Numerical methods for least squares problems:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Philadelphia
SIAM
1996
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 408 S. graph. Darst. |
| ISBN: | 0898713609 9780898713602 |
Internformat
MARC
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| 100 | 1 | |a Björck, Åke |d 1934- |e Verfasser |0 (DE-588)108090132 |4 aut | |
| 245 | 1 | 0 | |a Numerical methods for least squares problems |c Åke Björck |
| 264 | 1 | |a Philadelphia |b SIAM |c 1996 | |
| 300 | |a XVII, 408 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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| 650 | 7 | |a Numerieke wiskunde |2 gtt | |
| 650 | 7 | |a Vergelijkingen (wiskunde) |2 gtt | |
| 650 | 4 | |a Equations, Simultaneous -- Numerical solutions | |
| 650 | 4 | |a Least squares | |
| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-007218550 | ||
| 883 | 1 | |8 1\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
Datensatz im Suchindex
| _version_ | 1804125289360392192 |
|---|---|
| adam_text | Contents
Preface
xv
1.
Mathematical and Statistical Properties of Least Squares
Solutions
1
1.1
Introduction
............................... 1
1.1.1
Historical remarks
........................ 2
1.1.2
Statistical preliminaries
..................... 2
1.1.3
Linear models and the Gauss-Markoff theorem
........ 3
1.1.4
Characterization of least squares solutions
.......... 5
1.2
The Singular Value Decomposition
.................. 9
1.2.1
The singular value decomposition
............... 9
1.2.2
Related eigenvalue decompositions
............... 11
1.2.3
Matrix approximations
..................... 12
1.2.4
The sensitivity of singular values and vectors
......... 13
1.2.5
The
SVD
and
pseudoinverse
.................. 15
1.2.6
Orthogonal projectors and angles between subspaces
..... 17
1.3
The QR Decomposition
........................ 19
1.3.1
The full rank case
........................ 19
1.3.2
Rank revealing QR decompositions
.............. 21
1.3.3
The complete orthogonal decomposition
............ 23
1.4
Sensitivity of Least Squares Solutions
................ 24
1.4.1
Vector and matrix norms
.................... 24
1.4.2
Perturbation analysis of
pseudoinverses
............ 26
1.4.3
Perturbation analysis of least squares solutions
........ 27
1.4.4
Asymptotic forms and derivatives
............... 32
1.4.5
Componentwise perturbation analysis
............. 32
1.4.6
A posteriori estimation of errors
................ 34
2.
Basic Numerical Methods
37
2.1
Basics of Floating Point Computation
................ 37
2.1.1
Rounding error analysis
..................... 37
2.1.2
Running rounding error analysis
................ 39
vii
x
Contents
4.4.4
Weighted problems by updating
................171
4.5
Minimizing the lp Norm
........:...............172
4.5.1
Introduction
........................... 172
4.5.2
Iteratively reweighted least squares
..............173
4.5.3
Robust linear regression
.....................175
4.5.4
Algorithms for l and l^ approximation
............175
4.6
Total Least Squares
..........................176
4.6.1
Errors-in-variables models.
..................176
4.6.2
Total least squares problem by
SVD
..............177
4.6.3
Relationship to the least squares solution
...........180
4.6.4
Multiple right-hand sides
....................181
4.6.5
Generalized TLS problems
...................182
4.6.6
Linear orthogonal distance regression
.............184
5.
Constrained Least Squares Problems
187
5.1
Linear Equality Constraints
......................187
5.1.1
Introduction
...........................187
5.1.2
Method of direct elimination
..................188
5.1.3
The nullspace method
......................189
5.1.4
Problem
LSE
by generalized
SVD
...............191
5.1.5
The method of weighting
.................... 192
5.1.6
Solving
LSE
problems by updating
..............194
5.2
Linear Inequality Constraints
.....................194
5.2.1
Classification of problems
....................194
5.2.2
Basic transformations of problem LSI
.............196
5.2.3
Active set algorithms for problem LSI
............. 198
5.2.4
Active set algorithms for BLS
................. 201
5.3
Quadratic Constraints
.........................203
5.3.1
Ill-posed problems
........................203
5.3.2
Quadratic inequality constraints
................205
5.3.3
Problem LSQI by GSVD
....................206
5.3.4
Problem LSQI by QR decomposition
.............208
5.3.5
Cross-validation
.........................211
6.
Direct Methods for Sparse Problems
215
6.1
Introduction
...............................215
6.2
Banded Least Squares Problems
...................217
6.2.1
Storage schemes for banded matrices.
............218
6.2.2
Normal equations for banded problems
............219
6.2.3
Givens QR
decomposition for banded problems
........221
6.2.4
Householder QR decomposition for banded problems.
. . . 222
6.3
Block Angular Least Squares Problems
................224
6.3.1
Block angular form
.......................224
6.3.2
QR methods for block angular problems
........... 225
6.4
Tools for General Sparse Problems
..................227
Contents xi
6.4.1
Storage schemes for general sparse matrices
..........227
6.4.2
Graph representation of sparse matrices
............230
6.4.3
Predicting the structure of ATA
............... . 231
6.4.4
Predicting the structure of
Л
..................232
6.4.5
Block triangular form of a sparse matrix
...........234
6.5
Fill Minimizing Column
Orderings
...................237
6.5.1
Bandwidth reducing ordering methods
............237
6.5.2
Minimum degree ordering
....................238
6.5.3
Nested dissection
orderings
...................240
6.6
The Numerical Cholesky and QR Decompositions
..........242
6.6.1
The Cholesky factorization
...................242
6.6.2
Row sequential QR decomposition
...............242
6.6.3
Row orderings for sparse QR decomposition
.........244
6.6.4 Multifrontal QR
decomposition
................245
6.6.5
Iterative refinement and
seminormal
equations
........250
6.7
Special Topics
..............................252
6.7.1
Rank revealing sparse QR decomposition
...........252
6.7.2
Updating sparse least squares solutions
............254
6.7.3
Partitioning for out-of-core solution
..............255
6.7.4
Computing selected elements of the covariance matrix.
. . . 256
6.8
Sparse Constrained Problems
.....................257
6.8.1
An active set method for problem BLS
............257
6.8.2
Interior point methods for problem BLS
............262
6.9
Software and Test Results
.......................264
6.9.1
Software for sparse direct methods.
..............264
6.9.2
Test results
............................266
7.
Iterative Methods For Least Squares Problems
269
7.1
Introduction
...............................269
7.1.1
Iterative versus direct methods
.................270
7.1.2
Computing sparse matrix-vector products
...........270
7.2
Basic Iterative Methods
........................274
7.2.1
General stationary iterative methods
.............274
7.2.2
Splittings of rectangular matrices
...............276
7.2.3
Classical iterative methods
...................276
7.2.4
Successive overrelaxation methods
...............279
7.2.5
Semi-iterative methods
.....................280
7.2.6
Preconditioning
.........................283
7.3
Block Iterative Methods
........................284
7.3.1
Block column preconditioners
.................284
7.3.2
The two-block case
.......................286
7.4
Conjugate Gradient Methods
.....................288
7.4.1
CGLS and variants
.......................288
7.4.2
Convergence properties of CGLS
................290
xii
Contents
7.4.3
The conjugate gradient method in finite precision
......292
7.4.4
Preconditioned CGLS
......................293
7.5
Incomplete Factorization Preconditioners
..............294
7.5.1
Incomplete Cholesky preconditioners
.............294
7.5.2
Incomplete orthogonal decompositions
.............297
7.5.3
Preconditioners based on
LU
factorization
..........299
7.6
Methods Based on Lanczos Bidiagonalization
............303
7.6.1
Lanczos bidiagonalization
....................303
7.6.2
Best approximation in the Krylov subspace
..........306
7.6.3
The LSQR algorithm
......................307
7.6.4
Convergence of singular values and vectors
..........309
7.6.5
Bidiagonalization and total least squares
...........310
7.7
Methods for Constrained Problems
..................312
7.7.1
Problems with upper and lower bounds
............312
7.7.2
Iterative regularization
.....................314
8.
Least Squares Problems with Special Bases
317
8.1
Least Squares Approximation and Orthogonal Systems
.......317
8.1.1
General formalism
........................317
8.1.2
Statistical aspects of the method of least squares
.......318
8.2
Polynomial Approximation
......................319
8.2.1
Triangle family of polynomials
.................319
8.2.2
General theory of orthogonal polynomials
...........320
8.2.3
Discrete least squares fitting
..................321
8.2.4
Vandermonde-like systems
...................323
8.2.5
Chebyshev polynomials
.....................325
8.3
Discrete Fourier Analysis
.......................328
8.3.1
Introduction
...........................328
8.3.2
Orthogonality relations
.....................329
8.3.3
The fast Fourier transform
...................330
8.4
Toeplitz Least Squares Problems
...................332
8.4.1
Introduction
........................... 332
8.4.2
QR decomposition of Toeplitz matrices
............333
8.4.3
Iterative solvers for Toeplitz systems.
............334
8.4.4
Preconditioners for Toeplitz systems
..............335
8.5 Kronecker
Product Problems
.....................336
9.
Nonlinear Least Squares Problems
339
9.1
The Nonlinear Least Squares Problem
................ 339
9.1.1
Introduction
...........................339
9.1.2
Necessary conditions for local minima
.............340
9.1.3
Basic numerical methods
.................... 341
9.2
Gauss-Newton-Type Methods
.....................342
9.2.1
The damped Gauss-Newton method
..............343
9.2.2
Local convergence of the Gauss-Newton method.
..... .345
Contents
xiii
9.2.3
Trust region methods
......................346
9.3 Newton-Type
Methods
.........................348
9.3.1
Introduction
...........................348
9.3.2
A hybrid Newton method
....................348
9.3.3
Quasi-Newton
methods
.....................349
9.4
Separable and Constrained Problems
.................351
9.4.1
Separable problems
.......................351
9.4.2
General constrained problems
.................353
9.4.3
Orthogonal distance regression
.................354
9.4.4
Least squares fit of geometric elements
............357
Bibliography
359
Index
401
|
| any_adam_object | 1 |
| author | Björck, Åke 1934- |
| author_GND | (DE-588)108090132 |
| author_facet | Björck, Åke 1934- |
| author_role | aut |
| author_sort | Björck, Åke 1934- |
| author_variant | å b åb |
| building | Verbundindex |
| bvnumber | BV010805943 |
| callnumber-first | Q - Science |
| callnumber-label | QA214 |
| callnumber-raw | QA214.B56 1996 |
| callnumber-search | QA214.B56 1996 |
| callnumber-sort | QA 3214 B56 41996 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 910 SK 915 |
| classification_tum | MAT 625f |
| ctrlnum | (OCoLC)34190229 (DE-599)BVBBV010805943 |
| dewey-full | 512.9/4220 512.9/42 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.9/42 20 512.9/42 |
| dewey-search | 512.9/42 20 512.9/42 |
| dewey-sort | 3512.9 242 220 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV010805943 |
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| indexdate | 2024-07-09T17:59:12Z |
| institution | BVB |
| isbn | 0898713609 9780898713602 |
| language | English |
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| physical | XVII, 408 S. graph. Darst. |
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| publisher | SIAM |
| record_format | marc |
| spelling | Björck, Åke 1934- Verfasser (DE-588)108090132 aut Numerical methods for least squares problems Åke Björck Philadelphia SIAM 1996 XVII, 408 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Moindres carrés ram Numerieke wiskunde gtt Vergelijkingen (wiskunde) gtt Equations, Simultaneous -- Numerical solutions Least squares Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Gleichungssystem (DE-588)4128766-6 gnd rswk-swf Methode der kleinsten Quadrate (DE-588)4038974-1 gnd rswk-swf Methode der kleinsten Quadrate (DE-588)4038974-1 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Gleichungssystem (DE-588)4128766-6 s 1\p DE-604 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007218550&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 | Björck, Åke 1934- Numerical methods for least squares problems Moindres carrés ram Numerieke wiskunde gtt Vergelijkingen (wiskunde) gtt Equations, Simultaneous -- Numerical solutions Least squares Numerisches Verfahren (DE-588)4128130-5 gnd Gleichungssystem (DE-588)4128766-6 gnd Methode der kleinsten Quadrate (DE-588)4038974-1 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4128766-6 (DE-588)4038974-1 |
| title | Numerical methods for least squares problems |
| title_auth | Numerical methods for least squares problems |
| title_exact_search | Numerical methods for least squares problems |
| title_full | Numerical methods for least squares problems Åke Björck |
| title_fullStr | Numerical methods for least squares problems Åke Björck |
| title_full_unstemmed | Numerical methods for least squares problems Åke Björck |
| title_short | Numerical methods for least squares problems |
| title_sort | numerical methods for least squares problems |
| topic | Moindres carrés ram Numerieke wiskunde gtt Vergelijkingen (wiskunde) gtt Equations, Simultaneous -- Numerical solutions Least squares Numerisches Verfahren (DE-588)4128130-5 gnd Gleichungssystem (DE-588)4128766-6 gnd Methode der kleinsten Quadrate (DE-588)4038974-1 gnd |
| topic_facet | Moindres carrés Numerieke wiskunde Vergelijkingen (wiskunde) Equations, Simultaneous -- Numerical solutions Least squares Numerisches Verfahren Gleichungssystem Methode der kleinsten Quadrate |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007218550&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT bjorckake numericalmethodsforleastsquaresproblems |