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References
Axelsson, O.: On preconditioning and convergence acceleration in sparse matrix problems. CERN Data division Tech.Rep. 74-10 (1974).
Bathe, K-J, and Wilson, E.L.: Solution methods for eigenvalue problems in structural mechanics. Int. J. Num. Meth. Engrg. 6, 213–226 (1973).
Betteridge, T.: An analytic storage allocation model. Acta Informatica 3, 101–122 (1974).
Birkhoff, G, and George, A.: Elimination by nested dissection, pp. 221–269 in Complexity of sequential and parallel numerical algorithms, ed. J.F. Traub, Academic Press, (1973).
Björck, Å.: Solving linear least squares problems by Gram-Schmidt orthogonalization. BIT 7, 1–21 (1967).
Bradbury, W.W, and Fletcher, R.: New iterative methods for solution of the eigenproblem. Num. Math. 9, 259–267 (1966).
Bussemaker, F.C, Cobeljić, S, Cvetković, D.M, Seidel, J.J.: Computer investigation of cubic graphs. Tech.Hog. Eindhoven Rep. 76-WSK-01 (1976).
Cline, A.K, Golub, G.H, Platzman, G.W.: Calculation of normal modes of oceans using a Lanczos method, pp 409–426 in Bunch and Rose eds. Sparse matrix comp. Acad. Press (1976).
Concus, P, and Golub, G.H.: Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations. SIAM J. Num. An. 10, 1103–1120 (1973).
Cullum, J, and Donath, W.E.: A block generalization of the symmetric s-step Lanczos algorithm. IBM Yorktown Heights Tech. Rep. RC 4845 pp. 1–77 (1974).
Cullum, J, Donath, W.E, Wolfe, P.: The minimization of certain non-differentiable sums of eigenvalues of symmetric matrices. Math. Progr. Studiy 3, 35–55 (1975).
Cvetković, D.M.: Graphs and their spectra. Univ. Beograd Publ. Elektr. Fak. Ser. Mat. Fiz 354, 1–50 (1971).
Duff, I.S, and Reid, J.K.: On the reduction of sparse matrices to condensed forms by similarity transformations. J. Inst. Math. Applics. 15, 217–224 (1975).
Faddeev, D.K, and Faddeeva, V.N.: Computational methods of linear algebra (Transl.) Freeman & Co., San Fransisco (1963).
Fried, I.: Gradient method for finite element eigenproblems. AIAA Jour. 7, 739–741 (1969).
Fried, I.: Optimal gradient minimization schema for finite element eigenproblems. Jour. Sound and Vibration 20, 333–342 (1972).
Geradin, M.: The computational efficiency of a new minimization algorithm for eigenvalue analysis. Jour. Sound and Vibration 19, 319–331 (1971).
Geradin, M.: Analyse dynamique duale des structures par la methode des elements finis. Diss. Univ. de Liege, Belgium, (1972).
Handbook for Automatic Computation vol II: Linear Algebra. Wilkinson, J.H, and Reinsch, C. ed. Springer-Verlag, Berlin Heidelberg New York (1971).
Householder, A.S.: The theory of matrices in numerical analysis. Blaisdell, New York (1964).
Jensen, P.S.: The solution of large symmetric eigenproblems by sectioning. SIAM J. Num. An. 9, 534–545 (1972).
Kahan, W, and Parlett, B.: An analysis of Lanczos algorithms for symmetric matrices. Tech. Rep. Electronics Research Laboratory, University of California, Berkeley, (1974).
Kahan, W, and Parlett, B.N.: How far should you go with the Lanczos process? pp. 131–144 in Bunch and Rose eds. Sparse matrix computations. Acad. Press (1976).
Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. NBS J. Res. 45, 255–282 (1950).
Moler, C.B, and Stewart, G.W.: An algorithm for the generalized matrix eigenvalue problem Ax=λBx. SIAM J. Num. An. 10, 241–256 (1973).
Ostrowski, A.M.: On the convergence of the Rayleigh quotient iteration for the computation of characteristic roots and vectors Arch Rat. Mech. Anal. 1, 233–241, 2, 423–428, 3, 325–340, 3, 341–347, 3, 428–481, 4, 153–165 (1958).
Paige, C.C.: The computation of eigenvalues and eigenvectors of very large sparse matrices. Diss. London Univ. Institute of Computer Science, (1971).
Paige, C.C.: Computational variants of the Lanczos method for the eigenproblem. J. Inst. Maths. Applics. 10, 373–381 (1972).
Paige, C.C, and Saunders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Num. An. Vol. 12, 617–629 (1975).
Parlett, B.N, and Kahan, W.M.: On the convergence of a practical QR algorithm. Proc IFIP Congress (1968).
Reid, J.K.: Sparse matrices. Tech. Rep. CSS 31 Comp. Sc. Syst. Div. AERE Harwell (1976)
Rodrigue, G.: A gradient method for the matrix eigenvalue problem Ax=λ Bx. Num. Math. 22, 1–16 (1973).
Ruhe, A.: Iterative eigenvalue algorithms for large symmetric matrices, ISNM 24 Birkhäuser verlag, Basel und Stuttgart, pp. 97–112 (1974).
Ruhe, A.: SOR-methods for the eigenvalue problem with large sparse matrices. Math. Comp. 28, 695–710 (1974).
Ruhe, A.: Iterative eigenvalue algorithms based on convergent splittings. J. Comp. Phys. 19, 110–120 (1975).
Schwarz, H.R.: The eigenvalue problem (A-λB)x=0 for symmetric matrices of high order. Comp. Meth. Appl. Mech. and Engineering 3, 11–28 (1974).
Shavitt, I, Bender, C.F, Pipano, A, Hosteny, R.P.: The iterative calculation of several of the lowest or highest eigenvalues and corresponding eigenvectors of very large symmetric matrices. Jour. Comp. Phys. 11, 90–108 (1973).
Stewart, G.W.: The numerical treatment of large eigenvalue problems. pp.666–672 in IFIP 74-North Holland (1974).
Stewart, G.W.: A bibliographical tour of the large sparse generalized eigenvalue problem. pp. 113–130 of Bunch and Rose eds. Sparse matrix computations, Acad. Press (1976).
Stewart, W.J.: Markov analysis of operating system techniques. Diss. Queen's Univ. Belfast (1974).
Underwood, R.: An iterative block Lanczos method for the solution of large sparse symmetric eigenproblems. Tech. Rep. STAN-CS-75-496 Stanford University (1975).
Weaver, W, and Yoshida, D.M.: The eigenvalue problem for banded matrices. Computers & Structures 1, 651–664 (1971).
Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Clarendon Press, Oxford (1965).
Wilkinson, J.H.: Inverse iteration in theory and in practice. Symp. Math. 10, 361–379 (1971/72).
Young, D.M.: Iterative solution of large linear systems. Academic Press, New York (1971).
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Ruhe, A. (1977). Computation of eigenvalues and eigenvectors. In: Barker, V.A. (eds) Sparse Matrix Techniques. Lecture Notes in Mathematics, vol 572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0116617
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DOI: https://doi.org/10.1007/BFb0116617
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