For the randomized residual error, distribution- free bounds exist and are provided in the paper. We also provide asymptotic bounds, as well as bounds depending on the ratio of the two largest eigenvalues. The bounds for the Lanczos algorithm may be helpful in a practical implementation and termination of this algorithm. It motivates the construction of improved multigrid versions for diffusion problems with discontinuous coefficients. A computational two-level analysis method for a class of separable problems is also available.
Numerical experiments show the advantage of the present methods for several examples.
The algebraic properties of the matrix arising for the three-dimensional Dirichlet problem for Lame equations in a rotational domain by the boundary element method are considered. The use of the special basis leads to a matrix having a block structure with sparse blocks. The possible strategies for the efficient solution of the above problem are discussed. An inexact Newton algorithm for large sparse equality constrained non-linear programming problems is proposed. This algorithm is based on an indefinitely preconditioned smoothed conjugate gradient method applied to the linear KKT system and uses a simple augmented Lagrangian merit function for Armijo type stepsize selection.