An introduction to numerical methods and analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Hoboken, NJ
Wiley-Interscience
2007
|
| Edition: | Rev. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XVII, 572 S. graph. Darst. |
| ISBN: | 0470049634 9780470049631 |
Staff View
MARC
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| 100 | 1 | |a Epperson, James F. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a An introduction to numerical methods and analysis |c James F. Epperson |
| 250 | |a Rev. ed. | ||
| 264 | 1 | |a Hoboken, NJ |b Wiley-Interscience |c 2007 | |
| 300 | |a XVII, 572 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Numerical analysis | |
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Record in the Search Index
| _version_ | 1804137219524395008 |
|---|---|
| adam_text | CONTENTS
Preface
xiii
1
Introductory Concepts and Calculus Review
1
1.1 Basic Tools of Calculus
2
1.1.1
Taylor s Theorem
2
1.1.2
Mean Value and Extreme Value Theorems
9
1.2
Error, Approximate Equality, and Asymptotic Order Notation
14
1.2.1
Error
15
1.2.2
Notation: Approximate Equality
16
1.2.3
Notation: Asymptotic Order
16
1.3
A Primer on Computer Arithmetic
21
1.4
A Word on Computer Languages and Software
29
1.5
Simple Approximations
30
1.6
Application: Approximating the Natural Logarithm
34
References
37
2
A Survey of Simple Methods and Tools
39
2.1
Homer s Rule and Nested Multiplication
39
2.2
Difference Approximations to the Derivative
44
2.3
Application: Euler s Method for Initial Value Problems
52
VÍ
CONTENTS
2.4
Linear
Interpolation
58
2.5
Application—
The Trapezoid
Rule
64
2.6
Solution of Tridiagonal Linear Systems
74
2.7
Application: Simple Two-Point Boundary Value Problems
81
3
Root-Finding 87
3.1
The Bisection Method
88
3.2
Newton s Method: Derivation and Examples
95
3.3
How to Stop Newton s Method
101
3.4
Application: Division Using Newton s Method
104
3.5
The Newton Error Formula
108
3.6
Newton s Method: Theory and Convergence
113
3.7
Application: Computation of the Square Root
118
3.8
The Secant Method: Derivation and Examples
120
3.9
Fixed Point Iteration
125
3.10
Special Topics in Root-finding Methods
135
3.10.1
Extrapolation and Acceleration
135
3.10.2
Variants of Newton s method
139
3.10.3
The Secant Method: Theory and Convergence
143
3.10.4
Multiple Roots
147
3.10.5
In Search of Fast Global Convergence: Hybrid Algorithms
151
3.11
Literature and Software Discussion
156
References
159
4
Interpolation and Approximation
161
4.1 Lagrange
Interpolation
161
4.2
Newton Interpolation and Divided Differences
167
4.3
Interpolation Error
176
4.4
Application: Muller s Method and Inverse Quadratic Interpolation
182
4.5
Application: More Approximations to the Derivative
186
4.6
Hermite Interpolation
188
4.7
Piecewise Polynomial Interpolation
192
4.8
An Introduction to Splines
200
4.8.1
Definition of the Problem
201
4.8.2
Cubic B-Splines
201
4.9
Application: Solution of Boundary Value Problems
214
4.10
Least Squares Concepts in Approximation
219
4.10.1
An Introduction to Data Fitting
219
4.10.2
Least Squares Approximation and Orthogonal Polynomials
223
4.11
Advanced Topics in Interpolation Error
237
CONTENTS
VII
4.11.1
Stability of Polynomial
Interpolation 238
4.11.2
The
Runge
Example
241
4.11.3
The Chebyshev Nodes
244
4.12
Literature and Software Discussion
250
References
251
5
Numerical Integration
253
5.1
A Review of the Definite Integral
254
5.2
Improving the
Trapezoid
Rule
256
5.3
Simpson s Rule and Degree of Precision
261
5.4
The Midpoint Rule
272
5.5
Application: Stirling s Formula
276
5.6
Gaussian Quadrature
278
5.7
Extrapolation Methods
290
5.8
Special Topics in Numerical Integration
297
5.8.1
Romberg Integration
297
5.8.2
Quadrature with Non-smooth Integrands
302
5.8.3
Adaptive Integration
307
5.8.4
Peano Estimates for the
Trapezoid
Rule
ЗІЗ
5.9
Literature and Software Discussion
319
References
321
6
Numerical Methods for Ordinary Differential Equations
323
6.1
The Initial Value Problem
—
Background
324
6.2
Euler s Method
329
6.3
Analysis of Euler s Method
333
6.4
Variants of Euler s Method
337
6.4.1
The Residual and Truncation Error
339
6.4.2
Implicit Methods and Predictor-Corrector Schemes
342
6.4.3
Starting Values and Multistep Methods
347
6.4.4
The Midpoint Method and Weak Stability
348
6.5
Single Step Methods—Runge-Kutta
354
6.6
Multi-step Methods
361
6.6.1
The Adams Families
361
6.6.2
The BDF Family
364
6.7
Stability Issues
367
6.7.1
Stability Theory for Multistep Methods
367
6.7.2
Stability Regions
370
6.8
Application to Systems of Equations
373
Vili
CONTENTS
6.8.1 Implementation
Issues
and Examples
373
6.8.2
Stiff Equations
377
6.8.3
A-Stability
378
6.9
Adaptive Solvers
381
6.10
Boundary Value Problems
393
6.10.1
Simple Difference Methods
394
6.10.2
Shooting Methods
399
6.11
Literature and Software Discussion
403
References
405
7
Numerical Methods for the Solution of Systems of Equations
407
7.1
Linear Algebra Review
407
7.2
Linear Systems and Gaussian Elimination
409
7.3
Operation Counts
417
7.4
The
LU
Factorization
419
7.5
Perturbation, Conditioning, and Stability
429
7.5.1
Vector and Matrix Norms
430
7.5.2
The Condition Number and Perturbations
431
7.5.3
Estimating the Condition Number
438
7.5.4
Iterative Refinement
441
7.6 SPD
Matrices and the Cholesky Decomposition
445
7.7
Iterative Methods for Linear Systems
—
A Brief Survey
448
7.8
Nonlinear Systems: Newton s Method and Related Ideas
457
7.8.1
Newton s Method
457
7.8.2
Fixed Point Methods
460
7.9
Application: Numerical Solution of Nonlinear Boundary Value
Problems
462
7.10
Literature and Software Discussion
465
References
467
8
Approximate Solution of the Algebraic Eigenvalue Problem
469
8.1
Eigenvalue Review
469
8.2
Reduction to
Hessenberg Form 476
8.3
Power Methods
483
8.4
An Overview of the
Q R
Iteration
501
8.5
Literature and Software Discussion
510
References
511
CONTENTS
ІХ
9
A
Survey of Finite Difference Methods for
Partial Differential Equations
513
9.1
Difference Methods for the Diffusion Equation
513
9.1.1
The Basic Problem
513
9.1.2
The Explicit Method and Stability
514
9.1.3
Implicit Methods and the Crank-Nicolson Method
519
9.2
Difference Methods for
Poisson
Equations
529
9.2.1
Discretization
529
9.2.2
Banded Cholesky Solvers
532
9.2.3
Iteration and the Method of Conjugate Gradients
535
9.3
Literature and Software Discussion
545
References
547
Appendix A: Proofs of Selected Theorems,
and Other Additional Material
549
A.
1
Proofs of the Interpolation Error Theorems
549
A.2 Proof of the Stability Result for Smooth and Uniformly Monotone
Decreasing Initial Value Problems
551
A.3 Stiff Systems of Differential Equations and Eigenvalues
552
A.4 The Matrix Perturbation Theorem
553
A.5 Answers to Selected Exercises
555
Index
569
|
| adam_txt |
CONTENTS
Preface
xiii
1
Introductory Concepts and Calculus Review
1
1.1 Basic Tools of Calculus
2
1.1.1
Taylor's Theorem
2
1.1.2
Mean Value and Extreme Value Theorems
9
1.2
Error, Approximate Equality, and Asymptotic Order Notation
14
1.2.1
Error
15
1.2.2
Notation: Approximate Equality
16
1.2.3
Notation: Asymptotic Order
16
1.3
A Primer on Computer Arithmetic
21
1.4
A Word on Computer Languages and Software
29
1.5
Simple Approximations
30
1.6
Application: Approximating the Natural Logarithm
34
References
37
2
A Survey of Simple Methods and Tools
39
2.1
Homer's Rule and Nested Multiplication
39
2.2
Difference Approximations to the Derivative
44
2.3
Application: Euler's Method for Initial Value Problems
52
VÍ
CONTENTS
2.4
Linear
Interpolation
58
2.5
Application—
The Trapezoid
Rule
64
2.6
Solution of Tridiagonal Linear Systems
74
2.7
Application: Simple Two-Point Boundary Value Problems
81
3
Root-Finding 87
3.1
The Bisection Method
88
3.2
Newton's Method: Derivation and Examples
95
3.3
How to Stop Newton's Method
101
3.4
Application: Division Using Newton's Method
104
3.5
The Newton Error Formula
108
3.6
Newton's Method: Theory and Convergence
113
3.7
Application: Computation of the Square Root
118
3.8
The Secant Method: Derivation and Examples
120
3.9
Fixed Point Iteration
125
3.10
Special Topics in Root-finding Methods
135
3.10.1
Extrapolation and Acceleration
135
3.10.2
Variants of Newton's method
139
3.10.3
The Secant Method: Theory and Convergence
143
3.10.4
Multiple Roots
147
3.10.5
In Search of Fast Global Convergence: Hybrid Algorithms
151
3.11
Literature and Software Discussion
156
References
159
4
Interpolation and Approximation
161
4.1 Lagrange
Interpolation
161
4.2
Newton Interpolation and Divided Differences
167
4.3
Interpolation Error
176
4.4
Application: Muller's Method and Inverse Quadratic Interpolation
182
4.5
Application: More Approximations to the Derivative
186
4.6
Hermite Interpolation
188
4.7
Piecewise Polynomial Interpolation
192
4.8
An Introduction to Splines
200
4.8.1
Definition of the Problem
201
4.8.2
Cubic B-Splines
201
4.9
Application: Solution of Boundary Value Problems
214
4.10
Least Squares Concepts in Approximation
219
4.10.1
An Introduction to Data Fitting
219
4.10.2
Least Squares Approximation and Orthogonal Polynomials
223
4.11
Advanced Topics in Interpolation Error
237
CONTENTS
VII
4.11.1
Stability of Polynomial
Interpolation 238
4.11.2
The
Runge
Example
241
4.11.3
The Chebyshev Nodes
244
4.12
Literature and Software Discussion
250
References
251
5
Numerical Integration
253
5.1
A Review of the Definite Integral
254
5.2
Improving the
Trapezoid
Rule
256
5.3
Simpson's Rule and Degree of Precision
261
5.4
The Midpoint Rule
272
5.5
Application: Stirling's Formula
276
5.6
Gaussian Quadrature
278
5.7
Extrapolation Methods
290
5.8
Special Topics in Numerical Integration
297
5.8.1
Romberg Integration
297
5.8.2
Quadrature with Non-smooth Integrands
302
5.8.3
Adaptive Integration
307
5.8.4
Peano Estimates for the
Trapezoid
Rule
ЗІЗ
5.9
Literature and Software Discussion
319
References
321
6
Numerical Methods for Ordinary Differential Equations
323
6.1
The Initial Value Problem
—
Background
324
6.2
Euler's Method
329
6.3
Analysis of Euler's Method
333
6.4
Variants of Euler's Method
337
6.4.1
The Residual and Truncation Error
339
6.4.2
Implicit Methods and Predictor-Corrector Schemes
342
6.4.3
Starting Values and Multistep Methods
347
6.4.4
The Midpoint Method and Weak Stability
348
6.5
Single Step Methods—Runge-Kutta
354
6.6
Multi-step Methods
361
6.6.1
The Adams Families
361
6.6.2
The BDF Family
364
6.7
Stability Issues
367
6.7.1
Stability Theory for Multistep Methods
367
6.7.2
Stability Regions
370
6.8
Application to Systems of Equations
373
Vili
CONTENTS
6.8.1 Implementation
Issues
and Examples
373
6.8.2
Stiff Equations
377
6.8.3
A-Stability
378
6.9
Adaptive Solvers
381
6.10
Boundary Value Problems
393
6.10.1
Simple Difference Methods
394
6.10.2
Shooting Methods
399
6.11
Literature and Software Discussion
403
References
405
7
Numerical Methods for the Solution of Systems of Equations
407
7.1
Linear Algebra Review
407
7.2
Linear Systems and Gaussian Elimination
409
7.3
Operation Counts
417
7.4
The
LU
Factorization
419
7.5
Perturbation, Conditioning, and Stability
429
7.5.1
Vector and Matrix Norms
430
7.5.2
The Condition Number and Perturbations
431
7.5.3
Estimating the Condition Number
438
7.5.4
Iterative Refinement
441
7.6 SPD
Matrices and the Cholesky Decomposition
445
7.7
Iterative Methods for Linear Systems
—
A Brief Survey
448
7.8
Nonlinear Systems: Newton's Method and Related Ideas
457
7.8.1
Newton's Method
457
7.8.2
Fixed Point Methods
460
7.9
Application: Numerical Solution of Nonlinear Boundary Value
Problems
462
7.10
Literature and Software Discussion
465
References
467
8
Approximate Solution of the Algebraic Eigenvalue Problem
469
8.1
Eigenvalue Review
469
8.2
Reduction to
Hessenberg Form 476
8.3
Power Methods
483
8.4
An Overview of the
Q R
Iteration
501
8.5
Literature and Software Discussion
510
References
511
CONTENTS
ІХ
9
A
Survey of Finite Difference Methods for
Partial Differential Equations
513
9.1
Difference Methods for the Diffusion Equation
513
9.1.1
The Basic Problem
513
9.1.2
The Explicit Method and Stability
514
9.1.3
Implicit Methods and the Crank-Nicolson Method
519
9.2
Difference Methods for
Poisson
Equations
529
9.2.1
Discretization
529
9.2.2
Banded Cholesky Solvers
532
9.2.3
Iteration and the Method of Conjugate Gradients
535
9.3
Literature and Software Discussion
545
References
547
Appendix A: Proofs of Selected Theorems,
and Other Additional Material
549
A.
1
Proofs of the Interpolation Error Theorems
549
A.2 Proof of the Stability Result for Smooth and Uniformly Monotone
Decreasing Initial Value Problems
551
A.3 Stiff Systems of Differential Equations and Eigenvalues
552
A.4 The Matrix Perturbation Theorem
553
A.5 Answers to Selected Exercises
555
Index
569 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Epperson, James F. |
| author_facet | Epperson, James F. |
| author_role | aut |
| author_sort | Epperson, James F. |
| author_variant | j f e jf jfe |
| building | Verbundindex |
| bvnumber | BV022968865 |
| callnumber-first | Q - Science |
| callnumber-label | QA297 |
| callnumber-raw | QA297 |
| callnumber-search | QA297 |
| callnumber-sort | QA 3297 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 900 SK 905 SK 910 |
| ctrlnum | (OCoLC)85851582 (DE-599)BSZ271781890 |
| dewey-full | 518 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518 |
| dewey-search | 518 |
| dewey-sort | 3518 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | Rev. ed. |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV022968865 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:07:44Z |
| indexdate | 2024-07-09T21:08:49Z |
| institution | BVB |
| isbn | 0470049634 9780470049631 |
| language | English |
| lccn | 2007010267 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016173141 |
| oclc_num | 85851582 |
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| owner | DE-703 DE-20 DE-355 DE-BY-UBR DE-29T DE-19 DE-BY-UBM |
| owner_facet | DE-703 DE-20 DE-355 DE-BY-UBR DE-29T DE-19 DE-BY-UBM |
| physical | XVII, 572 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Wiley-Interscience |
| record_format | marc |
| spelling | Epperson, James F. Verfasser aut An introduction to numerical methods and analysis James F. Epperson Rev. ed. Hoboken, NJ Wiley-Interscience 2007 XVII, 572 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Numerical analysis Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf 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 Numerisches Verfahren (DE-588)4128130-5 s 1\p DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016173141&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Epperson, James F. An introduction to numerical methods and analysis Numerical analysis Numerisches Verfahren (DE-588)4128130-5 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4042805-9 (DE-588)4123623-3 |
| title | An introduction to numerical methods and analysis |
| title_auth | An introduction to numerical methods and analysis |
| title_exact_search | An introduction to numerical methods and analysis |
| title_exact_search_txtP | An introduction to numerical methods and analysis |
| title_full | An introduction to numerical methods and analysis James F. Epperson |
| title_fullStr | An introduction to numerical methods and analysis James F. Epperson |
| title_full_unstemmed | An introduction to numerical methods and analysis James F. Epperson |
| title_short | An introduction to numerical methods and analysis |
| title_sort | an introduction to numerical methods and analysis |
| topic | Numerical analysis Numerisches Verfahren (DE-588)4128130-5 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Numerical analysis Numerisches Verfahren Numerische Mathematik Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016173141&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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