Book:Nicholas Higham/Functions of Matrices: Theory and Computation

Nicholas Higham: Functions of Matrices: Theory and Computation

Published $2008$.

List of Figures

List of Tables

Preface

1 Theory of Matrix Functions

1.1 Introduction

1.2 Definitions of $f(A)$

1.2.1 Jordan Canonical Form

1.2.2 Polynomial Interpolation

1.2.3 Cauchy Integral Theorem

1.2.4 Equivalence of Definitions

1.2.5 Example: Function of Identity Plus Rank-$1$ Matrix

1.2.6 Example: Functions of Discrete Fourier Transform Matrix

1.3 Properties

1.4 Nonprimary Matrix Functions

1.5 Existence of (Real) Matrix Squre Roots and Logarithms

1.6 Classification of Matrix Square Roots and Logarithms

1.7 Principal Square Roots and Logarithms

1.8 $f(AB)$ and $f(BA)$

1.9 Miscellany

1.10 A Brief History of Matrix Functions

1.11 Notes and References

Problems

2 Applications

2.1 Differential Equations

2.1.1 Exponential Integrators

2.2 Nuclear Magnetic Resonance

2.3 Markov Models

2.4 Control Theory

2.5 The Nonsymmetric Eigenvalue Problem

2.6 Orthogonalization and the Orthogonal Procrustes Problem

2.7 Theoretical Particle Physics

2.8 Other Matrix Functions

2.9 Nonlinear Matrix Equations

2.10 Geometric Mean

2.11 Pseudospectra

2.12 Algebras

2.13 Sensitivity Analysis

2.14 Other Applications

2.14.1 Boundary Value Problems

2.14.2 Semidefinite Programming

2.14.3 Matrix Sector Function

2.14.4 Matrix Disk Function

2.14.5 The Average Eye in Optics

2.14.6 Computer Graphics

2.14.7 Bregman Divergences

2.14.8 Structured Matrix Interpolation

2.14.9 The Lambert $W$ Function and Delay Differential Equations

2.15 Notes and References

Problems

3 Conditioning

3.1 Condition Numbers

3.2 Properties of the Fréchet Derivative

3.3 Bounding the Condition Number

3.4 Computing or Estimating the Condition Number

3.5 Notes and References

Problems

4 Techniques for General Functions

4.1 Matrix Powers

4.2 Polynomial Evaluation

4.3 Taylor Series

4.4 Rational Approximation

4.4.1 Best $L_{\infty}$ Approximation

4.4.2 Padé Approximation

4.4.3 Evaluating Rational Functions

4.5 Diagonalization

4.6 Schur Decomposition and Triangular Matrices

4.7 Block Diagonalization

4.8 Interpolating Polynomial and Characteristic Polynomial

4.9 Matrix Iterations

4.9.1 Order of Convergence

4.9.2 Termination Criteria

4.9.3 Convergence

4.9.4 Numerical Stability

4.10 Preprocessing

4.11 Bounds for $\lVert f(A) \rVert$

4.12 Notes and References

Problems

5 Matrix Sign Function

5.1 Sensitivity and Conditioning

5.2 Schur Method

5.3 Newton's Method

5.4 The Padé Family of Iterations

5.5 Scaling the Newton Iteration

5.6 Terminating the Iterations

5.7 Numerical Stability of Sign Iterations

5.8 Numerical Experiments and Algorithm

5.9 Best $L_{\infty}$ Approximation

5.10 Notes and References

Problems

6 Matrix Square Root

6.1 Sensitivity and Conditioning

6.2 Schur Method

6.3 Newton's Method and Its Variants

6.4 Stability and Limiting Accuracy

6.4.1 Newton Iteration

6.4.2 DB Iterations

6.4.3 CR Iteration

6.4.4 IN Iteration

6.4.5 Summary

6.5 Scaling the Newton Iteration

6.6 Numerical Experiments

6.7 Iterations via the Matrix Sign Function

6.8 Special Matrices

6.8.1 Binomial Iteration

6.8.2 Modified Newton Iterations

6.8.3 $M$-Matrices and $H$-Matrices

6.8.4 Hermitian Positive Definite Matrices

6.9 Computing Small-Normed Square Roots

6.10 Comparison of Methods

6.11 Involutory Matrices

6.12 Notes and References

Problems

7 Matrix $p$th Root

7.1 Theory

7.2 Schur Method

7.3 Newton's Method

7.4 Inverse Newton Method

7.5 Schur-Newton Method

7.6 Matrix Sign Method

7.7 Notes and References

Problems

8 The Polar Decomposition

8.1 Approximation and Properties

8.2 Sensitivity and Conditioning

8.3 Newton's Method

8.4 Obtaining Iterations via the Matrix Sign Function

8.5 The Padé Family of Methods

8.6 Scaling the Newton Iteration

8.7 Terminating the Iterations

8.8 Numerical Stability and Choice of $H$

8.9 Algorithm

8.10 Notes and References

Problems

9 Schur-Parlett Algorithm

9.1 Evaluating Functions of the Atomic Blocks

9.2 Evaluating the Upper Triangular Part of $f(T)$

9.3 Reordering and Blocking the Schur Function

9.4 Schur-Parlett Algorithm for $f(A)$

9.5 Preprocessing

9.6 Notes and References

Problems

10 Matrix Exponential

10.1 Basic Properties

10.2 Conditioning

10.3 Scaling and Squaring Method

10.4 Schur Algorithms

10.4.1 Newton Divided Difference Interpolation

10.4.2 Schur-Fréchet Algorithm

10.4.3 Schur-Parlett Aglorithm

10.5 Numerical Experiment

10.6 Evaluating the Fréchet Derivative and Its Norm

10.6.1 Quadrature

10.6.2 The Kronecker Formulae

10.6.3 Computing and Estimating the Norm

10.7 Miscellany

10.7.1 Hermitian Matrices and Best $L_{\infty}$ Approximation

10.7.2 Essentially Nonnegative Matrices

10.7.3 Preprocessing

10.7.4 The $\psi$ Functions

10.8 Notes and References

Problems