Skip to main content
SpringerLink
Log in

Generalized product-type methods based on bi-conjugate gradient (GPBiCG) for solving shifted linear systems

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

GPBiCG is a generalization of a class of product-type methods where the residual polynomials can be factored by the residual polynomial of BiCG and other polynomials with standard three-term recurrence relations. Actually this method generalizes CGS and BiCGStab methods. In this paper we use GPBiCG to present a new method for solving shifted linear systems. GPBiCG is faster than BiCGStab and its convergence is smoother than CGS. So here we are expecting to develop a method which is faster and its convergence is smoother than shifted BiCGStab and shifted CGS methods for solving shifted linear systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Abe K, Sleijpen GL (2013) Solving linear equations with a stabilized GPBiCG method. Appl Numer Math 67:4–16

    Article  MATH  MathSciNet  Google Scholar 

  • Datta BN, Saad Y (1991) Arnoldi methods for large Sylvester-like observer matrix equations, and an associated algorithm for partial spectrum assignment. Linear Algebra Appl 154:225–244

    Article  MATH  MathSciNet  Google Scholar 

  • Fletcher R (1976) Conjugate gradient methods for indefinite systems. Numer Anal, Springer 506:73–89

  • Freund RW (1993) Solution of shifted linear systems by quasi-minimal residual iterations. In: Numerical linear algebra: proceedings of the conference in numerical linear algebra and scientific computation, pp 101–121 (1993)

  • Freund RW, Nachtigal NM (1991) QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numerische Mathematik 60:315–339

    Article  MATH  MathSciNet  Google Scholar 

  • Frommer A (2003) BiCGStab(\(\ell \)) for families of shifted linear systems. Computing 70:87–109

    Article  MATH  MathSciNet  Google Scholar 

  • Frommer A, Glssner U (1998) Restarted GMRES for shifted linear systems. SIAM J Sci Comput 19:15–26

    Article  MATH  MathSciNet  Google Scholar 

  • Frommer A, Nckel B, Gsken S, Lippert T, Schilling K (1995) Many masses on one stroke: economic computation of quark propagators. Int J Mod Phys C 6:627–638

    Article  Google Scholar 

  • Fujino S (2002) GPBiCG (\(m, \ell \)): a hybrid of BiCGSTAB and GPBiCG methods with efficiency and robustness. Appl Numer Math 41:107–117

    Article  MATH  MathSciNet  Google Scholar 

  • Glssner U, Gsken S, Lippert T, Ritzenhfer G, Schilling K, Frommer A (1996) How to compute Green’s functions for entire mass trajectories within Krylov solvers. Int J Mod Phys C 7:635–644

    Article  Google Scholar 

  • Gu G (2005) Restarted GMRES augmented with harmonic Ritz vectors for shifted linear systems. Int J Comput Math 82:837–849

    Article  MATH  MathSciNet  Google Scholar 

  • Gutknecht MH (1993) Variants of BICGSTAB for matrices with complex spectrum. SIAM J Sci Comput 14:1020–1033

    Article  MATH  MathSciNet  Google Scholar 

  • Laub A (1985) Numerical linear algebra aspects of control design computations. IEEE Trans Autom Control 30:97–108

    Article  MATH  MathSciNet  Google Scholar 

  • National Institute of Standards and Technology. Matrix Market. http://math.nist.gov/MatrixMarket

  • Rllin S, Gutknecht MH (2002) Variations of Zhang’s Lanczos-type product method. Appl Numer Math 41:119–133

    Article  MATH  MathSciNet  Google Scholar 

  • Saad Y (2003) Iterative methods for sparse linear systems. SIAM, Philadelphia

  • Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7:856–869

    Article  MATH  MathSciNet  Google Scholar 

  • Simoncini V (2003) Restarted full orthogonalization method for shifted linear systems. BIT Numer Math 43:459–466

    Article  MATH  MathSciNet  Google Scholar 

  • Sleijpen GL, Fokkema DR (1993) BiCGstab(l) for linear equations involving unsymmetric matrices with complex spectrum. Electron Trans Numer Anal 1:11–32

    MATH  MathSciNet  Google Scholar 

  • Sonneveld P (1989) CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J Sci Stat Comput 10:36–52

    Article  MATH  MathSciNet  Google Scholar 

  • Soodhalter KM, Szyld DB, Xue F (2014) Krylov subspace recycling for sequences of shifted linear systems. Appl Numer Math 81:105–118

    Article  MATH  MathSciNet  Google Scholar 

  • Sweet RA (1988) A parallel and vector variant of the cyclic reduction algorithm. SIAM J Sci Stat Comput 9:761–765

    Article  MATH  MathSciNet  Google Scholar 

  • Van Den Eshof J, Sleijpen GL (2004) Accurate conjugate gradient methods for families of shifted systems. Appl Numer Math 49:17–37

    Article  MATH  MathSciNet  Google Scholar 

  • Van Der Vorst HA (1992) Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J Sci Stat Comput 13:631–644

    Article  MATH  MathSciNet  Google Scholar 

  • Van Der Vorst HA (2003) Iterative Krylov methods for large linear systems. Cambridge University Press, Cambridge

  • Wilson KG (1975) Quarks and strings on a lattice. In: Zichichi A (ed) New phenomena in subnuclear physics. Plenum, New York, pp 69–142 (1975)

  • Zhang S-L (1997) GPBi-CG: generalized product-type methods based on Bi-CG for solving nonsymmetric linear systems. SIAM J Sci Comput 18:537–551

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors are very grateful to the two referees for their useful comments and suggestions that helped us improve the presentation of this paper.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave, 15914, Tehran, Iran

    Mehdi Dehghan & Reza Mohammadi-Arani

Authors
  1. Mehdi Dehghan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Reza Mohammadi-Arani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mehdi Dehghan.

Additional information

Communicated by Jinyun Yuan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dehghan, M., Mohammadi-Arani, R. Generalized product-type methods based on bi-conjugate gradient (GPBiCG) for solving shifted linear systems. Comp. Appl. Math. 36, 1591–1606 (2017). https://doi.org/10.1007/s40314-016-0315-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40314-016-0315-y

Keywords

Mathematics Subject Classification

Search

Navigation