Software for Roundoff Analysis of Matrix Algorithms - 1st Edition - ISBN: 9780124972506, 9781483265155

Software for Roundoff Analysis of Matrix Algorithms

1st Edition

Authors: Webb Miller Celia Wrathall
eBook ISBN: 9781483265155
Imprint: Academic Press
Published Date: 28th January 1980
Page Count: 166
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Description

Computer Science and Applied Mathematics: A Series of Monographs and Textbooks: Software for Roundoff Analysis of Matrix Algorithms focuses on the presentation of techniques and software tools for analyzing the propagation of rounding error in matrix algorithms.

The publication looks into some elements of error analysis, concepts from linear algebra and analysis, and directed graphs. Discussions focus on arithmetic graphs, sums of path products, linear transformations, Minkowski sums and Cartesian products, and elementary concepts from analysis. The text then examines software for roundoff analysis, including rounding and perturbations of the computational problem, comparing rounding errors with problem sensitivity, reverse condition numbers, and comparing two algorithms. The book ponders on case studies, as well as Gaussian elimination with iterative improvement, Cholesky factorization, Gauss-Jordan elimination, variants of the Gram-Schmidt method, and Cholesky factors after rank-one modifications.

The text is a valuable reference for researchers interested in the techniques and software tools involved in the analysis of the propagation of rounding error in matrix algorithms.

Table of Contents


Preface

1 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 References

2 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 Differentiation

3 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 Derivatives

4 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 Package

5 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 Modifications

Appendix: Fast Givens Transformations

Bibliography

Index

Details

No. of pages:
166
Language:
English
Copyright:
© Academic Press 1980
Published:
Imprint:
Academic Press
eBook ISBN:
9781483265155

About the Author

Webb Miller

Celia Wrathall