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On prescribing the convergence behavior of the conjugate gradient algorithm

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Abstract

The conjugate gradient (CG) algorithm is the most frequently used iterative method for solving linear systems Ax = b with a symmetric positive definite (SPD) matrix. In this paper we construct real symmetric positive definite matrices A of order n and real right-hand sides b for which the CG algorithm has a prescribed residual norm convergence curve. We also consider prescribing as well the A-norms of the error. We completely characterize the tridiagonal matrices constructed by the Lanczos algorithm and their inverses in terms of the CG residual norms and A-norms of the error. This also gives expressions and lower bounds for the ℓ2 norm of the error. Finally, we study the problem of prescribing both the CG residual norms and the eigenvalues of A. We show that this is not always possible. Our constructions are illustrated by numerical examples.

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References

  1. Arioli, M., Pták, V., Strakoš, Z.: Krylov sequences of maximal length and convergence of GMRES. BIT Numer. Math. 38(4), 636–643 (1998)

    MathSciNet  MATH  Google Scholar 

  2. Arnoldi, W.E.: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quart. Appl. Math. 9, 17–29 (1951)

    MathSciNet  MATH  Google Scholar 

  3. Baranger, J., Duc-Jacquet, M.: Matrices tridiagonales symétriques et matrices factorisables. RIRO 3, 61–66 (1971)

    MATH  Google Scholar 

  4. Duintjer Tebbens, J., Meurant, G.: Any Ritz value behavior is possible for Arnoldi and for GMRES. SIAM J. Matrix Anal. Appl. 33(3), 958–978 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Duintjer Tebbens, J., Meurant, G.: Prescribing the behavior of early terminating GMRES and Arnoldi iterations. Numer. Algorithms 65(1), 69–90 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Duintjer Tebbens, J., Meurant, G.: On the convergence of Q-OR and Q-MR Krylov methods for solving linear systems. BIT Numer. Math. 56(1), 77–97 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Fiedler, M.: Polynomials and Hankel matrices. Linear Algebra Appl. 66, 235–248 (1985)

    MathSciNet  MATH  Google Scholar 

  8. Golub, G.H., Meurant, G: Matrices, Moments and Quadrature with Applications. Princeton University Press (2010)

  9. Greenbaum, A., Strakoš, Z.: Matrices that generate the same Krylov residual spaces. In: Golub, G.H., Greenbaum, A., Luskin, M. (eds.) Recent Advances in Iterative Methods, pp 95–118. Springer (1994)

  10. Greenbaum, A., Pták, V., Strakoš, Z.: Any convergence curve is possible for GMRES. SIAM J. Matrix Anal. Appl. 17(3), 465–470 (1996)

    MathSciNet  MATH  Google Scholar 

  11. Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Nat. Bur. Standards 49(6), 409–436 (1952)

    MathSciNet  MATH  Google Scholar 

  12. Ikebe, Y.: On inverses of Hessenberg matrices. Linear Algebra Appl. 24, 93–97 (1979)

    MathSciNet  MATH  Google Scholar 

  13. Juárez-Ruiz, E., Cortés-Maldonado, R., Pérez-Rodríguez, F.: Relationship between the inverses of a matrix and a submatrix. Computación y Sistemas 20(2), 251–262 (2016)

    Google Scholar 

  14. Lanczos, C.: Solution of systems of linear equations by minimized iterations. J. Res. Nat. Bur. Standards 49, 33–53 (1952)

    MathSciNet  Google Scholar 

  15. Liesen, J., Strakoš, Z: Krylov Subspace Methods, Principles and Analysis. Oxford University Press (2013)

  16. Meurant, G.: A review on the inverse of tridiagonal and block tridiagonal symmetric matrices. SIAM J. Matrix Anal. Appl. 13(3), 707–728 (1992)

    MathSciNet  MATH  Google Scholar 

  17. Meurant, G.: Numerical experiments in computing bounds for the norm of the error in the preconditioned conjugate gradient algorithm. Numer. Algorithms 22, 353–365 (1999)

    MathSciNet  MATH  Google Scholar 

  18. Meurant, G.: The Lanczos and Conjugate Gradient Algorithms from Theory to Finite Precision Computations. SIAM, Philadelphia (2006)

    MATH  Google Scholar 

  19. Meurant, G., Strakoš, Z.: The Lanczos and conjugate gradient algorithms in finite precision arithmetic. Acta Numerica 15, 471–542 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Paige, C.C.: The Computation of Eigenvalues and Eigenvectors of Very Large Sparse Matrices. University of London, Ph.D. thesis (1971)

    Google Scholar 

  21. Paige, C.C., Saunders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12(4), 617–629 (1975)

    MathSciNet  MATH  Google Scholar 

  22. Saad, Y.: Krylov subspace methods for solving large nonsymmetric linear systems. Math. Comp. 37, 105–126 (1981)

    MathSciNet  MATH  Google Scholar 

  23. Saad, Y.: Practical use of some Krylov subspace methods for solving indefinite and nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 5(1), 203–228 (1984)

    MathSciNet  MATH  Google Scholar 

  24. Schweitzer, M.: Any finite convergence curve is possible in the initial iterations of restarted FOM. Electron. Trans. Numer. Anal. 45, 133–145 (2016)

    MathSciNet  MATH  Google Scholar 

  25. Strakoš, Z., Tichý, P.: On error estimates in the conjugate gradient method and why it works in finite precision computations. Electron. Trans. Numer. Anal. 13, 56–80 (2002)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

Many thanks to Erin Carson and Jurjen Duintjer Tebbens for interesting comments and suggestions and to Petr Tichý for comments and suggesting to use relation (2). The author thanks the referees for their detailed comments.

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    Gérard Meurant

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  1. Gérard Meurant
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Correspondence to Gérard Meurant.

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Meurant, G. On prescribing the convergence behavior of the conjugate gradient algorithm. Numer Algor 84, 1353–1380 (2020). https://doi.org/10.1007/s11075-019-00851-2

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