Numerical analysis: a pract. approach
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York
Macmillan
1987
|
| Ausgabe: | 2. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 534 S. |
| ISBN: | 0023762101 |
Internformat
MARC
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| 100 | 1 | |a Maron, Melvin J. |e Verfasser |0 (DE-588)172249538 |4 aut | |
| 245 | 1 | 0 | |a Numerical analysis |b a pract. approach |c M. J. Maron |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York |b Macmillan |c 1987 | |
| 300 | |a XVIII, 534 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 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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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
Chapter 1
Algorithms, Errors, and Digital Devices 1
1.0 Introduction 1
1.1 Implementing Numerical Methods on Digital Devices 1
1.1 A What Is a Numerical Method? 2
1.1B Why Use a Program for a Numerical Method? 7
1.1C Do Digital Devices Make Mistakes? 8
1.2 How Do Digital Devices Store Real Numbers? 9
1.2A Normalized Floating-Point Representation of Storable Numbers 9
1.2B Significant-Digit Accuracy of Fixed-Precision Devices 12
1.2C Inherent Error on a Digital Device 14
1.3 . Errors in Fixed-Precision Arithmetic 15
1.3A The Error and Relative Error of an Approximation 16
1.3B Error Propagation When Doing Fixed-Precision Arithmetic 17
1.3C The Errors of Approximating f (.x) by Af(x)lh Explained 21
1.3D Ill-Conditioned Calculations 23
1.3E A Quantitative Analysis of Propagated Roundoff (Optional) 24
1.4 Some Practical Strategies for Minimizing Roundoff Error 25
1.4A Strategies for Minimizing Inherent Error 26
1.4B Strategies for Evaluating Expressions Accurately 26
1.4C Strategies for Storing Intermediate Values 28
1.4D Strategies for Avoiding Subtractive Cancellation 31
Problems 34
xi
XJj Contents
Chapter 2
Numerical Methods for Solving Equations in One
Variable 40
2.0 Introduction 40
2.1 Iterative Algorithms 41
2.1 A Accuracy and Convergence Rates of Iterative Algorithms 42
2.1B The Repeated Substitution (RS) Algorithm 45
2.1C Finding Initial Guesses for RS Graphically 48
2.ID Visualizing the RS Algorithm Graphically 49
2. IE When Should the RS Algorithm Converge? Theorem RS 50
2.1F Accelerating Linear Convergence: Aitken s Formula 51
2.1G Proof of Theorem RS (Section 2. IE) 54
2.2 Implementing an Iterative Algorithm on a Computer 55
2.2A Pseudoprogram for a General Iterative Algorithm 55
2.2B Terminating Iterative Algorithms on a Computer 56
2.2C A Subprogram for the Repeated Substitution Algorithm 58
2.3 Numerical Methods for Solving /(x) = 0 61
2.3A Using Root-Finding Methods to Solve Equations 61
2.3B The Slope Method Strategy 62
2.3C The Newton-Raphson (NR) Method: mk = mtm(xk) = f(xk) 63
2.3D The Secant (SEC) Method: mk = msec = Ay^/Ax*., 66
2.3E Finding Initial Guesses of Solutions of Equations 68
2.3F Using a Root-Finding Method to Find Fixed Points 70
2.3G Bracketing Methods 71
2.4 Some Practical Aspects of Root Finding 74
2.4A Roots That Are Hard to Find Accurately 74
2.4B Convergence of NR and SEC to Multiple Roots 76
2.4C Overshoot, Cycling, and Wandering of Slope Methods 80
2.4D Root-Finding Methods Compared 83
2.4E A Subroutine for the Secant Method 83
2.5 Finding Roots of Real Polynomials; Bairstow s Method 85
2.5A Deflating Real Polynomials Using Quadratic Factors 86
2.5B Synthetic Division by q{x) 87
2.5C Bairstow s Method 88
2.5D Some Practical Aspects of Polynomial Root Finding 92
2.5E Derivation of the Bairstow s Method Formulas 94
Problems 95
Chapter 3
Direct Methods for Solving Linear Systems 102
3.0 Introduction 102
Contents XJjj
3.1 Basic Properties of Matrices 103
3.1A Terminology and Notation 103
3.IB Addition, Subtraction, and Scalar Multiplication 104
3.1C Matrix Multiplication 105
3.ID The Inverse A 1 of a Nonsingular Matrix A 107
3. IE The Matrix Form of an n X n Linear System 109
3.2 Solving Linear Systems Directly 111
3.2A Triangular Systems 111
3.2B Efficient Forward and Backward Substitution 112
3.2C Basic Gaussian Elimination 114
3.2D Performing Basic Gaussian Elimination in Place 117
3.2E Upper Triangulation = Li/-Factorization + Forward Substitution 118
3.2F Compact Forward/Backward Substitution 120
3.2G Counting Arithmetic Operations 122
3.2H Gaussian Elimination 123
3.21 Gauss—Jordan Elimination 127
3.3 LU-Decomposition = LU-Factorization + Pivoting 128
3.3A The Need for Pivoting (Exact Arithmetic) 128
3.3B The Need for Pivoting (Fixed-Precision Arithmetic) 129
3.3C Pivoting Strategies for the Lf/-Decomposition Algorithm 131
3.3D Lf/-Decompositions of a Given AnXn 135
3.3E What if amm, am+Um, ..., anm Are All Zero? 137
3.3F Scaled Full Pivoting (Optional) 139
3.4 Reusing an LU-Decomposition 140
3.4A Efficient Solution of Ax, = b,, ...,Axk = bk 140
3.4B Finding A^1 142
3.4C Counting Arithmetic Operations 144
3.5 The Determinant of an n x n Matrix 144
3.5A Definition and Basic Properties 145
3.5B Getting det(A) from an /.{/-Decomposition 146
3.5C Geometric Interpretation of det(A) 147
Problems 149
Chapter 4
Solving Systems of Equations in Fixed-Precision
Arithmetic 156
4.0 Introduction 156
4.1 Recognizing and Dealing with III Conditioning 156
4.1A Ill-Conditioned Linear Systems 157
4.IB Using Condition Numbers to Indicate 111 Conditioning 158
4.1C The Residual and Iterative Improvement 161
XJV Contents
4. ID Matrix Norms and the Condition Number cond A 163
4.2 Solving Linear Systems Directly on a Computer 166
4.2A Suggestions for Selecting and Using a Canned Program 167
4.2B Programming the L£/-Decomposition Algorithm 167
4.2C Formulas for L£/-Factorizations of A 172
4.2D The Crout Decomposition Algorithm with SPP (Optional) 173
4.2E Choleski s Factorization for Positive Definite Matrices 176
4.3 Iterative Methods for Solving Linear Systems 177
4.3A The Jacobi and Gauss-Seidel Methods 178
4.3B Improving the Likelihood of Convergence 181
4.3C Accelerating Convergence of Gauss-Seidel Iteration 183
4.4 Solving Nonlinear Systems of Equations 185
4.4A Introduction to 2 X 2 Nonlinear Systems 186
4.4B The Newton-Raphson Method for 2 x 2 Systems (NR2) 187
4.4C The Newton-Raphson Method for n x n Systems (NRn) 190
4.4D Implementing NRn on a Computer 193
4.4E Other Methods for Solving Nonlinear Systems 194
Problems 196
Chapter 5
Curve Fitting and Function Approximation 201
5.0 Introduction 201
5.1 Curve Fitting Using the Least-Square-Error Criterion 202
5.1 A The Discrete Curve-Fitting Problem 202
5. IB The Least-Square-Error Criterion 203
5.1C The General Linear Fit: L(x) = mx + b 206
5. ID Using the Determination Index R(g) to Assess Good Fit 208
5.2 Fitting Monotone, Convex Data 210
5.2A The Normal Equations for a Two-Parameter Guess Function 210
5.2B Linearizing Monotone, Convex Data 213
5.2C Fitting Monotone, Convex Data on a Computer 217
5.3 Fitting an n-Parameter Linear Model 218
5.3A The Normal Equations for g(x) = 2?=1 7/ty*) 219
5.3B Fitting Polynomials; The Polynomial Wiggle Problem 222
5.3C The Importance of Sketching the Data 227
5.3D Orthogonality and Weighted Least Squares 229
5.4 Approximating a Known f(x) by a Simpler g(x) on an Interval / 235
5.4A The Approximation Problem 236
5.4B Continuous Least Square Approximation of f(x) on / = [a, b] 237
Contents yy
5.4C Taylor Polynomial Approximation 239
5.4D Rational Function Approximation for x ~ 0 240
5.4E Chebyshev Economization 243
Problems 248
Chapter 6
Interpolation 259
6.0 Introduction 259
6.1 The Unique Interpolating Polynomial for p^,..., pk+n 260
6.1 A Notation for Interpolating Polynomials 260
6.IB Existence of pkk+n(x): Lagrange s Form 263
6.1C Uniqueness of pkk + n(x) 266
6.1D Lagrange s Form for ft-Spaced Points 267
6. IE Finding Best nth Interpolants p,,(z) for f(z) 269
6. IF Uniform Approximation Using Chebyshev Nodes 274
6.1G Derivation of the Formula for Ekk + n(z) 276
6.2 Difference Tables of Tabulated Functions 277
6.2A Using Forward Differences A yk When jc/s Are Equispaced 277
6.2B Using Divided Differences A yk 281
6.2C Formulas for Forward and Backward Interpolation 285
6.2D Recursive Formulas for /?-Spaced .v, s: Newton s Form of pk k + n(x) 288
6.2E Efficient Evaluation of Best Interpolants pn(z) 290
6.2F Error Propagation on Difference Tables 292
6.2G Interpolation in the Presence of Error 294
6.2H Inverse Interpolation 299
6.21 Proof That A yk Is the Leading Coefficient of pk.k+n(x) 301
6.3 Interpolation Using Piecewise Cubic Splines 302
6.3A Piecewise Linear Interpolation 302
6.3B Piecewise Cubic Splines 303
6.3C Equations for Finding yo(.Y), .... qn _i(.v) 303
6.3D Endpoint Strategies 306
6.3E Finding a,, a,, 308
6.3F Summary: Algorithm for Piecewise Cubic Spline Interpolation 310
6.3G Polynomial Versus Cubic Spline Interpolation 312
Problems 312
Chapter 7
Numerical Methods for Differentiation
and Integration 319
7.0 Introduction 319
XVJ_ Contents
7.1 Approximating fk z) and J* /(x) dx as 2?=0 w(/(x,) 320
7.1 A Formulas Involving One or Two Sampled Nodes 321
7. IB Using Lagrange Polynomials to Get Polynomial Exactness 324
7.1C Truncation Error Formulas 327
7.2 Numerical Differentiation and Richardson s Formula 329
7.2A Convergence of Af(z)/h and 8/(z)/2/z 330
7.2B The Order of an Approximation; Big O Notation 331
7.2C Truncation Errors of Af(z)/h, hf(z)/2h, and 82/(z)//j2 333
7.2D The Stepsize Dilemma 335
7.2E Richardson s Improvement Formula 337
7.2F Using Richardson s Formula Iteratively 338
7.2G Approximating/ (z), ...,/(iv)(z) Accurately 340
7.2H Approximating Derivatives of Tabulated Functions 343
7.3 Composite Rules and Romberg Integration 344
7.3A Quadrature Formulas for fc-Spaced Nodes 345
7.3B Composite Trapezoidal and Simpson Rules: T[h] and SU3[h] 352
7.3C Romberg Integration 356
7.3D Integrating Tabulated Functions 359
7.3E The Truncation Error of SV3[h] and M]ba 360
7.4 Methods Based on Unevenly Spaced Nodes 361
7.4A Gauss Quadrature on [- 1, 1] 362
7.4B Gauss Quadrature on [a, b] 364
7.4C Choosing a Method for Estimating a Proper Integral 366
7.4D Integrals with Infinite Discontinuities on [a, b) 366
7.4E Integrals over [a, =«), (-=, b], and (- », =») 369
7.4F Adaptive Quadrature 371
7.5 Multiple Integration 378
7.5A General Procedure for Approximating / = /* /£ * f(x, y) dy dx 379
7.5B Approximating / = J« Jpful /(* v) dy dx Accurately on a Computer 382
Problems 383
Chapter 8
Numerical Methods for Ordinary Differential
Equations 392
8.0 Introduction 392
8.1 The Initial Value Problem (IVP) 393
8.1 A Existence and Uniqueness of Solutions 393
8.IB Euler s Method 396
Contents XVU
8.1C The Order of a Numerical Method 399
8. ID The Modified Euler and Huen Methods 400
8.2 Self-Starting Methods: Runge-Kutta and Taylor 403
8.2A The Fourth-Order Runge-Kutta (RK4) Method 404
8.2B The Fourth-Order Runge-Kutta-Fehlberg (RKF4) Method 406
8.2C Taylor s Method 409
8.2D General Runge-Kutta Formulas 412
8.2E Stability 413
8.2F Stiffness 417
8.3 Multistep (Predictor-Corrector) Methods 419
8.3A Predictor-Corrector (PC) Strategies 419
8.3B Stepsize Control and Stability of PC Methods 424
8.3C Choosing a Method for Solving IVP 425
8.4 First-Order Systems and nth Order IVP s 426
8.4A Notation and Terminology 426
8.4B Numerical Methods for Solving (IVP),, 428
8.4C Solving an rcth-Order IVP; Degree Reduction 430
8.4D Solving (IVP)n on a Computer 433
8.4E Linearity 436
8.5 Boundary Value Problems 437
8.5A The Shooting Method 438
8.5B Importance of Linearity for the Shooting Method 440
8.5C The Finite Difference Method 443
8.5D The Finite Difference Method When (BVP)2 Is Linear 446
8.5E Comparison of the Shooting Method and Finite Difference Method 448
Problems 448
Chapter 9
Eigenvalues 457
9.0 Introduction 457
9.1 Basic Properties of Eigenvalues and Eigenvectors 458
9.1 A The Characteristic Polynomial pA(k) 458
9. IB Similar Matrices and Diagonalizability 460
9.1C Using Eigenvectors to Uncouple Linear IVP s 462
9.2 The Power Method 466
9.2A The Power Method for Finding Dominant Eigenvalues 466
9.2B Convergence Considerations 468
Xyjjj Contents
9.2C The Inverse Power Method for Finding Smallest Eigenvalues 469
9.2D Shifting Eigenvalues 470
9.2E Practical Considerations When Using the Power Method 471
9.3 Methods for Finding All Eigenpairs of a Matrix 472
9.3A Orthogonal Matrices 472
9.3B Jacobi s Method for Symmetric Matrices 475
9.3C Jacobi s Method with Thresholds 479
9.3D Factorization Methods 481
9.4 Characteristic Values and Solutions of Homogeneous BVP s 482
9.4A Buckling of Axially Loaded Beams 482
9.4B Numerical Procedure for Estimating Characteristic Values 484
9.4C Improving the Accuracy of Estimates of X 485
9.4D The Sturm-Liouville Equation 487
9.5 Using Eigenvalues to Uncover the Structure of A 489
9.5A The Principal Axis Theorem 489
9.5B Describing ||A|| and cond A When A Is Symmetric 491
9.5C Positive Definite Matrices 492
9.5D Relating cond A to Eigenvalues When A Is Not Symmetric 493
9.5E The Generalized Eigenvalue Problem 494
Problems 494
Appendix I
Using Pseudoprograms to Describe Algorithms 499
Appendix II
Review of the Basic Results of Calculus 505
Bibliography 512
Answers to Selected Problems 515
Index 527
|
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| author_GND | (DE-588)172249538 |
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| classification_tum | MAT 650f |
| ctrlnum | (OCoLC)631949178 (DE-599)BVBBV003527910 |
| discipline | Mathematik |
| edition | 2. ed. |
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| id | DE-604.BV003527910 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T16:01:21Z |
| institution | BVB |
| isbn | 0023762101 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002239661 |
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| physical | XVIII, 534 S. |
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| publishDateSearch | 1987 |
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| spelling | Maron, Melvin J. Verfasser (DE-588)172249538 aut Numerical analysis a pract. approach M. J. Maron 2. ed. New York Macmillan 1987 XVIII, 534 S. txt rdacontent n rdamedia nc rdacarrier 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=002239661&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Maron, Melvin J. Numerical analysis a pract. approach Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4042805-9 |
| title | Numerical analysis a pract. approach |
| title_auth | Numerical analysis a pract. approach |
| title_exact_search | Numerical analysis a pract. approach |
| title_full | Numerical analysis a pract. approach M. J. Maron |
| title_fullStr | Numerical analysis a pract. approach M. J. Maron |
| title_full_unstemmed | Numerical analysis a pract. approach M. J. Maron |
| title_short | Numerical analysis |
| title_sort | numerical analysis a pract approach |
| title_sub | a pract. approach |
| topic | Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Numerische Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002239661&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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