Abstract
In their original paper, Golub and Meurant (BIT 37:687–705, 1997) suggest to compute bounds for the A-norm of the error in the conjugate gradient (CG) method using Gauss, Gauss-Radau and Gauss-Lobatto quadratures. The quadratures are computed using the (1,1)-entry of the inverse of the corresponding Jacobi matrix (or its rank-one or rank-two modifications). The resulting algorithm called CGQL computes explicitly the entries of the Jacobi matrix and its modifications from the CG coefficients. In this paper, we use the fact that CG computes the Cholesky decomposition of the Jacobi matrix which is given implicitly. For Gauss-Radau and Gauss-Lobatto quadratures, instead of computing the entries of the modified Jacobi matrices, we directly compute the entries of the Cholesky decompositions of the (modified) Jacobi matrices. This leads to simpler formulas in comparison to those used in CGQL.
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References
Arioli, M.: A stopping criterion for the conjugate gradient algorithms in a finite element method framework. Numer. Math. 97, 1–24 (2004)
Auchmuty, G.: A posteriori error estimates for linear equations. Numer. Math. 61, 1–6 (1992)
Brezinski, C.: Error estimates for the solution of linear systems. SIAM J. Sci. Comput. 21, 764–781 (1999)
Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Computable error bounds and estimates for the conjugate gradient method. Numer. Algor. 25, 75–88 (2000)
Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: An iterative method with error estimators. J. Comput. Appl. Math. 127, 93–119 (2001)
Dahlquist, G., Eisenstat, S.C., Golub, G.H.: Bounds for the error of linear systems of equations using the theory of moments. J. Math. Anal. Appl. 37, 151–166 (1972)
Dahlquist, G., Golub, G.H., Nash, S.G.: Bounds for the error in linear systems. In: Semi-infinite Programming (Proc. Workshop, Bad Honnef, 1978). Lecture Notes in Control and Information Sci., vol. 15, pp. 154–172. Springer, Berlin (1979)
Fischer, B., Golub, G.H.: On the error computation for polynomial based iteration methods. In: Recent Advances in Iterative Methods. IMA Vol. Math. Appl., vol. 60, pp. 59–67. Springer, New York (1994)
Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Oxford University Press, UK (2004)
Golub, G.H.: Some modified matrix eigenvalue problems. SIAM Rev. 15, 318–334 (1973)
Golub, G.H., Meurant, G.: Matrices, moments and quadrature. In: Numerical Analysis 1993 (Dundee, 1993). Pitman Res. Notes Math. Ser., Longman Sci. Tech., vol. 303, pp. 105–156. Harlow (1994)
Golub, G.H., Meurant, G.: Matrices, moments and quadrature. II. How to compute the norm of the error in iterative methods. BIT 37, 687–705 (1997)
Golub, G.H., Meurant, G.: Matrices, Moments and Quadrature with Applications. Princeton University Press, USA (2010)
Golub, G.H., Strakoš, Z.: Estimates in quadratic formulas. Numer. Algor. 8, 241–268 (1994)
Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences, 3rd edn. Johns Hopkins University Press, Baltimore, MD (1996)
Golub, G.H., Welsch, J.H.: Calculation of Gauss quadrature rules. Math. Comp. 23, 221–230 (1969) (addendum, ibid., 23, A1–A10 (1969))
Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Research Nat. Bur. Standards 49, 409–436 (1952)
Jiránek, P., Strakoš, Z., Vohralík, M.: A posteriori error estimates including algebraic error and stopping criteria for iterative solvers. SIAM J. Sci. Comput. 32, 1567–1590 (2010)
Lanczos, C.: Solution of systems of linear equations by minimized iterations. J. Research Nat. Bur. Standards 49, 33–53 (1952)
Laurie, D.P.: Anti-Gaussian quadrature formulas. Math. Comp. 65, 739–747 (1996)
Meurant, G.: The computation of bounds for the norm of the error in the conjugate gradient algorithm. Numer. Algor. 16, 77–87 (1998)
Meurant, G.: Numerical experiments in computing bounds for the norm of the error in the preconditioned conjugate gradient algorithm. Numer. Algor. 22, 353–365 (1999)
Meurant, G.: The Lanczos and Conjugate Gradient Algorithms, from Theory to Finite Precision Computations. Software, Environments, and Tools, vol. 19. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2006)
Meurant, G., Strakoš, Z.: The Lanczos and conjugate gradient algorithms in finite precision arithmetic. Acta Numer. 15, 471–542 (2006)
Parlett, B.N.: The new qd algorithms. Acta Numer. 15, 459–491 (1995)
Parlett, B.N., Dhillon, I.S.: Relatively robust representations of symmetric tridiagonals. In: Proceedings of the International Workshop on Accurate Solution of Eigenvalue Problems (University Park, PA, 1998), vol. 309, pp. 121–151 (2000)
Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 2nd edn. Springer, Berlin, Germany (1983)
Strakoš, Z., Tichý, P.: On error estimation in the conjugate gradient method and why it works in finite precision computations. Electron. Trans. Numer. Anal. 13, 56–80 (2002)
Strakoš, Z., Tichý, P.: Error estimation in preconditioned conjugate gradients. BIT 45, 789–817 (2005)
Wilf, H.S.: Mathematics for the Physical Sciences. Wiley, New York, USA (1962)
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This work was supported by the project No. IAA100300802 of the Grant Agency of the Academy of Sciences of the Czech Republic. It was finalized in 2011 during a visit of G. Meurant to the Institute of Computer Science of the Academy of Sciences of the Czech Republic.
An erratum to this article is available at http://dx.doi.org/10.1007/s11075-014-9868-2.
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Meurant, G., Tichý, P. On computing quadrature-based bounds for the A-norm of the error in conjugate gradients. Numer Algor 62, 163–191 (2013). https://doi.org/10.1007/s11075-012-9591-9
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DOI: https://doi.org/10.1007/s11075-012-9591-9