Fundamentals of computer numerical analysis:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton, Fla. u.a.
CRC Press
1994
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 587 S. graph. Darst. 1 Diskette, 5,25" |
| ISBN: | 0849386373 |
Internformat
MARC
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| 100 | 1 | |a Friedman, Menahem |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Fundamentals of computer numerical analysis |c Menahem Friedman ; Abraham Kandel |
| 264 | 1 | |a Boca Raton, Fla. u.a. |b CRC Press |c 1994 | |
| 300 | |a XI, 587 S. |b graph. Darst. |e 1 Diskette, 5,25" | ||
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| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Electronic data processing | |
| 650 | 4 | |a Numerical analysis |x Data processing | |
| 650 | 0 | 7 | |a Numerische Mathematik |0 (DE-588)4042805-9 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Numerische Mathematik |0 (DE-588)4042805-9 |D s |
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| 700 | 1 | |a Kandel, Abraham |d 1941- |e Verfasser |0 (DE-588)12865242X |4 aut | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-006295238 | ||
Datensatz im Suchindex
| _version_ | 1804123870805884928 |
|---|---|
| adam_text | Contents
I INTRODUCTION 1
1.1 What is Numerical Analysis? 1
1.2 Algebraic Representation and Arithmetic with Complex Numbers 2
1.3 Geometric Interpretation 7
1.4 Polynomials 16
1.5 Taylor Series 19
1.6 Elementary Functions of a Complex Variable 29
1.7 Functions of Several Variables and Partial Derivatives 33
2. ERROR 39
2.1 Representation of Numbers and Conversions 39
2.1.1. Conversion 41
2.1.2. Special Bases 43
2.2 Floating Point Numbers 45
2.2.1. Rounding 46
2.2.2. Floating Point Arithmetic 48
2.3 Definitions and Sources 52
2.3.1. Sources of Error 54
2.3.2. Losing Significant Digits 56
2.3.3. Function Evaluation 59
2.3.4. Overflow and Underflow 59
2.4 Error Propagation 61
2.4.1. Error Propagation in Function Evaluation 63
2.5 Summation 65
D ITERATION 71
3.1 Definition of an Iteration Problem and the Standard Iteration Method 71
3.2 Rate of Convergence 84
3.3 Aitken s Method for Acceleration 87
3.4 Steffensen s Modification (STM) 95
vii
viii
3.5 Newton s Method: Advantages and Limitations 102
3.6 Related Schemes and Modifications to Newton s Method 116
3.6.1. The Bisection Method 116
3.6.2. The Secant Method 119
3.6.3. The Cubic Newton s Method 123
3.6.4. Muller s Method 125
3.6.5. Which Algorithm to Choose? 128
3.7 Extensions to Multivariable Systems 130
3.7.1. Error Estimate for Systems 135
3.7.2. Newton s Method for a System 136
4 LINEAR DIFFERENCE EQUATIONS 141
4.1 General Concepts 142
4.2 Homogeneous Difference Equations of Order 2 143
4.3 The General Solution to AX = B 144
4.4 Linear Difference Equations of Order N 152
4.5 The Backward Difference Operator 159
4.6 Application: Bernoulli s Method 164
4.6.1. A Single Dominant Zero 164
4.6.2. A Multiple Dominant Zero 166
4.6.3. Accelerating the Convergence 169
4.6.4. Two Conjugate Complex Dominant Zeroes 171
J INTERPOLATION AND APPROXIMATION 177
5.1 Lagrange s Interpolator 178
5.2 Error Estimate 182
5.2.1. On the Error Behavior 184
5.3 Convergence to the Interpolated Function 187
5.4 Divided Differences 190
5.5 Interpolation by Splines 197
5.5.1. Spline Interpolation 199
5.5.2. Error Estimate 205
5.6 Approximations of Functions 210
5.6.1. Best Approximations 211
5.7 Chebyshev Polynomials 216
5.8 Near Minimax Approximation 220
5.9 Least Squares Approximation 224
ix
5.10 Other Approximations 233
5.10.1. Approximations: Using the Least Squares Technique and Orthogonality 233
5.10.2. The Gram Schmidt Process 243
5.10.3. Approximation by Rational Functions 244
D NUMERICAL INTEGRATION AND DIFFERENTIATION 249
6.1 Numerical Integration: The Trapezoidal and Simpson Rules 249
6.1.1. The Trapezoidal Rule 250
6.1.2. Simpson s Rule 257
6.1.3. Richardson s Extrapolation 263
6.2 Gaussian Integration 265
6.2.1. The General Gaussian Formula 269
6.2.2. Infinite Discontinuities 273
6.3 The Romberg Method 276
6.4 Multiple Integrals 282
6.4.1. A Two Dimensional Simpson s Rule for a Rectangle 282
6.4.2. A Two Dimensional Gaussian Integration Scheme 286
6.5 Numerical Differentiation 289
6.5.1. Differentiating the Lagrange Interpolator 291
6.5.2. Undetermined Coefficients 293
/ LINEAR EQUATIONS 297
7.1 System of Linear Equations: Matrices 297
7.1.1. Matrices: Definition and Arithmetic 300
7.1.2. Determinants 301
7.2 Gaussian Elimination 307
7.2.1. Pivoting 312
7.2.2. Operations Count 313
7.2.3. The Inverse Matrix 314
7.3 LU Factorization 318
7.3.1. Doolittle s Method 320
7.3.2. Tridiagonal Systems 326
7.4 Iterative Methods 333
7.4.1. Norms of Vectors and Matrices 333
7.4.2. Jacobi Iterative Method 340
7.4.3. Gauss Seidel Iterative Method 341
7.4.4. Speed of Convergence 345
7.4.5. Relaxation Methods 346
X
7.5 Error and Stability 353
7.5.1. The Residual Correction Method 355
7.5.2. Stability 356
7.6 The Eigenvalue Problem 364
7.6.1. The Characteristic Polynomial 364
7.6.2. Eigenvalues of Symmetric Matrices 366
7.6.3. The Power Method 368
7.6.4. Error Analysis 371
7.6.5. The Deflation Method 375
7.6.6. Wielandt s Deflation Technique 376
O NUMERICAL SOLUTIONS OF DIFFERENTIAL EQUATIONS 381
8.1 Preliminaries 381
8.1.1. Stability 384
8.1.2. Differential Equations of Higher Order 386
8.1.3. Boundary Value Problems 388
8.2 Euler s Method 390
8.2.1. Error Analysis 394
8.3 Taylor s Method 401
8.4 Runge Kutta Methods 409
8.4.1. A Second Order Method 409
8.4.2. The Classical Runge Kutta Method 415
8.5 Error Control: Runge Kutta Fehlberg Method 417
8.5.1. The Runge Kutta Fehlberg Method 420
8.6 Multistep Methods 425
8.6.1. The Adams Bashforth Method 426
8.6.2. The Adams Moulton Method 429
8.6.3. A Predictor Corrector Method 431
8.7 Stability of Numerical Methods 435
8.7.1. Stability Regions 439
8.8 Systems of Differential Equations 440
8.8.1. Numerical Methods for Systems 441
!y NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS 447
9.1 Various Types of Partial Differential Equations 448
9.1.1. Poisson and Laplace Equations 448
9.1.2. The Heat Equation 451
xi
9.1.3. The Wave Equation 453
9.2 Analytical Solutions I: Fourier Series 455
9.2.1. Fourier Series 455
9.2.2. The Fourier Theorem 458
9.2.3. Even and Odd Functions 462
9.3 Analytical Solutions II: Separation of Variables 468
9.3.1. Heat Conduction 468
9.3.2. The Wave Equation 473
9.3.3. Dirichlet Problem for a Rectangle 476
9.3.4. Dirichlet Problem for a Circle 478
9.4 Finite Differences: Parabolic Partial Differential Equations 484
9.4.1. Forward Difference Method 484
9.4.2. An Unconditionally Stable Method 487
9.4.3. Crank Nicolson Method 490
9.5 Finite Differences: Hyperbolic Partial Differential Equations 494
9.6 Finite Differences: Elliptic Partial Differential Equations 500
9.7 Finite Elements 511
9.7.1. Rayleigh Ritz Method 511
9.7.2. Extension to Two Dimensional Problems 517
9.7.3. The Functional 520
9.7.4. The Finite Element Method 521
9.7.5. Triangulation 521
9.7.6. Shape Functions 522
BIBLIOGRAPHY 527
List of Algorithms and Numerical Methods 531
Answers and Solutions to Selected Problems 533
INDEX 575
|
| any_adam_object | 1 |
| author | Friedman, Menahem Kandel, Abraham 1941- |
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| indexdate | 2024-07-09T17:36:39Z |
| institution | BVB |
| isbn | 0849386373 |
| language | English |
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| spelling | Friedman, Menahem Verfasser aut Fundamentals of computer numerical analysis Menahem Friedman ; Abraham Kandel Boca Raton, Fla. u.a. CRC Press 1994 XI, 587 S. graph. Darst. 1 Diskette, 5,25" txt rdacontent n rdamedia nc rdacarrier Datenverarbeitung Electronic data processing Numerical analysis Data processing Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 s DE-604 Kandel, Abraham 1941- Verfasser (DE-588)12865242X aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006295238&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Friedman, Menahem Kandel, Abraham 1941- Fundamentals of computer numerical analysis Datenverarbeitung Electronic data processing Numerical analysis Data processing Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4042805-9 |
| title | Fundamentals of computer numerical analysis |
| title_auth | Fundamentals of computer numerical analysis |
| title_exact_search | Fundamentals of computer numerical analysis |
| title_full | Fundamentals of computer numerical analysis Menahem Friedman ; Abraham Kandel |
| title_fullStr | Fundamentals of computer numerical analysis Menahem Friedman ; Abraham Kandel |
| title_full_unstemmed | Fundamentals of computer numerical analysis Menahem Friedman ; Abraham Kandel |
| title_short | Fundamentals of computer numerical analysis |
| title_sort | fundamentals of computer numerical analysis |
| topic | Datenverarbeitung Electronic data processing Numerical analysis Data processing Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Datenverarbeitung Electronic data processing Numerical analysis Data processing Numerische Mathematik |
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