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a b s t r a c tSparse solutions for an underdetermined system of linear equations Φx = u can be found more accurately by l 1 -minimization type algorithms, such as the reweighted l 1 -minimization and l 1 greedy algorithms, than with analytical methods, in particular in the presence of noisy data. Recently, a generalized l 1 greedy algorithm was introduced and applied to signal and image recovery. Numerical experiments have demonstrated the convergence of the new algorithm and the superiority of the algorithm over the reweighted l 1 -minimization and l 1 greedy algorithms although the convergence has not yet been proven theoretically. In this paper, we provide an error bound for the reweighted l 1 greedy algorithm, a type of the generalized l 1 greedy algorithm, in the noisy case and show its improvement over the reweighted l 1 -minimization.

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Abstract. The Chebyshev semiiterative method (CHSIM) is a powerful method for finding the iterative solution of a nonsymmetric real linear system Ax = b if an ellipse excluding the origin well fits the spectrum of A. The asymptotic rate of convergence of the CHSIM for solving the above system under a perturbation of the foci of the optimal ellipse is studied. Several formulae to approximate the asymptotic rates of convergence, up to the first order of a perturbation, are derived. These generalize the results about the sensitivity of the asymptotic rate of convergence to a perturbation of a real-line segment spectrum by Hageman and Young, and by the first author. A numerical example is given to illustrate the theoretical results.Key words. Chebyshev semiiterative method, Asymptotic rate of convergence.

Rook placements and rook polynomials have been studied by mathematicians since the early 1970's. Since then many relationships between rook placements and other subjects have been discovered (cf. [1], [6-15]). In [2] and [3], K. Ding introduced the rook length polynomials and the $ \gamma - $compatible rook length polynomials. In [3] and [4], he used these polynomials to establish a connection between rook placements and algebraic geometry for the first time. In this paper, we give explicit formulas for the $ \gamma - $compatible rook length polynomials in more general cases than considered in [3]. In particular, we generalize the formula for the rook length polynomial in the parabolic case in [2] to the $ \gamma -$compatible rook length polynomial.