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2013, 3(2): 261-270. doi: 10.3934/naco.2013.3.261

The stationary iterations revisited

Xuzhou Chen 1, , Xinghua Shi 2, and  Yimin Wei 3,

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

Received  February 2012 Revised  January 2013 Published  April 2013

In this paper, we first present a necessary and sufficient conditions for the weakly and strongly convergence of the general stationary iterations $x^{(k+1)} = T x^{(k)} +c$ with initial iteration matrix $T$ and vectors $c$ and $x^{(0)}$. Then we apply these general results and present convergence conditions for the stationary iterations for solving singular linear system $A x = b$. We show that our convergence conditions are weaker and more general than the known results.
Keywords: Stationary iterations, Picard iterations, quotient convergence, singular linear systems, matrix splitting.
Mathematics Subject Classification: Primary: 65F10, 65F15; Secondary: 15A0.
Citation: Xuzhou Chen, Xinghua Shi, Yimin Wei. The stationary iterations revisited. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 261-270. doi: 10.3934/naco.2013.3.261
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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