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| Wilkinson J.H. — The algebraic eigenvalue problem |
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| Ïðåäìåòíûé óêàçàòåëü |
Laguerre’s method 443—445 478 479 482 484
Laguerre’s method, failure of convergence of 481
Lambda matrices 18—24
Lambda matrices, determinants of 34 432
Lanczos, C., on Lanczos’ method 388 (see also “Rosser J.B.”)
Lanczos’ method 388—394
Lanczos’ method, error analysis of 391—395
Lanczos’ method, need for re-orthogonalization in 391—392
Lanczos’ method, relation to Hessenberg’s method 402—403
Lanczos’ method, symmetric 394—395
latent root 2
Latent vector 3
Le Verrier’s method 434—435
Lotkin, M., on Jacobi-type method 568
Lower-triangular matrix 24
LR algorithm for band matrices, symmetric 553—556
LR algorithm for band matrices, unsymmetric 562
LR algorithm, complex conjugate eigenvalues 494—497
LR algorithm, convergence of 489—493 521—523
LR algorithm, convergence of acceleration of 505—509 511
LR algorithm, cubically convergent version 551—553
LR algorithm, deflation technique associated with 509—510
LR algorithm, positive definite matrices 493—494
LR algorithm, relation of symmetric, with QR 545—546
LR algorithm, symmetric Cholesky 544—547
LR algorithm, tri-diagonal matrices 562—565
Maehly, H.J., on acceleration of Laguerre’s method 484
Maehly, H.J., on acceleration of Newton’s method 480
Magnier, A., on improvement of an eigensystem 547
Mantissa of floating-point number 112
Matrix squaring 615—617
Minimax theorem 99—104
Minimum polynomial of a matrix 37—38
Minimum polynomial of a vector 36—37
Monic polynomial 20
Muller, D.E., on quadratic interpolation 435
Muller’s method 435—438
Muller’s method, accuracy of, for complex zeros 481 483
Muller’s method, accuracy of, for double zeros 458—459
Muller’s method, improved formulation of 484
Murray, F.J. see “Goldstine H.H.”
Newton’s method 441—443
Newton’s method, comparison with interpolation methods 442—443
Newton’s method, comparison with Laguerre’s method 445
Newton’s method, limitation of root variation with 483
Newton’s method, limiting behaviour of 459—461
Newton’s method, use in connexion with suppression of zeros 475 478
Non-derogatory matrices 13—15
Non-unitary elementary matrices 162—166
Norm of matrix 55—61
Norm of vector 55
Norm, compatible 56
Norm, Euclidean 57
Norm, Schur 57
Norm, spectral 57
Norm, subordinate 56
Normal matrices 51—52
Normal matrices, a posteriori bounds for 170—174
Normal matrices, Jacobi’s method for 486
Normalized vector 3
Ortega, J.M., on error analysis of Householder’s method 297—298 344
Orthogonal matrix 26
Orthogonalization see “Schmidt orthogonalization.”
Osborne, E.E., iteration for eigenvectors 637
Osborne, E.E., on equilibration 357 411
Ostrowski, A.M., inverse interpolation 439
Ostrowski, A.M., inverse iteration 637
Ostrowski, A.M., on continuity of eigenvalues 63—64
Parlett, B., on use of Laguerre’s method for eigenvalues 479 481 484
Permutation matrices 44
Perturbation theory 62—109
Perturbation theory for linear equations 189—190
Perturbation theory for multiple eigenvalues and linear divisors 75—77
Perturbation theory for non-linear divisors 77—81
Perturbation theory for real symmetric matrices 93—109
Perturbation theory for simple eigenvalues 66—70 72—75
Pivot 200
Pivoting 206
Pivoting, complete 212—213
Pivoting, necessity for 215—216
Pivoting, partial 212—213
Plane rotations 47—48
Plane rotations, fixed-point error analysis 143—151
Plane rotations, floating-point error analysis 131—143
Plane rotations, triangularization by 239—241
Plane rotations, use of in Jacobi’s method 266—269
Pope, D.A., and Tompkins, O., on the threshold Jacobi method 277—278 343.
Positive definite form 28—30
Positive definite matrix 28—30
Positive definite matrix, Cholesky decomposition of 229—233
Positive definite matrix, LR algorithm for 493—494 544
Power method 570—571
Principal vector 42—43
Principal vector, grade of 43
Proper value 3
Proper vector 3
Purification process, Richardson’s 614—615
qd algorithm 564
QR algorithm 515
QR algorithm for symmetric band matrix 557—562
QR algorithm, convergence of 516—521
QR algorithm, cubically convergent form of 548—549
QR algorithm, relation with symmetric LR algorithm 545—546
| Quadratic form 27—28
Quasi-symmetric tri-diagonal matrices 335—337
Rational canonical form 15—18
Rayleigh quotient 172—178 595 629
Rayleigh quotient, generalized 179 182
Rayleigh quotient, inverse iteration using 636
Re-orthogonalization in Arnoldi’s method 383—388
Re-orthogonalization in Lanczos’ method 391—395
Refinement of coincident and pathologically close eigenvalues 644—646
Refinement of complex conjugate eigensystem 643—644
Refinement of eigenvalues 637—641
Refinement of eigenvectors 641—642
Refinement, solution of linear equations 255—263
Residual vector in eigenvalue problem 171 639
Residual vector in linear equations problem 248—249 252 255
Residual vector, use in iterative refinement of eigensystem 637—646
Residual vector, use in iterative refinement of solution of linear system 257—261
Richardson’s purification process 614—615
Rollett, J.S., and Wilkinson, J.H., on Givens’ method on machine with two-level store 284—286
Rosser, J.B., Lanczos, C., Hestenes, M.R., and Karush, W., on eigenvectors of tri-diagonal matrices 344
Rosser, J.B., method of Lanczos 412
Rotations see “Plane rotations”
Rutishauser, H., and Schwarz, H.R., on stratagems for cubic convergence of symmetric LR algorithm 553
Rutishauser, H., on accuracy of cubic convergence of version of LR algorithm 485 487 569
Rutishauser, H., on accuracy of cubic convergence of version of symmetric LR algorithm 550 556 562
Rutishauser, H., on accuracy of eigenvalues in LR algorithm 556
Rutishauser, H., on accuracy of LR algorithm 499—500
Rutishauser, H., on accuracy of width of band matrix 567—568
Schmidt orthogonalization 243—244 383 386 517 607 612
Schoenhage, A., on convergence of Jacobi’s method 270—271 343
Schur norm 57
Schwarz, H.R. see “Rutishauser H.”
Separation theorem 103—104
Similarity transformation 6
Similarity transformation, elementary 43—47
Singular values 57
Smith’s canonical form 19—24
Spectral norm 57
Spectral radius 59
Square roots, rounding errors in 118
Stabilized elementary matrices 164
Stabilized elementary transformations 469 587
Standardized vector 3
Stiefel, E., on orthogonal polynomials 618 (see also “Engeli M.”)
Sturm sequence property 300 307 344
Sumner, F.H. see “Brooker R.A.”
Suppression of quadratic factors 475—476
Suppression of zeros 474—475
Suppression, stability of 476—477
SWAC 278
Sylvester’s theorem 496
Taussky, O., on Gerschgorin’s theorems 109
Temple, G., on the Rayleigh quotient 188
Todd, J., on finite segments of the Hilbert matrix 233—234
Tompkins, C. see “Pope D.A.”
Trapezoidal form 601 602 619
Traub, J.F., on formulation of Muller’s method 484
Treppeniteration 599 602—604
Treppeniteration, example of 610—612
Treppeniteration, using inverse matrix 647
Tri-diagonal matrix 121
Tri-diagonal matrix, deflation of 468
Tri-diagonal matrix, determinants of 423—426
Tri-diagonal matrix, Gaussian elimination of 312—315
Tri-diagonal matrix, inverse iteration with 628
Tri-diagonal matrix, LR algorithm for 562—565
Tri-diagonal matrix, QR algorithm for 565—567
Tri-diagonal matrix, quasi-symmetric 335—337
Tri-diagonal matrix, reduction of Hessenberg matrix to 388—404
Tri-diagonal matrix, symmetric, eigenvalues of 299—315
Tri-diagonal matrix, symmetric, eigenvectors of 315—332
Triangular canonical form 24
Triangular canonical form, reduction to 46—47 50—51 485
Triangular decomposition 201—204
Triangular decomposition, direct 222—223
Triangular decomposition, direct, error analysis of 227
Triangular decomposition, examples of failure and non-uniqueness of 224
Triangular decomposition, relation with Gaussian elimination 223
Triangular decomposition, relation with row interchanges 225—228
Triangularization by elementary Hermitians 233—236
Triangularization by elementary stabilized matrices 236
Triangularization by plane rotations 239—241
Turing, A.M., on rounding errors in matrix processes 264 344
Turnbull, H.W., and Aitken, A.C., on Frobenius canonical form 39
Unitary matrix 26
Unitary matrix, elementary 47—48
Upper triangular matrix 24
Varga, R.S., on iterative methods for solving equations 263
von Neumann, J., and Goldstine, H.H., on inversion of matrices 187 264 344 H.H.”)
Voyevodin, V.V., on an orthogonalization method for the general eigenvalue problem 647
Wasow, W.R. see “Forsythe G.E.”
White, P.A., on determinants of Hessenberg matrices 430
Wielandt iteration see “Inverse iteration”
Wielandt, H.W., on deflation 596 599 A.J.”)
Wilkinson, J.H. see “Rollett J.S.”
Wilkinson, J.H., error analysis of direct methods of matrix inversion 213 218 233 251
Wilkinson, J.H., error analysis of floating-point computation 187
Wilkinson, J.H., error analysis of orthogonal similarity transformations 298 344 594
Wilkinson, J.H., Householder’s method 344
Wilkinson, J.H., Lanczos’ method 412
Wilkinson, J.H., LR algorithm for Hessenberg matrices 569
Wilkinson, J.H., on eigenvectors of tri-diagonal matrices 326
Wilkinson, J.H., quadratic convergence of Jacobi’s method 270 271 343
Wilkinson, J.H., reduction of a general matrix to tri-diagonal form 399—402
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