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On the backward and forward error of approximations of analytic functions and applications to the computation of matrix functions

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2023年 419卷

作者： Sastre, Jorge Ibanez, Javier Inst Telecomunicac & Aplicac Multimedia Valencia Spain Univ Politecn Valencia Inst Instrumentac Imagen Mol Camino Vera S-N Valencia 46022 Spain

In this paper we give a new formula to write the forward error of Taylor approximations of analytical functions in terms of the backward error of those approximations, consider-ing exact arithmetic in both errors. Using this formula, a method to compute a backward error given by the power series centered in the same expansion point as the Taylor approximation is provided. The application of this method for Pade approximations is also shown. Based on the previous method, a MATLAB implementation for computing the first power series terms of the backward error for Taylor and Pade approximations of a given analytical function is provided, and examples of its use are given. Applications to the computation of matrix functions are given that overcome limitations of other backward error analyses use inverse compositional functions in the literature.(c) 2022 Elsevier B.V. All rights reserved.

关键词： Function approximation Forward error Backward error Taylor Pad? matrix functions MATLAB

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Solving Optimization Problems on Hermitian matrix functions with Applications

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JOURNAL OF APPLIED MATHEMATICS 2013年第unknown期2013卷 1-11页

作者： Zhang, Xiang Xiang, Shu-Wen Guizhou Univ Dept Comp Sci & Informat Guiyang 550025 Peoples R China

We consider the extremal inertias and ranks of the matrix expressions f(X, Y) = A(3) - B3X - (B3X)* - C3YD3 - (C3YD3)*, where A(3) = A(3)*, B-3, C-3, and D-3 are known matrices and Y and X are the solutions to the matrix equations A(1)Y = C-1, YB1 = D-1, and A(2)X = C-2, respectively. As applications, we present necessary and sufficient condition for the previous matrix function f(X, Y) to be positive (negative), non-negative (positive) definite or nonsingular. We also characterize the relations between the Hermitian part of the solutions of the above-mentioned matrix equations. Furthermore, we establish necessary and sufficient conditions for the solvability of the system of matrix equations A(1)Y = C-1, YB1 = D-1, A(2)X = C-2, and B3X + (B3X)* + C3YD3 + (C3YD3)* = A(3), and give an expression of the general solution to the above-mentioned system when it is solvable.

关键词： MATHEMATICAL optimization PROBLEM solving HERMITIAN operators matrix functions functions (Mathematics) MATHEMATICAL analysis

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Semismooth matrix-valued functions

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MATHEMATICS OF OPERATIONS RESEARCH 2002年第1期27卷 150-169页

作者： Sun, DF Sun, X Natl Univ Singapore Dept Math Singapore 117548 Singapore Natl Univ Singapore Fac Business Adm Singapore 117548 Singapore Natl Univ Singapore Singapore MIT Alliance Singapore 117548 Singapore

matrix-valued functions play an important role in the development of algorithms for semidefinite programming problems. This paper studies generalized differential properties of such functions related to nonsmooth-smoothing Newton methods. The first part of this paper discusses basic properties such as the generalized derivative, Rademacher's theorem, B-derivative, directional derivative, and semismoothness. The second part shows that the matrix absolute-value function, the matrix semidefinite-projection function, and the matrix projective residual function are strongly semismooth.

关键词： matrix functions Newton's method nonsmooth optimization semidefinite programming

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A restarted Lanczos approximation to functions of a symmetric matrix

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IMA JOURNAL OF NUMERICAL ANALYSIS 2010年第4期30卷 1044-1061页

作者： Ilic, M. Turner, I. W. Simpson, D. P. Queensland Univ Technol Sch Math Sci Brisbane Qld 4001 Australia

In this paper the error term in the 'first' method of Eiermann & Ernst (2006, SIAM J. Numer. Anal., 44, 2481-2504) for restarting the Lanczos approximation of the matrix-vector product f(A)b, where A is an element of R-n (x n) is symmetric, is re-derived and expressed as an explicit partial fraction expansion of the divided differences. The partial fraction representation makes the new variant slightly more stable (albeit still unstable) than the former method because it requires fewer finite-difference evaluations. We then present an error bound for the restarted Lanczos approximation of f(A)b for symmetric positive-definite A when f is in a particular class of completely monotone functions and illustrate for some important matrix function applications the usefulness of these bounds for terminating the restart process once the desired accuracy in the matrix function approximation has been achieved. Finally, in an attempt to overcome the inherent instability of our restart procedure, we propose a simple heuristic that identifies when to halt the iterations.

关键词： Krylov subspace methods matrix functions error bounds Stieltjes transforms Gaussian Markov random fields fractional Poisson equation

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functions of he Laplacian matrix With Application to Distributed Formation Control

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IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS 2022年第3期9卷 1459-1467页

作者： Morbidi, Fabio Univ Picardie Jules Verne Modeling Informat & Syst Lab F-80039 Amiens France

In this article, we study a class of matrix functions of the combinatorial Laplacian that preserve its structure, i.e., that define matrices, which are positive semidefinite and which have zero row-sum and nonpositive off-diagonal entries. This formulation has the merit of presenting different incarnations of the Laplacian matrix that have appeared in recent literature in a unified framework. For the first time, we apply this family of Laplacian functions to consensus theory, and we show that they leave the agreement value unchanged and offer distinctive advantages in terms of performance and design flexibility. The theory is illustrated via worked examples and numerical experiments featuring four representative Laplacian functions in a shape-based distributed formation control strategy for single-integrator robots.

关键词： Combinatorial Laplacian consensus protocol matrix functions network systems

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A matrix Exponential Generalization of the Laplace Transform of Poisson Shot Noise

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IEEE TRANSACTIONS ON INFORMATION THEORY 2025年第1期71卷 396-412页

作者： Olson, Nicholas R. Andrews, Jeffrey G. Univ Texas Austin 6GUT Austin TX 78712 USA Univ Texas Austin WNCG Austin TX 78712 USA

We consider a generalization of the Laplace transform of Poisson shot noise defined as an integral transform with respect to a matrix exponential. We denote this as the matrix Laplace transform and establish that it is in general a matrix function extension of the scalar Laplace transform. We show that the matrix Laplace transform of Poisson shot noise admits an expression analogous to that implied by Campbell's theorem. We demonstrate the utility of this generalization of Campbell's theorem in two important applications: the characterization of a Poisson shot noise process and the derivation of the complementary CDF (CCDF) and meta-distribution of signal-to-interference-and-noise (SINR) models in Poisson networks. In the former application, we demonstrate how the higher order moments of Poisson shot noise may be obtained directly from the elements of its matrix Laplace transform. We further show how the CCDF of this object may be bounded using a summation of the first row of its matrix Laplace transform. For the latter application, we show how the CCDF of SINR models with phase-type distributed desired signal power may be obtained via an expectation of the matrix Laplace transform of the interference and noise, analogous to the canonical case of SINR models with Rayleigh fading. Additionally, when the power of the desired signal is exponentially distributed, we establish that the meta-distribution may be obtained in terms of the limit of a sequence expressed in terms of the matrix Laplace transform of a related Poisson shot noise process.

关键词： Laplace equations Interference Signal to noise ratio Rayleigh channels Wireless networks Transforms Receivers Nakagami distribution Stochastic processes Measurement Stochastic geometry poisson shot noise laplace transform matrix functions phase-type distributions meta-distribution

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Forman-Ricci communicability curvature of graphs and networks

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EUROPEAN JOURNAL OF APPLIED MATHEMATICS 2025年

作者： Estrada, Ernesto UIB CSIC Inst Cross Disciplinary Phys & Complex Syst IFISC Palma De Mallorca Spain

Geometric parameters in general and curvature in particular play a fundamental role in our understanding of the structure and functioning of real-world networks. Here, the discretisation of the Ricci curvature proposed by Forman is adapted to capture the global influence of the network topology on individual edges of a graph. This is implemented mathematically by assigning communicability distances to edges in the Forman-Ricci definition of curvature. We study analytically both the edge communicability curvature and the global graph curvature and give mathematical characterisations of them. The Forman-Ricci communicability curvature is interpreted 'physically' on the basis of a non-conservative diffusion process taking place on the graph. We then solve analytically a toy model that allows us to understand the fundamental differences between edges with positive and negative Forman-Ricci communicability curvature. We complete the work by analysing three examples of applications of this new graph-theoretic invariant on real-world networks: (i) the network of airport flight connections in the USA, (ii) the neuronal network of the worm Caenorhabditis elegans and (iii) the collaboration network of authors in computational geometry, where we strengthen the many potentials of this new measure for the analysis of complex systems.

关键词： curvature matrix functions complex networks network communicability communicability distance

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Computing the action of the matrix generating function of Bernoulli polynomials on a vector with an application to non-local boundary value problems

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2025年第2期51卷 1-23页

作者： Aceto, Lidia Gemignani, Luca Univ Piemonte Orientale Dipartimento Sci & Innovaz Tecnolog Viale T Michel 11 I-15121 Alessandria Italy Univ Pisa Dipartimento Informat Largo Bruno Pontecorvo3 I-56127 Pisa Italy

This paper deals with efficient numerical methods for computing the action of the matrix generating function of Bernoulli polynomials, say q(tau,A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q(\tau ,A)$$\end{document}, on a vector when A is a large and sparse matrix. This problem occurs when solving some non-local boundary value problems. Methods based on the Fourier expansion of q(tau,w)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q(\tau ,w)$$\end{document} have already been addressed in the scientific literature. The contribution of this paper is twofold. First, we place these methods in the classical framework of Krylov-Lanczos (polynomial-rational) techniques for accelerating Fourier series. This allows us to apply the convergence results developed in this context to our function. Second, we design a new acceleration scheme. Some numerical results are presented to show the effectiveness of the proposed algorithms.

关键词： Rational approximation of Fourier series Convergence acceleration matrix functions Non-local boundary value problems

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Density-matrix functionals from Green's functions

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Physical Review B 2013年第20期88卷 205139-205139页

作者： Peter E. Blöchl Thomas Pruschke Michael Potthoff Clausthal University of Technology Institute for Theoretical Physics Leibnizstraße 10 D-38678 Clausthal-Zellerfeld Germany Georg-August Universität Göttingen Institute for Materials Physics Friedrich-Hund-Platz 1 D-37077 Göttingen Germany Georg-August Universität Göttingen Institute for Theoretical Physics Friedrich-Hund-Platz 1 D-37077 Göttingen Germany I. Institute for Theoretical Physics University of Hamburg Jungiusstraße 9 D-20355 Hamburg Germany

The exact reduced density-matrix functional is derived from the Luttinger-Ward functional of the single-particle Green's function. Thereby, a formal link is provided between diagrammatic many-body approaches using Green's functions on the one hand and theories based on many-body wave functions on the other. This link can be used to explicitly construct approximations for the density-matrix functional that are equivalent to standard diagrammatic resummation techniques and to nonperturbative dynamical mean field theory in particular. Contrary to functionals of the Green's function, the exact density-matrix functional is convex and thus provides a true minimum principle which facilitates the calculation of the grand potential and derived equilibrium properties. The benefits of the proposed Green's-function-based density-matrix functional theory for geometrical structure optimization of strongly correlated materials are discussed.

关键词： DENSITY functionals GREEN'S functions matrix functions MEAN field theory MATHEMATICAL physics

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Generalized spatial matrix specifications

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ECONOMETRIC REVIEWS 2025年第5期44卷 649-672页

作者： Leorato, Samantha Martinelli, Andrea Univ Milan Dept Econ Management & Quantitat Methods Milan Italy Insubria Univ Dept Sci & High Technol Como Italy

In this article, we propose a model that generalizes some of the most popular choices in spatial linear modeling, such as the SAR and MESS models. Our idea builds on their representation as link functions applied to a spatial weight W, corresponding to the uniform and exponential distributions, respectively. By allowing for more general families of distribution functions, one can encompass both models and capture different spatial patterns. We provide some insights into the difference between specifications, with emphasis on advantages and shortcomings as well as on interpretation of the parameters and correspondences between models. By exploiting the possibility to obtain a formal power series representation of the link family, we define the quasi maximum likelihood estimator and study its asymptotic properties under Gaussian and non Gaussian errors. By applying our approach to data on 2000 US election participation, as in LeSage and Pace (2007), we show that this model is able to capture a finite order neighboring spillover structure, as opposed to the infinite order implied by both the SAR and MESS models.

关键词： Generalized beta generalized gamma matrix exponential matrix functions quasi-ML estimation spatial linear regression models

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