Numerical Algorithms

, Volume 65, Issue 4, pp 759–782 | Cite as

Composite convergence bounds based on Chebyshev polynomials and finite precision conjugate gradient computations

  • Tomáš Gergelits
  • Zdeněk Strakoš
Original Paper


The conjugate gradient method (CG) for solving linear systems of algebraic equations represents a highly nonlinear finite process. Since the original paper of Hestenes and Stiefel published in 1952, it has been linked with the Gauss-Christoffel quadrature approximation of Riemann-Stieltjes distribution functions determined by the data, i.e., with a simplified form of the Stieltjes moment problem. This link, developed further by Vorobyev, Brezinski, Golub, Meurant and others, indicates that a general description of the CG rate of convergence using an asymptotic convergence factor has principal limitations. Moreover, CG is computationally based on short recurrences. In finite precision arithmetic its behaviour is therefore affected by a possible loss of orthogonality among the computed direction vectors. Consequently, any consideration concerning the CG rate of convergence relevant to practical computations must include analysis of effects of rounding errors. Through the example of composite convergence bounds based on Chebyshev polynomials, this paper argues that the facts mentioned above should become a part of common considerations on the CG rate of convergence. It also explains that the spectrum composed of small number of well separated tight clusters of eigenvalues does not necessarily imply a fast convergence of CG or other Krylov subspace methods.


Conjugate gradient method Stieltjes moment problem Chebyshev semi-iterative method Composite polynomial convergence bounds Finite precision computations Clusters of eigenvalues 

Mathematics Subject Classifications (2010)

65F10 65B99 65G50 65N15 


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic

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