Functions of matrices. Theory and computation.

*(English)*Zbl 1167.15001
Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-0-898716-46-7/hbk; 978-0-89871-777-8/ebook). xx, 425 p. (2008).

This book treats of matrix functions and both states pure theoretical fundamentals and presents applied numerical methods. To the reader benefit, the scope of the book is far beyond its title and it also comprises basic matrix theory and description of numerical packages procedures. Furthermore, it includes remarkable historic facts and also serves as a complete bibliographic reference. Surpassing his own style the author excels himself in details, presentation and style, providing essays, comments and a meticulous layout, which reflects care and dedication of a master to his work.

The book contains a preface, 14 chapters, 5 appendixes and a bibliography with an impressive number of 625 entries. Chapters include interesting examples and always contain abundant notes and references, ending with a list of problems with the solutions appearing in one of the appendixes. The first chapter introduces definitions for matrix functions, polynomial interpolation, Cauchy integral theorem, matrix square roots and logarithms. Chapter 2 is concerned with applications to differential equations, Markov models, control theory, eigenvalue problem and nonlinear matrix problems. The third chapter considers the conditioning and sensitivity of computations of matrix functions especially by means of the FrĂ©chet derivative. In Chapter 4 methods for matrix power, polynomial evaluation, Taylor series, rational approximation, diagonalization and others are discussed. Related theorems, algorithms, cost and stability studies are treated. The next three chapters focus on the matrix sign function, matrix square root and matrix \(p\)th root. Chapter 8 considers the polar decomposition and its relation with the singular value decomposition. Chapter 9 is devoted to Schur-Parlett algorithm by the use of the Taylor series expansion. Chapter 10 discusses the matrix exponential, “the most studied matrix function”, reflecting its importance in the solution of differential equations. A natural continuation is the matrix logarithm presented in Chapter 11, and matrix sine and cosine in Chapter 12. In Chapter 13 is given a substantial treatment of a function matrix times a vector, considering polynomial interpolation, Krylov subspace methods and numerical quadrature. The concluding Chapter 14 includes some additional topics under the title of miscellany. In the Appendixes are the notation, definitions, operation counts, toolbox references, and the solutions of the problems.

As a final word, this book should be a reference to anyone involved with matrix analysis and it appears to be a classic in the coming years, together with the ones of F. R. Gantmacher [The theory of matrices. (1953; Zbl 0050.24804)] and J. H. Wilkinson [The algebraic eigenvalue problem (1965; Zbl 0258.65037)].

The book contains a preface, 14 chapters, 5 appendixes and a bibliography with an impressive number of 625 entries. Chapters include interesting examples and always contain abundant notes and references, ending with a list of problems with the solutions appearing in one of the appendixes. The first chapter introduces definitions for matrix functions, polynomial interpolation, Cauchy integral theorem, matrix square roots and logarithms. Chapter 2 is concerned with applications to differential equations, Markov models, control theory, eigenvalue problem and nonlinear matrix problems. The third chapter considers the conditioning and sensitivity of computations of matrix functions especially by means of the FrĂ©chet derivative. In Chapter 4 methods for matrix power, polynomial evaluation, Taylor series, rational approximation, diagonalization and others are discussed. Related theorems, algorithms, cost and stability studies are treated. The next three chapters focus on the matrix sign function, matrix square root and matrix \(p\)th root. Chapter 8 considers the polar decomposition and its relation with the singular value decomposition. Chapter 9 is devoted to Schur-Parlett algorithm by the use of the Taylor series expansion. Chapter 10 discusses the matrix exponential, “the most studied matrix function”, reflecting its importance in the solution of differential equations. A natural continuation is the matrix logarithm presented in Chapter 11, and matrix sine and cosine in Chapter 12. In Chapter 13 is given a substantial treatment of a function matrix times a vector, considering polynomial interpolation, Krylov subspace methods and numerical quadrature. The concluding Chapter 14 includes some additional topics under the title of miscellany. In the Appendixes are the notation, definitions, operation counts, toolbox references, and the solutions of the problems.

As a final word, this book should be a reference to anyone involved with matrix analysis and it appears to be a classic in the coming years, together with the ones of F. R. Gantmacher [The theory of matrices. (1953; Zbl 0050.24804)] and J. H. Wilkinson [The algebraic eigenvalue problem (1965; Zbl 0258.65037)].

Reviewer: Edgar Pereira (Covilha)