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E-grāmata: Numerical Linear Algebra: An Introduction

3.62/5 (15 ratings by Goodreads)
(Universität Bayreuth, Germany)
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This self-contained introduction to numerical linear algebra provides a comprehensive, yet concise, overview of the subject. It includes standard material such as direct methods for solving linear systems and least-squares problems, error, stability and conditioning, basic iterative methods and the calculation of eigenvalues. Later chapters cover more advanced material, such as Krylov subspace methods, multigrid methods, domain decomposition methods, multipole expansions, hierarchical matrices and compressed sensing. The book provides rigorous mathematical proofs throughout, and gives algorithms in general-purpose language-independent form. Requiring only a solid knowledge in linear algebra and basic analysis, this book will be useful for applied mathematicians, engineers, computer scientists, and all those interested in efficiently solving linear problems.

This self-contained introduction to numerical linear algebra provides a comprehensive, yet concise, overview of the subject. This book will be of particular use to applied mathematicians, engineers, computer scientists, and to all those interested in efficiently solving linear problems.

Recenzijas

'Wendland delivers an introductory textbook on numerical linear algebra intended for advanced undergraduate and graduate students in applied mathematics. The book covers fairly standard material in this area; it includes error analysis and ill-conditioning, direct and iterative methods for the solution of linear systems of equations, least squares problems, and eigenvalue problems. Additional advanced topics that are not usually covered in introductory textbooks include multipole expansions, domain decomposition methods, and compressive sensing. The book is generally theoretical and mathematically rigorous in its approach. Algorithms are given only in pseudocode. The text will be of interest primarily to instructors and students in graduate numerical linear algebra courses.' B. Borchers, Choice 'Wendland's book provides the reader with rigorous and clean proofs throughout the text. There are a lot of new concepts being presented that can spark the interest of a student who wishes to take numerical linear algebra and can also serve as an excellent resource for an independent study. If you are considering a new text for your numerical linear algebra class or wish to supplement with another resource, I would recommend giving this book a review.' Peter Olszewski, MAA Reviews

Papildus informācija

This self-contained introduction to numerical linear algebra provides a comprehensive, yet concise, overview of the subject.
Preface ix
PART ONE PRELIMINARIES
1(56)
1 Introduction
3(27)
1.1 Examples Leading to Linear Systems
5(5)
1.2 Notation
10(3)
1.3 Landau Symbols and Computational Cost
13(4)
1.4 Facts from Linear Algebra
17(7)
1.5 Singular Value Decomposition
24(2)
1.6 Pseudo-inverse
26(4)
Exercises
29(1)
2 Error, Stability and Conditioning
30(27)
2.1 Floating Point Arithmetic
30(2)
2.2 Norms for Vectors and Matrices
32(14)
2.3 Conditioning
46(8)
2.4 Stability
54(3)
Exercises
55(2)
PART TWO BASIC METHODS
57(124)
3 Direct Methods for Solving Linear Systems
59(42)
3.1 Back Substitution
59(2)
3.2 Gaussian Elimination
61(4)
3.3 LU Factorisation
65(6)
3.4 Pivoting
71(5)
3.5 Cholesky Factorisation
76(2)
3.6 QR Factorisation
78(6)
3.7 Schur Factorisation
84(3)
3.8 Solving Least-Squares Problems
87(14)
Exercises
100(1)
4 Iterative Methods for Solving Linear Systems
101(31)
4.1 Introduction
101(1)
4.2 Banach's Fixed Point Theorem
102(4)
4.3 The Jacobi and Gauss-Seidel Iterations
106(10)
4.4 Relaxation
116(9)
4.5 Symmetric Methods
125(7)
Exercises
130(2)
5 Calculation of Eigenvalues
132(49)
5.1 Basic Localisation Techniques
133(8)
5.2 The Power Method
141(2)
5.3 Inverse Iteration by von Wielandt and Rayleigh
143(10)
5.4 The Jacobi Method
153(6)
5.5 Householder Reduction to Hessenberg Form
159(3)
5.6 The QR Algorithm
162(9)
5.7 Computing the Singular Value Decomposition
171(10)
Exercises
180(1)
PART THREE ADVANCED METHODS
181(214)
6 Methods for Large Sparse Systems
183(77)
6.1 The Conjugate Gradient Method
183(20)
6.2 GMRES and MINRES
203(23)
6.3 Biorthogonalisation Methods
226(18)
6.4 Multigrid
244(16)
Exercises
258(2)
7 Methods for Large Dense Systems
260(69)
7.1 Multipole Methods
261(21)
7.2 Hierarchical Matrices
282(25)
7.3 Domain Decomposition Methods
307(22)
Exercises
327(2)
8 Preconditioning
329(41)
8.1 Scaling and Preconditioners Based on Splitting
331(7)
8.2 Incomplete Splittings
338(8)
8.3 Polynomial and Approximate Inverse Preconditioners
346(11)
8.4 Preconditioning Krylov Subspace Methods
357(13)
Exercises
368(2)
9 Compressed Sensing
370(25)
9.1 Sparse Solutions
370(2)
9.2 Basis Pursuit and Null Space Property
372(6)
9.3 Restricted Isometry Property
378(6)
9.4 Numerical Algorithms
384(11)
Exercises
393(2)
Bibliography 395(8)
Index 403
Holger Wendland holds the Chair of Applied and Numerical Analysis at the Universität Bayreuth, Germany. He works in the area of Numerical Analysis and is the author of two books, Scattered Data Approximation (Cambridge, 2005) and Numerische Mathematik (2004, with Robert Schaback).
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