Convergence analysis of the ChebFilterCG algorithm

Summary The ChebFilterCG algorithm, proposed by Golub, Ruiz, and Touhami [SIAM J. Matrix Anal. Appl. 29 (2007), pp. 774‐795] is an iterative method that combines Chebyshev filter and conjugate gradient for solving symmetric positive definite linear systems with multiple right‐hand sides. The Chebyshev filter is used to produce initial residuals rich in eigenvectors corresponding to the smallest eigenvalues, which are then used in the initial phase of the conjugate gradient. This paper presents a convergence analysis of ChebFilterCG. In particular, it is shown theoretically and numerically that the algorithm yields an approximation of the invariant subspace associated with the smallest eigenvalues that can be recycled for solving several linear systems with the same matrix and different right‐hand sides. A refined error bound when solving these systems is also given. The choice and influence of the Chebyshev filtering steps is discussed. Numerical experiments are described to illustrate that the Chebyshev filter does not degrade the distribution of the smallest eigenvalues and highlight the effect of rounding errors when large outlying eigenvalues are present. Finally, it is shown that the method may become more effective when an additional Chebyshev filtering step is used in the initialization phase of ChebFilterCG.