Block-Iterative Algorithms with Diagonally Scaled Oblique Projections for the Linear Feasibility Problem

We formulate a block-iterative algorithmic scheme for the solution of systems of linear inequalities and/or equations and analyze its convergence. This study provides as special cases proofs of convergence of (i) the recently proposed component averaging (CAV) method of Censor, Gordon, and Gordon [Parallel Comput., 27 (2001), pp. 777–808], (ii) the recently proposed block-iterative CAV (BICAV) method of the same authors [IEEE Trans. Medical Imaging, 20 (2001), pp. 1050–1060], and (iii) the simultaneous algebraic reconstruction technique (SART) of Andersen and Kak [Ultrasonic Imaging, 6 (1984), pp. 81–94] and generalizes them to linear inequalities. The first two algorithms are projection algorithms which use certain generalized oblique projections and diagonal weighting matrices which reflect the sparsity of the underlying matrix of the linear system. The previously reported experimental acceleration of the initial behavior of CAV and BICAV is thus complemented here by a mathematical study of the convergence of the algorithms.