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Excerpt from Predicting the Behavior of Finite Precision Lanczos and Conjugate Gradient Computations
Finite precision CG computations for solving an n by n symmetric positive definite linear system Ar b sometimes fail to converge after n steps, especially when n is small. In such cases, it is demonstrated that exact CG applied to the corresponding large linear system A5: b also requires more than 11 iterations to converge. More commonly, finite precision CG computations converge in far fewer than n steps, and the same holds for the exact CG algorithm applied to any matrix A whose eigenvalues are clustered in tiny intervals about the eigenvalues of A. Frequently, finite precision CG computations go through several steps at which there is only a modest reduction in the error and then at the next step there is a very sharp decrease in the error. This same behavior is observed in the exact CG algorithm applied to matrices A whose eigenvalues are distributed in it tight clusters about the eigenvalues of A.
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