Solving linear and non-linear equations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Horwood
1992
|
| Edition: | 1. publ. |
| Series: | Ellis Horwood series in mathematics and its applications : statistics, operational research and computational mathematics section
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 190 S. |
| ISBN: | 0138304238 0138304157 |
Staff View
MARC
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| 245 | 1 | 0 | |a Solving linear and non-linear equations |c Chris Woodford |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a New York [u.a.] |b Horwood |c 1992 | |
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| 490 | 0 | |a Ellis Horwood series in mathematics and its applications : statistics, operational research and computational mathematics section | |
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Record in the Search Index
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| adam_text | Table of contents
Preface 9
1. LINEAR ALGEBRA 11
1.1 Introduction 11
1.2 Vectors 12
1.3 Vector addition 12
1.4 Vector multiplication by a real number 13
1.5 Vector multiplication 13
1.6 Vector spaces 14
1.7 Linear dependence and independence 15
1.8 Basis and dimension for a vector space 15
1.9 Matrices 17
1.10 Special matrices: the unit matrix, the inverse and the transpose ... .19
1.11 Linear transformations 20
1.12 Connection between linear transformations of Euclidean space
and matrices 21
1.13 Worked example 22
1.14 Exercises 22
2. LINEAR EQUATIONS 26
2.1 Introduction 26
2.2 Existence and multiplicity of solutions 27
2.3 Necessary condition for a unique solution Ax = b 29
2.4 Necessary condition for a unique solution of Ax — b for every
possible b 31
2.5 Necessary and sufficient condition for Ax = b
to have a unique solution for every possible b 31
2.6 Equivalent statements 32
2.7 Worked example 34
2.8 Exercises 38
6 Table of contents
3. GAUSSIAN ELIMINATION 41
3.1 Introduction 41
3.2 Upper triangular systems 41
3.3 Gaussian elimination 42
3.4 The proof of the method 45
3.5 Practical problems 46
3.6 Worked example 46
3.7 Exercises 48
4. PARTIAL PIVOTING 50
4.1 Introduction 50
4.2 Elementary lower triangular matrices 50
4.3 Gaussian elimination in terms of ELTs 51
4.4 Partial pivoting 53
4.5 Worked example . .55
4.6 Total pivoting 57
4.7 Scaling 57
4.8 Exercises 60
5. COMPUTER ARITHMETIC 62
5.1 Introduction 62
5.2 Integer representation 62
5.3 Real number representation 64
5.4 Floating point value of a real number 68
5.5 Floating point arithmetic 69
5.6 Conclusion 71
5.7 Exercises 72
6. VECTOR AND MATRIX NORMS 74
6.1 Introduction 74
6.2 Vector norms 74
6.3 Examples of vector norms 74
6.4 Matrix norms 77
6.5 The induced norm 78
6.6 Examples of consistent matrix norms 78
6.7 Banach s lemma 81
6.8 The fixed point theorem 83
6.9 Exercises 85
7. ERROR ANALYSIS OF GAUSSIAN ELIMINATION 86
7.1 Introduction 86
7.2 Condition number 87
7.3 Well conditioned and ill conditioned matrices 89
Table of contents 7
7.4 Worked examples 89
7.5 Error analysis of Gaussian elimination 91
7.6 Conclusion 95
7.7 Exercises 96
8. INTERPRETATION OF RESULTS 97
8.1 Introduction 97
8.2 The residual vector 97
8.3 Measuring the residual 97
8.4 Example of an ill conditioned system 98
8.5 Estimation of conditioning 99
8.6 Iterative refinement 102
8.7 Example 104
8.8 Conclusion 106
8.9 Exercises 107
9. NON LINEAR EQUATIONS IN GENERAL 109
9.1 Introduction 109
9.2 Iterative methods 110
9.3 Convergence 110
9.4 Convergence in general 112
9.5 Practical considerations 113
9.6 Deflation 114
9.7 Order of convergence 114
9.8 Guarantees of convergence, local and global theroems 115
9.9 Exercises 115
10. SINGLE NON LINEAR EQUATIONS 117
10.1 Introduction 117
10.2 Bisection 117
10.3 The rule of false position 120
10.4 The secant method 122
10.5 Newton s method 123
10.6 Worked example 125
10.7 Conclusion 128
10.8 Exercises 128
11. CONVERGENCE GUARANTEES 130
11.1 Introduction 130
11.2 Bisection 130
11.3 Secant method 131
11.4 Newton s method 139
11.5 Fixed point methods 141
11.6 Conclusion 143
11.7 Exercises 144
8 Table of contents
12. SECANT METHODS FOR SYSTEMS OF NON LINEAR
EQUATIONS 145
12.1 Introduction 145
12.2 A simple secant method 146
12.3 The method in detail 147
12.4 Worked example 150
12.5 Further secant methods 152
12.6 Broyden s secant method 153
12.7 Worked example, Broyden s method 155
12.8 The Jacobian matrix 156
12.9 Analysis of Broyden s secant method 158
12.10 Local convergence theorem for Broyden s secant method 160
12.11 Conclusion 162
12.12 Exercises 163
13. NEWTON S METHOD FOR SYSTEMS OF NON LINEAR
EQUATIONS 164
13.1 Introduction 164
13.2 Newton s method in outline 165
13.3 Newton s method in detail 166
13.4 Worked example 166
13.5 Guarantee of convergence 168
13.6 Order of convergence 173
13.7 Global convergence 174
13.8 Fixed point methods 175
13.9 Conclusion 176
13.10 Exercises 177
Hints and solutions to the exercises 178
Bibliography 187
Index 189
|
| adam_txt |
Table of contents
Preface 9
1. LINEAR ALGEBRA 11
1.1 Introduction 11
1.2 Vectors 12
1.3 Vector addition 12
1.4 Vector multiplication by a real number 13
1.5 Vector multiplication 13
1.6 Vector spaces 14
1.7 Linear dependence and independence 15
1.8 Basis and dimension for a vector space 15
1.9 Matrices 17
1.10 Special matrices: the unit matrix, the inverse and the transpose . .19
1.11 Linear transformations 20
1.12 Connection between linear transformations of Euclidean space
and matrices 21
1.13 Worked example 22
1.14 Exercises 22
2. LINEAR EQUATIONS 26
2.1 Introduction 26
2.2 Existence and multiplicity of solutions 27
2.3 Necessary condition for a unique solution Ax = b 29
2.4 Necessary condition for a unique solution of Ax — b for every
possible b 31
2.5 Necessary and sufficient condition for Ax = b
to have a unique solution for every possible b 31
2.6 Equivalent statements 32
2.7 Worked example 34
2.8 Exercises 38
6 Table of contents
3. GAUSSIAN ELIMINATION 41
3.1 Introduction 41
3.2 Upper triangular systems 41
3.3 Gaussian elimination 42
3.4 The proof of the method 45
3.5 Practical problems 46
3.6 Worked example 46
3.7 Exercises 48
4. PARTIAL PIVOTING 50
4.1 Introduction 50
4.2 Elementary lower triangular matrices 50
4.3 Gaussian elimination in terms of ELTs 51
4.4 Partial pivoting 53
4.5 Worked example . .55
4.6 Total pivoting 57
4.7 Scaling 57
4.8 Exercises 60
5. COMPUTER ARITHMETIC 62
5.1 Introduction 62
5.2 Integer representation 62
5.3 Real number representation 64
5.4 Floating point value of a real number 68
5.5 Floating point arithmetic 69
5.6 Conclusion 71
5.7 Exercises 72
6. VECTOR AND MATRIX NORMS 74
6.1 Introduction 74
6.2 Vector norms 74
6.3 Examples of vector norms 74
6.4 Matrix norms 77
6.5 The induced norm 78
6.6 Examples of consistent matrix norms 78
6.7 Banach's lemma 81
6.8 The fixed point theorem 83
6.9 Exercises 85
7. ERROR ANALYSIS OF GAUSSIAN ELIMINATION 86
7.1 Introduction 86
7.2 Condition number 87
7.3 Well conditioned and ill conditioned matrices 89
Table of contents 7
7.4 Worked examples 89
7.5 Error analysis of Gaussian elimination 91
7.6 Conclusion 95
7.7 Exercises 96
8. INTERPRETATION OF RESULTS 97
8.1 Introduction 97
8.2 The residual vector 97
8.3 Measuring the residual 97
8.4 Example of an ill conditioned system 98
8.5 Estimation of conditioning 99
8.6 Iterative refinement 102
8.7 Example 104
8.8 Conclusion 106
8.9 Exercises 107
9. NON LINEAR EQUATIONS IN GENERAL 109
9.1 Introduction 109
9.2 Iterative methods 110
9.3 Convergence 110
9.4 Convergence in general 112
9.5 Practical considerations 113
9.6 Deflation 114
9.7 Order of convergence 114
9.8 Guarantees of convergence, local and global theroems 115
9.9 Exercises 115
10. SINGLE NON LINEAR EQUATIONS 117
10.1 Introduction 117
10.2 Bisection 117
10.3 The rule of false position 120
10.4 The secant method 122
10.5 Newton's method 123
10.6 Worked example 125
10.7 Conclusion 128
10.8 Exercises 128
11. CONVERGENCE GUARANTEES 130
11.1 Introduction 130
11.2 Bisection 130
11.3 Secant method 131
11.4 Newton's method 139
11.5 Fixed point methods 141
11.6 Conclusion 143
11.7 Exercises 144
8 Table of contents
12. SECANT METHODS FOR SYSTEMS OF NON LINEAR
EQUATIONS 145
12.1 Introduction 145
12.2 A simple secant method 146
12.3 The method in detail 147
12.4 Worked example 150
12.5 Further secant methods 152
12.6 Broyden's secant method 153
12.7 Worked example, Broyden's method 155
12.8 The Jacobian matrix 156
12.9 Analysis of Broyden's secant method 158
12.10 Local convergence theorem for Broyden's secant method 160
12.11 Conclusion 162
12.12 Exercises 163
13. NEWTON'S METHOD FOR SYSTEMS OF NON LINEAR
EQUATIONS 164
13.1 Introduction 164
13.2 Newton's method in outline 165
13.3 Newton's method in detail 166
13.4 Worked example 166
13.5 Guarantee of convergence 168
13.6 Order of convergence 173
13.7 Global convergence 174
13.8 Fixed point methods 175
13.9 Conclusion 176
13.10 Exercises 177
Hints and solutions to the exercises 178
Bibliography 187
Index 189 |
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| index_date | 2024-07-02T16:05:22Z |
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| isbn | 0138304238 0138304157 |
| language | English |
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| series2 | Ellis Horwood series in mathematics and its applications : statistics, operational research and computational mathematics section |
| spelling | Woodford, Chris Verfasser aut Solving linear and non-linear equations Chris Woodford 1. publ. New York [u.a.] Horwood 1992 190 S. txt rdacontent n rdamedia nc rdacarrier Ellis Horwood series in mathematics and its applications : statistics, operational research and computational mathematics section Datenverarbeitung Equations Numerical solutions Data processing Nichtlineare Gleichung (DE-588)4455337-7 gnd rswk-swf Lineare Gleichung (DE-588)4234490-6 gnd rswk-swf Lineare Gleichung (DE-588)4234490-6 s DE-604 Nichtlineare Gleichung (DE-588)4455337-7 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015128389&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Woodford, Chris Solving linear and non-linear equations Datenverarbeitung Equations Numerical solutions Data processing Nichtlineare Gleichung (DE-588)4455337-7 gnd Lineare Gleichung (DE-588)4234490-6 gnd |
| subject_GND | (DE-588)4455337-7 (DE-588)4234490-6 |
| title | Solving linear and non-linear equations |
| title_auth | Solving linear and non-linear equations |
| title_exact_search | Solving linear and non-linear equations |
| title_exact_search_txtP | Solving linear and non-linear equations |
| title_full | Solving linear and non-linear equations Chris Woodford |
| title_fullStr | Solving linear and non-linear equations Chris Woodford |
| title_full_unstemmed | Solving linear and non-linear equations Chris Woodford |
| title_short | Solving linear and non-linear equations |
| title_sort | solving linear and non linear equations |
| topic | Datenverarbeitung Equations Numerical solutions Data processing Nichtlineare Gleichung (DE-588)4455337-7 gnd Lineare Gleichung (DE-588)4234490-6 gnd |
| topic_facet | Datenverarbeitung Equations Numerical solutions Data processing Nichtlineare Gleichung Lineare Gleichung |
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