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Solution of systems of nonlinear algebraic equations in three variables. Methods and algorithms. 3

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Abstract

An approach to constructing methods for solving systems of nonlinear algebraic equations in three variables (SNAEs-3) is suggested. This approach is based on the interrelationship between solutions of SNAEs-3, and solutions of spectral problems for two- and three-parameter polynomial matrices and for pencils of two-parameter matrices. Methods for computing all of the finite zero-dimensional roots of a SNAE-3 requiring no initial approximations of them are suggested. Some information on k-dimensional (k>0) roots of SNAEs-3 useful for a further analysis of them is obtained. Bibliography: 17 titles.

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References

  1. V. N. Kublanovskaya and V. N. Simonova, “An approach to solving nonlinear algebraic systems. 2”,Zap. Nauchn. Semin. POMI,202, 71–96 (1992).

    Google Scholar 

  2. V. N. Kublanovskaya and V. B. Khazanov, “Relative factorization of polynomials in several variables” to appear inZh. Vychisl. Mat. Mat. Fiz.

  3. V. Kublanovskaya, “Rank-division algorithms and their applications”,J. Numer. Linear Algebra Appl.,1, No. 2, 199–213 (1992).

    MathSciNet  Google Scholar 

  4. V. N. Kublanovskaya and V. N. Simonova, “Operations over scalar polynomials and their computer realization”,Zap. Nauchn. Semin. POMI,219, 157–175 (1994).

    Google Scholar 

  5. D. K. Faddeev, V. N. Kublanovskaya, and V. N. Faddeeva, “Linear algebraic systems with rectangular matrices”,Tr. Mat. Inst. Akad. Nauk SSSR,96, 16–75 (1968).

    Google Scholar 

  6. G. H. Golub and W. Kahan, “Calculating the singular values and pseudoinverse of a matrix”,SIAM J. Numer. Anal.,2, 205–224 (1965).

    Article  MathSciNet  Google Scholar 

  7. G. H. Golub and C. Reinsch, “Singular value decomposition and least squares solutions”,Numer. Math., 14, 409–420 (1970).

    Article  Google Scholar 

  8. V. Kublanovskaya, “Solving spectral problems for two-parameter polynomial matrices. Methods and algorithms”,J. Numer. Anal. Math. Model. 9, No. 2, 111–120 (1994).

    MATH  MathSciNet  Google Scholar 

  9. V. N. Kublanovskaya, “On some factorizations of two-parameter polynomial matrices”,Zap. Nauchn. Semin. POMI,219, 94–116 (1994).

    MATH  Google Scholar 

  10. T. Ja. Kon'kova and V. N. Simonova, “A library of MATLAB functions for polynomial matrices”,Zap. Nauchn. Semin. POMI,219, 53–80 (1994).

    Google Scholar 

  11. D. K. Faddeev and V. N. Faddeeva,Computational Methods of Linear Algebra [in Russian], Fizmatgiz. Moscow (1963).

    Google Scholar 

  12. T. Ya. Kon'kova and V. N. Kublanovskaya, “Inversion of polynomial and rational matrices”,Zap. Nauchn. Semin. POMI,202, 97–109 (1992).

    Google Scholar 

  13. V. N. Kublanovskaya and V. N. Simonova, “An approach to solving nonlinear algebraic systems”,Sov. J. Numer. Anal. Math. Model. 5, No. 4/5, 369–394 (1990).

    MathSciNet  Google Scholar 

  14. P. van Dooren, “The generalized eigenstructure problem. Applications to linear system theory”, Ph.D. Thesis, Univ. of Louvain (1979).

  15. V. N. Kublanovskaya and V. B. Khazanov, “Spectral problems for pencils of polynomial matrices. Methods and algorithms. V”,Jap. Nauchn. Semin. POMI,202, 26–70 (1992).

    Google Scholar 

  16. L. Kaufman, “Some thoughts on the QZ algorithm for solving the generalized eigenvalue problem”,ACM Trans. Math. Software,3, 65–75 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  17. V. N. Kublanovskaya, “TheAB-algorithm and its modifications for spectral problems of linear pencils of matrices”,Numer. Math.,43, 329–342 (1984).

    Article  MATH  MathSciNet  Google Scholar 

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Authors
  1. V. N. Kublanovskaya
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Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 159–190.

Translated by V. N. Kublanovskaya

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Kublanovskaya, V.N. Solution of systems of nonlinear algebraic equations in three variables. Methods and algorithms. 3. J Math Sci 89, 1694–1714 (1998). https://doi.org/10.1007/BF02355373

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