Influence of the Eigenvalue Spectrum on the Convergence Rate of the Conjugate Gradient Method
- 1 August 1977
- journal article
- research article
- Published by Oxford University Press (OUP) in IMA Journal of Applied Mathematics
- Vol. 20 (1) , 61-72
- https://doi.org/10.1093/imamat/20.1.61
Abstract
The analogy between the conjugate gradient method for the solution of linear simultaneous equations and a polynomial curve fitting problem provides a means of determining bounds for the convergence rate of the conjugate gradient method. Chebyshev polynomials are used to give bounds for the convergence rate associated with the main group of eigenvalues, assuming that the eigenvalues are closely spaced within the group. Additional penalty functions are developed to correct the convergence rate bounds when outlying eigenvalues are present.Keywords
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