We give a survey of numerical methods for evaluating matrix functions, along with a brief treatment of the underlying theory and a description of two recent applications. Society for industrial and applied mathematics, philadelphia, 2008. Matrix functions are of growing interest due to their fascinating theory and the many applications in which they provide insight and succinct solutions. Society for industrial and applied mathematics, philadelphia, pa, usa. March 27, 2008 this is the bibliography of the book 279, available as a bibtex database. A survey of condition number estimation for triangular matrices.
Caterina fenu postdoctoral research associate aices graduate school rwth aachen university germany april 10, 2017 the aim of the course is to introduce basic concepts related to the computation of matrix functions. The matrix sign function can be obtained from any of the definitions in chapter 1. I use emacs, latex and matlab on windowsbased desktops and mac laptops. The survey is organized by classes of methods, which are broadly those based on similarity transformations, those employing approximation by polynomial or rational functions.
The trigonometric functions especially sine and cosine for real or complex square matrices occur in solutions of secondorder systems of differential equations. Handbook of writing for the mathematical sciences, matlab guide. Most of the results in this chapter are for general functions. I also edited, and wrote about 100 pages of, the princeton companion to applied mathematics. From their origin in pure mathematics, matrix functions have broadened into a subject of study in. Theory and computation other titles in applied mathematics by nicholas j.
Functions of matrices have been studied for as long as matrix algebra itself. Matrix analysis by roger a horn download pdfepub ebook. Higham, accuracy and stability of numerical algorithms society for industrial. Theory and computation levesley 2009 bulletin of the london mathematical society wiley online library. Indeed, in his seminal a memoir on the theory of matrices 1858, cayley investigated the square root of a matrix, and it was not long before definitions of fa for general f were proposed by sylvester and others. Theory and computation other titles in applied mathematics nicholas j. Research matters nick higham february 25, 2009 school of.
They are defined by the same taylor series that hold for the trigonometric functions of real and complex numbers. We propose a simple preconditioning technique that, if incorporated into algorithms for computing functions of triangular matrices, can make them more efficient. A thorough and elegant treatment of the theory of matrix functions and numerical. This research article focus on the numerical computation of this function. Elementary matrices and some applications to dynamics and differential equations. All discounts are applied on final checkout screen. The best books on applied mathematics, as recommended by nick higham, professor of applied mathematics at the university of manchester. Two numerical algorithms for computing interval matrices containing the matrix exponential are proposed. In particular, we generalize 24, 25, which use quadrature to compute stable restarted approximations. Pdf functions of matrices theory and computation semantic. Functions of matrices with nick higham from the university of manchester. Pete stewart for nick higham s masterful summary of the theory and computation of functions of a matrix, and feel sure that this book will serve to inspire many others to work in this area of fundamental importance to all numerical scientists.
After the study of beta and gamma matrix functions, an integral representation of fa,b. Nineteen dubious ways to compute the exponential of a matrix, twenty. There are several techniques for lifting a real function to a square matrix function such that interesting properties are maintained. Pdf a preconditioning technique for computing functions. Higham the only book devoted exclusively to matrix functions, this research monograph gives a thorough treatment of the theory of matrix functions and numerical methods for computing them. Theory and computation ebook written by nicholas j. The aim of the course is to introduce basic concepts related to the computation of matrix functions and their use in the analysis of complex networks. In the spirit of course notes the article is not a comprehensive survey and does not cite all the. Basically, such a technique consists in a similarity transformation that reduces the departure from normality of a triangular matrix, thus decreasing its norm and in general its function condition number. Khan, a class of exponential matrices, this j ournal 34 2003. In mathematics, a matrix function is a function which maps a matrix to another matrix.
Download for offline reading, highlight, bookmark or take notes while you read functions of matrices. Wellknown techniques for the scalar gamma function. All of the following techniques yield the same matrix function, but the domains on which the function is defined may differ. The first algorithm is based on a numerical spectral decomposition and requires only cubic complexity under some assumptions.
The best books on applied mathematics five books expert. Proofs of the facts in this article can be found in one or more of hig, hj91 or lt85. Classic matlab help fun fun for matrix arguments x, the functions sin, cos, atan. We give a survey of numerical methods for evaluating matrix functions, along with a brief treatment of the underlying theory and a description of two recent. Pdf a preconditioning technique for computing functions of. In most applications it is primary matrix functions that are of interest. The material in it will allow me to expand some research i have done. The material in it will allow me to expand some research i have done in the topic functions of sets of matrices in new directions. Functions of matrices society for industrial and applied. Functions of matrices theory and computation a thorough treatment of the theory of matrix functions and numerical methods for computing them, including an overview of applications. Taylors theorem for matrix functions with applications to.
Computing matrix functions acta numerica cambridge core. Functions of matrices is a very useful topic, and this is the only book on it, and it is excellent. T betcke, nj higham, v mehrmann, c schroder, f tisseur. Higham, frs, is richardson professor of applied mathematics at. Verified computation of the matrix exponential springerlink.
Order functions of matrices from the siam bookstore. A thorough and elegant treatment of the theory of matrix functions and numerical methods for computing them, including an overview of applications, new and unpublished research results, and improved algorithms. One of the fundamental matrix functions, which is particularly important due to its connections with certain matrix differential equations and other special matrix functions, is the matrix gamma function. Matrix functions have a major role in science and engineering. Krylov methods for the computation of matrix functions. Theory and computation gives a thorough treatment of the theory of matrix functions and numerical methods for computing them, as well as an overview of applications. Theory and computation gives a thorough treatment of the. Key features include a detailed treatment of the matrix sign function and matrix roots. Block krylov subspace methods for computing functions of. The second algorithm is based on a numerical jordan decomposition and applicable even for defective matrices. Acm transactions on mathematical software toms 39 2. Note that in the case of the jordan canonical form and interpolating polynomial.
Theory and computation nick higham school of mathematics the university of manchester. The foundation of the book rests on highams development of the theory based on three equivalent definitions of fa, where f is a scalar function and a is an n x n complex matrix. Jordan blocks then a nonprimary matrix function is obtained. April 1986, newtons method for the matrix square root pdf, mathematics of computation, 46 174.
1072 601 1182 1216 128 635 171 64 637 1617 478 138 440 418 875 1598 825 756 178 916 1162 1532 1395 1392 714 941 1436 298 439 704 81 770 130 563 297 519 1290