A first course in numerical analysis
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| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY [u.a.]
McGraw-Hill
1978
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | International series in pure and applied mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
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| 245 | 1 | 0 | |a A first course in numerical analysis |c Anthony Ralston ; Philip Rabinowitz |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York, NY [u.a.] |b McGraw-Hill |c 1978 | |
| 300 | |a XIX, 556 S. |b graph. Darst. | ||
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| 490 | 0 | |a International series in pure and applied mathematics | |
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Datensatz im Suchindex
| DE-19_call_number | 1601/SK 900 R164(2) |
|---|---|
| DE-19_location | 95 |
| DE-BY-UBM_katkey | 2200354 |
| DE-BY-UBM_media_number | 41601948880016 |
| _version_ | 1823051861673902080 |
| adam_text | CONTENTS
Preface xiii
Notation xvii
CHAPTER ONE INTRODUCTION AND
PRELIMINARIES 1
1.1 What Is Numerical Analysis? 1
1.2 Sources of Error 2
1.3 Error Definitions and Related Matters 4
1.3 1 Significant Digits 1.3 2 Error in Functional Evaluation
1.3 3 Norms
1.4 Roundoff Error 9
1.4 1 The Probabilistic Approach to Roundoff; A Particular Example
1.5 Computer Arithmetic 12
1.5 1 Fixed Point Arithmetic 1.5 2 Floating Point Numbers
1.5 3 Floating Point Arithmetic 1.5 4 Overflow and Underflow
1.5 5 Single and Double Precision Arithmetic
v
Vi CONTENTS
1.6 Error Analysis 20
1.6 1 Backward Error Analysis
1.7 Condition and Stability 22
Bibliographic Notes 24
Bibliography 24
Problems 24
CHAPTER TWO APPROXIMATION AND
ALGORITHMS 31
2.1 Approximation 31
2.1 1 Classes of Approximating Functions 2.1 2 Types of
Approximations 2.1 3 The Case for Polynomial Approximation
2.2 Numerical Algorithms 39
2.3 Functionals and Error Analysis 42
2.4 The Method of Undetermined Coefficients 44
Bibliographic Notes 46
Bibliography 46
Problems 47
CHAPTER THREE INTERPOLATION 52
3.1 Introduction 52
3.2 Lagrangian Interpolation 54
3.3 Interpolation at Equal Intervals 56
3.3 1 Lagrangian Interpolation at Equal Intervals 3.3 2 Finite Differences
3.4 The Use of Interpolation Formulas 63
3.5 Iterated Interpolation 66
3.6 Inverse Interpolation 68
3.7 Hermite Interpolation 70
3.8 Spline Interpolation 73
3.9 Other Methods of Interpolation; Extrapolation 78
Bibliographic Notes 79
Bibliography 80
Problems 81
CONTENTS Vii
CHAPTER FOUR NUMERICAL DIFFERENTIATION,
NUMERICAL QUADRATURE, AND SUMMATION 89
4.1 Numerical Differentiation of Data 89
4.2 Numerical Differentiation of Functions 93
4.3 Numerical Quadrature: The General Problem 96
4.3 1 Numerical Integration of Data
4.4 Gaussian Quadrature 98
4.5 Weight Functions 102
4.6 Orthogonal Polynomials and Gaussian Quadrature 104
4.7 Gaussian Quadrature over Infinite Intervals 105
4.8 Particular Gaussian Quadrature Formulas 108
4.8 1 Gauss Jacobi Quadrature 4.8 2 Gauss Chebyshev Quadrature
4.8 3 Singular Integrals
4.9 Composite Quadrature Formulas 113
4.10 Newton Cotes Quadrature Formulas 118
4.10 1 Composite Newton Cotes Formulas 4.10 2 Romberg Integration
4.11 Adaptive Integration 126
4.12 Choosing a Quadrature Formula 130
4.13 Summation 136
4.13 1 The Euler Maclaurin Sum Formula 4.13 2 Summation of
Rational Functions; Factorial Functions 4.13 3 The Euler
Transformation
Bibliographic Notes 145
Bibliography 146
Problems 148
CHAPTER FIVE THE NUMERICAL SOLUTION OF
ORDINARY DIFFERENTIAL EQUATIONS 164
5.1 Statement of the Problem 164
5.2 Numerical Integration Methods 166
5.2 1 The Method of Undetermined Coefficients
5.3 Truncation Error in Numerical Integration Methods 171
5.4 Stability of Numerical Integration Methods 173
5.4 1 Convergence and Stability 5.4 2 Propagated Error
Bounds and Estimates
viii contents
5.5 Predictor Corrector Methods 183
5.5 1 Convergence of the Iterations 5.5 2 Predictors and Correctors
5.5 3 Error Estimation 5.5 4 Stability
5.6 Starting the Solution and Changing the Interval 195
5.6 1 Analytic Methods 5.6 2 A Numerical Method 5.6 3 Changing
the Interval
5.7 Using Predictor Corrector Methods 198
5.7 1 Variable Order Variable Step Methods 5.7 2 Some Illustrative
Examples
5.8 Runge Kutta Methods 208
5.8 1 Errors in Runge Kutta Methods 5.8 2 Second Order Methods
5.8 3 Third Order Methods 5.8 4 Fourth Order Methods
5.8 5 Higher Order Methods 5.8 6 Practical Error Estimation
5.8 7 Step Size Strategy 5.8 8 Stability 5.8 9 Comparison of
Runge Kutta and Predictor Corrector Methods
5.9 Other Numerical Integration Methods 224
5.9 1 Methods Based on Higher Derivatives 5.9 2 Extrapolation
Methods
5.10 Stiff Equations 228
Bibliographic Notes 233
Bibliography 234
Problems 236
CHAPTER SIX FUNCTIONAL APPROXIMATION:
LEAST SQUARES TECHNIQUES 247
6.1 Introduction 247
6.2 The Principle of Least Squares 248
6.3 Polynomial Least Squares Approximations 251
6.3 1 Solution of the Normal Equations 6.3 2 Choosing the Degree
of the Polynomial
6.4 Orthogonal Polynomial Approximations 254
6.5 An Example of the Generation of Least Squares
Approximations 260
6.6 The Fourier Approximation 263
6.6 1 The Fast Fourier Transform 6.6 2 Least Squares
Approximations and Trigonometric Interpolation
Bibliographic Notes 274
Bibliography 275
Problems 276
CONTENTS ix
CHAPTER SEVEN FUNCTIONAL APPROXIMATION:
MINIMUM MAXIMUM ERROR TECHNIQUES 285
7.1 General Remarks 285
7.2 Rational Functions, Polynomials, and Continued Fractions 287
7.3 Pade Approximations 293
7.4 An Example 295
7.5 Chebyshev Polynomials 299
7.6 Chebyshev Expansions 301
7.7 Economization of Rational Functions 307
7.7 1 Economization of Power Series 7.7 2 Generalization to
Rational Functions
7.8 Chebyshev s Theorem on Minimax Approximations 311
7.9 Constructing Minimax Approximations 315
7.9 1 The Second Algorithm of Remes 7.9 2 The Differential
Correction Algorithm
Bibliographic Notes 320
Bibliography 320
Problems 322
CHAPTER EIGHT THE SOLUTION OF
NONLINEAR EQUATIONS 332
8.1 Introduction 332
8.2 Functional Iteration 334
8.2 1 Computational Efficiency
8.3 The Secant Method 338
8.4 One Point Iteration Formulas 344
8.5 Multipoint Iteration Formulas 347
8.5 1 Iteration Formulas Using General Inverse Interpolation
8.5 2 Derivative Estimated Iteration Formulas
8.6 Functional Iteration at a Multiple Root 353
8.7 Some Computational Aspects of Functional Iteration 356
8.7 1 The 52 Process
8.8 Systems of Nonlinear Equations 359
8.9 The Zeros of Polynomials: The Problem 367
8.9 1 Sturm Sequences
X CONTENTS
8.10 Classical Methods 371
8.10 1 Bairstow s Method 8.10 2 Graeffe s Root squaring Method
8.10 3 Bernoulli s Method 8.10 4 Laguerre s Method
8.11 The Jenkins Traub Method 383
8.12 A Newton based Method 392
8.13 The Effect of Coefficient Errors on the Roots;
Ill conditioned Polynomials 395
Bibliographic Notes 397
Bibliography 399
Problems 400
CHAPTER NINE THE SOLUTION OF
SIMULTANEOUS LINEAR EQUATIONS 410
9.1 The Basic Theorem and the Problem 410
9.2 General Remarks 412
9.3 Direct Methods 414
9.3 1 Gaussian Elimination 9.3 2 Compact Forms of Gaussian
Elimination 9.3 3 The Doolittle, Crout, and Cholesky Algorithms
9.3 4 Pivoting and Equilibration
9.4 Error Analysis 430
9.4 1 Roundoff Error Analysis
9.5 Iterative Refinement 437
9.6 Matrix Iterative Methods 440
9.7 Stationary Iterative Processes and Related Matters 443
9.7 1 The Jacobi Iteration 9.7 2 The Gauss Seidel Method
9.7 3 Roundoff Error in Iterative Methods 9.7 4 Acceleration of
Stationary Iterative Processes
9.8 Matrix Inversion 450
9.9 Overdetermined Systems of Linear Equations 451
9.10 The Simplex Method for Solving Linear Programming
Problems 457
9.11 Miscellaneous Topics 465
Bibliographic Notes 468
Bibliography 470
Problems 472
CONTENTS Xi
CHAPTER TEN THE CALCULATION OF EIGEN¬
VALUES AND EIGENVECTORS OF MATRICES 483
10.1 Basic Relationships 483
10.1 1 Basic Theorems 10.1 2 The Characteristic Equation
10.1 3 The Location of, and Bounds on, the Eigenvalues
10.1 4 Canonical Forms
10.2 The Largest Eigenvalue in Magnitude by the Power Method 492
10.2 1 Acceleration of Convergence 10.2 2 The Inverse Power
Method
10.3 The Eigenvalues and Eigenvectors of Symmetric Matrices 501
10.3 1 The Jacobi Method 10.3 2 Givens Method
10.3 3 Householder s Method
10.4 Methods for Nonsymmetric Matrices 513
10.4 1 Lanczos Method 10.4 2 Supertriangularization
10.4 3 Jacobi Type Methods
10.5 The LR and QR Algorithms 521
10.5 1 The Simple QR Algorithm 10.5 2 The Double QR Algorithm
10.6 Errors in Computed Eigenvalues and Eigenvectors 536
Bibliographic Notes 538
Bibliography 539
Problems 541
Index 549
|
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| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| isbn | 0070511586 |
| language | English |
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| physical | XIX, 556 S. graph. Darst. |
| publishDate | 1978 |
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| publisher | McGraw-Hill |
| record_format | marc |
| series2 | International series in pure and applied mathematics |
| spellingShingle | Ralston, Anthony Rabinowitz, Philip A first course in numerical analysis Analyse mathématique ram Analyse numérique Numerieke wiskunde gtt Numerical analysis Analysis (DE-588)4001865-9 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4001865-9 (DE-588)4042805-9 (DE-588)4151278-9 |
| title | A first course in numerical analysis |
| title_auth | A first course in numerical analysis |
| title_exact_search | A first course in numerical analysis |
| title_full | A first course in numerical analysis Anthony Ralston ; Philip Rabinowitz |
| title_fullStr | A first course in numerical analysis Anthony Ralston ; Philip Rabinowitz |
| title_full_unstemmed | A first course in numerical analysis Anthony Ralston ; Philip Rabinowitz |
| title_short | A first course in numerical analysis |
| title_sort | a first course in numerical analysis |
| topic | Analyse mathématique ram Analyse numérique Numerieke wiskunde gtt Numerical analysis Analysis (DE-588)4001865-9 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Analyse mathématique Analyse numérique Numerieke wiskunde Numerical analysis Analysis Numerische Mathematik Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001496001&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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