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On convergence of the inner-outer iteration method for computing PageRank
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
A. Ben-Israel and T. N. E. Greville, "Generalized Inverses: Theory and Applications," Wiley, New York, 1974. |
[7] |
A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Science," Academic Press, New York, 1979. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
G. I. Marchuk and Y. Kuznetsov, "Iterative Methods and Quadratic Functionals," Science Press, Norvosibirsk, 1972 (in Russian). |
[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. |
[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. |
[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. |
[22] |
R. S. Varga, "Matrix Iterative Analysis," 2nd edition, Springer, Berlin, Heidelberg, 2000.
doi: 10.1007/978-3-642-05156-2. |
[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. |
[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. |
[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. |
[26] |
Y. Wei, Perturbation analysis of singular linear systems with index one, Int. J. Comput. Math., 74 (2000), 483-491.
doi: 10.1080/00207160008804956. |
[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. |
[28] |
D. M. Young, "Iterative Solution of Large Linear Systems," Academic Press, New York, 1971. |
show all references
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
A. Ben-Israel and T. N. E. Greville, "Generalized Inverses: Theory and Applications," Wiley, New York, 1974. |
[7] |
A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Science," Academic Press, New York, 1979. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
G. I. Marchuk and Y. Kuznetsov, "Iterative Methods and Quadratic Functionals," Science Press, Norvosibirsk, 1972 (in Russian). |
[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. |
[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. |
[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. |
[22] |
R. S. Varga, "Matrix Iterative Analysis," 2nd edition, Springer, Berlin, Heidelberg, 2000.
doi: 10.1007/978-3-642-05156-2. |
[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. |
[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. |
[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. |
[26] |
Y. Wei, Perturbation analysis of singular linear systems with index one, Int. J. Comput. Math., 74 (2000), 483-491.
doi: 10.1080/00207160008804956. |
[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. |
[28] |
D. M. Young, "Iterative Solution of Large Linear Systems," Academic Press, New York, 1971. |
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