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Wilkinson J.H. The algebraic eigenvalue problem
Wilkinson J.H.  The algebraic eigenvalue problem









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: The algebraic eigenvalue problem

: Wilkinson J.H.

:

This volume, which became a classic on first publication, is perhaps the most important and widely read book in the field of numerical analysis. It presents a distillation of the author's pioneering discoveries concerning the computation of matrix eigenvalues. The emphasis is on the transmission of knowledge rather than elaborate proofs. The book will be valued by all practising numerical analysts, students and researchers in the field, engineers, and scientists.


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Accumulation of inner-products in fixed-point      111112
Accumulation of inner-products in floating-point      116118
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      578579 (see also Turnbull H.W.)
Arithmetic block-floating      119
Arithmetic fixed-point      110112
Arithmetic floating-point      112118
Arithmetic floating-point, complex      447
Arithmetic floating-point, t-digit      120
Arnoldi, W.E., on reduction to Hessenberg form      382
Arnoldis method      244 382388
Back-substitution      199
Back-substitution, error analysis of      247251
Bairstow, L., on solution of algebraic equations      449
Bairstows method      449450 482
Bairstows method, generalized      450452 475476
Band matrices, LR and QR algorithms for, symmetric      553562
Band matrices, LR and QR algorithms for, unsymmetric      562565
Band matrices, reduction of band width of      567568
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      609610 647
Bauer, F.L., reduction to tri-diagonal form      412
Bauer, F.L., treppeniteration      599 602604 647
Bessel functions, zeros of      440
Bi-iteration      609610 613614 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      302306 453455
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      1011
Canonical form, Frobenius      1517
Canonical form, Jordan      1012
Canonical form, rational      1518
Canonical form, Smiths      1924
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      3839
Characteristic equation      2
Characteristic polynomial      2
Characteristic polynomial, direct determination of      410411
Characteristic value      2
Characteristic vector      3
Chartres, B.A., on a modification of the Jacobi method      282
Chebyshev polynomials      617618
Cholesky decomposition      229230
Cholesky decomposition, error analysis of      231232
Cholesky LR algorithm      544547
Cholesky LR algorithm for band matrices      553556
Classical canonical form      1011
Classical submatrix      10
Collar, A.R., on improving approximate eigensystem      647
Commuting matrices      5254
Companion matrix      1213
Complete pivoting      212213
Complex conjugate eigenvalues      579581 604606
Complex conjugate eigenvectors      581582 604606
Complex conjugate eigenvectors, inverse iteration for      629633
Complex conjugate shifts of origin in LR      512515
Complex conjugate shifts of origin in QR      528532
Complex conjugate zeros      447449
Complex Hermitian matrices, Householders method for      342
Complex non-Hermitian matrices      482483
Complex symmetric matrices      2627 265
Complex zeros      446447
Condition number      86
Condition number of explicit polynomials      416420
Condition number, n      8990
Condition number, spectral, with respect to eigenproblem      8789
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      99100
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      910
Deflation by plane rotations      469471
Deflation for explicit polynomials      464465
Deflation for Hessenberg matrices      465468
Deflation for tri-diagonal matrices      468469
Deflation in connexion with LR      509510
Deflation in connexion with matrix iteration      584602
Deflation, stability of      472474 477
Deflation, using non-similarity transformations      596602
Deflation, using similarity transformations      585596
Dekker, T.J., on eigenvectors of tri-diagonal matrices      330332
Dekker, T.J., successive linear interpolation and bisection      481
Departure from normality      167170 568569
Derogatory matrix      1516 41
Determinants of general matrices      431432
Determinants of Hessenberg matrices      426431
Determinants of lambda-matrices      34 432
Determinants of leading principal matrices      237239
Determinants of tri-diagonal matrices      423426
Determinants, evaluation of      228229
Deuce      278 342 376 625
Differential equations with constant coefficients      3035 443 483484 633635
Direct sum of matrices      1011
Disorder of eigenvalues      494 519 522
Dominant eigenvalue      571
Dominant eigenvector      571
Double shift techniques, LR      537538
Double shift techniques, QR      528537
Eberlein, P.J., on a Jacobi-like method      568
Eigenvalues      2
Eigenvalues of AB      5455
Eigenvalues of transposed matrix      3
Eigenvalues, distinct      4
Eigenvalues, multiple      711
Eigenvectors      3
Eigenvectors of transposed matrix      3
Eigenvectors, defective system of      910
Eigenvectors, independence of      5
Eigenvectors, left-hand      4
Eigenvectors, right-hand      4
Elementary divisors      12
Elementary Hermitian matrices      4850
Elementary Hermitian matrices in fixed-point      152
Elementary Hermitian matrices in floating-point      152162
Elementary Hermitian matrices, triangularization by      233236
Elementary Hermitian matrices, use of, for deflation      589590 594596
Elementary Hermitian matrices, use of, for similarity reduction to Hessenberg form      347353
Elementary Hermitian matrices, use of, for similarity reduction to tri-diagonal form      290299
Elementary matrices      4445
Elementary matrices, properties of      4546
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      357411
Equilibration in linear equations      192194 263
Equivalence transformation      1724
Error analysis      110187
Error analysis for complete eigensystem      180
Error analysis of Arnoldis method      382388
Error analysis, back-substitution      247251
Error analysis, bisection method      302306 453455
Error analysis, Choleskys decomposition      231232
Error analysis, deflation techniques      590596
Error analysis, direct reduction to Hessenberg form      363365
Error analysis, elementary Hermitians in fixed-point      152
Error analysis, elementary Hermitians in floating-point      152162
Error analysis, fixed-point      110112
Error analysis, floating-point      112118
Error analysis, Gaussian elimination      209212 214215
Error analysis, Givens method      286288
Error analysis, Givens triangularization      240
Error analysis, Householders method, symmetric      297
Error analysis, Householders method, unsymmetric      350
Error analysis, Householders triangularization      236
Error analysis, interpolation methods      435440
Error analysis, inverse iteration for complex conjugate eigenvectors      631633
Error analysis, inverse iteration for eigenvectors of general matrices      620621
Error analysis, inverse iteration for eigenvectors of tri-diagonal matrices      323325
Error analysis, Lanczos method      391395
Error analysis, matrix squaring      616
Error analysis, plane rotations in fixed-point      143151
Error analysis, plane rotations in floating-point      131143
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      9899
Feller, W., and Forsythe, G.E., on deflation using elementary Hermitian matrices      590
Fixed-point arithmetic      110112
Floating-point arithmetic      112118
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      1517
Frobenius canonical form, reduction to      405411
Frobenius canonical form, relation to Jordan form      1617
Frobenius canonical form, relation to minimal polynomial      3942
Frobenius canonical form, roots of      413423
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      370371
Gaussian elimination on high-speed computer      220221
Gaussian elimination, break-down of      204205 224
Gaussian elimination, error analysis of      209212 214215
Gaussian elimination, relation with triangular decomposition      221223
Gerschgorins theorems      7172 109
Gerschgorins theorems, application to perturbation theory      7281 638646
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      240241
Givens, J.W., Wielandt-Hoffman theorem      105
Givens method, relationship with Householders method      351353
Givens method, symmetric      282290
Givens method, symmetric, error analysis of      286290
Givens method, unsymmetric      345347
Givens method, unsymmetric, error analysis of      350351
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 Bairstows method for Hessenberg matrices      452 476
Henrici, P., on departure from normality      168 568
Henrici, P., serial Jacobi method      269 270
Hermitian matrix      2426 (see also Elementary Hermitian matrices)
Hermitian matrix, eigenvalue problem for      265344
Hermitian matrix, elementary      4850
Hermitian transposed matrix      25
Hessenberg matrix      218
Hessenberg matrix, determinants of      426431
Hessenberg matrix, direct reduction to      357365
Hessenberg matrix, further reduction of, to Frobenius form      405411
Hessenberg matrix, further reduction of, to tri-diagonal form      396403
Hessenberg matrix, invariance with respect to LR algorithm      502504
Hessenberg matrix, invariance with respect to QR algorithm      524
Hessenberg matrix, reduction to      345395
Hessenberg method      379382
Hessenberg method, applied to Hessenberg matrix      492505
Hessenberg method, generalized      377395
Hessenberg method, relation to Lanczos method      402403
Hessenberg, K., on reduction to Hessenberg form      379382
Hestenes, M.R.      see Rosser J.B.
Hilbert matrix      233234
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 Householders 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
Householders method, relationship with Givens method      351353
Householders method, symmetric      290299
Householders method, symmetric, error analysis of      297
Householders method, unsymmetric      347353
Householders method, unsymmetric, error analysis of      350
Hyman, M.A., on Hymens method      426431 469 627
Inner-product      4 25
Inner-product, accumulation in fixed-point      111112
Inner-product, accumulation in floating-point      116118
Interchanges      162 205207 399401 498501
Interpolation methods      435440
Interpolation methods, error analysis of      455459
Invariant factors      2224
Invariant factors, deflation using      587 599602
Invariant factors, subspaces      185
Inverse iteration for complex conjugate eigenvectors      629633
Inverse iteration for eigenvectors of band matrices      628629
Inverse iteration of general matrices      619633
Inverse iteration of generalized eigenvalue problem      633635
Inverse iteration of Hessenberg matrices      626627
Inverse iteration of symmetric tri-diagonal matrices      321330
Inversion of matrices      252254
Iterative method for the solution of the eigenvalue problem      570576
Iterative method, refinement of eigensystem      637646
Iterative method, refinement of solution of linear equations      255263
Jacobi, C.G.J., on the Jacobi method      266
Jacobis method      270282 486 568
Jacobis method for normal matrices      486
Jacobis method, classical      266269
Jacobis method, convergence of      270273
Jacobis method, error analysis of      279281
Jacobis method, serial      269
Jacobis method, threshold      277278
Jahn, H.A., on improvement of an eigensystem      647
Johansen, D.E., on Givens method for large matrices      284286
Jordan canonical form      1012
Jordan canonical form, perturbation theory based on      7781
Jordan canonical form, relation to Frobenius form      1617
Jordan canonical form, relation to minimum polynomial      4142
Kahan, W., on acceleration of convergence of Mullers method      481
Kahan, W., on acceleration of convergence of Newtons 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 Krylovs method      369
Krylovs method      369377 417 421 584
Kublanovskaya, V.N., on QR algorithm      569
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