Bounds on the largest singular value of a matrix and the convergence of simultaneous and block-iterative algorithms for sparse linear systems - Byrne, Charles (International transactions in operational research, 2009)
We obtain the following upper bounds for the eigenvalues of the matrix A†A. For any a in the interval [0, 2] let and ca and ra the maxima of the caj and rai, respectively. Then no eigenvalue of the matrix A†A exceeds the maximum of over all i, nor the maximum of over all j. Therefore, no eigenvalue of A†A exceeds cara. Using these bounds, it follows that, for the matrix G with entries no eigenvalue of G†G exceeds one, provided that, for some a in the interval [0, 2], we have and Using this result, we obtain convergence theorems for several iterative algorithms for solving the problem Ax=b, including the CAV, BICAV, CARP1, SART, SIRT, and the block‐iterative DROP and SART methods.