Numerical methods and scientific computing: using software libraries for problem solving
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford
Clarendon Press
1994
|
| Schriftenreihe: | Oxford science publications
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 328 S. graph. Darst. |
| ISBN: | 0198596987 |
Internformat
MARC
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| 245 | 1 | 0 | |a Numerical methods and scientific computing |b using software libraries for problem solving |c Norbert Köckler |
| 264 | 1 | |a Oxford |b Clarendon Press |c 1994 | |
| 300 | |a XVIII, 328 S. |b graph. Darst. | ||
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Datensatz im Suchindex
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| adam_text | Contents
Guide to Notation xv
List of Examples and Applications xvii
Introduction 1
The Structure of this Text 1
Libraries, Packages, Tools 2
Structured Programming 4
Calculus of Errors 5
Software Availability 9
Disclaimer of Warranty 9
1 Simultaneous Linear Equations 10
1.1 Nonsingular Systems 10
1.1.1 Introduction 11
1.1.2 Elimination and Backward Substitution 12
1.1.3 Pivoting Strategies 13
1.1.4 Scaling 14
1.1.5 Implicit Scaling = Relative Pivoting 15
1.1.6 Iterative Refinement 15
1.1.7 Multiple Right Hand Sides. Matrix Inversion 17
1.1.8 The Gaussian Algorithm 17
1.1.9 Program and Example 19
1.1.10 Application: Flow through a Plumbing Network 20
1.2 Positive Definite Systems 21
1.2.1 Introduction 22
1.2.2 The Cholesky Factorization 23
1.2.3 Program and Example 23
1.2.4 Application: Network Theory 25
1.3 Systems with Band Matrices 27
1.3.1 Introduction 27
1.3.2 Cholesky s Method for Variable Bandwidth 29
1.3.3 Cholesky s Method for Fixed Bandwidth 30
1.3.4 Gaussian Elimination for General Band Matrices 30
1.3.5 Program and Example 31
1.3.6 Application: A Plane Truss 32
vii
viii Contents
1.4 Large Sparse Systems 34
1.4.1 Introduction 34
1.4.2 Relaxation Methods 36
1.4.3 Method of Conjugate Gradients 37
1.4.4 Preconditioned Conjugate Gradients 38
1.4.5 Incomplete Cholesky Factorization 39
1.4.6 Nonsymmetric Sparse Systems 40
1.4.7 Program and Examples 40
1.5 Singular Systems 43
1.5.1 Introduction 43
1.5.2 Singular Value Decomposition and Pseudoinverse 44
1.5.3 Program and Examples 48
1.6 Norms, Condition Number, Error Estimates 52
1.6.1 Vector and Matrix Norms 52
1.6.2 Condition Number 54
1.6.3 Residual Vector and Error 55
1.7 Linear Systems Using IMSL or MATLAB 56
1.7.1 General Real Linear Systems 56
1.7.2 Symmetric Positive Definite Systems 56
1.7.3 Banded SPD Systems 56
1.7.4 Fixed Bandwidth 57
1.7.5 Sparse Linear Systems 57
1.7.6 Singular Linear Systems 57
1.7.7 An Example using MATLAB 57
1.8 Exercises 58
1.8.1 Electrical Networks 58
1.8.2 Ill Conditioning and Small Residual Norm 59
1.8.3 Determinant and Condition Number 60
1.8.4 Hilbert Matrices again 60
1.8.5 A Typical Positive Definite Matrix 60
1.8.6 Yet Another Algorithm for Sparse Systems 60
2 Linear Optimization 62
2.1 Introduction 62
2.2 The Simplex Algorithm 67
2.3 Program and Example 70
2.3.1 Application: Female Blue Whale Population 71
2.4 Linear Optimization with IMSL 74
2.5 Exercise: Optimal Mine Production 74
3 Interpolation and Approximation 75
3.1 Polynomial Interpolation 78
3.1.1 Introduction 78
Contents ix
3.1.2 Numerical Methods 80
3.1.3 Program and Example 85
3.2 Splines 87
3.2.1 Cubic Splines 88
3.2.2 Linear B splines 89
3.2.3 Cubic B splines 91
3.3 One Dimensional Interpolation: INTERDIM1 95
3.4 Curve Fitting with Polynomials 100
3.4.1 Least Squares Approximation of Functions 100
3.4.2 Least Squares Approximation of Data 101
3.4.3 Approximation with Chebyshev Polynomials 102
3.4.4 Program and Example 104
3.5 Trigonometric Approximation 104
3.5.1 Harmonic Analysis 104
3.5.2 Calculation of the Fourier Coefficients 106
3.5.3 Fast Fourier Transform 107
3.6 One Dimensional Approximation: APPROX 109
3.7 Two Dimensional Splines 111
3.7.1 Solution with Bicubic B Splines 112
3.7.2 Program and Examples 113
3.8 Interpolation of Curves 116
3.8.1 Introduction 116
3.8.2 Numerical Solution by Splines 117
3.8.3 Program and Examples 117
3.9 Interpolation and Approximation using IMSL 119
3.9.1 Polynomial Interpolation and Approximation 119
3.9.2 One Dimensional Spline Interpolation 119
3.9.3 Trigonometric Approximation 120
3.9.4 Multi Dimensional Splines 120
3.10 Exercises 120
3.10.1 Beloved B Splines 120
3.10.2 Square Wave Impulse 121
3.10.3 Design a Car 121
4 Nonlinear Equations 123
4.1 Introduction 123
4.1.1 The Problem 123
4.1.2 General Convergence Theory 125
4.1.3 Stability and Condition 127
4.2 A Single Nonlinear Equation 129
4.2.1 Newton s Method 129
4.2.2 Bisection Method 131
4.2.3 Brent s Method 133
x Contents
4.2.4 Determination of an Including Interval 134
4.2.5 Broyden s Method 135
4.3 Systems of Nonlinear Equations 136
4.3.1 Newton s Method 136
4.3.2 The Powell Hybrid Method 138
4.4 Programs and Examples 140
4.4.1 A Single Equation 140
4.4.2 Systems of Nonlinear Equations 142
4.5 Nonlinear Equations Using IMSL 147
4.5.1 Solution of a Single Nonlinear Equation 147
4.5.2 Systems of Nonlinear Equations 147
4.6 Exercises 147
4.6.1 Gamma Function 147
4.6.2 Calculation of Effective Annual Interest Rates 148
5 Matrix Eigenvalue Problems 149
5.1 The Standard Eigenvalue Problem 149
5.1.1 Introduction 149
5.1.2 Balancing 151
5.1.3 Householder s Tridiagonalization 152
5.1.4 The QR Method 154
5.2 The General Eigenvalue Problem 158
5.2.1 Introduction 158
5.2.2 The Symmetric Positive Definite Case 158
5.3 Singular Value Decomposition 160
5.4 Sparse Matrices 160
5.5 Programs Solving Different Eigenvalue Problems 165
5.5.1 STDEV: Standard Eigenvalue Problem 165
5.5.2 SYMEV: Symmetric Eigenvalue Problem 167
5.5.3 GEV: General Eigenvalue Problem 168
5.5.4 SPARSEEV: Sparse Eigenvalue Problems 170
5.5.5 SVD: Singular Value Decomposition 170
5.6 Applications 171
5.6.1 Vibrations of a Diaphragm 171
5.6.2 Axially Loaded Beam 172
5.7 Eigenvalue Problems with IMSL 175
5.7.1 The Standard Eigenvalue Problem 175
5.7.2 The General Eigenvalue Problem Ax = XBx 177
5.7.3 Singular Value Decomposition 177
5.8 Exercises 178
5.8.1 Singular Value Decomposition (SVD) of A 178
5.8.2 Eigenvalue/Eigenvector Approximations 178
5.8.3 The Sound of a Drum . 179
Contents xi
6 Numerical Integration 180
6.1 Quadrature Methods 180
6.1.1 Newton Cotes Rules 180
6.1.2 Romberg s Rule 182
6.1.3 Gaussian Quadrature 184
6.1.4 Interlacing Gauss Rules (Kronrod, Patterson) 186
6.1.5 Automatic Whole Interval and Adaptive Rules 187
6.1.6 Multi Dimensional Integration 188
6.2 Programs and Examples 188
6.2.1 Automatic One Dimensional Quadrature 189
6.2.2 Two Dimensional Patterson Quadrature 193
6.2.3 Multi Dimensional Quadrature over a Simplex 194
6.3 Numerical Integration Using IMSL 195
6.4 Exercises 196
6.4.1 Gauss Quadrature 196
6.4.2 Gauss Kronrod Quadrature 196
7 Initial Value Problems 197
7.1 Introduction 197
7.1.1 The Problem 197
7.1.2 A Model Problem 199
7.1.3 Three Example Methods 200
7.1.4 Application to the Model Equation 201
7.1.5 Discretization Error and Consistency 202
7.1.6 Convergence and Stability 203
7.1.7 Stiff Differential Equations 205
7.2 One Step Methods 206
7.2.1 Runge Kutta Methods 206
7.2.2 Classical Runge Kutta Method 208
7.2.3 Step Control Policy, Merson s Method 208
7.2.4 A Nonlinear Example 210
7.3 Multistep Methods 214
7.3.1 Adams Methods 215
7.3.2 Consistency of Linear Multistep Methods 216
7.3.3 Predictor Corrector Methods 217
7.3.4 Starting Methods 218
7.3.5 Variable Step Length and Order 218
7.4 Backward Differentiation Formulae (BDF) 218
7.4.1 Introduction 218
7.4.2 Consistency of BDF 219
7.4.3 BDF with Variable Step Length 220
7.4.4 Step Control Policy 221
7.4.5 Variable Step Length and Order 221
xii Contents
7.5 Programs 222
7.6 Applications 222
7.7 Initial Value Problems with IMSL 227
7.8 Exercises: Autonomous Initial Value Problems 227
8 Boundary Value Problems 229
8.1 Introduction 229
8.1.1 Boundary Value Problem 230
8.1.2 Eigenvalue Problem 230
8.1.3 Sturm Liouville Boundary Value Problem 230
8.1.4 Sturm Liouville Eigenvalue Problem 231
8.1.5 Deflection of a Beam 231
8.2 Finite Difference Methods 232
8.3 Shooting Methods 235
8.3.1 The Simple Shooting Method 235
8.3.2 Multiple Shooting 238
8.4 Programs 239
8.5 Applications 240
8.5.1 No or Infinitely Many Solutions 240
8.5.2 Several Solutions 241
8.5.3 Hanging Curve of an Elastic Rope 242
8.6 Boundary Value Problems with IMSL 245
8.6.1 Method of Finite Differences 246
8.6.2 Multiple Shooting 246
8.7 Exercises 246
8.7.1 Finite Differences 246
8.7.2 Temperature Distribution in a Rod 246
9 Partial Differential Equations 248
9.1 Introduction 248
9.1.1 Linear Partial Differential Equations of Order 2 249
9.1.2 Poisson s and Laplace s Equation 250
9.1.3 Time Dependent Heat Conduction 250
9.1.4 The Wave Equation 251
9.2 Discretization of Elliptic Problems 251
9.2.1 Difference Methods 251
9.2.2 The Finite Element Method 253
9.2.3 Multigrid Methods 256
9.3 Software 258
9.3.1 NAG Routines 258
9.3.2 PLTMG: Finite Element and Multigrid Method 259
9.4 PLTMG/PAN: An Example 260
Contents xiii
A PAN A Problem Solving Environment 267
A.I Working with PAN 269
A.2 Writing a H/TeX Document 274
A.3 The Template Library 275
A.4 Software 276
B The Programs 277
B.O Conventions 277
B.I Simultaneous Linear Equations 278
B.I.I GAUSS 279
B.1.2 CHOLESKY 279
B.1.3 BAND . 279
B.I.4 SPARSE 280
B.1.5 LSS 280
B.2 Linear Optimization 280
B.2.1 LOPT 280
B.3 Interpolation and Approximation 281
B.3.1 INTERDIM1 (GL, XGKS) 281
B.3.2 APPROX (GL, XGKS) 281
B.3.3 INTERDIM2 281
B.3.4 INTERPAR 282
B.4 Nonlinear Equations 282
B.4.1 DIRECT 282
B.4.2 NLIN 283
B.5 Matrix Eigenvalue Problems 283
B.5.1 STDEV 283
B.5.2 SYMEV 283
B.5.3 GEV 283
B.5.4 SPARSEEV 284
B.5.5 SVD 284
B.6 Numerical Integration 284
B.6.1 ADAPT 284
B.6.2 PATT 285
B.6.3 MULTIPATT 285
B.6.4 SIMPLEX 285
B.7 Initial Value Problems 285
B.7.1 RUKU (GL) ¦ 285
B.7.2 ADAMS (GL) 285
B.7.3 BDF (GL) 286
B.7.4 STIFFTEST 286
B.8 Boundary Value Problems: BVP 286
B.8.1 DIFFERENCE (GL) 287
B.8.2 SHOOTING (GL) 287
xiv Contents
B.9 Partial Differential Equations 287
B.9.1 PLTMG (GL, GraphEditor, XGKS) 287
B.10 Program Overview 288
C The NAG Fortran Library 290
C.I Introduction 290
C.I.I Supplementary Documentation 291
C.1.2 Marks of the Library 291
C.1.3 Notation 291
C.I.4 Conventions, Routine Names and Precision 293
C.I.5 Input/Output in the Library 294
C.1.6 Error Handling and the Parameter IFAIL 295
C.I.7 Array Parameters 295
C.I.8 Programming Advice 296
C.2 The Construction of the NAG Fortran Library 297
C.2.1 Systematics 297
C.2.2 The Routine Document Sections 297
C.2.3 The Library Chapters 298
C.3 Example: Concise Documentation of F04AEF 299
C.4 Concluding Remarks 301
C.4.1 Summary for New Users 301
C.4.2 Contact between Users and NAG 302
D The NAG Graphics Library 303
D.I An Example Program Using the NAG Graphics Library 304
D.2 The Contents of the NAG Graphics Library 306
D.2.1 J06A Axes, Grids, Borders and Titles 306
D.2.2 J06B Point Plotting and Straight Line Drawing 306
D.2.3 J06C Curve Drawing 306
D.2.4 J06D ODE Graphics 306
D.2.5 J06E General Function Drawing 307
D.2.6 J06F Special Function Drawing 307
D.2.7 J06G Contouring 307
D.2.8 J06H Surface Viewing 307
D.2.9 J06J Data Presentation 307
D.2.10 J06K Vector Field Plotting 308
D.2.11 J06S Statistical Graphics 308
D.2.12 J06T J06Z Elementary Routines 308
Bibliography 309
Index 315
|
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| id | DE-604.BV010425404 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:52:19Z |
| institution | BVB |
| isbn | 0198596987 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006946065 |
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| physical | XVIII, 328 S. graph. Darst. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | Clarendon Press |
| record_format | marc |
| series2 | Oxford science publications |
| spelling | Köckler, Norbert 1944- Verfasser (DE-588)110192133 aut Numerical methods and scientific computing using software libraries for problem solving Norbert Köckler Oxford Clarendon Press 1994 XVIII, 328 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oxford science publications Analyse numérique - Informatique ram Analyse numérique - Logiciels ram Datenverarbeitung Engineering mathematics Data processing Numerical analysis Data processing Algorithmus (DE-588)4001183-5 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 s Algorithmus (DE-588)4001183-5 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006946065&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Köckler, Norbert 1944- Numerical methods and scientific computing using software libraries for problem solving Analyse numérique - Informatique ram Analyse numérique - Logiciels ram Datenverarbeitung Engineering mathematics Data processing Numerical analysis Data processing Algorithmus (DE-588)4001183-5 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4001183-5 (DE-588)4042805-9 |
| title | Numerical methods and scientific computing using software libraries for problem solving |
| title_auth | Numerical methods and scientific computing using software libraries for problem solving |
| title_exact_search | Numerical methods and scientific computing using software libraries for problem solving |
| title_full | Numerical methods and scientific computing using software libraries for problem solving Norbert Köckler |
| title_fullStr | Numerical methods and scientific computing using software libraries for problem solving Norbert Köckler |
| title_full_unstemmed | Numerical methods and scientific computing using software libraries for problem solving Norbert Köckler |
| title_short | Numerical methods and scientific computing |
| title_sort | numerical methods and scientific computing using software libraries for problem solving |
| title_sub | using software libraries for problem solving |
| topic | Analyse numérique - Informatique ram Analyse numérique - Logiciels ram Datenverarbeitung Engineering mathematics Data processing Numerical analysis Data processing Algorithmus (DE-588)4001183-5 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Analyse numérique - Informatique Analyse numérique - Logiciels Datenverarbeitung Engineering mathematics Data processing Numerical analysis Data processing Algorithmus Numerische Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006946065&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kocklernorbert numericalmethodsandscientificcomputingusingsoftwarelibrariesforproblemsolving |