Methods of numerical mathematics:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1982
|
| Edition: | 2. ed. |
| Series: | Applications of mathematics
2 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus dem Russ. übers. - Literaturverz. S. 476 - 504 |
| Physical Description: | XIII, 510 S. graph. Darst. |
| ISBN: | 0387906142 3540906142 |
Staff View
MARC
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| 240 | 1 | 0 | |a Metody vyčislitel'noj matematiki |
| 245 | 1 | 0 | |a Methods of numerical mathematics |c G. I. Marchuk |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 1982 | |
| 300 | |a XIII, 510 S. |b graph. Darst. | ||
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| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Applications of mathematics |v 2 | |
| 500 | |a Aus dem Russ. übers. - Literaturverz. S. 476 - 504 | ||
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Record in the Search Index
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| adam_text | Contents
Introduction 1
Chapter 1
Fundamentals of the Theory of Difference Schemes 8
1.1. Basic Equations and Their Adjoints 8
1.1.1. Norm Estimates of Certain Matrices 12
1.1.2. Computing the Spectral Bounds of a Positive Matrix 13
1.1.3. Eigenvalues and Eigenfunctions of the Laplace Operator 21
1.1.4. Eigenvalues and Eigenvectors of the Finite-Difference Analog of
the Laplace Operator 24
1.2. Approximation 27
1.3. Countable Stability 36
1.4. The Convergence Theorem 44
Chapter 2
Methods of Constructing Difference Schemes for
Differential Equations 47
2.1. Variational Methods in Mathematical Physics 48
2.1.1. Some Problems of Variational Calculation 48
2.1.1. The Ritz Method 55
2.1.3. The Galerkin Method 60
2.1.4. The Method of Least Squares 63
2.2. The Method of Integral Identities 64
2.2.1. Method of Constructing Difference Equations for Problems with
Discontinuous Coefficients on the Basis of an Integral Identity 64
2.2.2. The Variational Form of an Integral Identity 72
ix
X Contents
2.3. Difference Schemes for Equations with Discontinuous
Coefficients Based on Variational Principles 84
2.3.1. Simple Difference Equations for a Diffusion Based on the Ritz Method 85
2.3.2. Constructions of Simple Difference Schemes Based on
the Galerkin (Finite Elements) Method 88
2.4. Principles for the Construction of Subspaces for the Solution of
One-Dimensional Problems by Variational Methods 90
2.4.1. A General Approach to the Construction of Subspaces of
Piecewise-Polynomial Functions 91
2.4.2. Constructing a Basis Using Trigonometric Functions and Applying It
in Variational Methods 94
2.5. Variational-Difference Schemes for Two-Dimensional Equations
of Elliptic Type 100
2.5.1. The Ritz Method 100
2.5.2. The Galerkin Method 106
2.5.3. Methods for Constructing Subspaces 109
2.6. Variational Methods for Multi-Dimensional Problems 112
2.6.1. Methods of Choosing the Subspaces 112
2.6.2. Coordinate-by-Coordinate Methods for Multi-Dimensional Problems 114
2.7. The Method of Fictive Domains 116
Chapter 3
Interpolation of Net Functions 122
3.1. Interpolation of Functions of One Variable 123
3.1.1. Interpolation of Functions of One Variable by Cubic Splines 123
3.1.2. Piecewise-Cubic Interpolation with Smoothing 127
3.1.3. Smooth Construction 129
3.1.4. The Convergence of Spline Functions 131
3.2. Interpolation of Functions of Two or More Variables 133
3.3. An /--Smooth Approximation to a Function of Several Variables 135
3.4. Elements of the General Theory of Splines 141
Chapter 4
Methods for Solving Stationary Problems of Mathematical
Physics 147
4.1. General Concepts of Iteration Theory 149
4.2. Some Iterative Methods and Their Optimization 150
4.2.1. The Simplest Iteration Method 150
4.2.2. Convergence and Optimization of Stationary Iterative Methods 153
4.2.3. The Successive Over-Relaxation Method 156
4.2.4. The Chebyshev Iteration Method 161
4.2.5. Comparison of the Convergence Rates of Various Iteration Methods
for a System of Finite-Difference Equations 169
4.3. Nonstationary Iteration Methods 171
4.3.1. Convergence Theorems 171
4.3.2. The Method of Minimizing the Residuals 173
4.3.3. The Conjugate Gradient Method 175
Contents xi
4.4. The Splitting-Up Method 180
4.4.1. The Commutative Case 183
4.4.2. The Noncommutative Case 188
4.4.3. Variational and Chebyshevian Optimization of Splitting-Up Methods 192
4.5. Iteration Methods for Systems with Singular Matrices 194
4.5.1. Consistent Systems 195
4.5.2. Inconsistent Systems 197
4.5.3. The Matrix Analog of the Method of Fictive Regions 199
4.6. Iterative Methods for Inaccurate Input Data 203
4.7. Direct Methods for Solving Finite-Difference Systems 205
4.7.1. The Fast Fourier Transform 205
4.7.2. The Cyclic Reduction Method 210
4.7.3. Factorization of Difference Equations 212
Chapter 5
Methods for Solving Nonstationary Problems 224
5.1. Second-Order Approximation Difference Schemes
with Time-Varying Operators 224
5.2. Nonhomogeneous Equations of the Evolution Type 227
5.3. Splitting-Up Methods for Nonstationary Problems 228
5.3.1. The Stabilization Method 229
5.3.2. The Predictor-Corrector Method 233
5.3.3. The Component-by-Component Splitting-Up Method 237
5.3.4. Some General Remarks 242
5.4. Multi-Component Splitting 243
5.4.1. The Stabilization Method 244
5.4.2. The Predictor-Corrector Method 245
5.4.3. The Component-by-Component Splitting-Up Method Based on
the Elementary Schemes 247
5.4.4. Splitting-Up of Quasi-Linear Problems 252
5.5. General Approach to Component-by-Component Splitting 253
5.6. Methods of Solving Equations of the Hyperbolic Type 257
5.6.1. The Stabilization Method 257
5.6.2. Reduction of the Wave Equation to an Evolution Problem 261
Chapter 6
Richardson s Method for Increasing the Accuracy of
Approximate Solutions 267
6.1. Ordinary First-Order Differential Equations 268
6.2. General Results 273
6.2.1. The Decomposition Theorem 273
6.2.2. Acceleration of Convergence 279
6.3. Simple Integral Equations 285
6.3.1. The Fredholm Equation of the Second Kind 285
6.3.2. The Volterra Equation of the First Kind 287
xii Contents
6.4. The One-Dimensional Diffusion Equation 290
6.4.1. The Difference Method 291
6.4.2. The Galerkin Method 293
6.5. Nonstationary Problems 299
6.5.1. The Heat Equation 299
6.5.2. The Splitting-Up Method for the Evolutionary Equation 305
6.6. Richardson s Extrapolation for Multi-Dimensional Problems 306
Chapter 7
Numerical Methods for Some Inverse Problems 312
7.1. Fundamental Definitions and Examples 313
7.2. Solution of the Inverse Evolution Problem
with a Constant Operator 321
7.2.1. The Fourier Method 322
7.2.2. Reduction to the Solution of a Direct Equation 324
7.3. Inverse Evolution Problems with Time-Varying Operators 327
7.4. Methods of Perturbation Theory for Inverse Problems 333
7.4.1. Some Problems of the Linear Theory of Measurements 333
7.4.2. Conjugate Functions and the Notion of Value 335
7.4.3. Perturbation Theory for Linear Functionals 337
7.4.4. Numerical Methods for Inverse Problems and Design of Experiment 339
7.5. Perturbation Theory for Complex Nonlinear Models 344
7.5.1. Fundamental and Adjoint Equations 345
7.5.2. The Adjoint Equation in Perturbation Theory 347
7.5.3. Perturbation Theory for Nonstationary Problems 348
7.5.4. Spectral Methods in Perturbation Theory 350
Chapter 8
Methods of Optimization 352
8.1. Convex Programming 352
8.2. Linear Programming 357
8.3. Quadratic Programming 362
8.4. Numerical Methods in Convex Programming Problems 366
8.5. Dynamic Programming 371
8.6. Pontrjagin s Maximum Principle 375
8.7. Extremal Problems with Constraints and Variational
Inequalities 381
8.7.1. Elements of the General Theory 382
8.7.2. Examples of Extremal Problems 384
8.7.3. Numerical Methods in Extremal Problems 390
Contents xiii
Chapter 9
Some Problems of Mathematical Physics 396
9.1. The Poisson Equation 396
9.1.1. The Dirichlet Problem for the One-Dimensional Poisson Equation 396
9.1.2. The One-Dimensional von Neumann Problem 398
9.1.3. The Two-Dimensional Poisson Equation 401
9.1.4. A Problem of Boundary Conditions 409
9.2. The Heat Equation 411
9.2.1. The One-Dimensional Problem of Heat Conduction 412
9.2.2. The Two-Dimensional Problem of Heat Conduction 416
9.3. The Wave Equation 417
9.4. The Equation of Motion 421
9.4.1. The Simplest Equations of Motion 422
9.4.2. The Two-Dimensional Equation of Motion with Variable Coefficients 428
9.4.3. The Multi-Dimensional Equation of Motion 434
9.5. The Neutron Transport Equation 439
9.5.1. The Nonstationary Equation 439
9.5.2. The Transport Equation in Self-Adjoint Form 451
Chapter 10
A Review of the Methods of Numerical Mathematics 456
10.1. The Theory of Approximation, Stability, and Convergence of
Difference Schemes 456
10.2. Numerical Methods for Problems of Mathematical Physics 459
10.3. Conditionally Well-Posed Problems 465
10.4. Numerical Methods in Linear Algebra 466
10.5. Optimization Problems in Numerical Methods 470
10.6. Optimization Methods 472
10.7. Some Trends in Numerical Mathematics 473
References 476
Index of Notation 505
Index 507
|
| any_adam_object | 1 |
| author | Marčuk, Gurij I. 1925-2013 |
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| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.4 |
| dewey-search | 519.4 |
| dewey-sort | 3519.4 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| indexdate | 2024-07-09T15:43:06Z |
| institution | BVB |
| isbn | 0387906142 3540906142 |
| language | English |
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| spelling | Marčuk, Gurij I. 1925-2013 Verfasser (DE-588)119060701 aut Metody vyčislitel'noj matematiki Methods of numerical mathematics G. I. Marchuk 2. ed. New York [u.a.] Springer 1982 XIII, 510 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applications of mathematics 2 Aus dem Russ. übers. - Literaturverz. S. 476 - 504 Analyse numérique Analyse numérique ram Numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 s DE-604 Applications of mathematics 2 (DE-604)BV000895226 2 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001491249&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Marčuk, Gurij I. 1925-2013 Methods of numerical mathematics Applications of mathematics Analyse numérique Analyse numérique ram Numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4042805-9 |
| title | Methods of numerical mathematics |
| title_alt | Metody vyčislitel'noj matematiki |
| title_auth | Methods of numerical mathematics |
| title_exact_search | Methods of numerical mathematics |
| title_full | Methods of numerical mathematics G. I. Marchuk |
| title_fullStr | Methods of numerical mathematics G. I. Marchuk |
| title_full_unstemmed | Methods of numerical mathematics G. I. Marchuk |
| title_short | Methods of numerical mathematics |
| title_sort | methods of numerical mathematics |
| topic | Analyse numérique Analyse numérique ram Numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Analyse numérique Numerical analysis Numerische Mathematik |
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