Matrix algorithms: 2 Eigensystems
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Philadelphia, Pa.
SIAM
2001
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009996211&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XIX, 469 S. |
| ISBN: | 0898715032 |
Internformat
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
Algorithms xv
Preface xvii
1 Eigensystems 1
1 The Algebra of Eigensystems 1
1.1 Eigenvalues and eigenvectors 2
Definitions. Existence of eigenpairs: Special cases. The
characteristic equation and the existence of eigenpairs. Matrices
of order two. Block triangular matrices. Real matrices.
1.2 Geometric multiplicity and defective matrices 6
Geometric multiplicity. Defectiveness. Simple eigenvalues.
1.3 Similarity transformations 8
1.4 Schur Decompositions 10
Unitary similarities. Schur form. Nilpotent matrices. Hermitian
matrices. Normal matrices.
1.5 Block diagonalization and Sylvester s equation 15
From block triangular to block diagonal matrices. Sylvester s
equation. An algorithm. Block diagonalization redux.
1.6 Jordan form 20
1.7 Notes and references 23
General references. History and Nomenclature. Schur form.
Sylvester s equation. Block diagonalization. Jordan canonical
form.
2 Norms, Spectral Radii, and Matrix Powers 25
2.1 Matrix and vector norms 25
Definition. Examples of norms. Operator norms. Norms and
rounding error. Limitations of norms. Consistency. Norms and
convergence.
2.2 The spectral radius 30
Definition. Norms and the spectral radius.
2.3 Matrix powers 33
Asymptotic bounds. Limitations of the bounds.
v
vi Contents
2.4 Notes and references 36
Norms. Norms and the spectral radius. Matrix powers.
3 Perturbation Theory 37
3.1 The perturbation of eigenvalues 37
The continuity of eigenvalues. Gerschgorin theory. Hermitian
matrices.
3.2 Simple eigenpairs 44
Left and right eigenvectors. First order perturbation expansions.
Rigorous bounds. Qualitative statements. The condition of an
eigenvalue. The condition of an eigenvector. Properties of sep.
Hermitian matrices.
3.3 Notes and references 52
General references. Continuity of eigenvalues. Gerschgorin
theory. Eigenvalues of Hermitian matrices. Simple eigenpairs.
Left and right eigenvectors. First order perturbation expansions.
Rigorous bounds. Condition numbers. Sep. Relative and
structured perturbation theory.
2 The QR Algorithm 55
1 The Power and Inverse Power Methods 56
1.1 The power method 56
Convergence. Computation and normalization. Choice of a
starting vector. Convergence criteria and the residual. Optimal
residuals and the Rayleigh quotient. Shifts and spectral
enhancement. Implementation. The effects of rounding error.
Assessment of the power method.
1.2 The inverse power method 66
Shift and invert enhancement. The inverse power method. An
example. The effects of rounding error. The Rayleigh quotient
method.
1.3 Notes and references 69
The power method. Residuals and backward error. The Rayleigh
quotient. Shift of origin. The inverse power and Rayleigh
quotient methods.
2 The Explicitly Shifted QR Algorithm 71
2.1 The QR algorithm and the inverse power method 72
Schur from the bottom up. The QR choice. Error bounds. Rates
of convergence. General comments on convergence.
2.2 The unshifted QR algorithm 75
The QR decomposition of a matrix polynomial. The
QR algorithm and the power method. Convergence of the
unshifted QR algorithm.
2.3 Hessenberg form 80
Contents vii
Householder transformations. Reduction to Hessenberg form.
Plane rotations. Invariance of Hessenberg form.
2.4 The explicitly shifted algorithm 94
Detecting negligible subdiagonals. Deflation. Back searching.
The Wilkinson shift. Reduction to Schur form. Eigenvectors.
2.5 Bits and pieces 104
Ad hoc shifts. Aggressive deflation. Cheap eigenvalues.
Balancing. Graded matrices.
2.6 Notes and references 110
Historical. Variants of the QR algorithm. The QR algorithm, the
inverse power method, and the power method. Local
convergence. Householder transformations and plane rotations.
Hessenberg form. The Hessenberg QR algorithm. Deflation
criteria. The Wilkinson shift. Aggressive deflation.
Preprocessing. Graded matrices.
3 The Implicitly Shifted QR Algorithm 113
3.1 Real Schur form 113
Real Schur form. A preliminary algorithm.
3.2 The uniqueness of Hessenberg reduction 116
3.3 The implicit double shift 117
A general strategy. Getting started. Reduction back to Hessenberg
form. Deflation. Computing the real Schur form. Eigenvectors.
3.4 Notes and references 128
Implicit shifting. Processing 2x2 blocks. The multishifted QR
algorithm.
4 The Generalized Eigenvalue Problem 129
4.1 The theoretical background 130
Definitions. Regular pencils and infinite eigenvalues.
Equivalence transformations. Generalized Schur form.
Multiplicity of eigenvalues. Projective representation of
eigenvalues. Generalized shifting. Left and right eigenvectors of
a simple eigenvalue. Perturbation theory. First order expansions:
Eigenvalues. The chordal metric. The condition of an eigenvalue.
Perturbation expansions: Eigenvectors. The condition of an
eigenvector.
4.2 Real Schur and Hessenberg triangular forms 143
Generalized real Schur form. Hessenberg triangular form.
4.3 The doubly shifted QZ algorithm 147
Overview. Getting started. The QZ step.
4.4 Important miscellanea 152
Balancing. Back searching. Infinite eigenvalues. The 2x2
problem. Eigenvectors.
VolII: Eigensystems
viii Contents
4.5 Notes and References 155
The algebraic background. Perturbation theory. The QZ
algorithm.
3 The Symmetric Eigenvalue Problem 157
1 The QR Algorithm 158
1.1 Reduction to tridiagonal form 158
The storage of symmetric matrices. Reduction to tridiagonal
form. Making Hermitian tridiagonal matrices real.
1.2 The symmetric tridiagonal QR algorithm 163
The implicitly shifted QR step. Choice of shift. Local
convergence. Deflation. Graded matrices. Variations.
1.3 Notes and references 169
General references. Reduction to tridiagonal form. The
tridiagonal QR algorithm.
2 A Clutch of Algorithms 171
2.1 Rank one updating 171
Reduction to standard form. Deflation: Small components of z.
Deflation: Nearly equal d» When to deflate. The secular
equation. Solving the secular equation. Computing eigenvectors.
Concluding comments.
2.2 A Divide and conquer algorithm 181
Generalities. Derivation of the algorithm. Complexity.
2.3 Eigenvalues and eigenvectors of band matrices 186
The storage of band matrices. Tridiagonalization of a symmetric
band matrix. Inertia. Inertia and the LU decomposition. The
inertia of a tridiagonal matrix. Calculation of a selected
eigenvalue. Selected eigenvectors.
2.4 Notes and References 201
Updating the spectral decomposition. The divide and conquer
algorithm. Reduction to tridiagonal form. Inertia and bisection.
Eigenvectors by the inverse power method. Jacobi s method.
3 The Singular Value Decomposition 203
3.1 Background 203
Existence and nomenclature. Uniqueness. Relation to the spectral
decomposition. Characterization and perturbation of singular
values. First order perturbation expansions. The condition of
singular vectors.
3.2 The cross product algorithm 210
Inaccurate singular values. Inaccurate right singular vectors.
Inaccurate left singular vectors. Assessment of the algorithm.
3.3 Reduction to bidiagonal form 215
The reduction. Real bidiagonal form.
Contents ix
3.4 The QR algorithm 217
The bidiagonal QR step. Computing the shift. Negligible
superdiagonal elements. Back searching. Final comments.
3.5 A hybrid QRD SVD algorithm 226
3.6 Notes and references 226
The singular value decomposition. Perturbation theory. The
cross product algorithm. Reduction to bidiagonal form and the
QR algorithm. The QRD SVD algorithm. The differential
qd algorithm. Divide and conquer algorithms. Downdating a
singular value decomposition. Jacobi methods.
4 The Symmetric Positive Definite (S/PD) Generalized Eigenvalue Problem 229
4.1 Theory 229
The fundamental theorem. Condition of eigenvalues.
4.2 Wilkinson s algorithm 231
The algorithm. Computation of R~TAR~1. Ill conditioned B.
4.3 Notes and references 234
Perturbation theory. Wilkinson s algorithm. Band matrices. The
definite generalized eigenvalue problem. The generalized
singular value and CS decompositions.
4 Eigenspaces and Their Approximation 239
1 Eigenspaces 240
1.1 Definitions 240
Eigenbases. Existence of eigenspaces. Deflation.
1.2 Simple eigenspaces 244
Definition. Block diagonalization and spectral representations.
Uniqueness of simple eigenspaces.
1.3 Notes and references 247
Eigenspaces. Spectral projections. Uniqueness of simple
eigenspaces. The resolvent.
2 Perturbation Theory 248
2.1 Canonical angles 248
Definitions. Computing canonical angles. Subspaces of unequal
dimensions. Combinations of subspaces.
2.2 Residual analysis 251
Optimal residuals. The Rayleigh quotient. Backward error.
Residual bounds for eigenvalues of Hermitian matrices. Block
deflation. The quantity sep. Residual bounds.
2.3 Perturbation theory 258
The perturbation theorem. Choice of subspaces. Bounds in terms
of E. The Rayleigh quotient and the condition of L. Angles
between the subspaces.
2.4 Residual bounds for Hermitian matrices 262
VolII: Eigensystems
x Contents
Residual bounds for eigenspaces. Residual bounds for
eigenvalues.
2.5 Notes and references 264
General references. Canonical angle. Optimal residuals and the
Rayleigh quotient. Backward error. Residual eigenvalue bounds
for Hermitian matrices. Block deflation and residual bounds.
Perturbation theory. Residual bounds for eigenspaces and
eigenvalues of Hermitian matrices.
3 Krylov Subspaces 266
3.1 Krylov sequences and Krylov spaces 266
Introduction and definition. Elementary properties. The
polynomial connection. Termination.
3.2 Convergence 269
The general approach. Chebyshev polynomials. Convergence
redux. Multiple eigenvalues. Assessment of the bounds.
Non Hermitian matrices.
3.3 Block Krylov sequences 277
3.4 Notes and references 279
Sources. Krylov sequences. Convergence (Hermitian matrices).
Convergence (non Hermitian matrices). Block Krylov sequences.
4 Rayleigh Ritz Approximation 280
4.1 Rayleigh Ritz methods 281
Two difficulties. Rayleigh Ritz and generalized eigenvalue
problems. The orthogonal Rayleigh Ritz procedure.
4.2 Convergence 284
Setting the scene. Convergence of the Ritz value. Convergence of
Ritz vectors. Convergence of Ritz values revisited. Hermitian
matrices. Convergence of Ritz blocks and bases. Residuals and
convergence.
4.3 Refined Ritz vectors 289
Definition. Convergence of refined Ritz vectors. The computation
of refined Ritz vectors.
4.4 Harmonic Ritz vectors 292
Computation of harmonic Ritz pairs.
4.5 Notes and references 295
Historical. Convergence. Refined Ritz vectors. Harmonic Ritz
vectors.
5 Krylov Sequence Methods 297
1 Krylov Decompositions 297
1.1 Arnoldi decompositions 298
The Arnoldi decomposition. Uniqueness. The Rayleigh quotient.
Reduced Arnoldi decompositions. Computing Arnoldi
Contents xi
decompositions. Reorthogonalization. Practical considerations.
Orthogonalization and shift and invert enhancement.
1.2 Lanczos decompositions 306
Lanczos decompositions. The Lanczos recurrence.
1.3 Krylov decompositions 308
Definition. Similarity transformations. Translation. Equivalence
to an Arnoldi decomposition. Reduction to Amoldi form.
1.4 Computation of refined and Harmonic Ritz vectors 314
Refined Ritz vectors. Harmonic Ritz vectors.
1.5 Notes and references 315
Lanczos and Arnoldi. Shift and invert enhancement and
orthogonalization. Krylov decompositions.
2 The Restarted Arnoldi Method 316
2.1 The implicitly restarted Arnoldi method 317
Filter polynomials. Implicitly restarted Arnoldi. Implementation.
The restarted Amoldi cycle. Exact shifts. Forward instability.
2.2 Krylov Schur restarting 325
Exchanging eigenvalues and eigenblocks. The Krylov Schur
cycle. The equivalence of Arnoldi and Krylov Schur.
Comparison of the methods.
2.3 Stability, convergence, and deflation 332
Stability. Stability and shift and invert enhancement.
Convergence criteria. Convergence with shift and invert
enhancement. Deflation. Stability of deflation. Nonorthogonal
bases. Deflation in the Krylov Schur method. Deflation in an
Arnoldi decomposition. Choice of tolerance.
2.4 Rational Krylov Transformations 342
Krylov sequences, inverses, and shifts. The rational Krylov
method.
2.5 Notes and references 344
Arnoldi s method, restarting, and filter polynomials. ARPACK.
Reverse communication. Exchanging eigenvalues. Forward
instability. The Krylov Schur method. Stability. Shift and invert
enhancement and the Rayleigh quotient. Convergence. Deflation.
The rational Krylov transformation.
3 The Lanczos Algorithm 347
3.1 Reorthogonalization and the Lanczos algorithm 348
Lanczos with reorthogonalization.
3.2 Full reorthogonalization and restarting 351
Implicit restarting. Krylov spectral restarting. Convergence and
deflation.
3.3 Semiorthogonal methods 352
VolII: Eigensystems
xii Contents
Semiorthogonal bases. The QR factorization of a semiorthogonal
basis. The size of the reorthogonalization coefficients. The effects
on the Rayleigh quotient. The effects on the Ritz vectors.
Estimating the loss of orthogonality. Periodic reorthogonalization.
Updating H . Computing residual norms. An example.
3.4 Notes and references 365
The Lanczos algorithm. Full reorthogonalization. Filter
polynomials. Semiorthogonal methods. No reorthogonalization.
The singular value decomposition. Block Lanczos algorithms.
The bi Lanczos algorithm.
4 The Generalized Eigenvalue Problem 367
4.1 Transformation and convergence 368
Transformation to an ordinary eigenproblem. Residuals and
backward error. Bounding the residual.
4.2 Positive definite B 370
The B inner product. Orthogonality. I? Orthogonalization. The
S Arnoldi method. The restarted B Lanczos method. The
S Lanczos method with periodic reorthogonalization.
Semidefinite B.
4.3 Notes and references 379
The generalized shift and invert transformation and the Lanczos
algorithm. Residuals and backward error. Semidefinite B.
6 Alternatives 381
1 Subspace Iteration 381
1.1 Theory 382
Convergence. The QR connection. Schur Rayleigh Ritz
refinement.
1.2 Implementation 386
A general algorithm. Convergence. Deflation. When to perform
an SRR step. When to orthogonalize. Real matrices.
1.3 Symmetric matrices 391
Economization of storage. Chebyshev acceleration.
1.4 Notes and references 394
Subspace iteration. Chebyshev acceleration. Software.
2 Newton Based Methods 395
2.1 Approximate Newton methods 396
A general approximate Newton method. The correction equation.
Orthogonal corrections. Analysis. Local convergence. Three
approximations.
2.2 The Jacobi Davidson method 404
The basic Jacobi Davidson method. Extending Schur
decompositions. Restarting. Harmonic Ritz vectors. Hermitian
Contents xiii
matrices. Solving projected systems. The correction equation:
Exact solution. The correction equation: Iterative solution.
2.3 Notes and references 418
Newton s method and the eigenvalue problem. Inexact Newton
methods. The Jacobi Davidson method. Hermitian matrices.
Appendix: Background 421
1 Scalars, vectors, and matrices 421
2 Linear algebra 424
3 Positive definite matrices 425
4 The LU decomposition 425
5 The Cholesky decomposition 426
6 The QR decomposition 426
7 Projectors 427
References 429
Index 451
VoIII: Eigensystems
|
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| id | DE-604.BV014761858 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T11:07:24Z |
| institution | BVB |
| isbn | 0898715032 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009996211 |
| oclc_num | 633143991 |
| open_access_boolean | |
| owner | DE-29T DE-703 DE-706 DE-83 |
| owner_facet | DE-29T DE-703 DE-706 DE-83 |
| physical | XIX, 469 S. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | SIAM |
| record_format | marc |
| spellingShingle | Stewart, Gilbert W. 1940- Matrix algorithms Matrix Mathematik (DE-588)4037968-1 gnd |
| subject_GND | (DE-588)4037968-1 |
| title | Matrix algorithms |
| title_auth | Matrix algorithms |
| title_exact_search | Matrix algorithms |
| title_full | Matrix algorithms 2 Eigensystems G. W. Stewart |
| title_fullStr | Matrix algorithms 2 Eigensystems G. W. Stewart |
| title_full_unstemmed | Matrix algorithms 2 Eigensystems G. W. Stewart |
| title_short | Matrix algorithms |
| title_sort | matrix algorithms eigensystems |
| topic | Matrix Mathematik (DE-588)4037968-1 gnd |
| topic_facet | Matrix Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009996211&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV012435940 |
| work_keys_str_mv | AT stewartgilbertw matrixalgorithms2 |