E-Book, Englisch, 216 Seiten, Web PDF
Todd / Blanc / Ghizzetti Numerical Algebra
1. Auflage 2014
ISBN: 978-1-4832-6941-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark
E-Book, Englisch, 216 Seiten, Web PDF
ISBN: 978-1-4832-6941-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark
Basic Numerical Mathematics, Volume II: Numerical Algebra focuses on numerical algebra, with emphasis on the ideas of 'controlled computational experiments' and 'bad examples'. The existence of an orthogonal matrix which diagonalizes a real symmetric matrix is highlighted, and partitioned or block matrices are discussed, along with induced norms and inversion problems. Comprised of 12 chapters, this volume begins with an overview of the manipulation of vectors and matrices, followed by an analysis of induced norms. The reader is then introduced to the direct solution of the inversion problem, first in the context of theoretical arithmetic (that is, when round-off is disregarded) and second in the context of practical computation. Various methods of handling the characteristic value problems are also considered, together with several iterative methods for the solution of a system of linear equations. Two applications are described: the solution of a two-point boundary value problem and the solution of least squares curve fitting. The book concludes with an account of the singular value decomposition and pseudo-inverses. This monograph will be of interest to mathematicians and students of mathematics.
Autoren/Hrsg.
Weitere Infos & Material
1;Front Cover;1
2;Numerical Algebra;4
3;Copyright Page;5
4;Table of Contents;6
5;Notations and Abbreviations;8
6;Preface;10
7;Basic Numerical Mathematics;14
8;CHAPTER 1. Manipulation of Vectors and Matrices;16
8.1;Chapter 1. Problems;16
9;CHAPTER 2. Norms of Vectors and Matrices;19
9.1;Chapter 2. Problems;20
10;CHAPTER 3. Induced Norms;22
10.1;1. THE CHEBYSHEV CASE;22
10.2;2. THE MANHATTAN CASE;23
10.3;3 . THE EUCLIDEAN CASE;23
10.4;4. CONTINUITY OF NORMS;24
10.5;5. PROOF OF THEOREM 3.1;25
10.6;6. SPECTRAL RADIUS AND CONVERGENCE;26
10.7;7. THE MATRIX NORM INDUCED BY THE p-NORM;28
10.8;Chapter 3, Problems;29
11;CHAPTER 4. The Inversion Problem I: Theoretical Arithmetic;32
11.1;1. DIAGONAL MATRIX;32
11.2;2. TRIANGULAR MATRIX;33
11.3;3. TRIPLE DIAGONAL MATRIX;33
11.4;4. BAND MATRICES;34
11.5;5. THE GENERAL CASE;35
11.6;6. COMMENTS;37
11.7;7. SOLUTION OF SYSTEMS, DETERMINANTS;39
11.8;8. INVERSION OF MATRICES;41
11.9;9. OPTIMALITY;42
11.10;Chapter 4, Problems;42
12;CHAPTER 5. The Inversion Problem II: Practical Computation;47
12.1;Chapter 5, Problems;50
13;CHAPTER 6. The Characteristic Value Problem—Generalities;56
13.1;1. ROW AND COLUMN CHARACTERISTIC VECTORS;56
13.2;2. LOCALIZATION OF CHARACTERISTIC VALUES;57
13.3;3. POSITIVE MATRICES, NON-NEGATIVE MATRICES;58
13.4;4. QUADRATIC FORMS;59
13.5;REMARKS;62
14;CHAPTER 7. The Power Method, Deflation, Inverse Iteration;68
14.1;1. THE POWER METHOD;68
14.2;2. DEFLATION;69
14.3;3. WIELANDT INVERSE ITERATION;70
14.4;Chapter 7, Problems;71
15;CHAPTER 8. Characteristic Values;74
15.1;1. ROTATION METHODS: JACOBI, GIVENS, HOUSEHOLDER;74
15.2;2. L R AND Q R METHODS;79
15.3;Chapter 8, Problems;83
16;CHAPTER 9. Iterative Methods for the Solution of Systems Ax - b;86
16.1;1. THE JACOBI AND GAUSS—SEIDEL METHODS;86
16.2;2. YOUNG OVERRELAXATION METHOD;90
16.3;3. GRADIENT OR STEEPEST DESCENT METHODS;93
16.4;4. ITERATIVE IMPROVEMENT OF APPROXIMATE INVERSES;99
16.5;Chapter 9, Problems;100
17;CHAPTER 10. Application: Solution of a Boundary Value Problem;102
17.1;1. METHOD I;103
17.2;2. METHOD II;104
17.3;3. METHOD III;106
17.4;Chapter 10. Problem;107
18;CHAPTER 11. Application: Least Squares Curve Fitting;108
18.1;Chapter 11. Problems;111
19;CHAPTER 12. Singular Value Decomposition and Pseudo-Inverses;113
19.1;Chapter 12. Problems;117
20;Solutions to Selected Problems;120
21;Bibliographical Remarks;215
22;Index;217
