Parallel Computing Developed an algorithm for orthogonal reduction of dense matrices to bidiagonal form on computers with distributed memory architectures. Created the code intended for bidiagonalization of square complex matrix. Worked on PSPARSLIB: a library of FORTRAN subroutines for solving large sparse linear systems in parallel in collaboration with Dr. Yousef Saad (http://www.cs.umn.edu/Research/arpa/p_sparslib/psp-abs.html). Krylov methods Proposed jointly with S. K. Godunov and J.-F. Carpraux the definition of condition numbers for the Krylov bases and subspaces. Obtained perturbation bounds of the orthogonal basis of the Krylov subspace as well as perturbation bounds of Hessenberg form of a matrix with respect to arbitrary perturbations in the matrix and in the initial vector. QR method Investigated the problem of convergence of the orthogonal power and QR algorithm (with prof. Godunov S. K.). The investigation is devoted to effect of round off errors on convergence of the orthogonal power method. Numerical methods for ordinary differential equations Developed the method of estimating the state of a process from measurements. The process is described by a system of ordinary differential equations with a vector stochastic process in the right-hand side. Suggested a new stable algorithm for finding efficient and some other estimates. Developed a new version of the orthogonal factorization method and proved its stability, which has been first proposed by S. K. Godunov Proposed the algorithm for calculating integral curves connecting a given two stationary points of a system of differential equations and its associated parameter values based on the new version of the orthogonal factorization method.
Main publications:
J. F. Carpraux, S. Godunov, S. Kuznetsov. Condition number of the Krylov bases and subspaces // Linear Algebra and its Applications, Elsevier Science, v. 248, p. 137–161, 1996.
S. Kuznetsov. Perturbation Bounds of The Krylov Bases and Associated Hessenberg Forms // Linear Algebra and its Applications, Elsevier Science, v. 265, p. 1–28, 1997.
S. Kuznetsov. Orthogonal reduction of dense matrices to bidiagonal form on computers with distributed memory architectures // Parallel Computing, Elsevier Science, v. 24/2, p. 305–313, 1998.
S. Kuznetsov, G. C. Lo, and Y. Saad. Parallel solution of general sparse linear systems // Domain Decomposition XI, editors: Choi-Hong Lai, Petter Bjorstad, Mark Cross, and Olof B. Widlund. Domain Decomposition Press, Bergen, Norway, p. 455–465, 1999.
Godunov S. K., Kuznetsov S. Estimates for the convergence of the orthogonal power method // Siberian Advances in Mathematics, v. 5, no. 1, p. 16–42, Alerton Press, New-York, 1995.
