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Solving the eigenvalue problem for matrices

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Abstract

One presents some algorithms related among themselves for solving the partial and the complete eigenvalue problem for an arbitrary matrix. Algorithm 1 allows us to construct the invariant subspaces and to obtain with their aid a matrix whose eigenvalues coincide with the eigenvalues of the initial matrix and belong to a given semiplane. Algorithm 2 solves the same problem for a given strip. The algorithms 3 and 4 reduces the complete eigenvalue problem of an arbitrary matrix to some problem for a quasitriangular matrix whose diagonal blocks have eigenvalues with identical real parts. Algorithm 4 finds also the unitary matrix which realizes this transformation. One gives Algol programs which realize the algorithms 1–3 for real matrices and testing examples.

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Literature cited

  1. A. N. Beavers and E. D. Denman, “A computational method for eigenvalues and eigenvectors of a matrix with real eigenvalues,” Numer. Math.,21, No. 5, 389–396 (1973).

    Google Scholar 

  2. A. N. Beavers and E. D. Denman, “A new similarity transformation method for eigenvalues and eigenvectors,” Math. Biosci.,21, 143–169 (1974).

    Google Scholar 

  3. J. D. Roberts, “Linear model reduction and solutions of algebraic Riccati equations by use of the sign function CUED/B-Control/TR 13,” Rpt. Cambridge University (1971).

  4. D. K. Faddeev, V. N. Kublanovskaya and V. N. Faddeeva, “Linear algebraic systems with rectangular matrices,” in: Modern Numerical Methods, No. 1 (Proc. Int. Summer School on Numerical Methods, Kiev, 1966), Moscow (1968), pp. 16–75.

  5. V. N. Kublanovskaya, “The Newton method for the determination of the eigenvalues and the eigenvectors of matrices,” Zh. Vychisl. Mat. Mat. Fiz.,12, No. 6, 1371–1380 (1972).

    Google Scholar 

  6. P. A. Businger and G. H. Golub, “Linear least squares solutions by Householder transformations,” Numer. Math.,7, No. 3, 269–276 (1965).

    Google Scholar 

  7. T. Ya. Kon'kova, “Algol procedures for solving certain problems of algebra, based on the application of a normalized process,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,35, 36–44 (1973).

    Google Scholar 

  8. R. S. Martin, G. Peters, and J. H. Wilkinson, “The QR algorithm for real Hessenberg matrices,” Numer. Math.,14, No. 3, 219–231 (1970).

    Google Scholar 

  9. P. J. Eberlein, “Solution to the complex eigenproblem by a norm reducing Jacobi type method,” Numer. Math.,14, No. 3, 232–245 (1970).

    Google Scholar 

  10. H. Rutishauser, “Simultaneous iteration method for symmetric matrices,” Numer. Math.,16, No. 3, 205–223 (1970).

    Google Scholar 

  11. R. S. Martin and J. H. Wilkinson, “Reduction of the symmetric eigenproblem Ax=ΛBx and related problems to standard form,” Numer. Math.,11, No. 2, 99–110 (1968).

    Google Scholar 

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Authors
  1. V. N. Kublanovskaya
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  2. L. T. Savinova
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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 70, pp. 124–139, 1977.

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Kublanovskaya, V.N., Savinova, L.T. Solving the eigenvalue problem for matrices. J Math Sci 23, 1966–1978 (1983). https://doi.org/10.1007/BF01093278

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  • DOI: https://doi.org/10.1007/BF01093278

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