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A QMR-Type Algorithm for Drazin-Inverse Solution of Singular Nonsymmetric Linear Systems

DOI: 10.4236/alamt.2016.64011, PP. 104-115

Keywords: Singular Linear Systems, DGMRES Method, Quasi-Minimal Residual Methods, Drazin-Inverse Solution, Krylov Subspace Methods

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Abstract:

In this paper, we propose DQMR algorithm for the Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax=b where \"\"?is a?singular and in general non-hermitian matrix that has an arbitrary index. DQMR algorithm for singular systems is analogous to QMR algorithm for non-singular systems. We compare this algorithm with DGMRES by numerical experiments.

References

[1]  Ben-Israel, A. and Grevile, T.N.E. (2003) Generalized Inverses: Theory and Applications. 2nd Edition, Springer-Verlag, New York.
[2]  Campell, S.L. and Meyer, C.D. (1979) Generalized Inverses of Linear Transformations. Pitman (Advanced Publishing Program), Boston.
[3]  Hartwig, R.E. and Hall, F. (1982) Applications of the Drazin Inverse to Cesaro-Neumann Iterations. In: Campbell, S.L., Ed., Recent Applications of Generalized Inverses, 66, 145-195.
[4]  Hartwig, R.E. and Levine, J. (1981) Applications of the Drazin Inverse to the Hill Cryptographic system. Part III, Cryptologia, 5, 67-77.
https://doi.org/10.1080/0161-118191855850
[5]  Eiermann, M., Marek, I. and Niethammer, W. (1988) On the Solution of Singular Linear Systems of Algebric Equations by Semiiterative Methods. Numerische Mathematik, 53, 265-283.
https://doi.org/10.1007/BF01404464
[6]  Freund, R.W. and Hochbruck, M. (1994) On the Use of Two QMR Algorithms for Solving Singular Systems and Applications in Markov Chain Modeling. Numerical Linear Algebra with Applications, 1, 403-420.
https://doi.org/10.1002/nla.1680010406
[7]  Simeon, B., Fuhrer, C. and Rentrop, P. (1993) The Drazin Inverse in Multibody System Dynamics. Numerische Mathematik, 64, 521-539.
https://doi.org/10.1007/BF01388703
[8]  Brown, P.N. and Walker, H.F. (1997) GMRES on (Nearly) Singular Systems. SIAM Journal on Matrix Analysis and Applications, 18, 37-51.
https://doi.org/10.1137/S0895479894262339
[9]  Ipsen, I.F. and Meyer, C.D. (1998) The Idea behind Krylov Methods. The American Mathematical Monthly, 105, 889-899.
https://doi.org/10.2307/2589281
[10]  Smoch, L. (1999) Some Result about GMRES in the Singular Case. Numerical Algorithms, 22, 193-212.
https://doi.org/10.1023/A:1019162908926
[11]  Wei, Y. and Wu, H. (2000) Convergence Properties of Krylov Subspace Methods for Singular System with Arbitrary Index. Journal of Computational and Applied Mathematics, 114, 305-318.
https://doi.org/10.1016/S0377-0427(99)90237-6
[12]  Sidi, A. (1999) A Unified Approach to Krylov Subspace Methods for the Drazin-Inverse Solution of Singular Non-Symmetric Linear Systems. Linear Algebra and Its Applications, 298, 99-113.
https://doi.org/10.1016/S0024-3795(99)00153-6
[13]  Sidi, A. and Kluzner, V. (1999) A Bi-CG Type Iterative Method for Drazin Inverse Solution of Singular Inconsistent Non-Symmetric Linear Systems of Arbitrary Index. The Electronic Journal of Linear Algebra, 6, 72-94.
[14]  Sidi, A. (2001) DGMRES: A GMRES-Type Algorithm for Drazin-Inverse Solution of Singular Non-Symmetric Linear Systems. Linear Algebra and Its Applications, 335, 189-204.
https://doi.org/10.1016/S0024-3795(01)00289-0
[15]  Zhou, J. and Wei, Y. (2004) DFOM Algorithm and Error Analysis for Projection Methods for Solving Singular Linear System. Applied Mathematics and Computation, 157, 313-329.
https://doi.org/10.1016/j.amc.2003.08.036
[16]  Lanczos, C. (1950) An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators. Journal of Research of the National Bureau of Standards, 45, 255-282.
https://doi.org/10.6028/jres.045.026
[17]  Saad, Y. (2003) Iterative Methods for Sparse Linear Systems. 2nd Edition, SIAM, Philadelphia.
https://doi.org/10.1137/1.9780898718003
[18]  Hank, M. and Hochbruck, M. (1993) A Chebyshev-Like Semiiteration for Inconsistent Linear Systems. Electronic Transactions on Numerical Analysis, 1, 315-339.

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