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A convergent gradient method for matrix eigenvector-eigentuple problems. (English) Zbl 0421.65024


MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices

Citations:

Zbl 0421.65025

References:

[1] Courant, R., Hilbert, D.: Methods of mathematical physics, Vol. 1. New York: Interscience 1953 · Zbl 0051.28802
[2] Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge: University Press 1927 · JFM 53.0180.04
[3] Collatz, L.: Multiparametric eigenvalue problems in inner-product spaces. J. Comput. System Sci.2, 333-341 (1968) · Zbl 0181.16901 · doi:10.1016/S0022-0000(68)80033-9
[4] Hadeler, K.P.: Mehrparametrige und nichtlineare Eigenwertaufgaben. Arch. Rational Mech. Anal. 306-328 (1967) · Zbl 0166.41701
[5] Fox, L., Hayes, L., Mayers, D.F.: The double eigenvalue problem. Topics in numerical analysis. Proc. Royal Irish Ac. Conference on Numerical Analysis, 1972 (J. Miller, ed.). London-New York: Academic Press 1973 · Zbl 0278.65095
[6] Arscott, F.M.: Two-parameter eigenvalue problems in differential equations. Proc. London Math. Soc.14, 459-470 (1964) · Zbl 0121.31102 · doi:10.1112/plms/s3-14.3.459
[7] Blum, E.K.: Numerical analysis and computation theory and practice. Addison-Wesley 1972 · Zbl 0273.65001
[8] Blum, E.K., Rodrigue, G.H.: Solution of eigenvalue problems in Hilbert spaces by a gradient method. J. Comput. System Sci.2, 220-237 (1974) · Zbl 0327.65052 · doi:10.1016/S0022-0000(74)80056-5
[9] Blum, E.K.: Unpublished note. November 1975
[10] Blum, E.K., Geltner, P.B.: Numerical solution of eigentuple-eigenvector problems in Hilbert spaces by a gradient method. U.S.C. Math. Dept. Report, July 1975. Also: Numer. Math.31, (1978) · Zbl 0421.65025
[11] Rodrigue, G.H.: A gradient method for the matrix eigenvalue problemAx-?Bx. Numer. Math.22, 1-16 (1973) · Zbl 0261.65031 · doi:10.1007/BF01436617
[12] Wilkinson, J.H.: The algebraic eigenvalue problem. Oxford: University Press 1965 · Zbl 0258.65037
[13] Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. New York: McGraw-Hill 1955 · Zbl 0064.33002
[14] Ince, E.L.: Ordinary differential equations, pp. 197-213. Dover 1956
[15] Faierman, M.: The completeness and expansion theorems associated with the multiparameter eigenvalue problem in ODE. J. Differential Equations5, 197-213 (1969) · Zbl 0165.10003 · doi:10.1016/0022-0396(69)90112-0
[16] Gregus, M., Neuman, F., Arscott, F.M.: Three-point boundary-value problems in differential equations. J. London Math. Soc.3, 429-436 (1971) · Zbl 0226.34010 · doi:10.1112/jlms/s2-3.3.429
[17] Sleeman, B.D.: The two-parameter Sturm-Liouville problem for ordinary differential equations. Proc. Roy. Soc. Edinburgh Sect. A,LXII, Part II, No. 10, 139-148 (1971) · Zbl 0235.34052
[18] Ikebe, Y.: Matrix-theoretic approach for the numerical computation of the eigenvalue of Mathieu’s equation. Report CNA-100, Center Num. Analysis, University of Texas, May 1975
[19] Campbell, J.B., Chartres, B.A.: An eigenvalue problem for rectangular matrices. Report NSF-E-782 App. Math. Div., Brown University, January 1964
[20] Blum, E.K.: Numerical solution of eigentuple-eigenvector problems in Hilbert spaces. In: Proceedings of the special session on functional analysis methods in numerical analysis. Lecture Notes Series. Berlin-Heidelberg-New York: Springer (to appear)
[21] Tables Relating to Mathieu Functions: NBS, Appl. Math. Series, Vol. 59. Washington, D.C.: 1976
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