Software for Roundoff Analysis of Matrix Algorithms

Preface1 Some Elements of Error Analysis 1.1 Introduction 1.2 Errors in Computation 1.3 The Machine Rounding Unit 1.4 Backward Roundoff Analysis 1.5 Case Study 1: Triangular Matrix Inversion 1.6 Case Study 2: The Normal Equations 1.7 Case Study 3: The Hat Matrix Notes and References2 Concepts from Linear Algebra and Analysis 2.1 Introduction 2.2 Elementary Concepts from Analysis 2.3 Wilkinson Numbers 2.4 Norms 2.5 Linear Transformations 2.6 Minkowski Sums and Cartesian Products 2.7 Computing Wilkinson Numbers 2.8 Operator Norms 2.9 Differentiation3 Directed Graphs 3.1 Introduction 3.2 Arithmetic Graphs 3.3 The Weak Composition Model 3.4 Sums of Path Products 3.5 Differentiation in Graphs 3.6 Improved Calculation of Derivatives4 Software for Roundoff Analysis 4.1 Introduction 4.2 Rounding and Perturbations of the Computational Problem 4.3 Comparing Rounding Errors with Problem Sensitivity 4.4 The Weak Composition Model 4.5 Reverse Condition Numbers 4.6 Comparing Two Algorithms 4.7 Using the Software Package5 Case Studies 5.1 Case Study 4: The Cholesky Factorization 5.2 Case Study 5: Gaussian Elimination 5.3 Case Study 6: Gaussian Elimination with Iterative Improvement 5.4 Case Study 7: Gauss-Jordan Elimination 5.5 Case Study 8: Householder Transformations for Least-Squares Problems 5.6 Case Study 9: Rational QR Methods 5.7 Case Study 10: Downdating the QR Factorization 5.8 Case Study 11: The Characteristic Polynomial 5.9 Case Study 12: Representations of Symmetric Matrices 5.10 Case Study 13: Variants of the Gram-Schmidt Method 5.11 Case Study 14: Cholesky Factors After Rank-One ModificationsAppendix: Fast Givens TransformationsBibliographyIndex