Numerical analysis:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Birkhäuser
2012
|
| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 543 - 569 |
| Physical Description: | XXVI, 588 S. graph. Darst. |
| ISBN: | 9780817682583 |
Staff View
MARC
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| 100 | 1 | |a Gautschi, Walter |d 1927- |e Verfasser |0 (DE-588)119264315 |4 aut | |
| 245 | 1 | 0 | |a Numerical analysis |c Walter Gautschi |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York, NY |b Birkhäuser |c 2012 | |
| 300 | |a XXVI, 588 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Literaturverz. S. 543 - 569 | ||
| 650 | 0 | 7 | |a Numerische Mathematik |0 (DE-588)4042805-9 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)4123623-3 |a Lehrbuch |2 gnd-content | |
| 689 | 0 | 0 | |a Numerische Mathematik |0 (DE-588)4042805-9 |D s |
| 689 | 0 | |5 DE-604 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-024741551 | ||
Record in the Search Index
| _version_ | 1804148827675951104 |
|---|---|
| adam_text | Contents
Prologue
.......................................................................... xix
Pl
Overview
.............................................................. xix
P2 Numerical Analysis Software
........................................ xxi
P3 Textbooks and Monographs
.......................................... xxi
P3.1 Selected Textbooks on Numerical Analysis
................. xxi
P3.2 Monographs and Books on Specialized Topics
............. xxiii
P4 Journals
................................................................ xxvi
1
Machine Arithmetic and Related Matters
............................... 1
1.1
Real Numbers, Machine Numbers, and Rounding
.................. 2
1.1.1
Real Numbers
................................................. 2
1.1.2
Machine Numbers
............................................ 3
1.1.3
Rounding
..................................................... 5
1.2
Machine Arithmetic
.................................................. 7
1.2.1
A Model of Machine Arithmetic
............................ 7
1.2.2
Error Propagation in Arithmetic Operations:
Cancellation Error
............................................. 8
1.3
The Condition of a Problem
.......................................... 11
1.3.1
Condition Numbers
.......................................... 13
1.3.2
Examples
....................................................... 16
1.4
The Condition of an Algorithm
...................................... 24
1.5
Computer Solution of a Problem; Overall Error
.................... 27
1.6
Notes to Chapter
1.................................................... 28
Exercises and Machine Assignments to Chapter
1 ......................... 31
Exercises
...................................................................... 31
Machine Assignments
........................................................ 39
Selected Solutions to Exercises
.............................................. 44
Selected Solutions to Machine Assignments
................................ 48
2
Approximation and Interpolation
......................................... 55
2.1
Least Squares Approximation
........................................ 59
2.1.1
Inner Products
................................................ 59
2.1.2
The Normal Equations
....................................... 61
xiii
X1V
Contents
2.1.3
Least Squares Error; Convergence
........................... 64
2.1.4
Examples of Orthogonal Systems
........................... 67
2.2
Polynomial Interpolation
............................................. 73
2.2.1 Lagrange
Interpolation Formula: Interpolation Operator
... 74
2.2.2
Interpolation Error
............................................ 77
2.2.3
Convergence
..................................................
SI
2.2.4
Chebyshev Polynomials and Nodes
......................... 86
2.2.5
Barycentnc Formula
......................................... 91
2.2.6
Newton s Formula
............................................ 93
2.2.7
Hermite Interpolation
........................................ 97
2.2.8
Inverse Interpolation
......................................... 100
2.3
Approximation and Interpolation by Spline Functions
............. 101
2.3.1
Interpolation by Piecewise Linear Functions
............... 102
2.3.2
A Basis for
Ѕ?(Л)
............................................ 104
2.3.3
Least Squares Approximation
............................... 106
2.3.4
Interpolation by Cubic Splines
.............................. 107
2.3.5
Minimality Properties of Cubic Spline
Interpolants
........ 110
2.4
Notes to Chapter
2.................................................... 112
Exercises and Machine Assignments to Chapter
2 ......................... 118
Exercises
...................................................................... 118
Machine Assignments
........................................................ 134
Selected Solutions to Exercises
.............................................. 138
Selected Solutions to Machine Assignments
................................ 150
3
Numerical Differentiation and Integration
.............................. 159
3.1
Numerical Differentiation
............................................ 159
3.1.1
A General Differentiation Formula for Unequally
Spaced Points
................................................. 159
3.1.2
Examples
........................................................ 161
3.1.3
Numerical Differentiation with Perturbed Data
............. 163
3.2
Numerical Integration
................................................. 165
3.2.1
The Composite Trapezoidal and Simpson s Rules
.......... 165
3.2.2
(Weighted) Newton-Cotes and Gauss Formulae
-........... 169
3.2.3
Properties of Gaussian Quadrature Rules
................... 175
3.2.4
Some Applications of the Gauss Quadrature Rule
.......... 178
3.2.5
Approximation of Linear Functionals: Method
of Interpolation vs. Method of Undetermined
Coefficients
................................................... 182
3.2.6
Peano Representation of Linear Functionals
................ 187
3.2.7
Extrapolation Methods
........................................ 190
3.3
Notes to Chapter
3.................................................... 195
Exercises and Machine Assignments to Chapter
3 ......................... 200
Exercises
...................................................................... 200
Machine Assignments
........................................................ 214
Selected Solutions to Exercises
.............................................. 219
Selected Solutions to Machine Assignments
................................ 232
Contente
xv
4
Nonlinear Equations
........................................................ 253
4.1
Examples
.............................................................. 254
4.1.1
A Transcendental Equation
.................................. 254
4.1.2
A Two-Point Boundary Value
Problern ..................... 254
4.1.3
A Nonlinear Integral Equation
............................... 256
4.1.4
s-Orthogonal Polynomials
................................... 257
4.2
Iteration, Convergence, and Efficiency
.............................. 258
4.3
The Methods of Bisection and Sturm Sequences
................... 261
4.3.1
Bisection Method
............................................. 261
4.3.2
Method of Sturm Sequences
................................. 264
4.4
Method of False Position
............................................. 266
4.5
Secant Method
........................................................ 269
4.6
Newton s Method
..................................................... 274
4.7
Fixed Point Iteration
.................................................. 278
4.8
Algebraic Equations
.................................................. 280
4.8.1
Newton s Method Applied to an Algebraic Equation
...... 280
4.8.2
An Accelerated Newton Method for Equations
with Real Roots
............................................... 282
4.9
Systems of Nonlinear Equations
..................................... 284
4.9.1
Contraction Mapping Principle
.............................. 284
4.9.2
Newton s Method for Systems of Equations
................ 285
4.10
Notes to Chapter
4.................................................... 287
Exercises and Machine Assignments to Chapter
4 ......................... 292
Exercises
...................................................................... 292
Machine Assignments
........................................................ 302
Selected Solutions to Exercises
.............................................. 306
Selected Solutions to Machine Assignments
................................ 318
5
Initial Value Problems for ODEs: One-Step Methods
.................. 325
5.1
Examples
.............................................................. 326
5.2
Types of Differential Equations
...................................... 328
5.3
Existence and Uniqueness
............................................ 331
5.4
Numerical Methods
................................................... 332
5.5
Local Description of One-Step Methods
............................. 333
5.6
Examples of One-Step Methods
...................................... 335
5.6.1
Euler s Method
............................................... 335
5.6.2
Method of Taylor Expansion
................................. 336
5.6.3
Improved
Euler
Methods
..................................... 337
5.6.4
Second-Order Two-Stage Methods
.......................... 339
5.6.5
Runge^Kutta Methods
....................................... 341
5.7
Global Description of One-Step Methods
........................... 343
5.7.1
Stability
........................................................ 344
5.7.2
Convergence
............................-..................... 347
5.7.3
Asymptotics of Global Error
................................. 348
XVj
Contents
5.8
Error Monitoring and Step Control
.................................. 352
5.8.1
Estimation of Global Error
................................... 352
5.8.2
Truncation Error Estimates
.................................. 354
5.8.3
Step Control
.................................................. 357
5.9
Stiff Problems
......................................................... 360
5.9.1
А
-Stability
.....................................................
361
5.9.2
Padé
Approximation
......................................... 362
5.9.3
Examples of
A-Stable
One-Step Methods
.................. 367
5.9.4
Regions of Absolute Stability
................................ 370
5.10
Notes to Chapter
5.................................................... 371
Exercises and Machine Assignments to Chapter
5 ......................... 378
Exercises
...................................................................... 378
Machine Assignments
........................................................ 383
Selected Solutions to Exercises
.............................................. 387
Selected Solutions to Machine Assignments
................................ 392
6
Initial Value Problems for ODEs: Multistep Methods
399
6.1
Local Description of Multistep Methods
............................ 399
6.1.1
Explicit and Implicit Methods
............................... 399
6.1.2
Local Accuracy
............................................... 401
6.1.3
Polynomial Degree vs. Order
................................ 405
6.2
Examples of
Mulüstep
Methods
..................................... 408
6.2.1
Adams-Bashforth Method
................................... 409
6.2.2
Adams—Moulton Method
.................................... 412
6.2.3
Predictor—Corrector Methods
................................ 413
6.3
Global Description of Multistep Methods
........................... 416
6.3.1
Linear Difference Equations
................................. 416
6.3.2
Stability and Root Condition
................................ 420
6.3.3
Convergence
...................................................... 424
6.3.4
Asymptotics of Global Error
................................. 426
6.3.5
Estimation of Global Error
.................................... 430
6.4
Analytic Theory of Order and Stability
............................... 433
6.4.1
Analytic Characterization of Order
.......................... 433
6.4.2
Stable Methods of Maximum Order
......................... 441
6.4.3
Applications
.................................................. 446
6.5
Stiff Problems
......................................................... 450
6.5.1
A-Stability
.................................................... 450
6.5.2
A (a) -Stability
................................................ 452
6.6
Notes to Chapter
6.................................................... 453
Exercises and Machine Assignments to Chapter
6 ......................... 456
Exercises
...................................................................... 456
Machine Assignments
........................................................ 459
Selected Solutions to Exercises
.............................................. 461
Selected Solutions to Machine Assignments
................................ 466
Contents
xvii
7
Two-Point Boundary Value Problems for ODEs
........................ 471
7.1
Existence and Uniqueness
............................................ 474
7.1.1
Examples
...................................................... 474
7.1.2
A Scalar Boundary Value Problem
.......................... 476
7.1.3
General Linear and Nonlinear Systems
..................... 481
7.2
Initial Value Techniques
.............................................. 482
7.2.1
Shooting Method for a Scalar Boundary Value Problem
... 483
7.2.2
Linear and Nonlinear Systems
............................... 485
7.2.3
Parallel Shooting
............................................. 490
7.3
Finite Difference Methods
........................................... 494
7.3.1
Linear Second-Order Equations
............................. 494
7.3.2
Nonlinear Second-Order Equations
......................... 500
7.4
Vanational Methods
.................................................. 503
7.4.1
Variational Formulation
...................................... 503
7.4.2
The Extremal Problem
........................................ 506
7.4.3
Approximate Solution of the Extremal Problem
............ 507
7.5
Notes to Chapter
7.................................................... 509
Exercises and Machine Assignments to Chapter
7 ......................... 512
Exercises
...................................................................... 512
Machine Assignments
........................................................ 518
Selected Solutions to Exercises
.............................................. 521
Selected Solutions to Machine Assignments
................................ 532
References
........................................................................ 543
Index
.............................................................................. 571
|
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| author | Gautschi, Walter 1927- |
| author_GND | (DE-588)119264315 |
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| indexdate | 2024-07-10T00:13:19Z |
| institution | BVB |
| isbn | 9780817682583 |
| language | English |
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| spelling | Gautschi, Walter 1927- Verfasser (DE-588)119264315 aut Numerical analysis Walter Gautschi 2. ed. New York, NY Birkhäuser 2012 XXVI, 588 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 543 - 569 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 Erscheint auch als Online-Ausgabe 978-0-8176-8259-0 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024741551&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gautschi, Walter 1927- Numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4123623-3 |
| title | Numerical analysis |
| title_auth | Numerical analysis |
| title_exact_search | Numerical analysis |
| title_full | Numerical analysis Walter Gautschi |
| title_fullStr | Numerical analysis Walter Gautschi |
| title_full_unstemmed | Numerical analysis Walter Gautschi |
| title_short | Numerical analysis |
| title_sort | numerical analysis |
| topic | Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Numerische Mathematik Lehrbuch |
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