Matrix algorithms: 1 Basic decompositions
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Philadelphia, Pa.
SIAM
1998
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIX, 458 S. |
| ISBN: | 0898714141 |
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MARC
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| 245 | 1 | 0 | |a Matrix algorithms |n 1 |p Basic decompositions |c G. W. Stewart |
| 264 | 1 | |a Philadelphia, Pa. |b SIAM |c 1998 | |
| 300 | |a XIX, 458 S. | ||
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| 650 | 7 | |a Algorithmes |2 ram | |
| 650 | 7 | |a Matrices |2 ram | |
| 650 | 7 | |a algèbre |2 inriac | |
| 650 | 7 | |a calcul matriciel |2 inriac | |
| 650 | 7 | |a décomposition QR |2 inriac | |
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| 650 | 7 | |a méthode moindre carré |2 inriac | |
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Datensatz im Suchindex
| _version_ | 1804127071782305792 |
|---|---|
| adam_text | Contents
Algorithms xiii
Notation xv
Preface xvii
1 Matrices, Algebra, and Analysis 1
1 Vectors 2
1.1 Scalars 2
Real and complex numbers. Sets and Minkowski sums.
1.2 Vectors 3
1.3 Operations with vectors and scalars 5
1.4 Notes and references 7
Representing vectors and scalars. The scalar product. Function
spaces.
2 Matrices 7
2.1 Matrices 8
2.2 Some special matrices 9
Familiar characters. Patterned matrices.
2.3 Operations with matrices 13
The scalar matrix product and the matrix sum. The matrix
product. The transpose and symmetry. The trace and the
determinant.
2.4 Submatrices and partitioning 17
Submatrices. Partitions. Northwest indexing. Partitioning and
matrix operations. Block forms.
2.5 Some elementary constructions 21
Inner products. Outer products. Linear combinations. Column
and row scaling. Permuting rows and columns. Undoing a
permutation. Crossing a matrix. Extracting and inserting
submatrices.
2.6 LU decompositions 23
2.7 Homogeneous equations 25
v
vi Contents
2.8 Notes and references 26
Indexing conventions. Hyphens and other considerations.
Nomenclature for triangular matrices. Complex symmetric
matrices. Determinants. Partitioned matrices. The
LU decomposition.
3 Linear Algebra 28
3.1 Subspaces, linear independence, and bases 28
Subspaces. Linear independence. Bases. Dimension.
3.2 Rank and nullity 33
A full rank factorization. Rank and nullity.
3.3 Nonsingularity and inverses 36
Linear systems and nonsingularity. Nonsingularity and inverses.
3.4 Change of bases and linear transformations 39
Change of basis. Linear transformations and matrices.
3.5 Notes and references 42
Linear algebra. Full rank factorizations.
4 Analysis 42
4.1 Norms 42
Componentwise inequalities and absolute values. Vector norms.
Norms and convergence. Matrix norms and consistency. Operator
norms. Absolute norms. Perturbations of the identity. The
Neumann series.
4.2 Orthogonality and projections 55
Orthogonality. The QR factorization and orthonormal bases.
Orthogonal projections.
4.3 The singular value decomposition 61
Existence. Uniqueness. Unitary equivalence. Weyl s theorem and
the min max characterization. The perturbation of singular
values. Low rank approximations.
4.4 The spectral decomposition 70
4.5 Canonical angles and the CS decomposition 73
Canonical angles between subspaces. The CS decomposition.
4.6 Notes and references 75
Vector and matrix norms. Inverses and the Neumann series. The
QR factorization. Projections. The singular value decomposition.
The spectral decomposition. Canonical angles and the
CS decomposition.
5 Addenda 77
5.1 Historical 77
On the word matrix. History.
5.2 General references 78
Linear algebra and matrix theory. Classics of matrix
Contents vii
computations. Textbooks. Special topics. Software. Historical
sources.
2 Matrices and Machines 81
1 Pseudocode 82
1.1 Generalities 82
1.2 Control statements 83
The if statement. The for statement. The while statement.
Leaving and iterating control statements. The goto statement.
1.3 Functions 85
1.4 Notes and references 86
Programming languages. Pseudocode.
2 Triangular Systems 87
2.1 The solution of a lower triangular system 87
Existence of solutions. The forward substitution algorithm.
Overwriting the right hand side.
2.2 Recursive derivation 89
2.3 A new algorithm 90
2.4 The transposed system 92
2.5 Bidiagonal matrices 92
2.6 Inversion of triangular matrices 93
2.7 Operation counts 94
Bidiagonal systems. Full triangular systems. General
observations on operations counts. Inversion of a triangular
matrix. More observations on operation counts.
2.8 BLAS for triangular systems 99
2.9 Notes and references 100
Historical. Recursion. Operation counts. Basic linear algebra
subprograms (BLAS).
3 Matrices in Memory 101
3.1 Memory, arrays, and matrices 102
Memory. Storage of arrays. Strides.
3.2 Matrices in memory 104
Array references in matrix computations. Optimization and the
BLAS. Economizing memory—Packed storage.
3.3 Hierarchical memories 109
Virtual memory and locality of reference. Cache memory. A
model algorithm. Row and column orientation. Level two BLAS.
Keeping data in registers. Blocking and the level three BLAS.
3.4 Notes and references 119
The storage of arrays. Strides and interleaved memory. The
BLAS. Virtual memory. Cache memory. Large memories and
matrix problems. Blocking.
viii Contents
4 Rounding Error 121
4.1 Absolute and relative error 121
Absolute error. Relative error.
4.2 Floating point numbers and arithmetic 124
Floating point numbers. The IEEE standard. Rounding error.
Floating point arithmetic.
4.3 Computing a sum: Stability and condition 129
A backward error analysis. Backward stability. Weak stability.
Condition numbers. Reenter rounding error.
4.4 Cancellation 136
4.5 Exponent exceptions 138
Overflow. Avoiding overflows. Exceptions in the IEEE standard.
4.6 Notes and references 141
General references. Relative error and precision. Nomenclature
for floating point numbers. The rounding unit. Nonstandard
floating point arithmetic. Backward rounding error analysis.
Stability. Condition numbers. Cancellation. Exponent exceptions.
3 Gaussian Elimination 147
1 Gaussian Elimination 148
1.1 Four faces of Gaussian elimination 148
Gauss s elimination. Gaussian elimination and elementary row
operations. Gaussian elimination as a transformation to triangular
form. Gaussian elimination and the LU decomposition.
1.2 Classical Gaussian elimination 153
The algorithm. Analysis of classical Gaussian elimination. LU
decompositions. Block elimination. Schur complements.
1.3 Pivoting 165
Gaussian elimination with pivoting. Generalities on pivoting.
Gaussian elimination with partial pivoting.
1.4 Variations on Gaussian elimination 169
Sherman s march. Pickett s charge. Crout s method. Advantages
over classical Gaussian elimination.
1.5 Linear systems, determinants, and inverses 174
Solution of linear systems. Determinants. Matrix inversion.
1.6 Notes and references 180
Decompositions and matrix computations. Classical Gaussian
elimination. Elementary matrix. The LU decomposition. Block
LU decompositions and Schur complements. Block algorithms
and blocked algorithms. Pivoting. Exotic orders of elimination.
Gaussian elimination and its variants. Matrix inversion.
Augmented matrices. Gauss Jordan elimination.
2 A Most Versatile Algorithm 185
Contents ix
2.1 Positive definite matrices 185
Positive definite matrices. The Cholesky decomposition. The
Cholesky algorithm.
2.2 Symmetric indefinite matrices 190
2.3 Hessenberg and tridiagonal matrices 194
Structure and elimination. Hessenberg matrices. Tridiagonal
matrices.
2.4 Band matrices 202
2.5 Notes and references 207
Positive definite matrices. Symmetric indefinite systems. Band
matrices.
3 The Sensitivity of Linear Systems 208
3.1 Normwise bounds 209
The basic perturbation theorem. Normwise relative error and the
condition number. Perturbations of the right hand side. Artificial
ill conditioning.
3.2 Componentwise bounds 217
3.3 Backward perturbation theory 219
Normwise backward error bounds. Componentwise backward
error bounds.
3.4 Iterative refinement 221
3.5 Notes and references 224
General references. Normwise perturbation bounds. Artificial
ill conditioning. Componentwise bounds. Backward perturbation
theory. Iterative refinement.
4 The Effects of Rounding Error 225
4.1 Error analysis of triangular systems 226
The results of the error analysis.
4.2 The accuracy of the computed solutions 227
The residual vector.
4.3 Error analysis of Gaussian elimination 229
The error analysis. The condition of the triangular factors. The
solution of linear systems. Matrix inversion.
4.4 Pivoting and scaling 235
On scaling and growth factors. Partial and complete pivoting.
Matrices that do not require pivoting. Scaling.
4.5 Iterative refinement 242
A general analysis. Double precision computation of the residual.
Single precision computation of the residual. Assessment of
iterative refinement.
4.6 Notes and references 245
General references. Historical. The error analyses. Condition of
x Contents
the L and U factors. Inverses. Growth factors. Scaling. Iterative
refinement.
4 The QR Decomposition and Least Squares 249
1 The QR Decomposition 250
1.1 Basics 250
Existence and uniqueness. Projections and the pseudoinverse.
The partitioned factorization. Relation to the singular value
decomposition.
1.2 Householder triangularization 254
Householder transformations. Householder triangularization.
Computation of projections. Numerical stability. Graded
matrices. Blocked reduction.
1.3 Triangularization by plane rotations 270
Plane rotations. Reduction of a Hessenberg matrix. Numerical
properties.
1.4 The Gram Schmidt algorithm 277
The classical and modified Gram Schmidt algorithms. Modified
Gram Schmidt and Householder triangularization. Error analysis
of the modified Gram Schmidt algorithm. Loss of orthogonality.
Reorthogonalization.
1.5 Notes and references 288
General references. The QR decomposition. The pseudoinverse.
Householder triangularization. Rounding error analysis. Blocked
reduction. Plane rotations. Storing rotations. Fast rotations. The
Gram Schmidt algorithm. Reorthogonalization.
2 Linear Least Squares 292
2.1 The QR approach 293
Least squares via the QR decomposition. Least squares via the
QR factorization. Least squares via the modified Gram Schmidt
algorithm.
2.2 The normal and seminormal equations 298
The normal equations. Forming cross product matrices. The
augmented cross product matrix. The instability of cross product
matrices. The seminormal equations.
2.3 Perturbation theory and its consequences 305
The effects of rounding error. Perturbation of the normal
equations. The perturbation of pseudoinverses. The perturbation
of least squares solutions. Accuracy of computed solutions.
Comparisons.
2.4 Least squares with linear constraints 312
The null space method. The method of elimination. The
weighting method.
Contents xi
2.5 Iterative refinement 320
2.6 Notes and references 323
Historical. The QR approach. Gram Schmidt and least squares.
The augmented least squares matrix. The normal equations. The
seminormal equations. Rounding error analyses. Perturbation
analysis. Constrained least squares. Iterative refinement.
3 Updating 326
3.1 Updating inverses 327
Woodbury s formula. The sweep operator.
3.2 Moving columns 333
A general approach. Interchanging columns.
3.3 Removing a column 337
3.4 Appending columns 338
Appending a column to a QR decomposition. Appending a
column to a QR factorization.
3.5 Appending a row 339
3.6 Removing a row 341
Removing a row from a QR decomposition. Removing a row
from a QR factorization. Removing a row from an R factor
(Cholesky downdating). Downdating a vector.
3.7 General rank one updates 348
Updating a factorization. Updating a decomposition.
3.8 Numerical properties 350
Updating. Downdating.
3.9 Notes and references 353
Historical. Updating inverses. Updating. Exponential
windowing. Cholesky downdating. Downdating a vector.
5 Rank Reducing Decompositions 357
1 Fundamental Subspaces and Rank Estimation 358
1.1 The perturbation of fundamental subspaces 358
Superior and inferior singular subspaces. Approximation of
fundamental subspaces.
1.2 Rank estimation 363
1.3 Notes and references 365
Rank reduction and determination. Singular subspaces. Rank
determination. Error models and scaling.
2 Pivoted Orthogonal Triangularization 367
2.1 The pivoted QR decomposition 368
Pivoted orthogonal triangularization. Bases for the fundamental
subspaces. Pivoted QR as a gap revealing decomposition.
Assessment of pivoted QR.
2.2 The pivoted Cholesky decomposition 375
xii Contents
2.3 The pivoted QLP decomposition 378
The pivoted QLP decomposition. Computing the pivoted
QLP decomposition. Tracking properties of the
QLP decomposition. Fundamental subspaces. The matrix Q and
the columns of X. Low rank approximations.
2.4 Notes and references 385
Pivoted orthogonal triangularization. The pivoted Cholesky
decomposition. Column pivoting, rank, and singular values.
Rank revealing QR decompositions. The QLP decomposition.
3 Norm and Condition Estimation 387
3.1 A 1 norm estimator 388
3.2 LINPACK style norm and condition estimators 391
A simple estimator. An enhanced estimator. Condition estimation.
3.3 A 2 norm estimator 397
3.4 Notes and references 399
General. LINPACK style condition estimators. The 1 norm
estimator. The 2 norm estimator.
4 UTV decompositions 400
4.1 Rotations and errors 401
4.2 Updating URV decompositions 402
URV decompositions. Incorporation. Adjusting the gap.
Deflation. The URV updating algorithm. Refinement. Low rank
splitting.
4.3 Updating ULV decompositions 412
ULV decompositions. Updating a ULV decomposition.
4.4 Notes and references 416
UTV decompositions.
References 417
Index 441
|
| any_adam_object | 1 |
| author | Stewart, Gilbert W. 1940- |
| author_GND | (DE-588)135799422 |
| author_facet | Stewart, Gilbert W. 1940- |
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| dewey-full | 519.2 511.8 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics 511 - General principles of mathematics |
| dewey-raw | 519.2 511.8 |
| dewey-search | 519.2 511.8 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| id | DE-604.BV012435945 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:27:31Z |
| institution | BVB |
| isbn | 0898714141 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008437272 |
| oclc_num | 495546960 |
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| physical | XIX, 458 S. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | SIAM |
| record_format | marc |
| spelling | Stewart, Gilbert W. 1940- Verfasser (DE-588)135799422 aut Matrix algorithms 1 Basic decompositions G. W. Stewart Philadelphia, Pa. SIAM 1998 XIX, 458 S. txt rdacontent n rdamedia nc rdacarrier Algorithmes ram Matrices ram algèbre inriac calcul matriciel inriac décomposition QR inriac matrice inriac méthode moindre carré inriac élimination Gauss inriac Matrix Mathematik (DE-588)4037968-1 gnd rswk-swf Matrizenrechnung (DE-588)4126963-9 gnd rswk-swf Algorithmus (DE-588)4001183-5 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Matrix Mathematik (DE-588)4037968-1 s Algorithmus (DE-588)4001183-5 s DE-604 Matrizenrechnung (DE-588)4126963-9 s Numerisches Verfahren (DE-588)4128130-5 s (DE-604)BV012435940 1 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008437272&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Stewart, Gilbert W. 1940- Matrix algorithms Algorithmes ram Matrices ram algèbre inriac calcul matriciel inriac décomposition QR inriac matrice inriac méthode moindre carré inriac élimination Gauss inriac Matrix Mathematik (DE-588)4037968-1 gnd Matrizenrechnung (DE-588)4126963-9 gnd Algorithmus (DE-588)4001183-5 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4037968-1 (DE-588)4126963-9 (DE-588)4001183-5 (DE-588)4128130-5 |
| title | Matrix algorithms |
| title_auth | Matrix algorithms |
| title_exact_search | Matrix algorithms |
| title_full | Matrix algorithms 1 Basic decompositions G. W. Stewart |
| title_fullStr | Matrix algorithms 1 Basic decompositions G. W. Stewart |
| title_full_unstemmed | Matrix algorithms 1 Basic decompositions G. W. Stewart |
| title_short | Matrix algorithms |
| title_sort | matrix algorithms basic decompositions |
| topic | Algorithmes ram Matrices ram algèbre inriac calcul matriciel inriac décomposition QR inriac matrice inriac méthode moindre carré inriac élimination Gauss inriac Matrix Mathematik (DE-588)4037968-1 gnd Matrizenrechnung (DE-588)4126963-9 gnd Algorithmus (DE-588)4001183-5 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Algorithmes Matrices algèbre calcul matriciel décomposition QR matrice méthode moindre carré élimination Gauss Matrix Mathematik Matrizenrechnung Algorithmus Numerisches Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008437272&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV012435940 |
| work_keys_str_mv | AT stewartgilbertw matrixalgorithms1 |