Numerical Linear Algebra for Applications in Statistics

ISBN-13: 9781461268420 / Angielski / Miękka / 2012 / 221 str.

James E. Gentle; James E. Gentle

Opis

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 book constitute 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

Recenzje

From a review:

JOURNAL OF AMERICAN STATISTICAL ASSOCIATION

"Gentle brings to this book (as well as his other recent books on further aspects of statistical computing) his vast knowledge and experience in the mathematics of scientific computing, the practical aspects of software development, and teaching. The presentation is exceptionally clear and well-sign-boarded. ...The writing style, though very precise, conveys a warmth and enthusiasm that will appeal to students."

 

Spis treści

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.- 2.1.1 Operations on Vectors; Vector Spaces.- 2.1.2 Vectors and Matrices.- 2.1.3 Operations on Vectors and Matrices.- 2.1.4 Partitioned Matrices.- 2.1.5 Matrix Rank.- 2.1.6 Identity Matrices.- 2.1.7 Inverses.- 2.1.8 Linear Systems.- 2.1.9 Generalized Inverses.- 2.1.10 Other Special Vectors and Matrices.- 2.1.11 Eigenanalysis.- 2.1.12 Similarity Transformations.- 2.1.13 Norms.- 2.1.14 Matrix Norms.- 2.1.15 Orthogonal Transformations.- 2.1.16 Orthogonalization Transformations.- 2.1.17 Condition of Matrices.- 2.1.18 Matrix Derivatives.- 2.2 Computer Representations and Basic Operations.- 2.2.1 Computer Representation of Vectors and Matrices.- 2.2.2 Multiplication of Vectors and Matrices.- Exercises.- 3 Solution of Linear Systems.- 3.1 Gaussian Elimination.- 3.2 Matrix Factorizations.- 3.2.1 LU and LDU Factorizations.- 3.2.2 Cholesky Factorization.- 3.2.3 QR Factorization.- 3.2.4 Householder Transformations (Reflections).- 3.2.5 Givens Transformations (Rotations).- 3.2.6 Gram-Schmidt Transformations.- 3.2.7 Singular Value Factorization.- 3.2.8 Choice of Direct Methods.- 3.3 Iterative Methods.- 3.3.1 The Gauss-Seidel Method with Successive Overrelaxation.- 3.3.2 Solution of Linear Systems as an Optimization Problem; Conjugate Gradient Methods.- 3.4 Numerical Accuracy.- 3.5 Iterative Refinement.- 3.6 Updating a Solution.- 3.7 Overdetermined Systems; Least Squares.- 3.7.1 Full Rank Coefficient Matrix.- 3.7.2 Coefficient Matrix Not of Full Rank.- 3.7.3 Updating a Solution to an Overdetermined System.- 3.8 Other Computations for Linear Systems.- 3.8.1 Rank Determination.- 3.8.2 Computing the Determinant.- 3.8.3 Computing the Condition Number.- 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.- 5.1.1 BLAS.- 5.1.2 Fortran and C Libraries.- 5.1.3 Fortran 90 and 95.- 5.2 Interactive Systems for Array Manipulation.- 5.2.1 Matlab.- 5.2.2 S, S-Plus.- 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.- 6.2.1 The Normal Equations and the Sweep Operator.- 6.2.2 Linear Least Squares Subject to Linear Equality Constraints.- 6.2.3 Weighted Least Squares.- 6.2.4 Updating Linear Regression Statistics.- 6.2.5 Tests of Hypotheses.- 6.2.6 D-Optimal Designs.- 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.

O autorze

Linear algebra is one of the most imprortant mathematical and computational tools in the sciences. A knowledge of linear algebra is essential for research in statistics or the application of statistics. This book presents the aspects of numerical linear algebra that are important to statisticians.