A concise introduction to numerical analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton ; London ; New York
CRC Press, Taylor & Francis Group
[2016]
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| Series: | A Chapman & Hall book
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | xv, 292 pages Illustrationen, Diagramme |
| ISBN: | 9781498712187 1498712185 |
Staff View
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| 245 | 1 | 0 | |a A concise introduction to numerical analysis |c A.C. Faul, University of Cambridge, UK |
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Record in the Search Index
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| adam_text | Contents
List of Figures xi
Preface xiii
Acknowledgments XV
Chapter 1 ■ Fundamentals 1
1.1 Floating Point Arithmetic 1
1.2 Overflow and Underflow 3
1.3 Absolute, Relative Error, Machine Epsilon 4
1.4 Forward and Backward Error Analysis 6
1.5 Loss of Significance 8
1.6 Robustness 11
1.7 Error Testing and Order of Convergence 12
1.8 Computational Complexity 14
1.9 Condition 15
1.10 Revision Exercises 19
Chapter 2 ■ Linear Systems 23
2.1 Simultaneous Linear Equations 23
2.2 Gaussian Elimination and Pivoting 25
2.3 LU Factorization 27
2.4 Cholesky Factorization 32
2.5 QR Factorization 34
2.6 The Gram---Schmidt Algorithm 36
2.7 Givens Rotations 38
2.8 Householder Reflections 41
2.9 Linear Least Squares 42
2.10 Singular Value Decomposition 43
2.11 Iterative Schemes and Splitting 46
2.12 Jacobi and Gauss-Seidel Iterations 48
vii
viii ■ Contents
2.13 Relaxation 51
2.14 Steepest Descent Method 52
2.15 Conjugate Gradients 56
2.16 Krylov Subspaces and Pre-Conditioning 59
2.17 Eigenvalues and Eigenvectors 63
2.18 The Power Method 63
2.19 Inverse Iteration 67
2.20 Deflation 69
2.21 Revision Exercises 72
Chapter 3 ■ Interpolation and Approximation Theory________________79
3.1 Lagrange Form of Polynomial Interpolation 79
3.2 Newton Form of Polynomial Interpolation 84
3.3 Polynomial Best Approximations 90
3.4 Orthogonal polynomials 91
3.5 Least-Squares Polynomial Fitting 94
3.6 The Peano Kernel Theorem 95
3.7 Splines 98
3.8 B-Spline 105
3.9 Revision Exercises 110
Chapter 4 ■ Non-Linear Systems____________________________________113
4.1 Bisection, Régula Falsi, and Secant Method 113
4.2 Newton’s Method 116
4.3 Broyden’s Method 119
4.4 Householder Methods 121
4.5 Müller’s Method 122
4.6 Inverse Quadratic Interpolation 123
4.7 Fixed Point Iteration Theory 124
4.8 Mixed Methods 126
4.9 Revision Exercises 127
Chapter 5 ■ Numerical Integration_________________________________131
5.1 Mid-Point and Trapezium Rule 131
5.2 The Peano Kernel Theorem 133
5.3 Simpson’s Rule 135
Contents ■ ix
5.4 Newton-Cotes Rules 137
5.5 Gaussian Quadrature 138
5.6 Composite Rules 145
5.7 Multi-Dimensional Integration 150
5.8 Monte Carlo Methods 152
5.9 Revision Exercises 153
Chapter 6 ■ ODEs_________________________________________________157
6.1 One-Step Methods 157
6.2 Multistep Methods, Order, and Consistency 159
6.3 Order Conditions 162
6.4 Stiffness and A-Stability 164
6.5 Adams Methods 169
6.6 Backward Differentiation Formulae 172
6.7 The Milne and Zadunaisky Device 174
6.8 Rational Methods 177
6.9 Runge-Kutta Methods 179
6.10 Revision Exercises 201
Chapter 7 ■ Numerical Differentiation____________________________205
7.1 Finite Differences 206
7.2 Differentiation of Incomplete or Inexact Data 209
Chapter 8 ■ PDEs________________________________________________211
8.1 Classification of PDEs 211
8.2 Parabolic PDEs 213
8.2.1 Finite Differences 214
8.2.2 Stability and Its Eigenvalue Analysis 217
8.2.3 Cauchy Problems and the Fourier Analysis
of Stability 222
8.3 Elliptic PDEs 227
8.3.1 Computational Stencils 228
8.3.2 Sparse Algebraic Systems Arising from
Computational Stencils 233
8.3.3 Hockney Algorithm 235
8.3.4 Multigrid Methods 238
8.4 Parabolic PDEs in Two Dimensions 243
x ■ Contents
8.4.1 Splitting 246
8.5 Hyperbolic PDEs 253
8.5.1 Advection Equation 253
8.5.2 The Wave Equation 256
8.6 Spectral Methods 258
8.6.1 Spectral Solution to the Poisson Equation 267
8.7 Finite Element Method 269
8.8 Revision Exercises 277
Bibliography 285
Index
287
Numerical Analysis Mathematical Computation
This textbook provides an accessible and concise introduction to nu-
merical analysis for upper undergraduate and beginning graduate stu-
dents from various backgrounds. It was developed from the lecture
notes of four successful courses on numerical analysis taught within
the MPhil of Scientific Computing at the University of Cambridge. The
book is easily accessible, even to those with limited knowledge of
mathematics. 1
Students will get a concise, but thorough introduction to numerical
analysis. In addition the algorithmic principles are emphasized to en-
courage a deeper understanding of why an algorithm is suitable, and
sometimes unsuitable, for a particular problem.
A Concise Introduction to Numerical Analysis strikes a balance be-
tween being mathematically comprehensive, but not overwhelming
with mathematical detail. In some places where further detail was felt
to be out of scope of the book, the reader is referred to further reading.
The book uses MATLAB® implementations to demonstrate the work-
ings of the method and thus MATLAB’s own implementations are
avoided, unless they are used as building blocks of an algorithm. In
some cases the listings are printed in the book, but all are available
online on the book’s page at www.crcpress.com.
Most implementations are in the form of functions returning the out-
come of the algorithm. Also, examples for the use of the functions are
given. Exercises are included in line with the text where appropriate,
and each chapter ends with a selection of revision exercises. Solutions
to odd-numbered exercises are also provided on the book’s page at
www.crcpress.com.
This textbook is also an ideal resource for graduate students coming
from other subjects who will use numerical techniques extensively in
their graduate studies.
ISBN: B70-l-4eIS?-iaia-7
|
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| discipline | Mathematik |
| format | Book |
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| spelling | Faul, Anita Verfasser (DE-588)1150897651 aut A concise introduction to numerical analysis A.C. Faul, University of Cambridge, UK Boca Raton ; London ; New York CRC Press, Taylor & Francis Group [2016] © 2016 xv, 292 pages Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier A Chapman & Hall book Includes bibliographical references and index Numerical analysis Textbooks Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029052124&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029052124&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Faul, Anita A concise introduction to numerical analysis Numerical analysis Textbooks Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4042805-9 |
| title | A concise introduction to numerical analysis |
| title_auth | A concise introduction to numerical analysis |
| title_exact_search | A concise introduction to numerical analysis |
| title_full | A concise introduction to numerical analysis A.C. Faul, University of Cambridge, UK |
| title_fullStr | A concise introduction to numerical analysis A.C. Faul, University of Cambridge, UK |
| title_full_unstemmed | A concise introduction to numerical analysis A.C. Faul, University of Cambridge, UK |
| title_short | A concise introduction to numerical analysis |
| title_sort | a concise introduction to numerical analysis |
| topic | Numerical analysis Textbooks Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Numerical analysis Textbooks Numerische Mathematik |
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