(t+jh/2-h/4). \end{align*} We also showed that these formulas have spectral accuracy; that is, $$\widehat{T}{}^{(s)}_n[f]-I[f]=O(n^{-μ})\quad\text{as $n\to\infty$}\quad \forall μ>0.$$ In the present work, we continue our study of these formulas for the special case in which $f(x)=\frac{\cos\frac{π(x-t)}{T}}{\sin^3\frac{π(x-t)}{T}}\,u(x)$, where $u(x)$ is in $C^\infty(\mathbb{R})$ and is $T$-periodic. Actually, we prove that $\widehat{T}{}^{(s)}_n[f]$, $s=0,1,2,$ are exact for a class of singular integrals involving $T$-periodic trigonometric polynomials of degree at most $n-1$; that is, $$ \widehat{T}{}^{(s)}_n[f]=I[f]\quad\text{when\ \ $f(x)=\frac{\cos\frac{π(x-t)}{T}}{\sin^3\frac{π(x-t)}{T}}\,\sum^{n-1}_{m=-(n-1)} c_m\exp(\mrm{i}2mπx/T)$.}$$ We also prove that, when $u(z)$ is analytic in a strip $\big|\text{Im}\,z\big|<σ$ of the complex $z$-plane, the errors in all three $\widehat{T}{}^{(s)}_n[f]$ are $O(e^{-2nπσ/T})$ as $n\to\infty$, for all practical purposes.

math.NA↗

arXiv2021

PVTSI$^{\boldmath(m)}$: A Novel Approach to Computation of Hadamard Finite Parts of Nonperiodic Singular Integrals

We consider the numerical computation of $I[f]=\intBar^b_a f(x)\,dx$, the Hadamard Finite Part of the finite-range singular integral $\int^b_a f(x)\,dx$, $f(x)=g(x)/(x-t)^{m}$ with $a<t<b$ and $m\in\{1,2,\ldots\},$ assuming that (i)\,$g\in C^\infty(a,b)$ and (ii)\,$g(x)$ is allowed to have arbitrary integrable singularities at the endpoints $x=a$ and $x=b$. We first prove that $\intBar^b_a f(x)\,dx$ is invariant under any suitable variable transformation $x=ψ(ξ)$, $ψ:[α,β]\rightarrow[a,b]$, hence there holds $\intBar^β_αF(ξ)\,dξ=\intBar^b_a f(x)\,dx$, where $F(ξ)=f(ψ(ξ))\,ψ'(ξ)$. Based on this result, we next choose $ψ(ξ)$ such that the transformed integrand $F(ξ)$ is sufficiently periodic with period $\T=β-α$, and prove, with the help of some recent extension/generalization of the Euler--Maclaurin expansion, that we can apply to $\intBar^β_αF(ξ)\,dξ$ the quadrature formulas derived for periodic singular integrals developed in an earlier work of the author. We give a whole family of numerical quadrature formulas for $\intBar^β_αF(ξ)\,dξ$ for each $m$, which we denote $\widehat{T}^{(s)}_{m,n}[{\cal F}]$, where ${\cal F}(ξ)$ is the $\T$-periodic extension of $F(ξ)$.

math.NA↗

arXiv2020

Acceleration of Convergence of Some Infinite Sequences $\boldsymbol{\{A_n\}}$ Whose Asymptotic Expansions Involve Fractional Powers of $\boldsymbol{n}$ via the $\tilde{d}^{(m)}$ transformation

In this paper, we discuss the application of the author's $\tilde{d}^{(m)}$ transformation to accelerate the convergence of infinite series $\sum^\infty_{n=1}a_n$ when the terms $a_n$ have asymptotic expansions that can be expressed in the form $$ a_n\sim(n!)^{s/m}\exp\left[\sum^{m}_{i=0}q_in^{i/m}\right]\sum^\infty_{i=0}w_i n^{γ-i/m}\quad\text{as $n\to\infty$},\quad s\ \text{integer.}$$ We discuss the implementation of the $\tilde{d}^{(m)}$ transformation via the recursive W-algorithm of the author. We show how to apply this transformation and how to assess in a reliable way the accuracies of the approximations it produces, whether the series converge or they diverge. We classify the different cases that exhibit unique numerical stability issues in floating-point arithmetic. We show that the $\tilde{d}^{(m)}$ transformation can also be used efficiently to accelerate the convergence of infinite products $\prod^\infty_{n=1}(1+v_n)$, where $v_n\sim \sum^\infty_{i=0}e_in^{-t/m-i/m}$ as $n\to\infty$,\ $t\geq m+1$ an integer. Finally, we give several numerical examples that attest the high efficiency of the $\tilde{d}^{(m)}$ transformation for the different cases.

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arXiv2020

Generalization of the Secant Method for Nonlinear Equations (extended version)

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form $f(x)=0$. It is derived via a linear interpolation procedure and employs only values of $f(x)$ at the approximations to the root of $f(x)=0$, hence it computes $f(x)$ only once per iteration. In this note, we generalize it by replacing the relevant linear interpolant by a suitable $(k+1)$-point polynomial of interpolation, where $k$ is an integer at least 2. Just as the secant method, this generalization too enjoys the property that it computes $f(x)$ only once per iteration. We provide its error in closed form and analyze its order of convergence $s_k$. We show that this order of convergence is greater than that of the secant method, and it increases towards $2$ as $k\to \infty$. (Indeed, $s_7=1.9960\cdots$, for example.) This is true for the efficiency index of the method too. We also confirm the theory via an illustrative example.

math.NA↗

arXiv2020

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 ${s}_k=\sum^k_{i=0}γ_i{x}_i$ with $\sum^k_{i=0}γ_i=1$, for some scalars $γ_i$. 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 $$μ_k{s}_k^\text{RRE} = μ_{k-1}{s}_{k-1}^\text{RRE} + ν_k{s}_{k}^\text{MPE} \quad \text{with} \quad μ_k = μ_{k-1} + ν_k,$$ for some positive scalars $μ_k$, $μ_{k-1}$, and $ν_k$ 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.

math.NA↗

arXiv2020

On a Vectorized Version of a Generalized Richardson Extrapolation Process

Let $\{\xx_m\}$ be a vector sequence that satisfies $$ \xx_m\sim \sss+\sum^\infty_{i=1}α_i \gg_i(m)\quad\text{as $m\to\infty$},$$ $\sss$ 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 $$ \sum^k_{i=1}\braket{\yy,Δ\gg_{i}(m)}\widetildeα_i=\braket{\yy,Δ\xx_m},\quad n\leq m\leq n+k-1;\quad \sss_{n,k}=\xx_n+\sum^k_{i=1}\widetildeα_i\gg_{i}(n),$$ $\sss_{n,k}$ being the approximation to $\sss$. Here $\yy$ is some nonzero vector, $\braket{\cdot\,,\cdot}$ is an inner product, such that $\braket{α\aaa,β\bb}=\barαβ\braket{\aaa,\bb}$, and $Δ\xx_m=\xx_{m+1}-~\xx_m$ and $Δ\gg_i(m)=\gg_i(m+1)-\gg_i(m)$. 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.

math.NA↗

arXiv2020

A Convergence Study for Reduced Rank Extrapolation on Nonlinear Systems

Reduced Rank Extrapolation (RRE) is a polynomial type method used to accelerate the convergence of sequences of vectors $\{\boldsymbol{x}_m\}$. It is applied successfully in different disciplines of science and engineering in the solution of large and sparse systems of linear and nonlinear equations of very large dimension. If $\boldsymbol{s}$ is the solution to the system of equations $\boldsymbol{x}=\boldsymbol{f}(\boldsymbol{x})$, first, a vector sequence $\{\boldsymbol{x}_m\}$ is generated via the fixed-point iterative scheme $\boldsymbol{x}_{m+1}=\boldsymbol{f}(\boldsymbol{x}_m)$, $m=0,1,\ldots,$ and next, RRE is applied to this sequence to accelerate its convergence. RRE produces approximations $\boldsymbol{s}_{n,k}$ to $\boldsymbol{s}$ that are of the form $\boldsymbol{s}_{n,k}=\sum^k_{i=0}γ_i\boldsymbol{x}_{n+i}$ for some scalars $γ_i$ depending (nonlinearly) on $\boldsymbol{x}_n, \boldsymbol{x}_{n+1},\ldots,\boldsymbol{x}_{n+k+1}$ and satisfying $\sum^k_{i=0}γ_i=1$. The convergence properties of RRE when applied in conjunction with linear $\boldsymbol{f}(\boldsymbol{x})$ have been analyzed in different publications. In this work, we discuss the convergence of the $\boldsymbol{s}_{n,k}$ obtained from RRE with nonlinear $\boldsymbol{f}(\boldsymbol{x})$ (i)\,when $n\to\infty$ with fixed $k$, and (ii)\,in two so-called {\em cycling} modes.

math.NA↗

arXiv2017

A New Algorithm for the Higher-Order $G$-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.

math.NA↗

arXiv2017

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.

math.NA↗

arXiv2015

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

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 $g\in{\bf A}^{(2)}$.

math.NA↗

arXiv2015

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.

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