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Modify Levinson algorithm for symmetric positive definite Toeplitz system

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Abstract

This paper describes a new \(O(N^{\frac {3}{2}}\log (N))\) solver for the symmetric positive definite Toeplitz system T N x N = b N . The method is based on the block QR decomposition of T N accompanied with Levinson algorithm and its generalized version for solving Schur complements S m of size m. In our algorithm we use a formula for displacement rank representation of the S m in terms of generating vectors of the matrix T N , and we assume that N = l m with \(l, m\in \mathbb {N}\). The new algorithm is faster than the classical O(N 2)-algorithm for N > 29. Numerical experiments confirm the good computational properties of the new method.

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References

  1. Ammar, G.S., Gragg, W.B.: Numerical experience with a superfast real Toeplitz solver. Linear Algebra Appl. 121, 185–206 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bitmead, R.R., Anderson, B.D.O.: Asymptotically fast solution of Toeplitz and related systems of linear equations. Linear Algebra Appl. 34, 103–116 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brigham, E.O., Brigham, E.O.: The fast Fourier transform and its applications, vol. 1. Prentice Hall, Englewood Cliffs, NJ (1988)

    MATH  Google Scholar 

  4. Musicus, B.R.: Levinson and Fast Choleski Algorithm for Toeplitz and Almost Toeplitz Matrices, Massachusetts Institute of Technology Cambridge, MA 02139 USA

  5. Bunch, J.R.: Stability of methods for solving Toeplitz systems of equations. SIAM J. Sci. Stat. Comput. 6.2, 349–364 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chan, T.F., Hansen, P.C.: A look-ahead Levinson algorithm for general Toeplitz systems. IEEE Trans. Signal Process. 40.5, 1079–1090 (1992)

    Article  MATH  Google Scholar 

  7. Chan, T.F., Hansen, P.C.: A look-ahead Levinson algorithm for indefinite Toeplitz systems. SIAM J. Matrix Anal. Appl. 13.2, 490–506 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cybenko, G.: The numerical stability of the Levinson-Durbin algorithm for Toeplitz systems of equations. SIAM J. Sci. Stat. Comput. 1.3, 303–319 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  9. Favati, P., Lotti, G., Menchi, O.: A divide and conquer algorithm for the superfast solution of Toeplitz-like systems. SIAM J. Matrix Anal. Appl. 33.4, 1039–1056 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Favati, P., Lotti, G., Menchi, O.: Stability of the Levinson algorithm for Toeplitz-like systems. SIAM J. Matrix Anal. Appl. 31.5, 2531–2552 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huckle, T.: Computations with Gohberg-Semencul-type formulas for Toeplitz matrices. Linear Algebra Appl. 273.1, 169–198 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Huckle, T.: Superfast solution of linear equations with low displacement rank. High Performance Algorithms for Structured Matrix Problems 2, 149–162 (1998)

    MathSciNet  MATH  Google Scholar 

  13. Stewart, M.: A superfast Toeplitz solver with improved numerical stability. SIAM J. Matrix Anal. Appl. 25.3, 669–693 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Toutounian, F., Akhoundi, N.: Recursive self preconditioning method based on Schur complement for Toeplitz matrices. Numer. Algorithms 62.3, 505–525 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Computer Sciences, Damghan University, P.O.Box 36715-364, Damghan, Iran

    Nasser Akhoundi & Iman Alimirzaei

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  1. Nasser Akhoundi
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  2. Iman Alimirzaei
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Correspondence to Iman Alimirzaei.

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Akhoundi, N., Alimirzaei, I. Modify Levinson algorithm for symmetric positive definite Toeplitz system. Numer Algor 71, 907–913 (2016). https://doi.org/10.1007/s11075-015-0029-z

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  • DOI: https://doi.org/10.1007/s11075-015-0029-z

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