Numerical computing and mathematical analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Chicago, Ill. u.a.
Science Research Assoc.
1975
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| Series: | Computer science series
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXVII, 529 S. |
| ISBN: | 0574191550 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Titel: Numerical computing and mathematical analysis
Autor: Pizer, Stephen M
Jahr: 1975
LIST OF PROGRAMS xiv PREFACE xvi 1 NUMERICAL COMPUTING-ACCURACY AND EFFICIENCY 1.1 Introduction 3 1.2 Quality of Numerical Methods 4 1.3 Errors in Numerical Methods 4 1.3.1 Definitions and Context 4 1.3.2 Propagation of Errors 6 1.3.3 Generation of Errors in Arithmetic Operations 12 1.3.4 Error Estimates vs. Error Bounds 15 Problems 17 1.4 Probability 19 1.4.1 Basic Concepts 19 1.4.2 Probability Distribution Measures 26 1.4.3 Functions of Random Variables; Estimators of Random Variables 34 1.4.4 Common Probability Distributions 37 1.4.5 Probabilistic Analysis of Error 40 Problems 45 1.5 Error Due to Subtraction of Approximately Equal Numbers 48 Problems 49 1.6 Summary 49 References 50 vii
TABLE OF CONTENTS viii 2 SYSTEMS OF LINEAR EQUATIONS 2.1 Introduction 53 2.2 Linear Algebra 54 2.2.1 Determinants 54 2.2.2 Vector Spaces and Linear Transformations 56 2.2.3 Existence and Uniqueness of Solutions to Systems of Linear Equations 67 Problems 68 2.3 An Algorithm for Solving Systems of Linear Equations 70 2.3.1 Gaussian Elimination 70 2.3.2 Pivoting and Scaling 72 2.3.3 Singular Matrices 78 2.3.4 Iterative Improvement 82 2.3.5 Triangular Factorization 83 Problems 86 2.4 Error Propagation in Solving Linear Equations 87 2.4.1 Norms 87 2.4.2 A Determinant Measure of Conditioning 91 2.4.3 The Condition Number 94 2.4.4 Evaluation of Conditioning 96 2.4.5 Effectiveness of Iterative Improvement 98 Problems 98 2.5 Inner Products 100 2.5.1 Definitions and Basic Properties 100 2.5.2 Orthogonality 101 2.5.3 Choice of Norms 104 Problems 104 2.6 Eigentheory 105 2.6.1 Usefulness of Eigenvectors and Eigenvalues 105 2.6.2 Theory for Eigenvectors and Eigenvalues 107 2.6.3 Similarity Transformations 118 2.6.4 Normal and Positive Definite Matrices 120 2.6.5 Conditioning in Terms of Eigenvalues 121 Problems 122
TABLE OF CONTENTS IX 2.7 Numerical Methods to Find Eigenvectors and Eigenvalues 124 2.7.1 Computing the Eigenvector Corresponding to a Given Eigenvalue 124 2.7.2 Power Method for Computing Eigenvectors and Eigenvalues 130 2.7.3 Acceleration of Convergence 137 2.7.4 Reduction to Allow Finding Other Eigenvalues and Eigenvectors 141 Problems 145 2.8 Iterative Methods for Solution of Systems of Linear Equations 147 2.8.1 The Iterative Form and Its Convergence 148 2.8.2 Jacobi Method 149 2.8.3 Gauss-Seidel Method 153 2.8.4 Acceleration of Convergence 158 2.8.5 Overrelaxation 161 2.8.6 A Convergent Iterative Solution Method 162 Problems 168 References 172 3 NONLINEAR EQUATIONS 3.1 Introduction 175 3.2 Graphing Functions of One Variable 176 Problems 179 3.3 Iterative Methods for Solving One Nonlinear Equation in One Unknown 179 3.3.1 Properties of Roots and Sequences 179 3.3.2 Bisection Method 187 3.3.3 Method of False Position 189 3.3.4 Acceleration of Convergence 196 3.3.5 Picard Iteration 199 3.3.6 Newton’s Method 205 3.3.7 Variations on Newton’s Method: Secant Method and Method of Constant Slope 211
X TABLE OF CONTENTS 3.3.8 Hybrid and Modified Methods 216 3:3.9 Choice of Method 219 Problems 220 3.4 Multiple Roots 223 Problems 225 3.5 Simultaneous Equations in Many Unknowns 225 3.5.1 Picard Iteration for Many Unknowns 226 3.5.2 Newton’s Method for Many Unknowns 230 Problems 232 3.6 Solution of Polynomial Equations 233 3.6.1 Properties of Polynomials and Polynomial Algorithms 233 3.6.2 Newton’s Method Applied to Polynomials 237 3.6.3 Complex Roots of Polynomials 239 3.6.4 Graeffe’s Root Squaring 243 3.6.5 Errors in Polynomial Equations 244 Problems 245 References 247 4 APPROXIMATION 4.1 Goals and Strategy 251 4.2 Polynomial Exact Matching 256 4.2.1 Interpolation Methods 256 a) Lagrange Polynomials 258 b) Truncation Error Analysis 260 c) Barycentric Lagrange Formulation 265 d) Newton Divided-Difference Formula 267 e) Iterated Linear Interpolation 276 f) Choosing Tabular Points 280 g) Equal-Interval Methods 291 h) Splines 307 i) Economization 312
TABLE OF CONTENTS XI 4.2.2 Extrapolation 313 4.2.3 Numerical Differentiation 314 4.2.4 Numerical Integration 319 a) Newton-Cotes Methods 320 b) Richardson Extrapolation ; Romberg Integration 330 c) Gaussian Integration; Orthogonal Functions 335 d) Weighted Integration 341 Problems 343 4.3 Least-Squares Approximation 358 4.3.1 Estimation Theory 358 4.3.2 Linear Least-Squares Approximation 362 4.3.3 Error Propagation ; Estimation of Parameters 363 4.3.4 Polynomial Least-Squares Approximation ; Linear Least-Squares Solution Algorithm 370 4.3.5 Constraints 381 Problems 383 4.3.6 Orthogonal Functions 388 4.3.7 Fourier Approximation 393 a) Concepts and Methods 393 b) Fourier Analysis and Filtering 414 c) Continuous vs. Discrete Operators ; Finite vs. Infinite Operators ; Sampling 434 Problems 441 4.4 Summary 451 Problems 453 References 454 5 NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS 5.1 Goals and Strategy 457 5.2 First-Order Initial-Value Problems 460 5.2.1 Rejection of Long Taylor Series 5.2.2 Euler’s Method 460 462
XU TABLE OF CONTENTS 5.2.3 Truncation Error and Stability 464 5.2.4 Heun Corrector 468 5.2.5 Modified Euler Predictor; with Heun Corrector to Form Predictor-Corrector 470 5.2.6 Automatic Error Estimation and Mop-Up 473 5.2.7 Multistep Predictor-Corrector Methods 476 5.2.8 Runge-Kutta Methods 481 5.2.9 Overall Solution Algorithm 484 Problems 486 5.3 Higher-Order Initial-Value Problems 489 Problems 491 5.4 Boundary-Value Problems 491 5.5 Summary 493 References 493 APPENDIX A. SOLUTION OF HOMOGENEOUS DIFFERENCE EQUATIONS 496 APPENDIX B. NOTATION 500 APPENDIX C. SPECIFICATIONS AND CONVENTIONS OF THE PROGRAMMING LANGUAGE USED 510 INDEX 516
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| dewey-full | 519.4 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.4 |
| dewey-search | 519.4 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| indexdate | 2024-07-09T15:38:01Z |
| institution | BVB |
| isbn | 0574191550 |
| language | English |
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| physical | XXVII, 529 S. |
| publishDate | 1975 |
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| spelling | Pizer, Stephen M. Verfasser aut Numerical computing and mathematical analysis Stephen M. Pizer Chicago, Ill. u.a. Science Research Assoc. 1975 XXVII, 529 S. txt rdacontent n rdamedia nc rdacarrier Computer science series Analyse numérique Análisis matemático - Procesamiento de datos Análisis numérico - Procesamiento de datos Calculo Numerico - Elementar larpcal Computers gtt Numerieke wiskunde gtt Toepassingen gtt Datenverarbeitung Mathematical analysis Data processing Numerical analysis Data processing Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001281044&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pizer, Stephen M. Numerical computing and mathematical analysis Analyse numérique Análisis matemático - Procesamiento de datos Análisis numérico - Procesamiento de datos Calculo Numerico - Elementar larpcal Computers gtt Numerieke wiskunde gtt Toepassingen gtt Datenverarbeitung Mathematical analysis Data processing Numerical analysis Data processing Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4042805-9 |
| title | Numerical computing and mathematical analysis |
| title_auth | Numerical computing and mathematical analysis |
| title_exact_search | Numerical computing and mathematical analysis |
| title_full | Numerical computing and mathematical analysis Stephen M. Pizer |
| title_fullStr | Numerical computing and mathematical analysis Stephen M. Pizer |
| title_full_unstemmed | Numerical computing and mathematical analysis Stephen M. Pizer |
| title_short | Numerical computing and mathematical analysis |
| title_sort | numerical computing and mathematical analysis |
| topic | Analyse numérique Análisis matemático - Procesamiento de datos Análisis numérico - Procesamiento de datos Calculo Numerico - Elementar larpcal Computers gtt Numerieke wiskunde gtt Toepassingen gtt Datenverarbeitung Mathematical analysis Data processing Numerical analysis Data processing Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Analyse numérique Análisis matemático - Procesamiento de datos Análisis numérico - Procesamiento de datos Calculo Numerico - Elementar Computers Numerieke wiskunde Toepassingen Datenverarbeitung Mathematical analysis Data processing Numerical analysis Data processing Numerische Mathematik |
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