Adaptive Chebyshev Iteration Based on Modified Moments

Abstract

The problem of solving a linear system of equations $$Ax = b\quad A \in {\mathbb{R}^{N \times N}},\quad x,b \in {\mathbb{R}^N}$$ (1) , with a large, sparse and nonsymmetric matrix A arises in many applications. A Chebyshev iterative method based on scaled Chebyshev polynomials p n for an interval in the complex plane can be used to solve (1) when the spectrum of A lies in the right half plane. Manteuffel [3] discusses such Chebyshev iterative schemes and shows that the iterations depend on two parameters only, the center d and the focal length c of an ellipse in the complex plane with foci at d±c. In these schemes, the p n are Chebyshev polynomials for the interval between the foci, and are scaled so that p n(0) = 1. Let x 0 denote a given initial approximate solution of (1). The three-term recurrence relation for the p n yields an inexpensive recurrence relation for computing a sequence of approximate solutions x 1, x 2, ... of (1). The iterates x n determined by the Chebyshev iterative method are such that the error vectors e n ≔A -1 b - x n satisfy e n = p n(A)e 0, n ≥ 0. Let λ(A) denote the spectrum of A. If the parameters d and c are chosen so that $$\mathop {\max }\limits_{x \in \lambda (A)} \left| {{p_n}(z)} \right|$$ decreases rapidly as n increases, then the Euclidean norm of e n decreases rapidly as n increases as well. Chebyshev iteration is an attractive solution method if parameters d and c exist, such that there is an ellipse with foci at d±c which contains λ(A) and is not very close to the origin. Assuming that such an ellipse exists, its center d and focal length c can be determined if λ(A) is explicitly known.