# American Institute of Mathematical Sciences

2012, 2(4): 863-873. doi: 10.3934/naco.2012.2.863

## On the convergence of generalized parallel multisplitting iterative methods for semidefinite linear systems

 1 Department of Mathematics, Changzhi University, Changzhi 046011, Shanxi Province, China 2 Department of Mathematics, Taiyuan Normal University, Taiyuan 030012, Shanxi Province, China, China

Received  January 2012 Revised  October 2012 Published  November 2012

In this paper, we present the generalized stationary and nonstationary multisplitting iterative methods for positive semidefinite linear systems. We study the convergence theories of new methods and show that the quotient convergence and convergence of stationary parallel multisplitting method are equivalent under a concise condition. Finally, we prove that the generalized nonstationary parallel multisplitting method is quotient convergence with general weighting matrices.
Citation: Yanxing Cui, Chuanlong Wang, Ruiping Wen. On the convergence of generalized parallel multisplitting iterative methods for semidefinite linear systems. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 863-873. doi: 10.3934/naco.2012.2.863
##### References:
 [1] Z.-Z. Bai and D.-R. Wang, Generalized matrix multisplitting relaxation methods and their convergence, Numer. Math. J. Chinese Univ. (English Ser.), 2 (1993), 87-100.  Google Scholar [2] Z.-Z. Bai, On the convergence of the generalized matrix multisplitting relaxed methods, Commun. Numer. Methods Engrg., 11 (1995), 363-371. doi: 10.1002/cnm.1640110410.  Google Scholar [3] Z.-Z. Bai, J.-C. Sun and D.-R. Wang, A unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations, Comput. Math. Appl., 32 (1996), 51-76. doi: 10.1016/S0898-1221(96)00207-6.  Google Scholar [4] Z.-Z. Bai and C.-L. Wang, On the convergence of nonstationary multisplitting two-stage iteration methods for Hermitian positive definite linear systems, J. Comput. Appl. Math., 138 (2002), 287-296. doi: 10.1016/S0377-0427(01)00376-4.  Google Scholar [5] Z.-Z. Bai, L. Wang and J.-Y. Yuan, Weak-convergence theory of quasi-nonnegative splittings for singular matrices, Appl. Numer. Math., 47 (2003), 75-89. doi: 10.1016/S0168-9274(03)00057-6.  Google Scholar [6] A. Ben-Israel and T. N. E. Greville, "Generalized Inverses: Theory and Applications," Wiley, New York, 1974.  Google Scholar [7] A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Science," Academic Press, New York, 1979.  Google Scholar [8] Z. Cao and Z. Liu, Symmetric multisplitting of a symmetric positive definite matrix, Linear Algebra Appl., 285 (1998), 309-319. doi: 10.1016/S0024-3795(98)10151-9.  Google Scholar [9] Z. Cao, On the convergence of nonstationary iterative methods for symmetric positive (semi)defnite systems, Appl. Numer. Math., 37 (2001), 319-330. doi: 10.1016/S0168-9274(00)00047-7.  Google Scholar [10] Z. Cao, On the convergence of general stationary linear iterative methods for singular linear systems, SIAM J. Matrix Anal. Appl., 29 (2007), 1382-1388. doi: 10.1137/060671243.  Google Scholar [11] Z. Cao, On the convergence of iterative methods for solving singular linear systems, J. Comput. Appl. Math., 145 (2002), 1-9. doi: 10.1016/S0377-0427(01)00531-3.  Google Scholar [12] M. J. Castel, V. Migallón and J. Penadés, Convergence of non-stationary parallel multisplitting methods for hermitian positive definite matrices, Math. Comput., 67 (1998), 209-220. doi: 10.1090/S0025-5718-98-00893-X.  Google Scholar [13] X. Cui, Y. Wei and N. Zhang, Quotient convergence and multisplitting methods for solving singular linear equations, Calcolo, 44 (2007), 21-31. doi: 10.1007/s10092-007-0127-y.  Google Scholar [14] A. Frommer, R. Nabben and D. B.Szyld, Convergence of stationary iterative methods for hermitian semidefinite linear systems and applications to schwarz methods, SIAM J. Matrix Anal. Appl., 30 (2008), 925-938. doi: 10.1137/080714038.  Google Scholar [15] H. B. Keller, On the solution of singular and semidefinite linear systems by iteration, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2 (1965), 281-290.  Google Scholar [16] Y.-J. Lee, J. Wu, Jinchao Xu and L. Zikatanov, On the convergence of iterative methods for semidefinite linear systems, SIAM J. Matrix Anal. Appl., 28 (2006), 634-641. doi: 10.1137/050644197.  Google Scholar [17] L. Lin, Y. Wei and N. Zhang, Convergence and quotient convergence of iterative methods for solving singular linear equations with index one, Linear Algebra Appl., 430 (2009), 1665-1674. doi: 10.1016/j.laa.2008.06.019.  Google Scholar [18] G. I. Marchuk and Y. Kuznetsov, "Iterative Methods and Quadratic Functionals," Science Press, Norvosibirsk, 1972 (in Russian). Google Scholar [19] V. Migallón, J. Penadés and D. B. Szyld, Nonstationary multisplittings with general weighting matrices, SIAM J. Matrix Anal. Appl., 22 (2001), 1089-1094. doi: 10.1137/S0895479800367038.  Google Scholar [20] D. P. O'Leary and R. E. White, Multisplittings of matrices and parallel solution of linear systems, SIAM J. on Alg. and Disc. Meth., 6 (1985), 630-640. doi: 10.1137/0606062.  Google Scholar [21] D. B. Szyld, Equivalence of conditions for convergence of iterative methods for singular equations, Numer. Linear Algebra Appl., 1 (1994), 151-154. doi: 10.1002/nla.1680010206.  Google Scholar [22] R. S. Varga, "Matrix Iterative Analysis," 2nd edition, Springer, Berlin, Heidelberg, 2000. doi: 10.1007/978-3-642-05156-2.  Google Scholar [23] C.-L. Wang, Nonstationary multisplitting with general weighting matrices for non-Hermitian positive definite systems, Appl. Math. Lett., 16 (2003), 919-924. doi: 10.1016/S0893-9659(03)90017-6.  Google Scholar [24] D.-R. Wang and Z.-Z. Bai, Asynchronous parallel matrix multisplitting multiparameter relaxation methods, (Chinese) Numer. Math. J. Chinese Univ., 16 (1994), 107-115.  Google Scholar [25] Y. Wei, Index splitting for the Drazin inverse and the singular linear system, Appl. Math. Comput., 95 (1998), 115-124. doi: 10.1016/S0096-3003(97)10098-4.  Google Scholar [26] Y. Wei, Perturbation analysis of singular linear systems with index one, Int. J. Comput. Math., 74 (2000), 483-491. doi: 10.1080/00207160008804956.  Google Scholar [27] J. Wu, Y.-J. Lee, J. Xu and Ludmil Zikatanov, Convergence analysis on iterative methods for semidefinite systems, J. Comput. Math., 26 (2008), 797-815.  Google Scholar [28] D. M. Young, "Iterative Solution of Large Linear Systems," Academic Press, New York, 1971.  Google Scholar

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##### References:
 [1] Z.-Z. Bai and D.-R. Wang, Generalized matrix multisplitting relaxation methods and their convergence, Numer. Math. J. Chinese Univ. (English Ser.), 2 (1993), 87-100.  Google Scholar [2] Z.-Z. Bai, On the convergence of the generalized matrix multisplitting relaxed methods, Commun. Numer. Methods Engrg., 11 (1995), 363-371. doi: 10.1002/cnm.1640110410.  Google Scholar [3] Z.-Z. Bai, J.-C. Sun and D.-R. Wang, A unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations, Comput. Math. Appl., 32 (1996), 51-76. doi: 10.1016/S0898-1221(96)00207-6.  Google Scholar [4] Z.-Z. Bai and C.-L. Wang, On the convergence of nonstationary multisplitting two-stage iteration methods for Hermitian positive definite linear systems, J. Comput. Appl. Math., 138 (2002), 287-296. doi: 10.1016/S0377-0427(01)00376-4.  Google Scholar [5] Z.-Z. Bai, L. Wang and J.-Y. Yuan, Weak-convergence theory of quasi-nonnegative splittings for singular matrices, Appl. Numer. Math., 47 (2003), 75-89. doi: 10.1016/S0168-9274(03)00057-6.  Google Scholar [6] A. Ben-Israel and T. N. E. Greville, "Generalized Inverses: Theory and Applications," Wiley, New York, 1974.  Google Scholar [7] A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Science," Academic Press, New York, 1979.  Google Scholar [8] Z. Cao and Z. Liu, Symmetric multisplitting of a symmetric positive definite matrix, Linear Algebra Appl., 285 (1998), 309-319. doi: 10.1016/S0024-3795(98)10151-9.  Google Scholar [9] Z. Cao, On the convergence of nonstationary iterative methods for symmetric positive (semi)defnite systems, Appl. Numer. Math., 37 (2001), 319-330. doi: 10.1016/S0168-9274(00)00047-7.  Google Scholar [10] Z. Cao, On the convergence of general stationary linear iterative methods for singular linear systems, SIAM J. Matrix Anal. Appl., 29 (2007), 1382-1388. doi: 10.1137/060671243.  Google Scholar [11] Z. Cao, On the convergence of iterative methods for solving singular linear systems, J. Comput. Appl. Math., 145 (2002), 1-9. doi: 10.1016/S0377-0427(01)00531-3.  Google Scholar [12] M. J. Castel, V. Migallón and J. Penadés, Convergence of non-stationary parallel multisplitting methods for hermitian positive definite matrices, Math. Comput., 67 (1998), 209-220. doi: 10.1090/S0025-5718-98-00893-X.  Google Scholar [13] X. Cui, Y. Wei and N. Zhang, Quotient convergence and multisplitting methods for solving singular linear equations, Calcolo, 44 (2007), 21-31. doi: 10.1007/s10092-007-0127-y.  Google Scholar [14] A. Frommer, R. Nabben and D. B.Szyld, Convergence of stationary iterative methods for hermitian semidefinite linear systems and applications to schwarz methods, SIAM J. Matrix Anal. Appl., 30 (2008), 925-938. doi: 10.1137/080714038.  Google Scholar [15] H. B. Keller, On the solution of singular and semidefinite linear systems by iteration, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2 (1965), 281-290.  Google Scholar [16] Y.-J. Lee, J. Wu, Jinchao Xu and L. Zikatanov, On the convergence of iterative methods for semidefinite linear systems, SIAM J. Matrix Anal. Appl., 28 (2006), 634-641. doi: 10.1137/050644197.  Google Scholar [17] L. Lin, Y. Wei and N. Zhang, Convergence and quotient convergence of iterative methods for solving singular linear equations with index one, Linear Algebra Appl., 430 (2009), 1665-1674. doi: 10.1016/j.laa.2008.06.019.  Google Scholar [18] G. I. Marchuk and Y. Kuznetsov, "Iterative Methods and Quadratic Functionals," Science Press, Norvosibirsk, 1972 (in Russian). Google Scholar [19] V. Migallón, J. Penadés and D. B. Szyld, Nonstationary multisplittings with general weighting matrices, SIAM J. Matrix Anal. Appl., 22 (2001), 1089-1094. doi: 10.1137/S0895479800367038.  Google Scholar [20] D. P. O'Leary and R. E. White, Multisplittings of matrices and parallel solution of linear systems, SIAM J. on Alg. and Disc. Meth., 6 (1985), 630-640. doi: 10.1137/0606062.  Google Scholar [21] D. B. Szyld, Equivalence of conditions for convergence of iterative methods for singular equations, Numer. Linear Algebra Appl., 1 (1994), 151-154. doi: 10.1002/nla.1680010206.  Google Scholar [22] R. S. Varga, "Matrix Iterative Analysis," 2nd edition, Springer, Berlin, Heidelberg, 2000. doi: 10.1007/978-3-642-05156-2.  Google Scholar [23] C.-L. Wang, Nonstationary multisplitting with general weighting matrices for non-Hermitian positive definite systems, Appl. Math. Lett., 16 (2003), 919-924. doi: 10.1016/S0893-9659(03)90017-6.  Google Scholar [24] D.-R. Wang and Z.-Z. Bai, Asynchronous parallel matrix multisplitting multiparameter relaxation methods, (Chinese) Numer. Math. J. Chinese Univ., 16 (1994), 107-115.  Google Scholar [25] Y. Wei, Index splitting for the Drazin inverse and the singular linear system, Appl. Math. Comput., 95 (1998), 115-124. doi: 10.1016/S0096-3003(97)10098-4.  Google Scholar [26] Y. Wei, Perturbation analysis of singular linear systems with index one, Int. J. Comput. Math., 74 (2000), 483-491. doi: 10.1080/00207160008804956.  Google Scholar [27] J. Wu, Y.-J. Lee, J. Xu and Ludmil Zikatanov, Convergence analysis on iterative methods for semidefinite systems, J. Comput. Math., 26 (2008), 797-815.  Google Scholar [28] D. M. Young, "Iterative Solution of Large Linear Systems," Academic Press, New York, 1971.  Google Scholar
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