The Computation of Eigenvalues and Eigenvectors of a Matrix
1958; Society for Industrial and Applied Mathematics; Volume: 6; Issue: 4 Linguagem: Inglês
10.1137/0106027
ISSN2168-3484
Autores Tópico(s)Mathematics and Applications
ResumoPrevious article Next article The Computation of Eigenvalues and Eigenvectors of a MatrixPaul A. WhitePaul A. Whitehttps://doi.org/10.1137/0106027PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Cornelius Lanczos, Applied analysis, Prentice Hall Inc., Englewood Cliffs, N. J., 1956xx+539 MR0084175 0111.12403 Google Scholar[2] E. Bodewig, Matrix calculus, North-Holland Publishing Company, Amsterdam, 1956xii+334 MR0080363 0086.32501 Google Scholar[3] C. G. J. Jacobi, Uber ein leichtes Verfahren, die in der Theorie der Säkularstöorungen vorkommenden Gleichungen numerisch aufzulösen, J. Reine Angew. Math., 30 (1846), 51–95 CrossrefGoogle Scholar[4] H. H. Goldstine, 1951, unpublished paper presented to UCLA symposium, Aug. Google Scholar[5] G. E. Forsythe and , P. Henrici, The cyclic Jacobi method for computing the principal values of a complex matrix, Tech. Report No. 74, Appl. Math. and Statist. Lab., Stanford Univ., 1958 Google Scholar[6] Robert T. Gregory, Computing eigenvalues and eigenvectors of a symmetric matrix on the ILLIAC, Math. Tables and Other Aids to Computation, 7 (1953), 215–220 MR0057643 0052.35703 CrossrefGoogle Scholar[7] David A. Pope and , C. Tompkins, Maximizing functions of rotations. Experiments concerning speed of diagonalization of symmetric matrices using Jacobi's method, J. Assoc. Comput. Mach., 4 (1957), 459–466 MR0096355 CrossrefISIGoogle Scholar[8] L. J. Paige and , Olga Taussky, Simultaneous Linear Equations and the Determination of Eigenvalues, Nat. Bur. Standards. Appl. Math. Ser. No. 29, 1953 Google Scholar[9] Wallace Givens, Numerical conputation of the characteristic values of a real symmetric matrix, Rep. ORNL 1574, Oak Ridge National Laboratory, Oak Ridge, Tenn., 1954vi+107 MR0063771 0055.35005 CrossrefGoogle Scholar[10] R. H. Dykaar and , C. D. La Budde, The NYU matrix codes, 1956, Contract No. AT (30-1)-1480, AEC Computing Facility Google Scholar[11] Magnus R. Hestenes and , William Karush, A method of gradients for the calculation of the characteristic roots and vectors of a real symmetric matrix, J. Research Nat. Bur. Standards, 47 (1951), 45–61 MR0043552 CrossrefISIGoogle Scholar[12] Magnus R. Hestenes, Determination of eigenvalues and eigenvectors of matricesSimultaneous linear equations and the determination of eigenvalues, National Bureau of Standards Applied Mathematics Series, No. 29, U. S. Government Printing Office, Washington, D. C., 1953, 89–94 MR0059069 0052.35002 Google Scholar[13] R. von Mises and , H. Geiringer, Praktische Verfahren der Gleichungsauflosung, Z. Angew. Math. Mech., 9 (1929), 58–77, 152–64 CrossrefGoogle Scholar[14] A. C. Aitken, The evaluation of the latent roots and latent vectors of a matrix, Proe. Roy. Soc. Edinburgh, Sect. A, 57 (1937), 268–304 0017.14701 Google Scholar[15] J. H. Wilkinson, The use of iterative methods for finding the latent roots and vectors of matrices, Math. Tables Aids Comput., 9 (1955), 184–191 MR0079833 0068.10301 CrossrefGoogle Scholar[16] J. H. Wilkinson, The calculation of the latent roots and vectors of matrices on the pilot model of the A.C.E, Proc. Cambridge Philos. Soc., 50 (1954), 536–566 MR0063775 0056.12202 CrossrefISIGoogle Scholar[17] H. Wielandt, Bestimmung höherer eigenwerte durch gebrochene Iteration, Ber. Aerodynamischen Versuchsanstalt Göttingen, 44 (1944), Google Scholar[18] L. F. Richardson, A purification method for computing the latent columns of numerical matrices and some integrals of differential equations, Philos. Trans. Roy. Soc. London. Ser. A., 242 (1950), 439–491 MR0036591 0037.20903 CrossrefISIGoogle Scholar[19] H. Hotelling, Analysis of a complex of statistical variables into principal components, J. Educ. Psychol., 24 (1933), 417–41, 498–520 CrossrefGoogle Scholar[20] William Feller and , George E. Forsythe, New matrix transformations for obtaining characteristic vectors, Quart. Appl. Math., 8 (1951), 325–331 MR0039373 0043.01502 CrossrefISIGoogle Scholar[21] J. Greenstadt, A method for finding roots of arbitrary matrices, Math. Tables Aids Comput., 9 (1955), 47–52 MR0073283 0065.24801 CrossrefGoogle Scholar[22] Cornelius Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Research Nat. Bur. Standards, 45 (1950), 255–282 MR0042791 CrossrefISIGoogle Scholar[23] A. N. Krylov, Über die numerische auflösung einer Gleichung, durch die in technischen Fragen die Frequenz kleiner Schwingungen bestimmtest ist, Bull. Acad. Sci. U.R.S.S. (4), 7 (1931), 491– Google Scholar[24] R. A. Frazer, , W. J. Duncan and , A. R. Collar, Elementary Matrices, Macmillan, London, 1938 0021.22803 CrossrefGoogle Scholar[25] Harold Wayland, Expansion of determinantal equations into polynomial form, Quart. Appl. Math., 2 (1945), 277–306 MR0011799 0061.27302 CrossrefGoogle Scholar[26] U. J. J. LeVerrier, Sur les variations séculaires des éléments elliptiques des Sept planets principales, J. Math. Pures Appl., 5 (1840), 230– Google Scholar[27] P. S. Dwyer, Linear Computations, Wiley, New York, 1951 0044.12804 Google Scholar[28] George E. Forsythe and , Louise W. Straus, The Souriau-Frame characteristic equation algorithm on a digital computer, J. Math. Phys., 34 (1955), 152–156 MR0071121 0068.10602 CrossrefGoogle Scholar[29] E. L. Leppert, Jr., A fraction series solution for characteristic values useful in some problems of airplane dynamics, J. Aero. Sci., 22 (1955), 326–328 0064.19903 CrossrefISIGoogle Scholar[30] Wallace Givens, Computation of plane unitary rotations transforming a general matrix to triangular form, J. Soc. Indust. Appl. Math., 6 (1958), 26–50 10.1137/0106004 MR0092223 0087.11902 LinkISIGoogle Scholar[31] B. Dimsdale, The non-convergence of a characteristic root method, J. Soc. Indust. Appl. Math., 6 (1958), 23–25 10.1137/0106003 MR0093905 0097.11502 LinkISIGoogle Scholar[32] Robert L. Causey, Computing eigenvalues of non-Hermitian matrices by methods of Jacobi type, J. Soc. Indust. Appl. Math., 6 (1958), 172–181 10.1137/0106010 MR0103583 0097.32701 LinkISIGoogle Scholar[33] Robert T. Gregory, Results using Lanczos' method for finding eigenvalues of arbitrary matrices, J. Soc. Indust. Appl. Math., 6 (1958), 182–188 10.1137/0106011 MR0103584 0085.10805 LinkISIGoogle Scholar[34] Elmer E. Osborne, On acceleration and matrix deflation processes used with the power method, J. Soc. Indust. Appl. Math., 6 (1958), 279–287 10.1137/0106019 MR0096354 0086.11001 LinkISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A Jacobi-Like Method for the Automatic Computation of Eigenvalues and Eigenvectors of an Arbitrary MatrixJournal of the Society for Industrial and Applied Mathematics, Vol. 10, No. 1 | 13 July 2006AbstractPDF (946 KB)The Design of a Computer Program to Determine the Natural Frequencies and Normal Modes of Vibration of an In-Line Mechanical System of Springs and MassesJournal of the Society for Industrial and Applied Mathematics, Vol. 9, No. 2 | 10 July 2006AbstractPDF (1469 KB)Newton’s Method for the Characteristic Value Problem $Ax = \lambda Bx$L. B. RallJournal of the Society for Industrial and Applied Mathematics, Vol. 9, No. 2 | 10 July 2006AbstractPDF (412 KB) Volume 6, Issue 4| 1958Journal of the Society for Industrial and Applied Mathematics History Submitted:13 February 1958Published online:10 July 2006 InformationCopyright © 1958 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0106027Article page range:pp. 393-437ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics
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