A greedy block Kaczmarz algorithm for solving largescale linear systems Abstract   The block Kaczmarz algorithm is an iterative scheme for solving largescale linear systems. At each iteration, the process of the algorithm is to orthogonally project the current iteration point onto the solution space of a subset of the constraints. In this paper, we propose a greedy block Kaczmarz algorithm by using a greedy strategy to construct control index set and choosing row submatrix in each iteration. This algorithm has an linear rate of convergence that can be expressed in terms of the geometric properties of the matrix and its row submatrices. The theoretical analysis and numerical results show that the developed algorithm can be more efficient than the greedy randomized Kaczmarz algorithm if parameter    xcexb7    is chosen appropriately.

A quadratically convergent local algorithm on minimizing sums of the largest eigenvalues of a symmetric matrix In this paper, we consider the problem on minimizing sums of the largest eigenvalues of a symmetric matrix which depends on the decision variable affinely. An important application of this problem is the graph partitioning problem, which arises in layout of circuit boards, computer logic partitioning, and paging of computer programs. Given xe2x88x88xe2x89xa50, we first derive an optimality condition which ensures that the objective function is within xe2x88x88 error bound of the solution. This condition may be used as a practical stopping criterion for any algorithm solving the underlying problem. We also show that, in a neighborhood of the minimizer, the optimization problem can be equivalently formulated as a smooth constrained problem. An existing algorithm on minimizing the largest eigenvalue of a symmetric matrix is shown to be applicable here. This algoritm enjoys the property that if started close enough to the minimizer, then it will converge quadratically. To implement a practical algorithm, one needs to incorporate some technique to generate a good starting point. Since the problem is convex, this can be done by using an algorithm for general convex optimization problems (e.g., Kelleyu0027s cutting plane method or ellipsoid methods), or an algorithm specific for the optimization problem under consideration (e.g., the algorithm developed by Cullum, Donath, and Wolfe). Such union ensures that the overall algorithm has global convergence with quadratic rate. Finally, the results presented in this paper are readily extended on minimizing sums of the largest eigenvalues of a Hermitian matrix.