A sequence of least squares problems of the form miny kG1=2(AT y h)k2, where G is an n n positive definite diagonal weight matrix, and A anm n (m n) sparse matrix with some dense columns; has many applications in linear programming, electrical networks, elliptic boundary value problems, and structural analysis. We suggest low-rank correction preconditioners for such problems, and a mixed solver...

Stretching is a new sparse matrix method that makes matrices sparser by making them larger. Stretching has implications for computational complexity theory and applications in scientific and parallel computing. It changes matrix sparsity patterns to render linear equations more easily solved by parallel and sparse techniques. Some stretchings increase matrix condition numbers only moderately, a...

This paper provides a new numerical strategy for solving fractional-in-space reactiondiffusion equations on bounded domains under homogeneous Dirichlet boundary conditions. Using the matrix transfer technique the fractional Laplacian operator is replaced by a matrix which, in general, is dense. The approach here presented is based on the approximation of this matrix by the product of two suitab...

in which A is a large sparse symmetric positive definite matrix, λ is an eigenvalue and u is a corresponding eigenvector. The evaluation of one or more smallest eigenpairs has much practical interest for describing the characteristics of physical phenomena. For example, smallest eigenvalues characterize the base frequences of vibrating mechanical structures. Typically, the matrix A is a discret...

We formulate new optimal quadratic spline collocation methods for the solution of various elliptic boundary value problems in the unit square. These methods are constructed so that the collocation equations can be solved using a matrix decomposition algorithm. The results of numerical experiments exhibit the expected optimal global accuracy as well as superconvergence phenomena. AMS subject cla...

The singularly perturbed parabolic boundary value problem is considered. Difference scheme is obtained by using cubic spline difference scheme on Shishkin’s mesh in space and classical discretization on uniform mesh in time. To obtain better stability and simpler matrix the fitting factor in polynomial form is used. The uniform convergence is achieved. Numerical results are presented. AMS Mathe...

Parallel normalized preconditioned conjugate gradient type methods based on normalized approximate finite element inverse matrix techniques are investigated for the efficient solution of sparse linear systems. Application of the proposed methods on a three dimensional boundary value problems is discussed and numerical results are given. The parallel implementation of the normalized precondition...

A rigorous and complete solution of the initial-boundary-value (Goursat) problem for second harmonic generation (and its matrix analog) on the semi-strip is given in terms of the Weyl functions. A wide class of the explicit solutions and their Weyl functions is obtained also.