Introduction to numerical analysis:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English German |
| Published: |
New York [u.a.]
Springer
2002
|
| Edition: | 3. ed. |
| Series: | Texts in applied mathematics
12 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XV, 744 S. |
| ISBN: | 038795452X |
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MARC
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| 100 | 1 | |a Stoer, Josef |d 1934- |e Verfasser |0 (DE-588)105888052 |4 aut | |
| 240 | 1 | 0 | |a Einführung in die numerische Mathematik |
| 245 | 1 | 0 | |a Introduction to numerical analysis |c J. Stoer ; R. Bulirsch |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 2002 | |
| 300 | |a XV, 744 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Texts in applied mathematics |v 12 | |
| 500 | |a Includes bibliographical references and index | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
Préface
to the Third Edition
VII
Preface to the First Edition IX
1
Error Analysis
1
1.1
Representation of Numbers
2
1.2
Roundoff Errors and Floating-Point Arithmetic
!
1.3
Error Propagation
9
1.4
Examples
21
1.5
Interval Arithmetic: Statistical Roundoff Estimation
27
Exercises for Chapter
1 33
References for Chapter
1 36
2
Interpolation
37
2.1
Interpolation by Polynomials
38
2.1.1
Theoretical Foundation: The Interpolation Formula of
Lagrange 38
2.1.2
Neville s Algorithm
40
2.1.3 Newtons
Interpolation Formula: Divided Differences
43
2.1.4
The Error in Polynomial Interpolation
48
2.1.5
Hermite Interpolation
51
2.2
Interpolation by Rational Functions
59
2.2.1
General Properties of Rational Interpolation
59
2.2.2
Inverse and Reciprocal Differences.
Thiele
s
Continued Fraction
04
2.2.3
Algorithms of the Neville Type f>8
2.2.4
Comparing Rational and Polynomial Interpolation
73
2.3
Trigonometric Interpolation
71
2.3.1
Basic Facts
74
2.3.2
Fast Fourier Transforms
Ы)
2.3.3
The Algorithms of Goertzel and Reinsch 4<S
2.3.4
The Calculation of Fourier Coefficients. Attenuation Factors
92
xi
xii Contents
2.4 Interpolation
by Spline Functions
97
2.4.1
Theoretical Foundations
97
2.4.2
Determining Interpolating Cubic Spline Functions
101
2.4.3
Convergence Properties of Cubic Spline Functions
107
2.4.4
B-Splines 111
2.4.5
The Computation of B-Splines
117
2.4.6
Multi-Resolution Methods and B-Splines
121
Exercises for Chpater
2 134
References for Chapter2
143
3
Topics in Integration
145
3.1
The Integration Formulas of Newton and Cotes
146
3.2
Peano s Error Representation
151
3.3
The Euler-Maclaurin Summation Formula
156
3.4
Integration by Extrapolation
160
3.5
About Extrapolation Methods
165
3.6
Gaussian Integration Methods
171
3.7
Integrals with Singularities
181
Exercises for Chapter
3 184
References for Chapter
3 188
4
Systems of Linear Equations
190
4.1
Gaussian Elimination. The Triangular Decomposition of a Matrix
190
4.2
The Gauss-Jordan Algorithm
200
4.3
The Choleski Decompostion
204
4.4
Error Bounds
207
4.5
Roundoff-Error Analysis for Gaussian Elimination
215
4.6
Roundoff Errors in Solving Triangular Systems
221
4.7
Orthogonalization Techniques of Householder and Gram-Schmidt
223
4.8
Data Fitting
231
4.8.1
Linear Least Squares. The Normal Equations
232
4.8.2
The Use of Orthogonalization in Solving Linear Least-Squares
Problems
235
4.8.3
The Condition of the Linear Least-Squares Problem
236
4.8.4
Nonlinear Least-Squares Problems
241
4.8.5
The
Pseudoinverse
of a Matrix
243
4.9
Modification Techniques for Matrix Decompositions
247
4.10
The Simplex Method
256
4.11
Phase One of the Simplex Method
268
4.12
Appendix: Elimination Methods for Sparse Matrices
272
Exercises for Chapter
4 280
References for Chapter
4 286
Contents xiii
5
Finding Zeros and Minimum Points by Iterative
Methods
289
5.1
The Development of Iterative Methods
290
5.2
General Convergence Theorems
293
5.3
The Convergence of Newton s Method in Several Variables
298
5.4
A Modified Newton Method
302
5.4.1
On the Convergence of Minimization Methods
303
5.4.2
Application of the Convergence Criteria to the Modified
Newton Method
308
5.4.3
Suggestions for a Practical Implementation of the Modified
Newton Method. A Rank-One Method Due to Broyden
313
5.5
Roots of Polynomials. Application of Newton s Method
316
5.6
Sturm Sequences and Bisection Methods
328
5.7
Bairstow s Method
333
5.8
The Sensitivity of Polynomial Roots
335
5.9
Interpolation Methods for Determining Roots
338
5.10
The A2-Method of Aitken
344
5.11
Minimization Problems without Constraints
349
Exercises for Chapter
5 358
References for Chapter
5 361
6
Eigenvalue Problems
364
6.0
Introduction
364
6.1
Basic Facts on Eigenvalues
366
6.2
The Jordan Normal Form of a Matrix
369
6.3
The Frobenius Normal Form of a Matrix
375
6.4
The
Schur
Normal Form of a Matrix; Hermitian and
Normal Matrices; Singular Values of Matrixes
379
6.5
Reduction of Matrices to Simpler Form
386
6.5.1
Reduction of a Hermitian Matrix to Tridiagonal Form:
The Method of Householder
388
6.5.2
Reduction of a Hermitian Matrix to Tridiagonal or Diagonal
Form: The Methods of
Givens
and Jacobi
394
6.5.3
Reduction of a Hermitian Matrix to Tridiagonal Form:
The Method of Lanczos
398
6.5.4
Reduction to
Hessenberg Form 402
6.6
Methods for Determining the Eigenvalues and Eigenvectors
405
6.6.1
Computation of the Eigenvalues of a Hermitian
Tridiagonal Matrix
405
6.6.2
Computation of the Eigenvalues of
a Hessenberg
Matrix.
The Method of Hyman
407
6.6.3
Simple Vector Iteration and Inverse Iteration of Wielandt
408
6.6.4
The LR and QR Methods
415
6.6.5
The Practical Implementation of the
Q
Л
Method
425
xiv Contents
6.7
Computation of the Singular Values of a Matrix
436
6.8
Generalized Eigenvalue Problems
440
6.9
Estimation of Eigenvalues
441
Exercises for Chapter
6 455
References for Chapter
6 462
7
Ordinary Differential Equations
465
Introduction
465
Some Theorems from the Theory of Ordinary Differential
Equations
467
Initial-Value Problems
471
One-Step Methods: Basic Concepts
471
Convergence of One-Step Methods
477
Asymptotic Expansions for the Global Discretization Error
of One-Step Methods
480
The Influence of Rounding Errors in One-Step Methods
483
Practical Implementation of One-Step Methods
485
Multistep Methods: Examples
492
General Multistep Methods
495
An Example of Divergence
498
Linear Difference Equations
501
Convergence of Multistep Methods
504
Linear Multistep Methods
508
Asymptotic Expansions of the Global Discretization Error for
Linear Multistep Methods
513
Practical Implementation of Multistep Methods
517
Extrapolation Methods for the Solution of the Initial-Value
Problem
521
Comparison of Methods for Solving Initial-Value Problems
524
Stiff Differential Equations
525
Implicit Differential Equations. Differential-Algebraic Equations
531
Handling Discontinuities in Differential Equations
536
Sensitivity Analysis of Initial-Value Problems
538
Boundary-Value Problems
539
Introduction
539
The Simple Shooting Method
542
The Simple Shooting Method for Linear Boundary-Value
Problems
548
7.3.3
An Existence and Uniqueness Theorem for the Solution of
Boundary-Value Problems
550
7.3.4
Difficulties in the Execution of the Simple Shooting
Method
552
7.3.5
The Multiple Shooting Method
557
7.
0
7.
1
7.
2
7.
2.1
7.
2.2
7.
2.3
7.
2.4
7.
2.5
7.2.6
7.
.2.7
7.
.2.8
7.
.2.9
7.2.10
7
.2.11
7
.2.12
7.2.13
7.2.14
7
.2.15
7
.2.16
7
.2.17
7
.2.18
7.2.19
7
.3
7
.3.0
7
.3.1
7
.3.2
Contents xv
7.3.6
Hints for the Practical Implementation of the Multiple
Shooting Method
561
7.3.7
An Example: Optimal Control Program for a Lifting Reentry
Space Vehicle
565
7.3.8
Advanced Techniques in Multiple Shooting
572
7.3.9
The Limiting Case
m
—
y
oc
of the Multiple Shooting Method
(General Newton s Method. Quasilinearization)
577
7.4
Difference Methods
582
7.5
Variational Methods
586
7.6
Comparison of the Methods for Solving Boundary-Value Problems
for Ordinary Differential Equations
596
7.7
Variational Methods for Partial Differential Equations
The Finite-Element Method
600
Exercises for Chapter
7 607
References for Chapter
7 613
8
Iterative Methods for the Solution of
Large Systems of Linear Equations.
Additional Methods
619
8.0
Introduction
619
8.1
General Procedures for the Construction of Iterative Methods
621
8.2
Convergence Theorems
623
8.3
Relaxation Methods
629
8.4
Applications to Difference Methods- An Example
639
8.5
Block Iterative Methods
645
8.6
The ADI-Method of Peaceman and
Rachford 647
8.7
Krylov
Space Methods for Solving Linear Equations
657
8.7.1
The Conjugate-Gradient Method of Hestenes and
Stiefel 658
8.7.2
The GMRES Algorithm
667
8.7.3
The Biorthogonalization Method of Lanczos and the Q.MR
algorithm
680
8.7.4
The Bi-CG and BI-CGSTAB Algorithms
686
8.8
Buneman s Algorithm and Fourier Methods for Solving the
Discretized
Poisson
Equation
691
8.9
Multigrid Methods
702
8.10
Comparison of Iterative Methods
712
Exercises for Chapter
8 719
References for Chapter
8 727
General Literature on Numerical Methods
730
Index
732
|
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| indexdate | 2024-07-09T19:02:09Z |
| institution | BVB |
| isbn | 038795452X |
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| spelling | Stoer, Josef 1934- Verfasser (DE-588)105888052 aut Einführung in die numerische Mathematik Introduction to numerical analysis J. Stoer ; R. Bulirsch 3. ed. New York [u.a.] Springer 2002 XV, 744 S. txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 12 Includes bibliographical references and index Analyse numérique Análise numérica larpcal Análisis numérico Numerieke wiskunde gtt Numerical analysis 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 Bulirsch, Roland Sonstige oth Texts in applied mathematics 12 (DE-604)BV002476038 12 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009844800&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Stoer, Josef 1934- Introduction to numerical analysis Texts in applied mathematics Analyse numérique Análise numérica larpcal Análisis numérico Numerieke wiskunde gtt Numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4123623-3 |
| title | Introduction to numerical analysis |
| title_alt | Einführung in die numerische Mathematik |
| title_auth | Introduction to numerical analysis |
| title_exact_search | Introduction to numerical analysis |
| title_full | Introduction to numerical analysis J. Stoer ; R. Bulirsch |
| title_fullStr | Introduction to numerical analysis J. Stoer ; R. Bulirsch |
| title_full_unstemmed | Introduction to numerical analysis J. Stoer ; R. Bulirsch |
| title_short | Introduction to numerical analysis |
| title_sort | introduction to numerical analysis |
| topic | Analyse numérique Análise numérica larpcal Análisis numérico Numerieke wiskunde gtt Numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Analyse numérique Análise numérica Análisis numérico Numerieke wiskunde Numerical analysis Numerische Mathematik Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009844800&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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