Skip to main content
Log in

On certain methods for expanding the characteristic polynomial

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  • Arnoldi, W. E.: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quart. Appl. Math.9, 17–29 (1951).

    Google Scholar 

  • Bauer, F. L.: Zusammenhänge zwischen einigen Iterationsverfahren der linearen Algebra. Intern. Koll. über Probl. d. Rechentechnik, S. 99–111. Dresden 1955.

  • Bauer, F. L.: Beiträge zum Danjelewskij-Verfahren. Intern. Koll. über Probl. d. Rechentechnik, S. 133–139. Dresden 1955.

  • Bauer, F. L.: Sequential reduction to tridiagonal form. To appear in J. Soc. Indust. Appl. Math.

  • Danilevskiî, A.: O číslennom rešenii vekovogo uravneniya. Mat. Sbornik2 (44), 169–171 (1937).

    Google Scholar 

  • Faddeev, D. K., andI. S. Sominskiî: Sbornik zadač po vysšeî algebre. Moskow-Leningrad 1949.

  • Frame, J. S.: A simple recursion formula for inverting a matrix (abstract). Bull. Am. Math. Soc.55, 1045 (1949).

    Google Scholar 

  • Frazer, R. A., W. J. Duncan andA. R. Collar: Elementary matrices and some applications to dynamics and differential equations. New York 1946.

  • Givens, J. W.: Numerical computation of the characteristic values of a real symmetric matrix. Oak Ridge National Laboratory, ORNL-1574. 1954.

  • Givens, J. W.: The characteristic value-vector problem. J. Assoc. Comp-Mach4, 298–307 (1957).

    Google Scholar 

  • Hessenberg, K.: Auflösung linearer Eigenwertaufgaben mit Hilfe der Hamilton-Cayleyschen Gleichung. Diss. T. H. Darmstadt 1941.

  • Hestenes, Magnus R., andEduard Stiefel: Methods of conjugate gradients for solving linear systems. J. National Bureau of Standards49, 409–436 (1952).

    Google Scholar 

  • Hlodovskiî, I. N.: K teorii obščego slučaya preobrazovaniya vekovogo uravneniya metodoî Akademika A. N. Krylova. Izv. Akad. Nauk, Otd. m.e.n1933, 1077–1102.

  • Horst, Paul: A method of determining the coefficients of a characteristic equation. Ann. Math. Stat.6, 83–84 (1935).

    Google Scholar 

  • Householder, A. S.: Unitary triangularization of a nonsymmetric matrix. J. Assoc. Comp. Mach.5, 339–342 (1958).

    Google Scholar 

  • Krylov, A. N.: O čislennom rešenii uravneniya, kotorym v techničeskih voprosah opredelyayutsya častoty malyh kolebaniî material'nyh sistem. Izv. Akad. Nauk SSSR., Otd. m.e.n.1931, 491–539.

  • Lanczos, Cornelius: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Research, National Bureau of Standards45, 255–282 (1950).

    Google Scholar 

  • Leverrier, U. J. J.: Sur les variations séculaires des éléments des orbites pour les sept planètes principales. J. de Math. (1)5, 230 (1840).

    Google Scholar 

  • Lopšic, A. M.: Čislennyî metod nahoždeniya sobstvennyh značeniî i sobstvennyh ploskostei lineînogo operatora. Trud. Sem. Vekt. Tenz. An. c ih Pril. k Geom. Meh. i Fiz.7, 233–259 (1949).

    Google Scholar 

  • Luzin, N. N.: O metode Akademika A. N. Krylova sostavleniya vekovogo uravneniya. Izv. Akad. Nauk SSSR., Otd. m.e.n.1931, 903–958.

  • Luzin, N. N.: O nekotoryh svoistvah peremeščayuščego množitelya v metode Akademika A. N. Krylova. Izv. Akad. Nauk SSSR., Otd. m.e.n.1932, 595–638 735–762, 1065–1102.

  • Rutishauser, Heinz: Beiträge zur Kenntnis des Biorthogonalisierungsalgorithmus vonC. Lanczos. Z. angew. Math. Phys.,4, 35–56 (1953).

    Google Scholar 

  • Saibel, Edward, andW. J. Berger: On finding the characteristic equation of a square matrix. M.T.A.C.7, 228–236 (1953).

    Google Scholar 

  • Samuelson, P. A.: A method of determining explicitly the coefficients of the characteristic equation. Ann. Math. Stat.13, 424–429 (1942).

    Google Scholar 

  • Schacknow, Arnold: Solution of(I-qK)z=y″ on the IBM Type 650 Computer. Republic Aviation Corp. Internal Memorandum. 1956.

  • Schwarz, Hans Rudolf: Critère de stabilité pour les systèmes à coefficients constants. Comptes Rend.241, 15–16 (1955).

    Google Scholar 

  • Schwarz, Hans Rudolf: Ein Verfahren zur Stabilitätsfrage bei Matrizen-Eigenwert-problemen. Z. angew. Math. Phys.7, 473–500 (1956).

    Google Scholar 

  • Souriau, Jean-Marie: Une méthode pour la décomposition spectrale et l'inversion des matrices. Comptes Rend.227, 1010–1011 (1948).

    Google Scholar 

  • Stiefel, Eduard: Über einige Methoden der Relaxationsrechnung. Z. angew. Math. Phys.3, 1–33 (1952).

    Google Scholar 

  • Turnbull, H. W., andA. C. Aitken: An introduction to the theory of canonical matrices. pp. xiii+192. London and Glasgow: Blackie and Son Ltd. 1930.

    Google Scholar 

  • Unger, Heinz: Über direkte Verfahren bei Matrizeneigenwertproblemen. Wiss. Z. Techn. Hochsch. Dresden2, 449–456 (1952).

    Google Scholar 

  • Unger, Heinz: Zur Praxis der Biorthonormierung von Eigen- und Hauptvektoren. Z. angew. Math. Mech.33, 319–331 (1953).

    Google Scholar 

  • Wegner, U.: Bemerkungen zur Matrizentheorie. Z. angew. Math. Med.33, 262–264 (1953).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Operated by Union Carbide Corporation for U. S. Atomic Energy Commission.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Householder, A.S., Bauer, F.L. On certain methods for expanding the characteristic polynomial. Numer. Math. 1, 29–37 (1959). https://doi.org/10.1007/BF01386370

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01386370

Keywords

Navigation