On a Modified Kovarik Algorithm for Symmetric Matrices

In some of his scientific papers and university courses, professor Silviu Sburlan has studied integral equations (see the list of references). Beside the theoretical qualitative analysis concerning the existence, uniqueness and other properties of the solution, he was also interested in its numerical approximation. In the case of first kind integral equations with smooth kernel (e.g. continuous) it is well known that, by applying classical discretization techniques (as collocation or projection methods) we get (very) ill-conditioned symmetric positive semidefinite linear systems. This will cause big troubles for both direct or iterative solvers. Moreover, the system matrix is usually dense, thus classical preconditioning techniques, as Incomplete Decomposition can not be used. One possibility to overcome this difficulty is to use orthogonalization algorithms, which also “compress” the spectrum of the system matrix, by transforming it into a well-conditioned one. Unfortunately, the well known Gram-Schmidt method fails in the case of singular matrices or is totally unstable for ill-conditioned ones. In a previous paper, the author extended an iterative approximate orthogonalization algorithm due to Z. Kovarik, to the case of arbitrary rectangular matrices. In the present one, we adapt this algorithm to the class of symmetric (positive semi-definite) matrices. The new algorithm has similar convergence properties as the initial one, but requires much less computational effort per iteration. Some numerical experiments are also described for a “model problem” first kind integral equation.