Functions of matrices - theory and computation

  title={Functions of matrices - theory and computation},
  author={Nicholas John Higham},
  • N. Higham
  • Published 2008
  • Computer Science, Mathematics
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the theory of matrix functions and numerical methods for computing them, including an overview of applications, new and unpublished research results, and improved algorithms. Key features include a detailed treatment of the matrix sign function and matrix roots; a development of the theory of conditioning and properties of the Frechet derivative; Schur decomposition; block Parlett recurrence; a thorough analysis of the accuracy, stability, and computational… Expand
The Computation of Matrix Functions in Particular, The Matrix Exponential
Matrix functions in general are an interesting area in matrix analysis and are used in many areas of linear algebra and arise in numerous applications in science and engineering. We consider how toExpand
Computing matrix functions
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An efficient computation of generalized inverse of a matrix
We propose a hyperpower iteration for numerical computation of the outer generalized inverse of a matrix which achieves 18th order of convergence by using only seven matrix multiplications perExpand
Efficient approximation of functions of some large matrices by partial fraction expansions
The use and the convergence of a recent technique for generating sequences of incomplete factorizations of matrices in order to face with both issues of solving several linear systems and approximate matrix inversions are studied. Expand
A Technique for Improving the Computation of Functions of Triangular Matrices
  • Joao R. Cardoso, A. Sadeghi
  • Mathematics, Computer Science
  • 2020
We propose a simple technique that, if combined with algorithms for computing functions of triangular matrices, can make them more efficient. Basically, such a technique consists in a specificExpand
A Secant Method for Nonlinear Matrix Problems
This paper proposes a specialized secant method for the special problem of computing the inverse or the pseudoinverse of a given matrix, for which stability and q-superlinear convergence are established, and for which some numerical results are presented. Expand
Symbolic spectral decomposition of 3x3 matrices
This paper introduces an alternative form for the computation of the involved matrix invariants (in particular the discriminant) in terms of sum-of-products expressions as function of the matrix elements that leads to increased floating point accuracy. Expand
Matrix Arithmetic-Geometric Mean and the Computation of the Logarithm
An algorithm is developed, whose main block is an optimized AGM scheme, for the computation of the logarithm of a matrix, which is shown to be competitive, in terms of accuracy, with the state-of-the-art methods. Expand
Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection
Matrix functions are a central topic of linear algebra, and problems of their numerical approximation appear increasingly often in scientific computing. We review various rational Krylov methods forExpand
Matrix Functions of Exponential Order
Both the theoretical and practical investigations of various dynamical systems need to extend the definitions of various functions defined on the real axis to the set of matrices. To this end oneExpand


Accuracy and stability of numerical algorithms
This book gives a thorough, up-to-date treatment of the behavior of numerical algorithms in finite precision arithmetic by combining algorithmic derivations, perturbation theory, and rounding error analysis. Expand
Linear model reduction and solution of the algebraic Riccati equation by use of the sign function
The sign function of a square matrix can be defined in terms of a contour integral or as the result of an iterated map $. Application of this function enables a matrix to be decomposed into twoExpand
SIAM Review
  • SIAM Review
  • 1997
Accuracy and stability of numerical algorithms (Society for Industrial and Applied
  • 1996
3 Padé Approximation of Cosine . . . . . . . . . . . . . . . . . . . . . . 290 12.4 Double Angle Algorithm
  • 3 Padé Approximation of Cosine . . . . . . . . . . . . . . . . . . . . . . 290 12.4 Double Angle Algorithm
302 13.2.2 Arnoldi Approximation of f (A)b . . . . . . . . . . . . . . . . . 304 13
  • 302 13.2.2 Arnoldi Approximation of f (A)b . . . . . . . . . . . . . . . . . 304 13
313 14.1.1 Algebras and
  • 1.5 Computing Structured f (A) for Structured A . . . . . . . . . 316 14.2 Exponential Decay of Functions of Banded Matrices . . . . . . . . . . 317 14.3 Approximating Entries of Matrix Functions
321 B.2 Eigenvalues and
  • 328 B.9 Perturbation Expansions for Matrix Inverse . . . . . . . . . . . . . . 328 Copyright ©2008 by the Society for Industrial and Applied Mathematics. This electronic version is for personal use and may not be duplicated or distributed. From "Functions of Matrices: Theory and Computation
8 (a) is for ℓ > m and what can be said about convergence for ℓ < m−1
  • 8 (a) is for ℓ > m and what can be said about convergence for ℓ < m−1