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| Ïîèñê ïî óêàçàòåëÿì |
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| Wilkinson J.H. — The algebraic eigenvalue problem |
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| Ïðåäìåòíûé óêàçàòåëü |
Accumulation of inner-products in fixed-point 111—112
Accumulation of inner-products in floating-point 116—118
ACE 116 118 260 324 329 342 357 404 412 462 477 484 595 629 638 642 646
Adjoint matrix 38
Aitken, A.C., on acceleration of convergence of eigenvectors 578—579 (see also “Turnbull H.W.”)
Arithmetic block-floating 119
Arithmetic fixed-point 110—112
Arithmetic floating-point 112—118
Arithmetic floating-point, complex 447
Arithmetic floating-point, t-digit 120
Arnoldi, W.E., on reduction to Hessenberg form 382
Arnoldi’s method 244 382—388
Back-substitution 199
Back-substitution, error analysis of 247—251
Bairstow, L., on solution of algebraic equations 449
Bairstow’s method 449—450 482
Bairstow’s method, generalized 450—452 475—476
Band matrices, LR and QR algorithms for, symmetric 553—562
Band matrices, LR and QR algorithms for, unsymmetric 562—565
Band matrices, reduction of band width of 567—568
Bauer, F.L. see “Householder A.S.”
Bauer, F.L., and Fike, C.T., on the spectral condition number 87
Bauer, F.L., and Householder, A.S., on error in Rayleigh quotient 188
Bauer, F.L., equilibration 263 357 411
Bauer, F.L., on bi-iteration 609—610 647
Bauer, F.L., reduction to tri-diagonal form 412
Bauer, F.L., treppeniteration 599 602—604 647
Bessel functions, zeros of 440
Bi-iteration 609—610 613—614 647
Bi-orthogonal system of eigenvectors 6
Bi-orthogonal system of Lanczos vectors 391
Bilinear form 27
Binet — Cauchy theorem 18
Bisection method 302 440
Bisection method, error analysis of 302—306 453—455
Block-floating arithmetic 119
Bolzano — Weierstrass theorem 105
Brooker, R.A., and Sumner, F.H., Lanczos’ method 412
Brooker, R.A., and Sumner, F.H., on eigenvectors of tri-diagonal form 344
Canonical form, classical 10—11
Canonical form, Frobenius 15—17
Canonical form, Jordan 10—12
Canonical form, rational 15—18
Canonical form, Smith’s 19—24
Canonical form, triangular 24
Causey, R., and Gregory, R.T., on Lanczos’ method 412
Causey, R., on methods of Jacobi type for non-Hermitian matrices 568
Cayley — Hamilton theorem 38—39
Characteristic equation 2
Characteristic polynomial 2
Characteristic polynomial, direct determination of 410—411
Characteristic value 2
Characteristic vector 3
Chartres, B.A., on a modification of the Jacobi method 282
Chebyshev polynomials 617—618
Cholesky decomposition 229—230
Cholesky decomposition, error analysis of 231—232
Cholesky LR algorithm 544—547
Cholesky LR algorithm for band matrices 553—556
Classical canonical form 10—11
Classical submatrix 10
Collar, A.R., on improving approximate eigensystem 647
Commuting matrices 52—54
Companion matrix 12—13
Complete pivoting 212—213
Complex conjugate eigenvalues 579—581 604—606
Complex conjugate eigenvectors 581—582 604—606
Complex conjugate eigenvectors, inverse iteration for 629—633
Complex conjugate shifts of origin in LR 512—515
Complex conjugate shifts of origin in QR 528—532
Complex conjugate zeros 447—449
Complex Hermitian matrices, Householder’s method for 342
Complex non-Hermitian matrices 482—483
Complex symmetric matrices 26—27 265
Complex zeros 446—447
Condition number 86
Condition number of explicit polynomials 416—420
Condition number, n 89—90
Condition number, spectral, with respect to eigenproblem 87—89
Condition number, spectral, with respect to inversion 191
Corresponding Matrices, Theorem of 18 21 491 523
Courant — Fischer theorem 101
Courant, R., and Hilbert, D., on minimax characterization of eigenvalues 99—100
Danilewski, A., on reduction of a general matrix to Frobenius form 409
Dean, P., on eigenvalues of large tri-diagonal matrices 307
Defective matrix 12
Defective system of eigenvectors 9—10
Deflation by plane rotations 469—471
Deflation for explicit polynomials 464—465
Deflation for Hessenberg matrices 465—468
Deflation for tri-diagonal matrices 468—469
Deflation in connexion with LR 509—510
Deflation in connexion with matrix iteration 584—602
Deflation, stability of 472—474 477
Deflation, using non-similarity transformations 596—602
Deflation, using similarity transformations 585—596
Dekker, T.J., on eigenvectors of tri-diagonal matrices 330—332
Dekker, T.J., successive linear interpolation and bisection 481
Departure from normality 167—170 568—569
Derogatory matrix 15—16 41
Determinants of general matrices 431—432
Determinants of Hessenberg matrices 426—431
Determinants of lambda-matrices 34 432
Determinants of leading principal matrices 237—239
Determinants of tri-diagonal matrices 423—426
Determinants, evaluation of 228—229
Deuce 278 342 376 625
Differential equations with constant coefficients 30—35 443 483—484 633—635
Direct sum of matrices 10—11
Disorder of eigenvalues 494 519 522
Dominant eigenvalue 571
Dominant eigenvector 571
Double shift techniques, LR 537—538
Double shift techniques, QR 528—537
Eberlein, P.J., on a Jacobi-like method 568
Eigenvalues 2
Eigenvalues of AB 54—55
Eigenvalues of transposed matrix 3
Eigenvalues, distinct 4
Eigenvalues, multiple 7—11
Eigenvectors 3
Eigenvectors of transposed matrix 3
Eigenvectors, defective system of 9—10
Eigenvectors, independence of 5
Eigenvectors, left-hand 4
Eigenvectors, right-hand 4
Elementary divisors 12
Elementary Hermitian matrices 48—50
Elementary Hermitian matrices in fixed-point 152
Elementary Hermitian matrices in floating-point 152—162
Elementary Hermitian matrices, triangularization by 233—236
Elementary Hermitian matrices, use of, for deflation 589—590 594—596
Elementary Hermitian matrices, use of, for similarity reduction to Hessenberg form 347—353
Elementary Hermitian matrices, use of, for similarity reduction to tri-diagonal form 290—299
Elementary matrices 44—45
Elementary matrices, properties of 45—46
Elementary matrices, stabilized 164
Elementary operations 19
Engeli, M., Ginsburg, T., Rutishauser, H., and Stiefel, E., on iteration using orthogonal polynomials 618
Equilibration in eigenvalue problem 357—411
Equilibration in linear equations 192—194 263
Equivalence transformation 17—24
Error analysis 110—187
Error analysis for complete eigensystem 180
Error analysis of Arnoldi’s method 382—388
Error analysis, back-substitution 247—251
Error analysis, bisection method 302—306 453—455
Error analysis, Cholesky’s decomposition 231—232
Error analysis, deflation techniques 590—596
Error analysis, direct reduction to Hessenberg form 363—365
| Error analysis, elementary Hermitians in fixed-point 152
Error analysis, elementary Hermitians in floating-point 152—162
Error analysis, fixed-point 110—112
Error analysis, floating-point 112—118
Error analysis, Gaussian elimination 209—212 214—215
Error analysis, Givens’ method 286—288
Error analysis, Givens’ triangularization 240
Error analysis, Householder’s method, symmetric 297
Error analysis, Householder’s method, unsymmetric 350
Error analysis, Householder’s triangularization 236
Error analysis, interpolation methods 435—440
Error analysis, inverse iteration for complex conjugate eigenvectors 631—633
Error analysis, inverse iteration for eigenvectors of general matrices 620—621
Error analysis, inverse iteration for eigenvectors of tri-diagonal matrices 323—325
Error analysis, Lanczos’ method 391—395
Error analysis, matrix squaring 616
Error analysis, plane rotations in fixed-point 143—151
Error analysis, plane rotations in floating-point 131—143
Error analysis, QR algorithm 538
Error analysis, QR algorithm for band matrices 561
Error analysis, reduction of Hessenberg matrix to Frobenius form 405
Error analysis, reduction of Hessenberg matrix to tri-diagonal form 399
Error analysis, triangular decomposition 227
Euclidean norm 57
Exponent of floating-point number 112
Extremal properties of eigenvalues 98—99
Feller, W., and Forsythe, G.E., on deflation using elementary Hermitian matrices 590
Fixed-point arithmetic 110—112
Floating-point arithmetic 112—118
Floating-point arithmetic, complex 447
Floating-point arithmetic, t-digit 120
Forsythe, G.E. see “Feller W.”
Forsythe, G.E., and Straus, E.G., on equilibration 263
Forsythe, G.E., and Wasow, W.R., on iterative methods for solving equations 263
Forsythe, G.E., on eigenvectors of tri-diagonal matrices 344
Fox, L., on numerical linear algebra 189 263
Francis, J.G.F., on QR algorithm 486 515 525 526 569
Frobenius canonical form 15—17
Frobenius canonical form, reduction to 405—411
Frobenius canonical form, relation to Jordan form 16—17
Frobenius canonical form, relation to minimal polynomial 39—42
Frobenius canonical form, roots of 413—423
Gantmacher, F.R., on the theory of matrices 61 496
Gantmacher, F.R., Theorem of Corresponding Matrices 18
Garwick, J., on termination of iterative processes 463
Gastinel, N., on error analysis 264
Gaussian elimination 200 484
Gaussian elimination by columns 370—371
Gaussian elimination on high-speed computer 220—221
Gaussian elimination, break-down of 204—205 224
Gaussian elimination, error analysis of 209—212 214—215
Gaussian elimination, relation with triangular decomposition 221—223
Gerschgorin’s theorems 71—72 109
Gerschgorin’s theorems, application to perturbation theory 72—81 638—646
Ginsburg, T. see “Engeli M.”
Givens, J.W., on error analysis 343
Givens, J.W., reduction to tri-diagonal form 282
Givens, J.W., Sturm sequences 300 307 344
Givens, J.W., triangularization 240—241
Givens, J.W., Wielandt-Hoffman theorem 105
Givens’ method, relationship with Householder’s method 351—353
Givens’ method, symmetric 282—290
Givens’ method, symmetric, error analysis of 286—290
Givens’ method, unsymmetric 345—347
Givens’ method, unsymmetric, error analysis of 350—351
Goldstine, H.H. see “von Neumann J.”
Goldstine, H.H., and Horwitz, L.P., on the Jacobi method for normal matrices 486
Goldstine, H.H., and Murray, F.J., and Von Neumann, J., on the Jacobi method 343
Goursat, E., on algebraic functions 64
Grade of vector 37
Greenstadt, J., on Jacobi-type methods 568
Gregory, R.T. see “Causey R.”
Gregory, R.T., on the Jacobi method 343
Gregory, R.T., on the Lanczos’ method 412
Handscomb, D.C., on Bairstow’s method for Hessenberg matrices 452 476
Henrici, P., on departure from normality 168 568
Henrici, P., serial Jacobi method 269 270
Hermitian matrix 24—26 (see also “Elementary Hermitian matrices”)
Hermitian matrix, eigenvalue problem for 265—344
Hermitian matrix, elementary 48—50
Hermitian transposed matrix 25
Hessenberg matrix 218
Hessenberg matrix, determinants of 426—431
Hessenberg matrix, direct reduction to 357—365
Hessenberg matrix, further reduction of, to Frobenius form 405—411
Hessenberg matrix, further reduction of, to tri-diagonal form 396—403
Hessenberg matrix, invariance with respect to LR algorithm 502—504
Hessenberg matrix, invariance with respect to QR algorithm 524
Hessenberg matrix, reduction to 345—395
Hessenberg method 379—382
Hessenberg method, applied to Hessenberg matrix 492—505
Hessenberg method, generalized 377—395
Hessenberg method, relation to Lanczos’ method 402—403
Hessenberg, K., on reduction to Hessenberg form 379—382
Hestenes, M.R. see “Rosser J.B.”
Hilbert matrix 233—234
Hilbert, D. see “Courant R.”
Hoffman, A.J., and Wielandt, H.W., on the Wielandt — Hoffman theorem 104
Hotelling, H., on matrix deflation 585 607
Householder, A.S., and Bauer, F.L., on Householder’s method 343
Householder, A.S., on the theory of matrices in numerical analysis 61 109 569 647
Householder, A.S., unitary triangularization 233
Householder, A.S., use of elementary Hermitian matrices 343
Householder’s method, relationship with Givens’ method 351—353
Householder’s method, symmetric 290—299
Householder’s method, symmetric, error analysis of 297
Householder’s method, unsymmetric 347—353
Householder’s method, unsymmetric, error analysis of 350
Hyman, M.A., on Hymen’s method 426—431 469 627
Inner-product 4 25
Inner-product, accumulation in fixed-point 111—112
Inner-product, accumulation in floating-point 116—118
Interchanges 162 205—207 399—401 498—501
Interpolation methods 435—440
Interpolation methods, error analysis of 455—459
Invariant factors 22—24
Invariant factors, deflation using 587 599—602
Invariant factors, subspaces 185
Inverse iteration for complex conjugate eigenvectors 629—633
Inverse iteration for eigenvectors of band matrices 628—629
Inverse iteration of general matrices 619—633
Inverse iteration of generalized eigenvalue problem 633—635
Inverse iteration of Hessenberg matrices 626—627
Inverse iteration of symmetric tri-diagonal matrices 321—330
Inversion of matrices 252—254
Iterative method for the solution of the eigenvalue problem 570—576
Iterative method, refinement of eigensystem 637—646
Iterative method, refinement of solution of linear equations 255—263
Jacobi, C.G.J., on the Jacobi method 266
Jacobi’s method 270—282 486 568
Jacobi’s method for normal matrices 486
Jacobi’s method, classical 266—269
Jacobi’s method, convergence of 270—273
Jacobi’s method, error analysis of 279—281
Jacobi’s method, serial 269
Jacobi’s method, threshold 277—278
Jahn, H.A., on improvement of an eigensystem 647
Johansen, D.E., on Givens’ method for large matrices 284—286
Jordan canonical form 10—12
Jordan canonical form, perturbation theory based on 77—81
Jordan canonical form, relation to Frobenius form 16—17
Jordan canonical form, relation to minimum polynomial 41—42
Kahan, W., on acceleration of convergence of Muller’s method 481
Kahan, W., on acceleration of convergence of Newton’s method 480
Kaiser, H.F. see “Ortega J.M.”
Karush, W. see “Rosser J.B.”
Kato, T., on inclusion theorems for eigenvalues of Hermitian matrices 188
Krylov, A.N., on Krylov’s method 369
Krylov’s method 369—377 417 421 584
Kublanovskaya, V.N., on QR algorithm 569
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