The palindromic cyclic reduction and related algorithms

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• 作者：Bruno Iannazzo (1)
  Beatrice Meini (2)
  
  1. Dipartimento di Matematica e Informatica ; Universit脿 di Perugia ; Via Vanvitelli 1 ; 06123聽 ; Perugia ; Italy
  2. Dipartimento di Matematica ; Universit脿 di Pisa ; Largo Bruno Pontecorvo 5 ; 56100聽 ; Pisa ; Italy
  
• 关键词：Cyclic reduction ; Trapezoidal rule ; Gauss鈥揅hebyshev quadrature ; Matrix geometric mean ; Matrix sign ; Matrix square root ; Polar decomposition ; Laurent matrix polynomial ; 65F30 ; 15A15
• 刊名：Calcolo
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：52
• 期：1
• 页码：25-43
• 全文大小：250 KB
• 参考文献：1. Anderson, W.N., Trapp, G.: Operator means and electrical networks. In: Proceedings of 1980 IEEE international symposium on circuits and systems, pp. 523鈥?27 (1980)
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  6. Bini, D.A., Latouche, G., Meini, B.: Numerical methods for structured Markov chains. Numerical mathematics and scientific computation. Oxford University Press, Oxford Science Publications, New York (2005) 27688.001.0001" target="_blank" title="It opens in new window">CrossRef
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  8. Bini, D.A., Meini, B.: The cyclic reduction algorithm: from Poisson equation to stochastic processes and beyond. In memoriam of Gene H. Golub. Numer. Algorithms 51(1), 23鈥?0 (2009) 253-0" target="_blank" title="It opens in new window">CrossRef
  9. Byers, R.: Solving the algebraic Riccati equation with the matrix sign function. Linear Algebra Appl. 85, 267鈥?79 (1987) 24-3795(87)90222-9" target="_blank" title="It opens in new window">CrossRef
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  11. Horn, R.A., Johnson, C.R.: Topics in matrix analysis. Cambridge University Press, Cambridge (1994). (Corrected reprint of the 1991 original)
  12. Iannazzo, B.: The geometric mean of two matrices from a computational viewpoint. arXiv. 201.0101" class="a-plus-plus">http://arxiv.org/abs/1201.0101 (2011)
  13. Iannazzo, B.: A note on computing the matrix square root. Calcolo 40(4), 273鈥?83 (2003) 2-003-0079-9" target="_blank" title="It opens in new window">CrossRef
  14. Iannazzo, B.: A family of rational iterations and its application to the computation of the matrix \(p\) th root. SIAM J. Matrix Anal. Appl. 30(4), 1445鈥?462 (2008/2009) CrossRef
  15. Iannazzo, B., Meini, B.: Palindromic matrix polyomials, matrix functions and integral representations. Linear Algebra Appl. 434(1), 174鈥?84 (2011) 2010.09.013" target="_blank" title="It opens in new window">CrossRef
  16. Kubo, F., Ando, T.: Means of positive linear operators. Math. Ann. 246(3), 205鈥?24 (1979/1980) 2" target="_blank" title="It opens in new window">CrossRef
  17. Lancaster, P., Tismenetsky, M.: The theory of matrices. Computer science and applied mathematics, 2nd edn. Academic Press Inc., Orlando (1985)
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Theory of Computation
  
• 出版者：Springer Milan
• ISSN：1126-5434

文摘

The cyclic reduction algorithm is specialized to palindromic matrix polynomials and a complete analysis of applicability and convergence is provided. The resulting iteration is then related to other algorithms as the evaluation/interpolation at the roots of unity of a certain Laurent matrix polynomial, the trapezoidal rule for a certain integral and an algorithm based on the finite sections of a tridiagonal block Toeplitz matrix.