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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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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.

The block cyclic projection method in the compressed sensing framework (BCPCS) was introduced for image reconstruction in computed tomography and its convergence had been proven in the case of unity relaxation (λ = 1). In this paper, we prove its convergence with underrelaxation parameters λ ∈ (0, 1). As a result, the convergence of compressed sensing based block component averaging algorithm (BCAVCS) and block diagonally-relaxed orthogonal projection algorithm (BDROPCS) with underrelaxation parameters under a certain condition are derived. Experiments are given to illustrate the convergence behavior of these algorithms with selected parameters.