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Computing eigenvalues and singular values of Toeplitz matrices

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Abstract

We present a review of the main specialized techniques for finding the eigenvalues of a Toeplitz matrix, based on fast evaluation of the characteristic polynomial, restricted rank projection methods or updating procedures. The latter approach is described in more detail, by presenting some recent extensions to rational Toeplitz matrices and to the computation of (generalized) Toeplitz singular values, which can be of great interest in image processing applications.

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References

  1. P. Arbenz,Computing eigenvalues of banded symmetric Toeplitz matrices, SIAM J. Sci. Stat. Comp.,12 (1991) 743–754.

    Article  MATH  MathSciNet  Google Scholar 

  2. D. Bini andM. Capovani,Spectral and computational properties of band symmetric Toeplitz matrices, Linear Algebra Appl.52, (1983) 99–126.

    MathSciNet  Google Scholar 

  3. D. Bini andF. Di Benedetto,Solving the generalized eigenvalue problem for rational Toeplitz matrices, SIAM J. Matrix Anal. Appl.11, (1990) 537–552.

    Article  MATH  MathSciNet  Google Scholar 

  4. D. Bini andL. Gemignani,Iteration schemes for the divide-and-conquer eigenvalue solver, Numer. Math.67, (1994) 403–425.

    Article  MATH  MathSciNet  Google Scholar 

  5. D. Bini andV. Pan,Efficient algorithms for the evaluation of the eigenvalues of (block) banded Toeplitz matrices, Math. Comp.50, (1988) 431–448.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. R. Bunch,Stability of methods for solving Toeplitz systems of equations, SIAM J. Sci. Stat. Comp.6, (1985) 349–364.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Chan andM. K. Ng,Conjugate Gradient Methods for Toeplitz Systems, SIAM Rev.38, (1996) 427–482.

    Article  MATH  MathSciNet  Google Scholar 

  8. F. Chiuppesi, G. Galati, andP. Lombardi,Optimisation of rejection filters, IEE Proc.127, (1980) 354–360.

    Google Scholar 

  9. F. Di Benedetto,Solution of nonsymmetric Toeplitz systems by preconditioning of the normal equations. Preprint #268, Dipartimento di Matematica, Università di Genova, 1994.

  10. F. Di Benedetto,Generalized updating problems and computation of the eigenvalues of rational Toeplitz matrices, Linear Algebra Appl., to appear.

  11. F. Di Benedetto,The use of discrete sine transform in computations with Toeplitz matrices, Lecture Notes Comp. Sci.1196, (1997) 126–133.

    Google Scholar 

  12. B. W. Dickinson,Solution of linear equations with rational Toeplitz matrices, Math. Comp.34, (1980) 227–233.

    Article  MATH  MathSciNet  Google Scholar 

  13. G. H. Golub, M. Heath andG. Wahba,Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics,21, (1979) 215–223.

    Article  MATH  MathSciNet  Google Scholar 

  14. G. H. Golub andC. Van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, MD, 1983.

    MATH  Google Scholar 

  15. S. L. Handy andJ. L. Barlow,Numerical solution of the eigenproblem for banded, symmetric Toeplitz matrices, SIAM J. Matrix Anal. Appl.15, (1994) 205–214.

    Article  MATH  MathSciNet  Google Scholar 

  16. T. Kailath, A. Sayed,Displacement structure: Theory and applications, SIAM Rev.37, (1995) 297–386.

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Mezzanotte, Analisi e implementazione di alcuni metodi numerici per il calcolo di autovalori di matrici di Toeplitz, Graduate thesis in Matematics, Università di Genova, A.A. 1993–94.

  18. W. F. Trench,On the eigenvalue problem for Toeplitz band matrices, Linear Algebra Appl.64, (1985) 199–214.

    Article  MATH  MathSciNet  Google Scholar 

  19. W. F. Trench,On the eigenvalue problem for Toeplitz matrices generated by rational functions, Lin. Multilin. Alg.17, (1985) 337–353.

    MATH  MathSciNet  Google Scholar 

  20. W. F. Trench,Numerical solution of the eigenvalue problem for Hermitian Toeplitz matrices, SIAM J. Matrix Anal. Appl.10, (1989) 135–146.

    Article  MATH  MathSciNet  Google Scholar 

  21. C. Van Loan,Generalizing the singular value decomposition, SIAM J. Numer. Anal.13, (1976) 76–83.

    Article  MATH  MathSciNet  Google Scholar 

  22. J. M. Varah,Pitfalls in the numerical solution of linear ill-posed problems, SIAM J. Sci. Stat. Comp.4, (1983) 164–176.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Dipartmento di Matematica, Università di Genova, Genova, Italia

    Fabio Di Benedetto

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  1. Fabio Di Benedetto
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Di Benedetto, F. Computing eigenvalues and singular values of Toeplitz matrices. Calcolo 33, 237–248 (1996). https://doi.org/10.1007/BF02576003

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  • DOI: https://doi.org/10.1007/BF02576003

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