An introduction to numerical analysis:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge Univ. Press
2007
|
| Ausgabe: | reprint. with corr. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | X, 433 S. graph. Darst. |
| ISBN: | 9780521007948 9780521810265 |
Internformat
MARC
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| 100 | 1 | |a Süli, Endre |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a An introduction to numerical analysis |c Endre Süli and David F. Mayers |
| 250 | |a reprint. with corr. | ||
| 264 | 1 | |a Cambridge |b Cambridge Univ. Press |c 2007 | |
| 300 | |a X, 433 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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Datensatz im Suchindex
| _version_ | 1804137922671149056 |
|---|---|
| adam_text | Contents
ťrej
асе
page
vii
1
Solution
of equations by iteration
1
1.1
Introduction
1
1.2
Simple iteration
2
1.3
Iterative solution of equations
17
1.4
Relaxation and Newton s method
19
1.5
The secant method
25
1.6
The bisection method
28
1.7
Global behaviour
29
1.8
Notes
32
Exercises
35
2
Solution of systems of linear equations
39
2.1
Introduction
39
2.2
Gaussian elimination
44
2.3
LU
factorisation
48
2.4
Pivoting
52
2.5
Solution of systems of equations
55
2.6
Computational work
56
2.7
Norms and condition numbers
58
2,8
Hilbert matrix
72
2.9
Least squares method
74
2.10
Notes
79
Exercises
82
3
Special matrices
87
3.1
Introduction
87
3.2
Symmetric positive definite matrices
87
3.3
Tridiagonal and band matrices
93
Ш
Contents
3.4
Monotone
matrices
98
3.5
Notes
101
Exercises
102
4
Simultaneous nonlinear equations
104
4.1
Introduction
104
4.2
Simultaneous iteration
106
4.3
Relaxation and Newton s method
116
4.4
Global convergence
123
4.5
Notes
124
Exercises
126
5
Eigenvalues and eigenvectors of a symmetric matrix
133
5.1
Introduction
133
5.2
The characteristic polynomial
137
5.3
Jacobi s method
137
5.4
The Gerschgorin theorems
145
5.5
Householder s method
150
5.6
Eigenvalues of a tridiagonal matrix
156
5.7
The QR algorithm
162
5.7.1
The QR factorisation revisited
162
5.7.2
The definition of the QR algorithm
164
5.8
Inverse iteration for the eigenvectors
166
5.9
The Rayleigh quotient
170
5.10
Perturbation analysis
172
5.11
Notes
174
Exercises
175
6
Polynomial interpolation
179
6.1
Introduction
179
6.2
Lagrange
interpolation
180
6.3
Convergence
185
6.4
Hermite interpolation
187
6.5
Differentiation
191
6.6
Notes
194
Exercises
195
7
Numerical integration
—
I
200
7.1
Introduction
200
7.2
Newton Cotes formulae
201
7.3
Error estimates
204
7.4
The
Runge
phenomenon revisited
208
7.5
Composite formulae
209
Contents
7.6
The Euler-Maclaurin expansion
211
7.7
Extrapolation methods
215
7.8
Notes
219
Exercises
220
8
Polynomial approximation in the oo-norm
224
8.1
Introduction
224
8.2
Normed linear spaces
224
8.3
Best approximation in the oo-norm
228
8.4
Chebyshev polynomials
241
8.5
Interpolation
244
8.6
Notes
247
Exercises
248
9
Polynomial approximation in the 2-norm
252
9.1
Introduction
252
9.2
Inner product spaces
253
9.3
Best approximation in the 2-norm
256
9.4
Orthogonal polynomials
259
9.5
Comparisons
270
9.6
Notes
272
Exercises
273
10
Numerical integration
—
II
277
10.1
Introduction
277
10.2
Construction of Gauss quadrature rules
277
10.3
Direct construction
280
10.4
Error estimation for Gauss quadrature
282
10.5
Composite Gauss formulae
285
10.6
Radau and Lobatto quadrature
287
10.7
Note
288
Exercises
288
11
Piecewise polynomial approximation
292
11.1
Introduction
292
11.2
Linear interpolating splines
293
11.3
Basis functions for the linear spline
297
11.4
Cubic splines
298
11.5
Hermite cubic splines
300
11.6
Basis functions for cubic splines
302
11.7
Notes
306
Exercises
307
vi
Contents
12 Initial
value problems for ODEs
310
12.1
Introduction
310
12.2
One-step methods
317
12.3
Consistency and convergence
321
12.4
An implicit one-step method
324
12.5
Runge-Kutta methods
325
12.6
Linear multistep methods
329
12.7
Zero-stability
331
12.8
Consistency
337
12.9
Dahlquisťs
theorems
340
12.10
Systems of equations
341
12.11
Stiff systems
343
12.12
Implicit Runge-Kutta methods
349
12.13
Notes
353
Exercises
355
13
Boundary value problems for ODEs
361
13.1
Introduction
361
13.2
A model problem
361
13.3
Error analysis
364
13.4
Boundary conditions involving a derivative
367
13.5
The general second-order self-adjoint problem
370
13.6
The Sturm-Liouville eigenvalue problem
373
13.7
The shooting method
375
13.8
Notes
380
Exercises
381
14
The finite element method
385
14.1
Introduction: the model problem
385
14.2
Rayleigh-Ritz and
Galer
kin principles
388
14.3
Formulation of the finite element method
391
14.4
Error analysis of the finite element method
397
14.5
A posteriori error analysis by duality
403
14.6
Notes
412
Exercises
414
Appendix A An overview of results from real analysis
419
Appendix
B WWW-resources
423
Bibliography
424
Index
429
|
| adam_txt |
Contents
ťrej
асе
page
vii
1
Solution
of equations by iteration
1
1.1
Introduction
1
1.2
Simple iteration
2
1.3
Iterative solution of equations
17
1.4
Relaxation and Newton's method
19
1.5
The secant method
25
1.6
The bisection method
28
1.7
Global behaviour
29
1.8
Notes
32
Exercises
35
2
Solution of systems of linear equations
39
2.1
Introduction
39
2.2
Gaussian elimination
44
2.3
LU
factorisation
48
2.4
Pivoting
52
2.5
Solution of systems of equations
55
2.6
Computational work
56
2.7
Norms and condition numbers
58
2,8
Hilbert matrix
72
2.9
Least squares method
74
2.10
Notes
79
Exercises
82
3
Special matrices
87
3.1
Introduction
87
3.2
Symmetric positive definite matrices
87
3.3
Tridiagonal and band matrices
93
Ш
Contents
3.4
Monotone
matrices
98
3.5
Notes
101
Exercises
102
4
Simultaneous nonlinear equations
104
4.1
Introduction
104
4.2
Simultaneous iteration
106
4.3
Relaxation and Newton's method
116
4.4
Global convergence
123
4.5
Notes
124
Exercises
126
5
Eigenvalues and eigenvectors of a symmetric matrix
133
5.1
Introduction
133
5.2
The characteristic polynomial
137
5.3
Jacobi's method
137
5.4
The Gerschgorin theorems
145
5.5
Householder's method
150
5.6
Eigenvalues of a tridiagonal matrix
156
5.7
The QR algorithm
162
5.7.1
The QR factorisation revisited
162
5.7.2
The definition of the QR algorithm
164
5.8
Inverse iteration for the eigenvectors
166
5.9
The Rayleigh quotient
170
5.10
Perturbation analysis
172
5.11
Notes
174
Exercises
175
6
Polynomial interpolation
179
6.1
Introduction
179
6.2
Lagrange
interpolation
180
6.3
Convergence
185
6.4
Hermite interpolation
187
6.5
Differentiation
191
6.6
Notes
194
Exercises
195
7
Numerical integration
—
I
200
7.1
Introduction
200
7.2
Newton Cotes formulae
201
7.3
Error estimates
204
7.4
The
Runge
phenomenon revisited
208
7.5
Composite formulae
209
Contents
7.6
The Euler-Maclaurin expansion
211
7.7
Extrapolation methods
215
7.8
Notes
219
Exercises
220
8
Polynomial approximation in the oo-norm
224
8.1
Introduction
224
8.2
Normed linear spaces
224
8.3
Best approximation in the oo-norm
228
8.4
Chebyshev polynomials
241
8.5
Interpolation
244
8.6
Notes
247
Exercises
248
9
Polynomial approximation in the 2-norm
252
9.1
Introduction
252
9.2
Inner product spaces
253
9.3
Best approximation in the 2-norm
256
9.4
Orthogonal polynomials
259
9.5
Comparisons
270
9.6
Notes
272
Exercises
273
10
Numerical integration
—
II
277
10.1
Introduction
277
10.2
Construction of Gauss quadrature rules
277
10.3
Direct construction
280
10.4
Error estimation for Gauss quadrature
282
10.5
Composite Gauss formulae
285
10.6
Radau and Lobatto quadrature
287
10.7
Note
288
Exercises
288
11
Piecewise polynomial approximation
292
11.1
Introduction
292
11.2
Linear interpolating splines
293
11.3
Basis functions for the linear spline
297
11.4
Cubic splines
298
11.5
Hermite cubic splines
300
11.6
Basis functions for cubic splines
302
11.7
Notes
306
Exercises
307
vi
Contents
12 Initial
value problems for ODEs
310
12.1
Introduction
310
12.2
One-step methods
317
12.3
Consistency and convergence
321
12.4
An implicit one-step method
324
12.5
Runge-Kutta methods
325
12.6
Linear multistep methods
329
12.7
Zero-stability
331
12.8
Consistency
337
12.9
Dahlquisťs
theorems
340
12.10
Systems of equations
341
12.11
Stiff systems
343
12.12
Implicit Runge-Kutta methods
349
12.13
Notes
353
Exercises
355
13
Boundary value problems for ODEs
361
13.1
Introduction
361
13.2
A model problem
361
13.3
Error analysis
364
13.4
Boundary conditions involving a derivative
367
13.5
The general second-order self-adjoint problem
370
13.6
The Sturm-Liouville eigenvalue problem
373
13.7
The shooting method
375
13.8
Notes
380
Exercises
381
14
The finite element method
385
14.1
Introduction: the model problem
385
14.2
Rayleigh-Ritz and
Galer
kin principles
388
14.3
Formulation of the finite element method
391
14.4
Error analysis of the finite element method
397
14.5
A posteriori error analysis by duality
403
14.6
Notes
412
Exercises
414
Appendix A An overview of results from real analysis
419
Appendix
B WWW-resources
423
Bibliography
424
Index
429 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T21:41:47Z |
| indexdate | 2024-07-09T21:20:00Z |
| institution | BVB |
| isbn | 9780521007948 9780521810265 |
| language | English |
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| spelling | Süli, Endre Verfasser aut An introduction to numerical analysis Endre Süli and David F. Mayers reprint. with corr. Cambridge Cambridge Univ. Press 2007 X, 433 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 s DE-604 Mayers, David F. Verfasser aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016674165&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Süli, Endre Mayers, David F. An introduction to numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4042805-9 |
| title | An introduction to numerical analysis |
| title_auth | An introduction to numerical analysis |
| title_exact_search | An introduction to numerical analysis |
| title_exact_search_txtP | An introduction to numerical analysis |
| title_full | An introduction to numerical analysis Endre Süli and David F. Mayers |
| title_fullStr | An introduction to numerical analysis Endre Süli and David F. Mayers |
| title_full_unstemmed | An introduction to numerical analysis Endre Süli and David F. Mayers |
| title_short | An introduction to numerical analysis |
| title_sort | an introduction to numerical analysis |
| topic | Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Numerische Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016674165&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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