The algebraic eigenvalue problem:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford
Clarendon Press
1992
|
| Ausgabe: | Reprinted |
| Schriftenreihe: | Monographs on numerical analysis
Oxford science publications |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 662 S. |
| ISBN: | 0198534183 |
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Datensatz im Suchindex
| _version_ | 1804141345008254976 |
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| adam_text | THE ALGEBRAIC EIGENVALUE PROBLEM BY J. H. WILKINSON M.A. (CANTAB.),
SC.D., D.TECH., F.R.S. CLARENDON PRESS * OXFORD / CONTENTS THEORETICAL
BACKGROUND PAGE INTRODUCTION 1 DEFINITIONS 2 EIGENVALUES AND
EIGENVECTORS OF THE TRANSPOSED MATRIX 3 DISTINCT EIGENVALUES 4
SIMILARITY TRANSFORMATIONS 6 MULTIPLE EIGENVALUES AND CANONICAL FORMA
FOR GENERAL MATRICES 7 DEFECTIVE SYSTEM OF EIGENVECTORS 9 THE JORDAN
(CLASSICAL) CANONICAL FORM 10 THE ELEMENTARY DIVISORS 12 COMPANION
MATRIX OF THE CHARACTERISTIC POLYNOMIAL OF A 12 NON-DEROGATORY MATRICES
13 THE FROBENIUS (RATIONAL) CANONICAL FORM 15 RELATIONSHIP BETWEEN THE
JORDAN AND FROBENIUS CANONICAL FORMS 16 EQUIVALENCE TRANSFORMATIONS 17
LAMBDA MATRICES 18 ELEMENTARY OPERATIONS 19 SMITH S CANONICAL FORM 19
THE HIGHEST COMMON FACTOR OF FC-ROWED MINORS OF A A-MATRIX 22 INVARIANT
FACTORS OF (A *AI) 22 THE TRIANGULAER CANONICAL FORM 24 HERMITIAN AND
SYMMETRIE MATRICES 24 ELEMENTARY PROPERTIES OF HERMITIAN MATRICES 25
COMPLEX SYMMETRIE MATRICES 26 REDUCTION TO TRIANGULAER FORM BY UNITARY
TRANSFORMATIONS 27 QUADRATIC FORMS 27 NECESSARY AND SUFFICIENT
CONDITIONS FOR POSITIVE DEFINITENESS 28 DIFFERENTIAL EQUATIONS WITH
CONSTANT COEFFICIENTS 30 SOLUTIONS CORRESPONDING TO NON-LINEAR
ELEMENTARY DIVISORS 31 DIFFERENTIAL EQUATIONS OF HIGHER ORDER 32
SECOND-ORDER EQUATIONS OF SPECIAL FORM 34 EXPLICIT SOLUTION OF BY = *AY
35 EQUATIONS OF THE FORM (AB *AI)* = 0 35 THE MINIMUM POLYNOMIAL OF A
VECTOR 36 THE MINIMUM POLYNOMIAL OF A MATRIX 37 CAYLEY-HAMILTON THEOREM
38 RELATION BETWEEN MINIMUM POLYNOMIAL AND CANONICAL FORMS 39 PRINCIPAL
VECTORS 42 ELEMENTARY SIMILARITY TRANSFORMATIONS 43 PROPERTIES OF
ELEMENTARY MATRICES 45 REDUCTION TO TRIANGULAER CANONICAL FORM BY
ELEMENTARY SIMILARITY TRANSFORMATIONS 46 ELEMENTARY UNITARY
TRANSFORMATIONS 47 ELEMENTARY UNITARY HERMITIAN MATRICES 48 REDUCTION TO
TRIANGULAER FORM BY ELEMENTARY UNITARY TRANSFORMATIONS 50 NORMAL MATRICES
51 COMMUTING MATRICES 52 CONTENTS EIGENVALUES OF AB 54 VECTOR AND MATRIX
NORMS 55 SUBORDINATE MATRIX NORMS 56 THE EUCLIDEAN AND SPECTRAL NORMS 57
NORMS AND LIMITS 58 AVOIDING USE OF INFINITE MATRIX SERIES 60 2.
PERTURBATION THEORY INTRODUCTION 62 OSTROWSKI S THEOREM ON CONTINUITY OF
THE EIGENVALUES 63 ALGEBRAIC FUNCTIONS 64 NUMERICAL EXAMPLES 65
PERTURBATION THEORY FOR SIMPLE EIGENVALUES 66 PERTURBATION OF
CORRESPONDING EIGENVECTORS 67 MATRIX WITH LINEAR ELEMENTARY DIVISORS 68
FIRST-ORDER PERTURBATIONS OF EIGENVALUES 68 FIRST-ORDER PERTURBATIONS OF
EIGENVECTORS 69 HIGHER-ORDER PERTURBATIONS 70 MULTIPLE EIGENVALUES 70
GERSCHGORIN S THEOREMS 71 PERTURBATION THEORY BASED ON GERSCHGORIN S
THEOREMS 72 CASE 1. PERTURBATION OF A SIMPLE EIGENVALUE X X OF A MATRIX
HAVING LINEAR ELEMENTARY DIVISORS. 72 CASE 2. PERTURBATION OF A MULTIPLE
EIGENVALUE A X OF A MATRIX HAVING LINEAR ELEMENTARY DIVISORS. 75 CASE 3.
PERTURBATION OF A SIMPLE EIGENVALUE OF A MATRIX HAVING ONE OR MORE
NON-LINEAR ELEMENTARY DIVISORS. 77 CASE 4. PERTURBATIONS OF THE
EIGENVALUES CORRESPONDING TO A NON-LINEAR ELEMENTARY DIVISOR OF A
NON-DEROGATORY MATRIX. 79 CASE 5. PERTURBATIONS OF EIGENVALUES X T WHEN
THERE IS MORE THAN ONE DIVISOR INVOLVING (X I *X) AND AT LEAST ONE OF
THEM IS NON-LINEAR. 80 PERTURBATIONS CORRESPONDING TO THE GENERAL
DISTRIBUTION OF NON-LINEAR DIVISORS 81 PERTURBATION THEORY FOR THE
EIGENVECTORS FROM JORDAN CANONICAL FORM 81 PERTURBATIONS OF EIGENVECTORS
CORRESPONDING TO A MULTIPLE EIGENVALUE (LINEAR ELEMENTARY DIVISORS) 83
LIMITATIONS OF PERTURBATION THEORY 84 RELATIONSHIPS BETWEEN THE S ( 85
THE CONDITION OF A COMPUTING PROBLEM 86 CONDITION NUMBERS 86 SPECTRAL
CONDITION NUMBER OF A WITH RESPECT TO ITS EIGENPROBLEM 87 PROPERTIES OF
SPECTRAL CONDITION NUMBER 88 INVARIANT PROPERTIES OF CONDITION NUMBERS
89 VERY ILL-CONDITIONED MATRICES 90 PERTURBATION THEORY FOR REAL
SYMMETRIE MATRICES 93 UNSYMMETRIC PERTURBATIONS 93 SYMMETRIE
PERTURBATIONS 94 CLASSICAL TECHNIQUES 94 SYMMETRIE MATRIX OF RANK UNITY
97 EXTREMAL PROPERTIES OF EIGENVALUES 98 MINIMAX CHARACTERIZATION OF
EIGENVALUES 99 EIGENVALUES OF THE SUM OF TWO SYMMETRIE MATRICES 101
CONTENTS PRACTICAL APPLICATIONS 102 FURTHER APPLICATIONS OF MINIMAX
PRINOIPLE 103 SEPARATION THEOREM 103 THE WIELANDT-HOFFMAN THEOREM 104 3.
ERROR ANALYSIS INTRODUCTION 110 FIXED-POINT OPERATIONS 110 ACCUMULATION
OF INNER-PRODUCTS 111 FLOATING-POINT OPERATIONS 112 SIMPLIFIED
EXPRESSIONS FOR ERROR BOUNDS 113 ERROR BOUNDS FOR SOME BASIC
FLOATING-POINT COMPUTATIONS 114 BOUNDS FOR NORMS OF THE ERROR MATRICES
115 ACCUMULATION OF INNER-PRODUCTS IN FLOATING-POINT ARITHMETIC 116
ERROR BOUNDS FOR SOME BASIC/I 2 ( ) COMPUTATIONS 117 COMPUTATION OF
SQUARE ROOTS 118 BLOCK-FLOATING VECTORS AND MATRICES 119 FUNDAMENTAL
LIMITATIONS OF T-DIGIT COMPUTATION 120 EIGENVALUE TECHNIQUES BASED ON
REDUCTION BY SIMILARITY TRANSFORMATIONS 123 ERROR ANALYSIS OF METHODS
BASED ON ELEMENTARY NON-UNITARY TRANS- FORMATIONS 124 ERROR ANALYSIS OF
METHODS BASED ON ELEMENTARY UNITARY TRANSFORMATIONS 126 SUPERIORITY OF
THE UNITARY TRANSFORMATION 128 REAL SYMMETRIE MATRICES 129 LIMITATIONS
OF UNITARY TRANSFORMATIONS 129 ERROR ANALYSIS OF FLOATING-POINT
COMPUTATION OF PLANE ROTATIONS 131 MULTIPLICATION BY A PLANE ROTATION
133 MULTIPLICATION BY A SEQUENCE OF PLANE ROTATIONS 134 ERROR IN PRODUET
OF APPROXIMATE PLANE ROTATIONS 139 ERRORS IN SIMILARITY TRANSFORMS 140
SYMMETRIE MATRICES 141 PLANE ROTATIONS IN FIXED-POINT ARITHMETIC 143
ALTERNATIVE COMPUTATION OF SIN 0 AND COS 6 145 PRE-MULTIPLICATION BY AN
APPROXIMATE FIXED-POINT ROTATION 145 MULTIPLICATION BY A SEQUENCE OF
PLANE ROTATIONS (FIXED-POINT) 147 THE COMPUTED PRODUET OF AN APPROXIMATE
SET OF PLANE ROTATIONS 148 ERRORS IN SIMILARITY TRANSFORMATIONS 148
GENERAL COMMENTS ON THE ERROR BOUNDS 151 ELEMENTARY HERMITIAN MATRICES
IN FLOATING-POINT 152 ERROR ANALYSIS OF THE COMPUTATION OF AN ELEMENTARY
HERMITIAN MATRIX 153 NUMERICAL EXAMPLE 156 PRE-MULTIPLICATION BY AN
APPROXIMATE ELEMENTARY HERMITIAN MATRIX 157 MULTIPLICATION BY A SEQUENCE
OF APPROXIMATE ELEMENTARY HERMITIANS 160 NON-UNITARY ELEMENTARY MATRICES
ANALOGOUS TO PLANE ROTATIONS 162 NON-UNITARY ELEMENTARY MATRICES
ANALOGOUS TO ELEMENTARY HERMITIAN MATRICES 163 PRE-MULTIPLICATION BY A
SEQUENCE OF NON-UNITARY MATRICES 165 A PRIORI ERROR BOUNDS 166 DEPARTURE
FROM NORMALITY 167 SIMPLE EXAMPLES 169 A POSTERIORI BOUNDS 170 A
POSTERIORI BOUNDS FOR NORMAL MATRICES 170 XU CONTENTS RAYLEIGH QUOTIENT
172 ERROR IN RAYLEIGH QUOTIENT 173 HERMITIAN MATRICES 174 PATHOLOGIOALLY
CLOSE EIGENVALUES 176 NON-NORMAL MATRICES 178 ERROR ANALYSIS FOR A
COMPLETE EIGENSYSTEM 180 NUMERICAL EXAMPLE 181 CONDITIONS LIMITING
ATTAINABLE AOCURACY 181 NON-LINEAR ELEMENTARY DIVISORS 182 APPROXIMATE
INVARIANT SUBSPACES 184 ALMOST NORMAL MATRICES 187 4. SOLUTION OF LINEAR
ALGEBRAIC EQUATIONS INTRODUCTION 189 PERTURBATION THEORY 189 CONDITION
NUMBERS 191 EQUILIBRATED MATRICES 192 SIMPLE PRACTICAL EXAMPLES 193
CONDITION OF MATRIX OF EIGENVECTORS 193 EXPLICIT SOLUTION 194 GENERAL
COMMENTS ON CONDITION OF MATRICES 195 RELATION OF ILL-CONDITIONING TO
NEAR-SINGULARITY 196 LIMITATIONS IMPOSED BY T-DIGIT ARITHMETIC 197
ALGORITHMS FOR SOLVING LINEAR EQUATIONS 198 GAUSSIAN ELIMINATION 200
TRIANGULAER DECOMPOSITION 201 STRUCTURE OF TRIANGULAER DECOMPOSITION
MATRICES 201 EXPLICIT EXPRESSIONS FOR CLEMENTS OF THE TRIANGLES 202
BREAKDOWN OF GAUSSIAN ELIMINATION 204 NUMERICAL STABILITY 205
SIGNIFICANCE OF THE INTERCHANGES 206 NUMERICAL EXAMPLE 207 ERROR
ANALYSIS OF GAUSSIAN ELIMINATION 209 UPPER BOUNDS FOR THE PERTURBATION
MATRICES USING FIXED-POINT ARITHMETIC 211 UPPER BOUND FOR ELEMENTS OF
REDUCED MATRICES 212 COMPLETE PIVOTING 212 PRACTICAL PROCEDURE WITH
PARTIAL PIVOTING 214 FLOATING-POINT ERROR ANALYSIS 214 FLOATING-POINT
DECOMPOSITION WITHOUT PIVOTING 215 LOSS OF SIGNIFICANT FIGURES 217 A
POPULAER FALLACY 217 MATRICES OF SPECIAL FORM 218 GAUSSIAN ELIMINATION ON
A HIGH-SPEED COMPUTER 220 SOLUTIONS CORRESPONDING TO DIFFERENT
RIGHT-HAND SIDES 221 DIRECT TRIANGULAER DECOMPOSITION 221 RELATIONS
BETWEEN GAUSSIAN ELIMINATION AND DIRECT TRIANGULAER DECOM- POSITION 223
EXAMPLES OF FAILURE AND NON-UNIQUENESS OF DECOMPOSITION 224 TRIANGULAER
DECOMPOSITION WITH ROW INTERCHANGES 225 ERROR ANALYSIS OF TRIANGULAER
DECOMPOSITION 227 EVALUATION OF DETERMINANTS 228 CHOLESKY DECOMPOSITION
229 CONTENTS XIII SYMMETRIE MATRICES WHICH ARE NOT POSITIVE DEFINITE 230
ERROR ANALYSIS OF CHOLESKY DECOMPOSITION IN FIXED-POINT ARITHMETIC 231
AN ILL-CONDITIONED MATRIX 233 TRIANGULARIZATION USING ELEMENTARY
HERMITIAN MATRICES 233 ERROR ANALYSIS OF HOUSEHOLDER TRIANGULARIZATION
236 TRIANGULARIZATION BY ELEMENTARY STABILIZED MATRICES OF THE TYPE M *
236 EVALUATION OF DETERMINANTS OF LEADING PRINCIPE! MINORE 237
TRIANGULARIZATION BY PLANE ROTATIONS 239 ERROR ANALYSIS OF GIVENS
REDUETION 240 UNIQUENESS OF ORTHOGONAL TRIANGULARIZATION 241 SCHMIDT
ORTHOGONALIZATION 242 COMPARISON OF THE METHODS OF TRIANGULARIZATION 244
BACK-SUBSTITUTION 247 HIGH ACCURACY OF COMPUTED SOLUTIONS OF TRIANGULAER
SETS OF EQUATIONS 249 SOLUTION OF A GENERAL SET OF EQUATIONS 251
COMPUTATION OF THE INVERSE OF A GENERAL MATRIX 252 ACCURACY OF .COMPUTED
SOLUTIONS 253 ILL-CONDITIONED MATRICES WHICH GIVE NO SMALL PIVOTS 254
ITERATIVE IMPROVEMENTS OF APPROXIMATE SOLUTION 255 EFFECT OF ROUNDING
ERRORS ON THE ITERATIVE PROCESS 256 THE ITERATIVE PROCEDURA IN
FIXED-POINT COMPUTATION 257 SIMPLE EXAMPLE OF ITERATIVE PROCEDURE 258
GENERAL COMMENTS ON THE ITERATIVE PROCEDURE 260 RELATED ITERATIVE
PROCEDURES 261 LIMITATIONS OF THE ITERATIVE PROCEDURE 261 RIGOROUS
JUSTIFICATION OF THE ITERATIVE METHOD 262 5. HERMITIAN MATRICES
INTRODUCTION 265 THE CLASSICAL JACOBI METHOD FOR REAL SYMMETRIE MATRICES
266 RATE OF CONVERGENCE 267 CONVERGENCE TO FIXED DIAGONAL MATRIX 268
SERIAL JACOBI METHOD 269 THE GERSCHGORIN DISES 269 ULTIMATE QUADRATIC
CONVERGENCE OF JACOBI METHODS 270 CLOSE AND MULTIPLE EIGENVALUES 271
NUMERICAL EXAMPLES 273 CALCULATION OF COS 0 AND SIN 9 274 SIMPLER
DETERMINATION OF THE ANGLES OF ROTATION 276 THE THRESHOLD JACOBI METHOD
277 CALCULATION OF THE EIGENVECTORS 278 NUMERICAL EXAMPLE 279 ERROR
ANALYSIS OF THE JACOBI METHOD 279 ACCURACY OF THE COMPUTED EIGENVECTORS
280 ERROR BOUNDS FOR FIXED-POINT COMPUTATION 281 ORGANIZATIONAL PROBLEMS
282 GIVENS METHOD 282 GIVENS* PROCESS ON A COMPUTER WITH A TWO-LEVOL
STOERE 284 FLOATING-POINT ERROR ANALYSIS OF GIVENS PROCESS 286
FIXED-POINT ERROR ANALYSIS 287 NUMERICAL EXAMPLE 288 HOUSEHOLDER S
METHOD 290 XIV CONTENTS TAKING ADVANTAGE OF SYMMETRY 292 STORAGE
CONSIDERATIONS 293 HOUSEHOLDER S PROCESS ON A COMPUTER WITH A TWO-LEVEL
STOERE 294 HOUSEHOLDER S METHOD IN FIXED-POINT ARITHMETIC 294 NUMERICAL
EXAMPLE 296 ERROR ANALYSES OF HOUSEHOLDER S METHOD 297 EIGENVALUES OF A
SYMMETRIE TRI-DIAGONAL MATRIX 299 STURM SEQUENCE PROPERTY 300 METHOD OF
BISECTION 302 NUMERICAL STABILITY OF THE BISECTION METHOD 302 NUMERICAL
EXAMPLE 305 GENERAL COMMENTS ON THE BISECTION METHOD 306 SMALL
EIGENVALUES 307 CLOSE EIGENVALUES AND SMALL /?,* 308 FIXED-POINT
COMPUTATION OF THE EIGENVALUES 312 COMPUTATION OF THE EIGENVECTORS OF A
TRI-DIAGONAL FORM 315 INSTABILITY OF THE EXPLICIT EXPRESSION FOR THE
EIGENVECTOR 316 NUMERICAL EXAMPLES 319 INVERSE ITERATION 321 CHOICE OF
INITIAL VECTOR 6 322 ERROR ANALYSIS 323 NUMERICAL EXAMPLE 325 CLOSE
EIGENVALUES AND SMALL SS { 327 INDEPENDENT VECTORS CORRESPONDING TO
COINCIDENT EIGENVALUES 328 ALTERNATIVE METHOD FOR COMPUTING THE
EIGENVECTORS 330 NUMERICAL EXAMPLE 331 COMMENTS ON THE EIGENPROBLEM FOR
TRI-DIAGONAL MATRICES 332 COMPLETION OF THE GIVENS AND HOUSEHOLDER
METHODS 333 COMPARISON OF METHODS 334 QUASI-SYMMETRIC TRI-DIAGONAL
MATRICES 335 CALCULATION OF THE EIGENVECTORS 336 EQUATIONS OF THE FORM
AX = KBX AND ABX = HC 337 NUMERICAL EXAMPLE 339 SIMULTANEOUS REDUCTION
OF A AND B TO DIAGONAL FORM 340 TRI-DIAGONAL A AND B 340 COMPLEX
HERMITIAN MATRICES 342 6. REDUCTION OF A GENERAL MATRIX TO CONDENSED
FORM INTRODUCTION 345 GIVENS METHOD 345 HOUSEHOLDER S METHOD 347
STORAGE CONSIDERATIONS 350 ERROR ANALYSIS 350 BELATIONSHIP BETWEEN THE
GIVENS AND HOUSEHOLDER METHODS 351 ELEMENTARY STABILIZED TRANSFORMATIONS
353 SIGNIFICANCE OF THE PERMUTATIONS 355 DIRECT REDUCTION TO HESSENBERG
FORM 357 INCORPORATION OF INTERCHANGES 359 NUMERICAL EXAMPLE 360 ERROR
ANALYSIS 363 RELATED ERROR ANALYSES 365 CONTENTS XV POOR DETERMINATION
OF THE HESSENBERG MATRIX 368 REDUCTION TO HESSENBERG FORM USING
STABILIZED MATRICES OF THE TYPE M TT 368 THE METHOD OF KRYLOV 369
GAUSSIAN ELIMINATION BY COLUMNS 370 PRACTICAL DIFFLCULTIES 371 CONDITION
OF G FOR SOME STANDARD DISTRIBUTIONS OF EIGENVALUES 372 INITIAL VECTORS
OF GRADE LESS THAN N 374 PRACTICAL EXPERIENCE 376 GENERALIZED HESSENBERG
PROCESSES 377 FAILURE OF THE GENERALIZED HESSENBERG PROCESS 378 THE
HESSENBERG METHOD 379 PRACTICAL PROCEDURE 380 RELATION BETWEEN THE
HESSENBERG METHOD AND EARLIER METHODS 381 THE METHOD OF ARNOLDI 382
PRACTICAL CONSIDERATIONS 383 SIGNIFICANCE OF RE-ORTHOGONALIZATION 385
THE METHOD OF LANCZOS 388 FAILURE OF PROCEDURE 389 NUMERICAL EXAMPLE 390
THE PRACTICAL LANCZOS PROCESS 391 NUMERICAL EXAMPLE 392 GENERAL COMMENTS
ON THE UNSYMMETRIC LANCZOS PROCESS 394 THE SYMMETRIE LANCZOS PROCESS 394
REDUCTION OF A HESSENBERG MATRIX TO A MORE COMPACT FORM 395 REDUCTION OF
A LOWER HESSENBERG MATRIX TO TRI-DIAGONAL FORM 396 THE USE OF
INTERCHANGES 397 EFFECT OF A SMALL PIVOTAL ELEMENT 398 ERROR ANALYSIS
399 THE HESSENBERG PROCESS APPLIED TO A LOWER HESSENBERG MATRIX 402
RELATIONSHIP BETWEEN THE HESSENBERG PROCESS AND THE LANCZOS PROCESS 402
REDUCTION OF A GENERAL MATRIX TO TRI-DIAGONAL FORM 403 COMPARISON WITH
LANCZOS METHOD 404 RE-EXAMINATION OF REDUCTION TO TRI-DIAGONAL FORM 404
REDUCTION FROM UPPER HESSENBERG FORM TO FROBENIUS FORM 405 EFFECT OF
SMALL PIVOT 407 NUMERICAL EXAMPLE 408 GENERAL COMMENTS ON THE STABILITY
408 SPECIALIZED UPPER HESSENBERG FORM 409 DIRECT DETERMINATION OF THE
CHARACTERISTIC POLYNOMIAL 410 7. EIGENVALUES OF MATRICES OF CONDENSED
FORMS INTRODUCTION 413 EXPLICIT POLYNOMIAL FORM 413 CONDITION NUMBERS OF
EXPLICIT POLYNOMIALS 416 SOME TYPICAL DISTRIBUTIONS OF ZEROS 417 FINAL
ASSESSMENT OF KRYLOV S METHOD 421 GENERAL COMMENTS ON EXPLICIT
POLYNOMIALS 421 TRI-DIAGONAL MATRICES 423 DETERMINANTS OF HESSENBERG
MATRICES 426 EFFECT OF ROUNDING ERRORS 427 FLOATING-POINT AECUMULATION
428 EVALUATION BY ORTHOGONAL TRANSFORMATIONS 429 XVI CONTENTS EVALUATION
OF DETERMINANTS OF GENERAL MATRICES 431 THE GENERALIZED EIGENVALUE
PROBLEM 432 INDIRECT DETERMINATIONS OF THE CHARACTERISTIC POLYNOMIAL 432
LE VERRIER S METHOD 434 ITERATIVE METHODS BASED ON INTERPOLATION 435
ASYMPTOTIC RATE OF CONVERGENCE 436 MULTIPLE ZEROS 437 INVERSION OF THE
FUNCTIONAL RELATIONSHIP 439 THE METHOD OF BISECTION 440 NEWTON S METHOD
441 COMPARISON OF NEWTON S METHOD WITH INTERPOLATION 442 METHODS GIVING
CUBIC CONVERGENCE 443 LAGUERRE S METHOD 443 COMPLEX ZEROS 446 COMPLEX
CONJUGATE ZEROS 447 BAIRSTOW S METHOD 449 THE GENERALIZED BAIRSTOW
METHOD 450 PRACTICAL CONSIDERATIONS 452 EFFECT OF ROUNDING ERRORS ON
ASYMPTOTIC CONVERGENCE 453 THE METHOD OF BISECTION 453 SUCCESSIVE LINEAR
INTERPOLATION 455 MULTIPLE AND PATHOLOGICALLY CLOSE EIGENVALUES 457
OTHER INTERPOLATION METHODS 458 METHODS INVOLVING THE USE OF A
DERIVATIVE 459 CRITERION FOR ACCEPTANCE OF A ZERO 461 EFFECT OF ROUNDING
ERRORS 462 SUPPRESSION OF COMPUTED ZEROS 464 DEFLATION FOR HESSENBERG
MATRICES 465 DEFLATION OF TRI-DIAGONAL MATRICES 468 DEFLATION BY
ROTATIONS OR STABILIZED ELEMENTARY TRANSFORMATIONS 469 STABILITY OF THE
DEFLATION 472 GENERAL COMMENTS ON DEFLATION 474 SUPPRESSION OF COMPUTED
ZEROS 474 SUPPRESSION OF COMPUTED QUADRATIC FACTORS 475 GENERAL COMMENTS
ON THE METHODS OF SUPPRESSION 476 ASYMPTOTIC RATES OF CONVERGENCE 478
CONVERGENCE IN THE LARGE 478 COMPLEX ZEROS 481 RECOMMENDATIONS 482
COMPLEX MATRICES 483 MATRICES CONTAINING AN INDEPENDENT PARAMETER 483 8.
THE LR AND QR ALGORITHMS INTRODUCTION 485 REAL MATRICES WITH COMPLEX
EIGENVALUES 486 THE LR ALGORITHM 487 PROOF OF THE CONVERGENCE OF THE A,
489 POSITIVE DEFINITE HERMITIAN MATRICES 493 COMPLEX CONJUGATE
EIGENVALUES 494 INTRODUCTION OF INTERCHANGES 498 NUMERICAL EXAMPLE 499
CONVERGENCE OF THE MODIFIED PROCESS 501 CONTENTS XVII PRELIMINARY
REDUCTION OF ORIGINAL MATRIX 501 INVARIANCE OF UPPER HESSENBERG FORM 502
SIMULTANEOUS ROW AND COLUMN OPERATIONS 504 ACCELERATION OF CONVERGENOE
505 INCORPORATION OF SHIFTS OF ORIGIN 506 CHOICE OF SHIFT OF ORIGIN 507
DEFLATION OF THE MATRIX 509 PRACTICA! EXPERIENCE OF CONVERGENCE 510
IMPROVED SHIFT STRATEGY 511 COMPLEX CONJUGATE EIGENVALUES 512 CRITICISMS
OF THE MODIFIED LR ALGORITHM 515 THE QR ALGORITHM 515 CONVERGENCE OF THE
QR ALGORITHM 516 FORMAL PROOF OF CONVERGENCE 517 DISORDER OF THE
EIGENVALUES 519 EIGENVALUES OF EQUAL MODULUS 520 ALTERNATIVE PROOF FOR
THE LR TECHNIQUE 521 PRACTICA! APPLICATION OF THE QR ALGORITHM 523
SHIFTS OF ORIGIN 524 DECOMPOSITION OF A, 525 NUMERICAL EXAMPLE 527
PRACTICAL PROCEDURE 527 AVOIDING COMPLEX CONJUGATE SHIFTS 528 DOUBLE QR
STEP USING ELEMENTARY HERMITIANS 532 COMPUTATIONAL DETAILS 534
DECOMPOSITION OF A, 535 DOUBLE-SHIFT TECHNIQUE FOR LR 537 ASSESSMENT OF
LR AND QR ALGORITHMS 538 MULTIPLE EIGENVALUES 540 SPECIAL USE OF THE
DEFLATION PROCESS 543 SYMMETRIE MATRICES 544 RELATIONSHIP BETWEEN LR AND
QR ALGORITHMS 545 CONVERGENCE OF THE CHOLESKY LR ALGORITHM 546 CUBIC
CONVERGENCE OF THE QR ALGORITHM 548 SHIFT OF ORIGIN IN CHOLESKY LR 549
FAILURE OF THE CHOLESKY DECOMPOSITION 550 CUBICALLY CONVERGENT LR
PROCESS 551 BAND MATRICES 553 QR DECOMPOSITIO N OF A BAND MATRIX 557
ERROR ANALYSIS 561 UNSYMMETRIC BAND MATRICES 562 SIMULTANEOUS
DECOMPOSITION AND REOOMBINATION IN QR ALGORITHM 565 REDUCTION OF BAND
WIDTH 567 9. ITERATIVE METHODS INTRODUCTION 570 THE POWER METHOD 570
DIRECT ITERATION WITH A SINGLE VECTOR 571 SHIFT OF ORIGIN 572 EFFECT OF
ROUNDING ERRORS 573 VARIATION OFP 576 AD HOC CHOICE OFP 577 XVIII
CONTENTS AITKEN S ACCELERATION TEEHNIQUE 578 COMPLEX CONJUGATE
EIGENVALUES 579 CALCULATION OF THE COMPLEX EIGENVECTOR 581 SHIFTOF
ORIGIN 582 NON-LINEAR DIVISORS I 582 SIMULTANEOUS DETERMINATION OF
SEVERAL EIGENVALUES 583 COMPLEX MATRICES 584 DEFLATION 584 DEFLATION
BASED ON SIMILARITY TRANSFORMATIONS 585 DEFLATION USING INVARIANT
SUBSPACES 587 DEFLATION USING STABILIZED ELEMENTARY TRANSFORMATIONS 587
DEFLATION USING UNITARY TRANSFORMATIONS 589 NUMERICAL STABILITY 590
NUMERICAL EXAMPLE 592 STABILITY OF UNITARY TRANSFORMATIONS 594 DEFLATION
BY NON-SIMILARITY TRANSFORMATIONS 596 GENERAL REDUCTION USING INVARIANT
SUBSPACES 599 PRACTICAL APPLICATION 601 TREPPEN-ITERATION 602 ACCURATE
DETERMINATION OF COMPLEX CONJUGATE EIGENVALUES 604 VERY CLOSE
EIGENVALUES 606 ORTHOGONALIZATION TECHNIQUES 606 ANALOGUE OF
TREPPEN-ITERATION USING ORTHOGONALIZATION 607 BI-ITERATION 609 NUMERICAL
EXAMPLE 610 RICHARDSON S PURIFICATION PROCESS 614 MATRIX SQUARING 615
NUMERICAL STABILITY 616 USE OF CHEBYSHEV POLYNOMIALS 617 GENERAL
ASSESSMENT OF METHODS BASED ON DIRECT ITERATION 618 INVERSE ITERATION
619 ERROR ANALYSIS OF INVERSE ITERATION 620 GENERAL COMMENTS ON THE
ANALYSIS 621 FURTHER REFINEMENT OF EIGENVECTORS 622 NON-LINEAR
ELEMENTARY DIVISORS 626 INVERSE ITERATION WITH HESSENBERG MATRICES 626
DEGENERATE CASES 627 INVERSE ITERATION WITH BAND MATRICES 628 COMPLEX
CONJUGATE EIGENVECTORS 629 ERROR ANALYSIS 631 NUMERICAL EXAMPLE 633 THE
GENERALIZED EIGENVALUE PROBLEM 633 VARIATION OF APPROXIMATE EIGENVALUES
635 REFINEMENT OF EIGENSYSTEMS 637 NUMERICAL EXAMPLE 639 REFINEMENT OF
THE EIGENVECTORS 641 COMPLEX CONJUGATE EIGENVALUES 643 COINCIDENT AND
PATHOLOGICALLY CLOSE EIGENVALUES 644 COMMENTS ON THE ACE PROGRAMMES 646
BIBLIOGRAPHY 649 INDEX 657
|
| any_adam_object | 1 |
| author | Wilkinson, James H. 1919-1986 |
| author_GND | (DE-588)118995006 |
| author_facet | Wilkinson, James H. 1919-1986 |
| author_role | aut |
| author_sort | Wilkinson, James H. 1919-1986 |
| author_variant | j h w jh jhw |
| building | Verbundindex |
| bvnumber | BV025885858 |
| ctrlnum | (OCoLC)258121621 (DE-599)BVBBV025885858 |
| dewey-full | 512.9'434 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.9'434 |
| dewey-search | 512.9'434 |
| dewey-sort | 3512.9 3434 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Reprinted |
| format | Book |
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| id | DE-604.BV025885858 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:14:23Z |
| institution | BVB |
| isbn | 0198534183 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-019132564 |
| oclc_num | 258121621 |
| open_access_boolean | |
| owner | DE-11 |
| owner_facet | DE-11 |
| physical | XVIII, 662 S. |
| publishDate | 1992 |
| publishDateSearch | 1992 |
| publishDateSort | 1992 |
| publisher | Clarendon Press |
| record_format | marc |
| series2 | Monographs on numerical analysis Oxford science publications |
| spelling | Wilkinson, James H. 1919-1986 Verfasser (DE-588)118995006 aut The algebraic eigenvalue problem by J.H. Wilkinson Reprinted Oxford Clarendon Press 1992 XVIII, 662 S. txt rdacontent n rdamedia nc rdacarrier Monographs on numerical analysis Oxford science publications Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Matrix Mathematik (DE-588)4037968-1 gnd rswk-swf Algebra (DE-588)4001156-2 gnd rswk-swf Eigenwertproblem (DE-588)4013802-1 gnd rswk-swf Eigenwert (DE-588)4151200-5 gnd rswk-swf Matrix Mathematik (DE-588)4037968-1 s Eigenwertproblem (DE-588)4013802-1 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Algebra (DE-588)4001156-2 s 1\p DE-604 Eigenwert (DE-588)4151200-5 s 2\p DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019132564&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Wilkinson, James H. 1919-1986 The algebraic eigenvalue problem Numerisches Verfahren (DE-588)4128130-5 gnd Matrix Mathematik (DE-588)4037968-1 gnd Algebra (DE-588)4001156-2 gnd Eigenwertproblem (DE-588)4013802-1 gnd Eigenwert (DE-588)4151200-5 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4037968-1 (DE-588)4001156-2 (DE-588)4013802-1 (DE-588)4151200-5 |
| title | The algebraic eigenvalue problem |
| title_auth | The algebraic eigenvalue problem |
| title_exact_search | The algebraic eigenvalue problem |
| title_full | The algebraic eigenvalue problem by J.H. Wilkinson |
| title_fullStr | The algebraic eigenvalue problem by J.H. Wilkinson |
| title_full_unstemmed | The algebraic eigenvalue problem by J.H. Wilkinson |
| title_short | The algebraic eigenvalue problem |
| title_sort | the algebraic eigenvalue problem |
| topic | Numerisches Verfahren (DE-588)4128130-5 gnd Matrix Mathematik (DE-588)4037968-1 gnd Algebra (DE-588)4001156-2 gnd Eigenwertproblem (DE-588)4013802-1 gnd Eigenwert (DE-588)4151200-5 gnd |
| topic_facet | Numerisches Verfahren Matrix Mathematik Algebra Eigenwertproblem Eigenwert |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019132564&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT wilkinsonjamesh thealgebraiceigenvalueproblem |