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Linearized alternating direction method of multipliers with Gaussian back substitution for separable convex programming
The stationary iterations revisited
1. | Department of Computer Science, Fitchburg State University, Fitchburg, MA 01420, United States |
2. | Institute of Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China |
3. | School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai 200433, China |
References:
[1] |
A. Berman and R. Plemmons, "Nonnegative Matrices in Mathematical Science," Academic Press, New York, 1979, SIAM, Philadelphia, 1994. |
[2] |
P. Bochev and R. B. Lehoucq, On the finite element solution of the pure Neumann problem, SIAM Review, 47 (2005), 50-66.
doi: 10.1137/S0036144503426074. |
[3] |
S. Campbell and C. Meyer, "Generalized Inverses of Linear Transformations," Pitman, London, 1979; Dover Publications, 1991; SIAM, Philadelphia, 2008. |
[4] |
Z. Cao, On the convergence of iterative methods for solving singular linear systems, Journal of Computational and Applied Mathematics, 145 (2002), 1-9.
doi: 10.1016/S0377-0427(01)00531-3. |
[5] |
Z. Cao, On the convergence of general stationary linear iterative methods for singular linear systems, SIAM Journal on Matrix Analysis and Applications, 29 (2008), 1382-1388.
doi: 10.1137/060671243. |
[6] |
X. Chen and R. E. Hartwig, The Picard iteration and its application, Linear and Multilinear Algebra, 54 (2006), 329-341.
doi: 10.1080/03081080500209703. |
[7] |
X. Cui, Y. Wei and N. Zhang, Quotient convergence and multi-splitting methods for solving singular linear equations, Calcolo, 44 (2007), 21-31.
doi: 10.1007/s10092-007-0127-y. |
[8] |
M. Eiermann, I. Marek and W. Niethammer, On the solution of singular linear systems of algebraic equations by semi-iterative methods, Numerische Mathematik, 53 (1988), 265-283.
doi: 10.1007/BF01404464. |
[9] |
A. Frommer, R. Nabben and D. Szyld, Convergence of stationary iterative methods for Hermitian semidefinite linear systems and applications to Schwarz methods, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 925-938.
doi: 2009m:65055. |
[10] |
F. R. Gantmacher, "The Theory of Matrices," Chelsea, New York, 1 (1960). |
[11] |
N. Higham and P. Knight, Finite precision behavior of stationary iteration for solving singular systems, Linear Algebra and Its Applications, 192 (1993), 165-186.
doi: 10.1016/0024-3795(93)90242-G. |
[12] |
H. Keller, On the solution of singular and semi-definite linear systems by iteration, SIAM Journal on Numerical Analysis, 2 (1965), 281-290. |
[13] |
Y. Lee, J. Wu and L. Zikatanov, On the convergence of iterative methods for semidefinite linear systems, SIAM Journal on Matrix Analysis and Applications, 28 (2006), 634-641.
doi: 10.1137/050644197. |
[14] |
L. Lin, Y. Wei, C. Woo and J. Zhou, On the convergence of splittings for semidefinite linear systems, Linear Algebra and its Applications, 429 (2008), 2555-2566.
doi: 10.1016/j.laa.2007.12.019. |
[15] |
L. Lin, Y. Wei and N. Zhang, Convergence and quotient convergence of iterative methods for solving singular linear equations with index one, Linear Algebra and its Applications, 430 (2009), 1665-1674.
doi: 10.1016/j.laa.2008.06.019. |
[16] |
G. I. Marchuk and Y. Kuznetzov, "Iterative Methods and Quadratic Functionals," Science Press, Norvosibirsk, in Russian, 1972. |
[17] |
M. Neumann, Subproper splitting for rectangular matrices, Linear Algebra and its Applications, 14 (1976), 41-51.
doi: 10.1016/0024-3795(76)90062-8. |
[18] |
Y. Song, Semiconvergence of nonnegative splittings for singular matrices, Numerische Mathematik, 85 (2000), 109-127.
doi: 10.1007/s002110050479. |
[19] |
G. Wang, Y. Wei, and S. Qiao, "Generalized Inverses: Theory and Computations," Science Press, Beijing, 2004. |
[20] |
N. Zhang and Y. Wei, On the convergence of general stationary iterative methods for range-Hermitian singular linear systems, Numerical Linear Algebra with Applications, 17 (2010), 139-154.
doi: 10.1002/nla.663. |
show all references
References:
[1] |
A. Berman and R. Plemmons, "Nonnegative Matrices in Mathematical Science," Academic Press, New York, 1979, SIAM, Philadelphia, 1994. |
[2] |
P. Bochev and R. B. Lehoucq, On the finite element solution of the pure Neumann problem, SIAM Review, 47 (2005), 50-66.
doi: 10.1137/S0036144503426074. |
[3] |
S. Campbell and C. Meyer, "Generalized Inverses of Linear Transformations," Pitman, London, 1979; Dover Publications, 1991; SIAM, Philadelphia, 2008. |
[4] |
Z. Cao, On the convergence of iterative methods for solving singular linear systems, Journal of Computational and Applied Mathematics, 145 (2002), 1-9.
doi: 10.1016/S0377-0427(01)00531-3. |
[5] |
Z. Cao, On the convergence of general stationary linear iterative methods for singular linear systems, SIAM Journal on Matrix Analysis and Applications, 29 (2008), 1382-1388.
doi: 10.1137/060671243. |
[6] |
X. Chen and R. E. Hartwig, The Picard iteration and its application, Linear and Multilinear Algebra, 54 (2006), 329-341.
doi: 10.1080/03081080500209703. |
[7] |
X. Cui, Y. Wei and N. Zhang, Quotient convergence and multi-splitting methods for solving singular linear equations, Calcolo, 44 (2007), 21-31.
doi: 10.1007/s10092-007-0127-y. |
[8] |
M. Eiermann, I. Marek and W. Niethammer, On the solution of singular linear systems of algebraic equations by semi-iterative methods, Numerische Mathematik, 53 (1988), 265-283.
doi: 10.1007/BF01404464. |
[9] |
A. Frommer, R. Nabben and D. Szyld, Convergence of stationary iterative methods for Hermitian semidefinite linear systems and applications to Schwarz methods, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 925-938.
doi: 2009m:65055. |
[10] |
F. R. Gantmacher, "The Theory of Matrices," Chelsea, New York, 1 (1960). |
[11] |
N. Higham and P. Knight, Finite precision behavior of stationary iteration for solving singular systems, Linear Algebra and Its Applications, 192 (1993), 165-186.
doi: 10.1016/0024-3795(93)90242-G. |
[12] |
H. Keller, On the solution of singular and semi-definite linear systems by iteration, SIAM Journal on Numerical Analysis, 2 (1965), 281-290. |
[13] |
Y. Lee, J. Wu and L. Zikatanov, On the convergence of iterative methods for semidefinite linear systems, SIAM Journal on Matrix Analysis and Applications, 28 (2006), 634-641.
doi: 10.1137/050644197. |
[14] |
L. Lin, Y. Wei, C. Woo and J. Zhou, On the convergence of splittings for semidefinite linear systems, Linear Algebra and its Applications, 429 (2008), 2555-2566.
doi: 10.1016/j.laa.2007.12.019. |
[15] |
L. Lin, Y. Wei and N. Zhang, Convergence and quotient convergence of iterative methods for solving singular linear equations with index one, Linear Algebra and its Applications, 430 (2009), 1665-1674.
doi: 10.1016/j.laa.2008.06.019. |
[16] |
G. I. Marchuk and Y. Kuznetzov, "Iterative Methods and Quadratic Functionals," Science Press, Norvosibirsk, in Russian, 1972. |
[17] |
M. Neumann, Subproper splitting for rectangular matrices, Linear Algebra and its Applications, 14 (1976), 41-51.
doi: 10.1016/0024-3795(76)90062-8. |
[18] |
Y. Song, Semiconvergence of nonnegative splittings for singular matrices, Numerische Mathematik, 85 (2000), 109-127.
doi: 10.1007/s002110050479. |
[19] |
G. Wang, Y. Wei, and S. Qiao, "Generalized Inverses: Theory and Computations," Science Press, Beijing, 2004. |
[20] |
N. Zhang and Y. Wei, On the convergence of general stationary iterative methods for range-Hermitian singular linear systems, Numerical Linear Algebra with Applications, 17 (2010), 139-154.
doi: 10.1002/nla.663. |
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