Classical and modern numerical analysis: theory, methods and practice
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Boca Raton, Fla. [u.a.]
CRC Press
2010
|
| Series: | Chapman & Hall/CRC numerical analysis and scientific computing
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XIX, 608 S. graph. Darst. |
| ISBN: | 9781420091571 |
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MARC
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| 245 | 1 | 0 | |a Classical and modern numerical analysis |b theory, methods and practice |c Azmy S. Ackleh ... |
| 264 | 1 | |a Boca Raton, Fla. [u.a.] |b CRC Press |c 2010 | |
| 300 | |a XIX, 608 S. |b graph. Darst. | ||
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| 490 | 0 | |a Chapman & Hall/CRC numerical analysis and scientific computing | |
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Record in the Search Index
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| adam_text | Contents
List of Figures
xvii
List of Tables
xix
1
Mathematical Review and Computer Arithmetic
1
1.1
Mathematical Review
...................... 1
1.1.1
Intermediate Value Theorem, Mean Value Theorems,
and Taylor s Theorem
.................. 1
1.1.2
Big O and Little o Notation
............ 5
1.1.3
Convergence Rates
.................... 7
1.2
Computer Arithmetic
...................... 8
1.2.1
Floating Point Arithmetic and Rounding Error
.... 10
1.2.2
Practicalities and the IEEE Floating Point Standard
. 17
1.3
Interval Computations
..................... 23
1.3.1
Interval Arithmetic
.................... 23
1.3.2
Application of Interval Arithmetic: Examples
..... 28
1.4
Exercises
............................. 29
2
Numerical Solution of Nonlinear Equations of One Variable
35
2.1
Introduction
........................... 35
2.2
Bisection Method
........................ 35
2.3
The Fixed Point Method
.................... 39
2.4
Newton s Method (Newton-Raphson Method)
........ 49
2.5
The Univariate Interval Newton Method
........... 54
2.5.1
Existence and Uniqueness Verification
......... 54
2.5.2
Interval Newton Algorithm
............... 55
2.5.3
Convergence Rate of the Interval Newton Method
. . 57
2.6
Secant Method and Miiller s Method
............. 58
2.6.1
The Secant Method
................... 59
2.6.2
Miiller s Method
..................... 63
2.7
Aitken Acceleration and Steffensen s Method
......... 64
2.7.1
Aitken Acceleration
................... 65
2.7.2
Steffensen s Method
................... 68
2.8
Roots of Polynomials
...................... 69
2.9
Additional Notes and Summary of Chapter
2......... 74
2.10
Exercises
............................. 75
Xl
Xli
Numerical Linear Algebra 85
3.1
Basic Results from Linear Algebra
............... 85
3.2
Normed Linear Spaces
......................
88
3.3
Direct Methods for Solving Linear Systems
.......... 105
3.3.1
Gaussian Elimination
..................
Ю5
3.3.2
Symmetric, Positive Definite Matrices
......... 117
3.3.3
Tridiagonal Matrices
................... 1^
3.3.4
Block Tridiagonal Matrices
............... 120
3.3.5
Roundoff Error and Conditioning in Gaussian Elimina¬
tion
............................
I21
3.3.6
Iterative Refinement
................... 127
3.3.7
Interval Bounds
...................... 130
3.3.8
Orthogonal Decomposition (QR Decomposition)
. . . 132
3.4
Iterative Methods for Solving Linear Systems
......... 141
3.4.1
The Jacobi Method
................... 143
3.4.2
The Gauss-Seidel Method
................ 143
3.4.3
Successive Overrelaxation
................ 145
3.4.4
Convergence of the
SOR,
Jacobi, and Gauss-Seidel Meth¬
ods
............................. 146
3.4.5
The Interval Gauss-Seidel Method
........... 153
3.4.6
Graph-Theoretic Properties of Matrices
........ 157
3.4.7
Convergence Rate of the
SOR
Method
......... 159
3.4.8
Convergence of General Matrix Fixed Point Iterations
160
3.4.9
The Block
SOR
Method
................. 161
3.4.10
The Conjugate Gradient Method
............ 163
3.4.11
Iterative Methods for Matrix Inversion
......... 167
3.4.12
Krylov Subspace Methods
................ 169
3.5,
The Singular Value Decomposition
............... 174
3.6
Exercises
............................. 180
Approximation Theory
191
4.1
Introduction
........................... 191
4.2
Norms, Projections, Inner Product Spaces, and Orthogonaliza-
tion in Function Spaces
..................... 191
4.2.1
Best Approximations
................... 192
4.2.2
Best Approximations in Inner Product Spaces
..... 198
4.2.3
The Gram-Schmidt Process (Formal Treatment)
... 201
4.3
Polynomial Approximation
................... 205
4.3.1
The
Weierstrass
Approximation Theorem
....... 205
4.3.2
Taylor Polynomial Approximations
........... 209
4.3.3 Lagrange
Interpolation
.................. 209
4.3.4
The Newton Form for the Interpolating Polynomial
. 213
4.3.5
Hermite Interpolation
.................. 220
4.3.6
Least Squares Polynomial Approximation
....... 222
4.3.7
Error in Least Squares Polynomial Approximation
. . 228
Х1П
4.3.8 Minimax (Best Uniform) Approximation........ 229
4.3.9
Interval
(Rigorous) Bounds on the Errors
....... 236
4.4
Piecewise Polynomial Approximation
............. 238
4.4.1
Piecewise Linear Interpolation
............. 238
4.4.2
Cubic Spline Interpolation
................ 243
4.4.3
Cubic Spline
Interpolants
................ 246
4.5
Trigonometric Approximation
................. 249
4.5.1
Least Squares Trigonometric Approximation (Fourier
Series)
........................... 249
4.5.2
Trigonometric Interpolation on a Finite Point Set (the
FFT)
........................... 254
4.5.3
The FFT Procedure
................... 257
4.6
Rational Approximation
.................... 259
4.6.1
Introduction
....................... 259
4.6.2
Best Rational Approximations in the Uniform Norm
. 259
4.6.3
Padé
Approximation
................... 259
4.6.4
Rational Interpolation
.................. 266
4.7
Wavelet Bases
.......................... 267
4.7.1
Assumed Properties of the Scaling Function
...... 268
4.7.2
The Wavelet Basis and the Scaling Function
...... 272
4.7.3
Construction of Scaling Functions
........... 276
4.8
Least Squares Approximation on a Finite Point Set
..... 279
4.9
Exercises
............................. 284
Eigenvalue-Eigenvector Computation
291
5.1
Basic Results from Linear Algebra
............... 291
5.2
The Power Method
....................... 297
5.3
The Inverse Power Method
................... 303
5.4
Deflation
............................. 305
5.5
The QR Method
......................... 307
5.5.1
Reduction to
Hessenberg
or Tridiagonal Form
..... 307
5.5.2
The QR Method with Origin Shifts
........... 308
5.5.3
The Double QR Algorithm
............... 312
5.6
Jacobi Diagonalization (Jacobi Method)
............ 315
5.7
Simultaneous Iteration (Subspace Iteration)
......... 318
5.8
Exercises
............................. 319
Numerical Differentiation and Integration
323
6.1
Numerical Differentiation
.................... 323
6.1.1
Derivation with Taylor s Theorem
........... 323
6.1.2
Derivation via
Lagrange
Polynomial Representation
. 324
6.1.3
Error Analysis
...................... 325
6.2
Automatic (Computational) Differentiation
.......... 327
6.2.1
The Forward Mode
.................... 328
6.2.2
The Reverse Mode
.................... 330
XIV
6.2.3 Implementation
of Automatic Differentiation
..... 333
6.3
Numerical Integration
...................... 333
6.3.1
Introduction
....................... 333
6.3.2
Newton-Cotes Formulas
................. 336
6.3.3
Gaussian Quadrature
.................. 343
6.3.4
Romberg Integration
................... 353
6.3.5
Multiple Integrals, Singular Integrals, and Infinite In¬
tervals
........................... 364
6.3.6
Monte Carlo and
Quasi-Monte
Carlo Methods
.... 369
6.3.7
Adaptive Quadrature
.................. 373
6.3.8
Interval Bounds
...................... 375
6.4
Exercises
............................. 376
7
Initial Value Problems for Ordinary Differential Equations
381
7.1
.Introduction
........................... 381
7.2
Euler s method
.......................... 385
7.3
Single-Step Methods: Taylor Series and Runge-Kutta
.... 389
7.3.1
Order and Consistency of Euler s Method
....... 391
7.3.2
Order and Consistency of the Midpoint Method
.... 391
7.3.3
An Error Bound for Single-Step Methods
....... 392
7.3.4
Higher-Order Methods
.................. 395
7.4
Error Control and the Runge-Kutta-Fehlberg Method
... 401
7.5
Multistep Methods
....................... 405
7.6
Predictor-Corrector Methods
.................. 416
7.7
Stiff Systems
........................... 418
7.7.1
Absolute Stability of Methods for Systems
....... 421
7.7.2
Methods for Stiff Systems
................ 422
7.8
Extrapolation Methods
..................... 426
7.9
Application to Parameter Estimation in Differential Equations
430
7.10
Exercises
............................. 431
8
Numerical Solution of Systems of Nonlinear Equations
439
8.1
Introduction and
Fréchet
Derivatives
............. 439
8.2
Successive Approximation (Fixed Point Iteration) and the Con¬
traction Mapping Theorem
................... 442
8.3
Newton s Method and Variations
................ 447
8.3.1
Some Local Convergence Results for Newton s Method
447
8.3.2
Convergence Rate of Newton s Method
......... 450
8.3.3
Example of Newton s Method
.............. 452
8.3.4
Local, Semilocal, and Global Convergence
....... 454
8.3.5
The Newton-Kantorovich Theorem
........... 454
8.3.6
A Global Convergence Result for Newton s Method
. . 455
8.3.7
Practical Considerations
................. 458
8.4
Multivariate Interval Newton Methods
............ 463
8.4.1
The Nonlinear Interval Gauss-Seidel Method
..... 465
XV
8.4.2
The Multivariate Krawczyk Method
.......... 470
8.4.3
Using Interval Gaussian Elimination
.......... 472
8.4.4
Relationship to the Kantorovich Theorem
....... 472
8.5
Quasi-Newton
Methods (Broyden s Method)
......... 473
8.5.1
Practicalities
....................... 479
8.6
Methods for Finding All Solutions
............... 480
8.6.1
Homotopy Methods
................... 480
8.6.2
Branch and Bound Methods
............... 481
8.7
Exercises
............................. 482
9
Optimization
487
9.1
Local Optimization
....................... 489
9.1.1
Introduction to Unconstrained Local Optimization
. . 489
9.1.2
Golden Section Search
.................. 490
9.1.3
Relationship to Nonlinear Systems
........... 491
9.1.4
Steepest Descent
..................... 493
9.1.5
Quasi-Newton
Methods for Minimization
....... 495
9.1.6
The Nelder-Mead Simplex (Direct Search) Method
. . 497
9.1.7
Software for Unconstrained Local Optimization
.... 501
9.2
Constrained Local Optimization
................ 501
9.3
Constrained Optimization and Nonlinear Systems
...... 501
9.4
Linear Programming
...................... 503
9.4.1
Converting to Standard Form
.............. 505
9.4.2
The Fundamental Theorem of Linear Programming
. . 506
9.4.3
The Simplex Method
................... 507
9.4.4
Proof of the Fundamental Theorem of Linear Program¬
ming
............................ 511
9.4.5
Degenerate Cases
..................... 512
9.4.6
Failure of Linear Programming Solvers
......... 513
9.4.7
Software for Linear Programming
............ 514
9.4.8
Quadratic Programming and Convex Programming
. . 514
9.5
Dynamic Programming
..................... 515
9.6
Global (Nonconvex) Optimization
............... 518
9.6.1
Genetic Algorithms
.................... 520
9.6.2
Simulated Annealing
................... 523
9.6.3
Branch and Bound Algorithms
............. 523
9.7
Exercises
............................. 531
10
Boundary-Value Problems and Integral Equations
535
10.1
Boundary-Value Problems
................... 535
10.1.1
Shooting Method for Numerical Solution of
(10.1) . . 536
10.1.2
Finite-Difference Methods
................ 540
10.1.3
Galerkin Methods
.................... 544
10.2
Approximation of Integral Equations
............. 554
10.2.1
Volterra Integral Equations of Second Kind
...... 554
XVI
10.2.2
Fredholm
Integral
Equations
of Second Kind
..... 559
10.3
Exercises
............................. 566
A Solutions to Selected Exercises
571
References
591
Index
599
|
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| spelling | Classical and modern numerical analysis theory, methods and practice Azmy S. Ackleh ... Boca Raton, Fla. [u.a.] CRC Press 2010 XIX, 608 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC numerical analysis and scientific computing Literaturangaben Numerical analysis / Data processing Datenverarbeitung Numerical analysis Data processing Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Numerische Mathematik (DE-588)4042805-9 s DE-604 Ackleh, Azmy S. Sonstige oth Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017751776&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Classical and modern numerical analysis theory, methods and practice Numerical analysis / Data processing Datenverarbeitung Numerical analysis Data processing Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4123623-3 |
| title | Classical and modern numerical analysis theory, methods and practice |
| title_auth | Classical and modern numerical analysis theory, methods and practice |
| title_exact_search | Classical and modern numerical analysis theory, methods and practice |
| title_full | Classical and modern numerical analysis theory, methods and practice Azmy S. Ackleh ... |
| title_fullStr | Classical and modern numerical analysis theory, methods and practice Azmy S. Ackleh ... |
| title_full_unstemmed | Classical and modern numerical analysis theory, methods and practice Azmy S. Ackleh ... |
| title_short | Classical and modern numerical analysis |
| title_sort | classical and modern numerical analysis theory methods and practice |
| title_sub | theory, methods and practice |
| topic | Numerical analysis / Data processing Datenverarbeitung Numerical analysis Data processing Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Numerical analysis / Data processing Datenverarbeitung Numerical analysis Data processing Numerische Mathematik Lehrbuch |
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