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A new single-step iteration method for solving complex symmetric linear systems

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Abstract

For solving a class of complex symmetric linear systems, we introduce a new single-step iteration method, which can be taken as a fixed-point iteration adding the asymptotical error (FPAE). In order to accelerate the convergence, we further develop the parameterized variant of the FPAE (PFPAE) iteration method. Each iteration of the FPAE and the PFPAE methods requires the solution of only one linear system with a real symmetric positive definite coefficient matrix. Under suitable conditions, we derive the spectral radius of the FPAE and the PFPAE iteration matrices, and discuss the quasi-optimal parameters which minimize the above spectral radius. Numerical tests support the contention that the PFPAE iteration method has comparable advantage over some other commonly used iteration methods, particularly when the experimental optimal parameters are not used.

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References

  1. Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7, 197–218 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bai, Z.Z.: A class of two-stage iterative methods for systems of weakly nonlinear equations. Numer. Algorithms 14, 295–319 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai, Z.Z.: Several splittings for non-Hermitian linear systems. Sci. China (Ser. A: Math) 51, 1339–1348 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bai, Z.Z.: Optimal parameters in the HSS-like methods for saddle-point problems. Numer. Linear Algebra Appl. 16, 447–479 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bai, Z.Z.: On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems. Computing 89, 171–197 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bai, Z.Z.: On Hermitian and skew-Hermitian spliting iteration methods for continuous Sylvester equations. J. Comput. Math. 29, 185–198 (2011)

    MathSciNet  MATH  Google Scholar 

  7. Bai, Z.Z.: Rotated block triangular preconditioning based on PMHSS. Sci. China Math. 56, 2523–2538 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bai, Z.Z., Benzi, M., Chen, F.: Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87, 93–111 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bai, Z.Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algorithms 56, 297–317 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bai, Z.Z., Benzi, M., Chen, F., Wang, Z.Q.: Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems. IMA J. Numer. Anal. 33, 343–369 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bai, Z.Z., Deng, Y.B., Gao, Y.H.: Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations. Numer. Linear Algebra Appl. 13, 801–823 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bai, Z.Z., Golub, G.H., Li, C.K.: Convergence properties of preconditioned Hermitian and skew- Hermitian splitting methods for non-Hermitian positive semidefinite matrices. Math. Comput. 76, 287–298 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bai, Z.Z., Golub, G.H., Lu, L.Z., Yin, J.F.: Block triangular and skew-Hermitian splitting methods for positive-definite linear systems. SIAM J. Sci. Comput. 26, 844–863 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bai, Z.Z., Golub, G.H., Ng, M.K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24, 603–626 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bai, Z.Z., Golub, G.H., Ng, M.K.: On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations. Numer. Linear Algebra Appl. 14, 319–335 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bai, Z.Z., Golub, G.H., Ng, M.K.: On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. Linear Algebra Appl. 428, 413–440 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bai, Z.Z., Golub, G.H., Pan, J.Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98, 1–32 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bai, Z.Z., Guo, X.P.: On Newton-HSS methods for systems of nonlinear equations with positive definite Jacobian matrices. J. Comput. Math. 28, 235–260 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Bai, Z.Z., Huang, Y.M., Ng, M.K.: On preconditioned iterative methods for Burgers equations. SIAM J. Sci. Comput. 29, 415–439 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bai, Z.Z., Ng, M.K.: Preconditioners for nonsymmetric block Toeplitz-like-plus-diagonal linear systems. Numer. Math. 96, 197–220 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bai, Z.Z., Yin, J.F., Su, Y.F.: A shift-splitting preconditioner for non-Hermitian positive definite matrices. J. Comput. Math. 24, 539–552 (2006)

    MathSciNet  MATH  Google Scholar 

  22. Benzi, M.: A generalization of the Hermitian and skew-Hermitian splitting iteration. SIAM J. Matrix Anal. Appl. 31, 360–374 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Benzi, M., Bertaccini, D.: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal. 28, 598–618 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bertaccini, D.: Efficient solvers for sequences of complex symmetric linear systems. Electron. Trans. Numer. Anal. 18, 49–64 (2004)

    MathSciNet  MATH  Google Scholar 

  25. Feriani, A., Perotti, F., Simoncini, V.: Iterative system solvers for the frequency analysis of linear mechanical systems. Comput. Methods Appl. Mech. Eng. 190, 1719–1739 (2000)

    Article  MATH  Google Scholar 

  26. Hezari, D., Edalatpour, V., Salkuyeh, D.K.: Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations. Numer. Linear Algebra Appl. 22, 761–776 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, C.X., Wu, S.L.: A single-step HSS method for non-Hermitian positive definite linear systems. Appl. Math. Lett. 44, 26–29 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, W.W., Wang, X.: A modified GPSS method for non-Hermitian positive definite linear systems. Appl. Math. Comput. 234, 253–259 (2014)

    MathSciNet  MATH  Google Scholar 

  29. Li, X., Yang, A.L., Wu, Y.J.: Lopsided PMHSS iteration method for a class of complex symmetric linear systems. Numer. Algorithms 66, 555–568 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pour, H.N., Goughery, H.S.: New Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems. Numer. Algorithms 69, 207–225 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Salkuyeh, D.K., Hezari, D., Edalatpour, V.: Generalized SOR iterative method for a class of complex symmetric linear system of equations. Int. J. Comp. Math. 92, 802–815 (2015)

    Article  MATH  Google Scholar 

  32. Wang, X., Li, W.W., Mao, L.Z.: On positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equation A X + X B = C. Comput. Math. Appl. 66, 2352–2361 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, X., Li, Y., Dai, L.: On Hermitian and skew-Hermitian splitting iteration methods for the linear matrix equation A X B = C. Comput. Math. Appl. 65, 657–664 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Xiao, X.Y., Wang, X., Yin, H.W.: Efficient single-step preconditioned HSS iteration methods for complex symmetric linear systems. Comput. Math. Appl. In Press, 10.1016/j.camwa.2017.07.007 (2017)

  35. Xiao, X.Y., Yin, H.W.: Efficient parameterized HSS iteration methods for complex symmetric linear systems. Comput. Math. Appl. 73, 87–95 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zeng, M.L., Ma, C.F.: A parameterized SHSS iteration method for a class of complex symmetric system of linear equations. Comput. Math. Appl. 71, 2124–2131 (2016)

    Article  MathSciNet  Google Scholar 

  37. Zhou, R., Wang, X., Tang, X.B.: A generalization of the Hermitian and skew-Hermitian splitting iteration method for solving Sylvester equations. Appl. Math. Comput. 271, 609–617 (2015)

    MathSciNet  Google Scholar 

  38. Zhou, R., Wang, X., Tang, X.B.: Preconditioned positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equations A X + X B = C. East Asian J. Appl. Math. 7, 55–69 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhou, R., Wang, X., Zhou, P.: A modified HSS iteration method for solving the complex linear matrix equation A X B = C. J. Comput. Math. 34, 437–450 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, School of Sciences, Nanchang University, Nanchang, 330031, China

    X. Y. Xiao & X. Wang

  2. Numerical Simulation and High-Performance Computing Laboratory, School of Sciences, Nanchang University, Nanchang, 330031, China

    X. Y. Xiao & X. Wang

Authors
  1. X. Y. Xiao
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  2. X. Wang
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Correspondence to X. Y. Xiao.

Additional information

This work is supported by NNSF with Nos. 11461046, 61563033, and 11401293, NSF of Jiangxi Province with Nos. 20161ACB21005 and 20151BAB201009, and the Scientific Research Foundation of Graduate School of Nanchang University with Nos. YC2015-S018 and CX2016143.

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Xiao, X.Y., Wang, X. A new single-step iteration method for solving complex symmetric linear systems. Numer Algor 78, 643–660 (2018). https://doi.org/10.1007/s11075-017-0393-y

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  • DOI: https://doi.org/10.1007/s11075-017-0393-y

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