Numerical Linear Algebra for Applications in Statistics, 1998 Statistics and Computing Series

Numerical linear algebra is one of the most important subjects in the field of statistical computing. Statistical methods in many areas of application require computations with vectors and matrices. This book describes accurate and efficient computer algorithms for factoring matrices, solving linear systems of equations, and extracting eigenvalues and eigenvectors. Although the book is not tied to any particular software system, it describes and gives examples of the use of modern computer software for numerical linear algebra. An understanding of numerical linear algebra requires basic knowledge both of linear algebra and of how numerical data are stored and manipulated in the computer. The book begins with a discussion of the basics of numerical computations, and then describes the relevant properties of matrix inverses, matrix factorizations, matrix and vector norms, and other topics in linear algebra; hence, the book is essentially self- contained. The topics addressed in this bookconstitute the most important material for an introductory course in statistical computing, and should be covered in every such course. The book includes exercises and can be used as a text for a first course in statistical computing or as supplementary text for various courses that emphasize computations. James Gentle is University Professor of Computational Statistics at George Mason University. During a thirteen-year hiatus from academic work before joining George Mason, he was director of research and design at the world's largest independent producer of Fortran and C general-purpose scientific software libraries. These libraries implement many algorithms for numerical linear algebra. He is a Fellow of the American Statistical Association and member of the International Statistical Institute. He has held several national

1 Computer Storage and Manipulation of Data.- 1.1 Digital Representation of Numeric Data.- 1.2 Computer Operations on Numeric Data.- 1.3 Numerical Algorithms and Analysis.- Exercises.- 2 Basic Vector/Matrix Computations.- 2.1 Notation, Definitions, and Basic Properties.- 2.2 Computer Representations and Basic Operations.- Exercises.- 3 Solution of Linear Systems.- 3.1 Gaussian Elimination.- 3.2 Matrix Factorizations.- 3.3 Iterative Methods.- 3.4 Numerical Accuracy.- 3.5 Iterative Refinement.- 3.6 Updating a Solution.- 3.7 Overdetermined Systems; Least Squares.- 3.8 Other Computations for Linear Systems.- Exercises.- 4 Computation of Eigenvectors and Eigenvalues and the Singular Value Decomposition.- 4.1 Power Method.- 4.2 Jacobi Method.- 4.3 QR Method for Eigenanalysis.- 4.4 Singular Value Decomposition.- Exercises.- 5 Software for Numerical Linear Algebra.- 5.1 Fortran and C.- 5.2 Interactive Systems for Array Manipulation.- 5.3 High-Performance Software.- 5.4 Test Data.- Exercises.-6 Applications in Statistics.- 6.1 Fitting Linear Models with Data.- 6.2 Linear Models and Least Squares.- 6.3 Ill-Conditioning in Statistical Applications.- 6.4 Testing the Rank of a Matrix.- 6.5 Stochastic Processes.- Exercises.- Appendices.- A Notation and Definitions.- B Solutions and Hints for Selected Exercises.- Literature in Computational Statistics.- World Wide Web, News Groups, List Servers, and Bulletin Boards.- References.- Author Index.