Research Topics (Selected publications by area)

Bullet Matrix Perturbation theory

  • A.Minenkova, E. Nitch-Griffin, V Olshevsky, Backward Stability of the Schur Decomposition under small perturbations, Accepted on October 21, 2024, available online here: Linear Algebra and Its Applications.  
  • A.Minenkova, E. Nitch-Griffin, V Olshevsky, Forward Stability of the Schur Decomposition under small perturbations, Submitted.
  • V. Olshevsky. A condition for the closeness of the sets of invariant subspaces of the close matrices in terms of their Jordan structures, (in Russian), Siberian Math. Journal,   30 (No. 4) (1989), 102 – 110. English translation : Siberian Math. Journal, Plenum publishing corp., 30 (No. 4) (1989), 580 – 586.
  • V.Matsaev and V.Olshevsky. Cyclic dimensions, kernel multiplicities and Gohberg-Kaashoek numbers. Linear Algebra Appl., 239 (1996), 161-174 .

Bullet Perturbations in Indefinite Inner Product Spaces and Stability of Jordan Bases

  • S. Dogruer Akgul, A. Minekova, V.Olshevsky, Existence of flipped orthogonal and conjugate symmetric Jordan bases of real H-selfadjoint matrices, Linear and Multilinear Algebra, 72(7), 1160–1169. 
  • S. Dogruer Akgul, A. Minekova, V.Olshevsky, Lipschitz stability of  Ɣ-FOCS and RC canonical Jordan bases of real H-selfadjoint matrices under small perturbations,  arXiv:2204.04639o, Submitted.
  • T.Bella, V.Olshevsky and U.Prasad, Lipschitz stability of canonical Jordan bases of H-selfadjoint matrices under structure-preserving perturbations, Linear Algebra and its Applications, Volume 428, Issues 8-9, 15 April 2008, Pages 2130-2176.
  • V. Olshevsky. Change of Jordan structure of G-selfadjoint operators and selfadjoint operator functions under small perturbations, (in Russian), Izvestia Akad. nauk U.S.S.R.,   54 (No. 5) (1990), 1021 – 1048. English translation : AMS, Math. U.S.S.R. Izvestia, 37 (No. 2) (1991), 371 – 396.

Bullet Filtering and Estimation

  • V.Olshevsky and L.Sakhnovich, Matched filters for generalized stationary processes, IEEE Transactions on Information Theory 51(9): 3308-3313 (2005).
  • V.Olshevsky and L.Sakhnovich, Optimal Prediction of generalized stationary processes, In Recent Advances in Operator Theory and Its Applications The Israel Gohberg Anniversary Volume Series: Operator Theory: Advances and Applications, Vol. 160 Kaashoek, Marinus A.; Seatzu, Sebastiano; Mee, Cornelis van der (Eds.) 2005, p. 257-266.

Bullet  Coding Theory

  • US PATENT # 6,631,172. V.Olshevsky and A.Shokrollahi, “Efficient List Decoding of Reed-Solomon Codes for Message Recovery in the Presence of High Noise Levels.” Awarded in October 2003.
  • V. Olshevsky and M. A. Shokrollahi, A displacement approach to decoding algebraic codes, in Fast Algorithms for Structured Matrices: Theory and Applications, CONM/323, p. 265 – 292, AMS publications, May 2003.
  • T.Bella, V.Olshevsky, L. Sakhnovich, Ranks of Hadamard Matrices and Equivalence of Sylvester Hadamard and Pseudo-Noise Matrice, In Recent Advances in Matrix and Operator Theory, Operator Theory: Advances and Applications, Volume 179, Birkhauser Basel, 2008.

Bullet  Stability of Polynomials and Entire Functions. Generalized Kharitonov’s theorem. Generalized Bezoutians

  • Vadim Olshevsky, L Sakhnovich, A generalized Kharitonov theorem for quasi-polynomials and entire functions occurring in systems with multiple and distributed delays In Advanced Signal Processing Algorithms, Architectures, and Implementations XV. Editor(s): Franklin T. Luk, SPIE Publications, Aug 2005, p. 325-336.
  • A.Olshevsky and V.Olshevsky, Kharitonov’s theorem and Hermite’s criterion, Linear Algebra and its Applications, Volume 399, 1 April 2005, Pages 285-297
  • V.Olshevsky and L.Sakhnovich, An Operator Identities Approach to Bezoutians. A General Scheme and Examples, Proc. of the MTNS’04 conference, July 2004.

Bullet Quasiseparable Matrices and Eigenvalue Problems

  • T.Bella, Y.Eidelman, I.Gohberg, V.Olshevsky, Computations with quasiseparable matrices and polynomials , to appear in Theoretical Computer Science, vol 409(2), December 2008.
  • T.Bella, V.Olshevsky and P.Zhlobich, Classifications of recurrence relations via subclasses of (H,m)-quasiseparable matrices, In: Van Dooren, P., Bhattacharyya, S., Chan, R., Olshevsky, V., Routray, A. (eds) Numerical Linear Algebra in Signals, Systems and Control. Lecture Notes in Electrical Engineering, vol 80. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0602-6_2
  • T.Bella, V.Olshevsky, P.Zhlobich, A Quasiseparable approach to five-diagonal CMV and companion matrices, Linear Algebra and its Applications Volume 434, Issue 4, 15 February 2011, Pages 957-976
  • V.Olshevsky, G.Strang, P.Zhlobich, Green’s matrices, Linear Algebra and its Applications Volume 432, Issue 1, 1 January 2010, Pages 218-241
  • T.Bella, V.Olshevsky and P.Zhlobich, Signal Flow Graphs Approach to Inversion of (H,m)-Quasiseparable Vandermonde Matrices and New Filter Structures , Linear Algebra and its Applications, Volume 432, Issue 8, 1 April 2010, Pages 2032-2051
  • T.Bella, Y.Eidelman, I.Gohberg, V.Olshevsky, E.Tyrtyshnikov, P.Zhlobich, A Traub-like algorithm for Hessenberg-quasiseparable-Vandermonde matrices of arbitrary order , In: Bini, D.A., Mehrmann, V., Olshevsky, V., Tyrtyshnikov, E.E., van Barel, M. (eds) Numerical Methods for Structured Matrices and Applications. Operator Theory: Advances and Applications, vol 199. Birkhauser Basel. https://doi.org/10.1007/978-3-7643-8996-3_5
  • T.Bella, Y.Eidelman, I.Gohberg, V.Olshevsky, Characterization of (H,1) quasiseparable matrices and their subclasses via recurrence relations and signal flow graphs, preprint.
  • T.Bella, Y.Eidelman, I.Gohberg, V.Olshevsky, E.Tyrtyshnikov, Fast inversion of Hessenberg-quasiseparable-Vandermonde matrices and resulting recurrence relations and characterizations, preprint.
  • T.Bella, Y.Eidelman, I.Gohberg, I.Koltracht, V.Olshevsky, A fast Bjorck-Pereyra-type algorithm for solving Quasiseparable-Vandermonde systems, SIAM J.MatrixAnal.AndAppl.31(2):790-815 (2009).
  • T.Bella, Y.Eidelman, I.Gohberg, I.Koltracht, V.Olshevsky, A Bjorck-Pereyra-type algorithm for Szego-Vandermonde matrices based on properties of unitary Hessenberg matrices, Linear Algebra and Applications, Volume 420, Issues 2-3 pp. 634-647 (2007).
  • Yu.Eidelman, I.Gohberg and V.Olshevsky, The QR iteration method for Hermitian quasiseparable matrices of an arbitrary order, Linear Algebra and its Applications, Volume 404, 15 July 2005, Pages 305-324
  • Yu.Eidelman, I.Gohberg and V.Olshevsky, Eigenstructure of Order-One-Quasiseparable Matrices. Three-term and Two-term Recurrence Relations, Linear Algebra and its Applications, Volume 405, 1 August 2005, Pages 1-40

Bullet  Fast Transforms

  • A. Olshevsky, V. Olshevsky, and J. Wang, A comrade-matrix-based derivation of the eight versions of fast cosine and sine transforms, in Fast Algorithms for Structured Matrices: Theory and Applications, CONM/323, p. 119-150, AMS publications, May 2003.

Bullet  Numerical Linear Algebra

  • V. Olshevsky, Pivoting for structured matrices and rational tangential interpolation, in Fast Algorithms for Structured Matrices: Theory and Applications, CONM/323, p. 1 – 75, AMS publications, May 2003.
  • T.Boros, T.Kailath and V.Olshevsky, Pivoting and Backward Stability of Fast Algorithms for Solving Cauchy Linear Equations, Linear Algebra and Its Applications, Vol 343/344, March 2002.
  • T.Boros, T.Kailath and V.Olshevsky, Fast Bjorck-Pereyra-type algorithm for parallel solution of Cauchy linear equations, Linear Algebra and Its Applications, 302-303 (1999), p.265-293.
  • V.Olshevsky and Michael Stewart. Stable Factorization of Hankel and Hankel-like Matrices. Numerical Linear Algebra, Vol 8, Issue 6/7, p 401-434, 2001. (Special Issue: Numerical Linear Algebra Techniques for Control and Signal Processing.)
  • Dario Fasino and V.Olshevsky. How bad are symmetric Pick matrices. in “Structured Matrices in Mathematics, Computer Science, and Engineering I,” Contemporary Mathematics series, Vol 280, 301-312, AMS, 2001.
  • Kailath and V.Olshevsky. Diagonal Pivoting for Partially Reconstructible Cauchy-like Matrices, With Applications to Toeplitz-like Linear Equations and to Boundary Rational Matrix Interpolation Problems. Linear Algebra and Appl, 254 (1997), 251-302.
  • I.Gohberg, T.Kailath and V. Olshevsky. Fast Gaussian elimination with partial pivoting for matrices with displacement structure. Math. of Computation,   64 No. 212 (1995), 1557-1576.
  • I.Gohberg and V.Olshevsky. The fast generalized Parker-Traub algorithm for inversion of Vandermonde and related matrices. J. of Complexity, 13(2) (1997), 208-234.

Bullet Theoretical Computer Science. FOCS and STOC publications

  • V.Olshevsky and A.Shokrollahi, Fast matrix-vector multiplication algorithms for confluent Cauchy-like matrices with applications, in Proc of of the Thirty Second ACM Symposium on Theory of Computing (STOC’00), p.573-581; ACM, New York, 2000.
  • V.Olshevsky and A.Shokrollahi. A displacement structure approach to efficient decoding of Reed-Solomon and algebraic geometric codes. A conference paper appeared in Proc of of the Thirty First ACM Symposium on Theory of Computing. (STOC’99), p.235-244; ACM, New York, 1999.
  • V.Olshevsky and V.Pan, Polynomial and rational evaluation and interpolation (with structured matrices), Automata, languages and programming (Prague, 1999), 585–594, Lecture Notes in Comput. Sci., 1644, Springer, Berlin, 1999.
  • V.Olshevsky and Victor Pan, A unified superfast algorithm for boundary rational tangential interpolation problems and for inversion and factorization of dense strauctured matrices. Proc. of 39th Annual Symposium on Foundations of Computer Science (FOCS’98), IEEE Computer Society, Los Alamitos, CA, 1998, 192-201.

Bullet  Structured Matrices and Fast Algorithms. Displacement Structure

  • G.Heinig, V.Olshevsky, The Schur algorithm for matrices with Hessenberg displacement structure, in “Structured Matrices in Mathematics, Computer Science, and Engineering II,” Contemporary Mathematics series, Vol 281, 3-16, AMS, 2001.
  • T. Kailath, and V. Olshevsky. Displacement structure approach to polynomial Vandermonde and related matrices. Linear Algebra and Appl, 261(1997), 49-90.
  • Gohberg and V.Olshevsky, Fast inversion of Vandermonde and Vandermonde-like matrices, pp. 205-221, in Communications, Computation, Control and Signal Processing: A tribute to Thomas Kailath, Eds. A.Paulraj, V Roychowdhury and C.Shaper, Kluwer Academic Publishing, 1996.
  • T.Kailath and V. Olshevsky. Displacement structure approach to Chebyshev-Vandermonde and related matrices. Integral Equations and Operator Theory,   22 (1995), 65-92.
  • I.Gohberg and V. Olshevsky. Fast algorithms with preprocessing for matrix-vector multiplication problems. Journal of Complexity,   10 (1994), 411-427.
  • I.Gohberg and V. Olshevsky. Fast inversion of Chebyshev-Vandermonde matrices. Numerische Mathematik,   67 (1994), 71-92.
  • I.Gohberg and V. Olshevsky. Complexity of multiplication with vectors for structured matrices. Linear Algebra Appl.,   202 (1994), 163-192.
  • I.Gohberg and V. Olshevsky. Circulants, displacements and decompositions of matrices. Integral Equations and Operator Theory,   15 (1992), 730-743.
    • PDF is available.

Bullet  Multilevel Structures

  • V.Olshevsky, I. Oseledets, E.Tyrtyshnikov Superfast inversion of two-level Toeplitz matrices using Newton iteration and tensor-displacement structure preprint.
  • V.Olshevsky, I.Oseledets and E.Tyrtyshnikov, Tensor properties of multilevel Toeplitz and related matrices, Linear Algebra and its Applications, Volume 412, Issue 1, 1 January 2006, Pages 1-21

Bullet  Preconditioners

  • T.Kailath and V.Olshevsky. Displacement structure approach to discrete trigonometric transform based preconditioners of G.Strang and T.Chan types. SIAM Journal on Matrix Analysis and Applications Volume 26, Number 3 pp. 706-734, 2005.

    Bullet  Rational interpolation

    • I.Gohberg and V. Olshevsky. Fast state space algorithms for matrix Nehari and Nehari-Takagi interpolation problems. Integral Equations and Operator Theory,   20 (1994), 44-83.

      A conference paper appeared in Proc. of MTNS-93, Systems and Networks: Mathematical Theory and Applications, v.2, Invited and Contributed Papers, edited by U. Helmke, R. Mennicken and J. Sauers, Academy Verlag,1994, p. 687-690.

    Bullet  Orthogonal Polynomials and unitary Hessenberg matrices

    • V.Olshevsky, Unitary Hessenberg matrices and the generalized Parker-Forney-Traub algorithm for inversion of Szego-Vandermonde matrices. invited chapter in the book “Structured Matrices: Recent Developments in Theory and Computation,” 67-78, (D.Bini, E. Tyrtyshnikov, P. Yalamov., Eds.), 2001, NOVA Science Publ., USA.
    • V.Olshevsky. Eigenvector computation for almost unitary Hessenberg matrices and inversion of Szego-Vandermonde matrices via Discrete Transmission lines. Linear Algebra and Its Applications, 285 (1998), 37-67.

    Bullet  Some Old Papers on Complete Controllability and Perturbation Theory

    • A.Markus and V. Olshevsky. Complete controllability and assignment of the spectrum in infinite dimensional spaces. Integral Equations and Operator Theory,   17 (1993), 107-122.
    • A.Markus and V. Olshevsky. Exact controllability and spectrum assignment in infinite dimensional spaces, (in Russian), Matematicheskie Issledovania,   108 (1989), 97 -114.
    • V. Olshevsky. On distance between invariant subspaces of operators in finite dimensional spaces, (in Russian), Izvestia Akad. nauk M.S.S.R.,   1 (1989), 65-66.
    • V. Olshevsky. Variation of the Jordan structure of G-selfadjoint operators and selfadjoint operator-functions under small perturbations, (in Russian), Functional Analysis and its Applications,   22 (No. 3) (1988), 79-80; English translation : Functional Analysis and its Applications, Plenum publishing corp., 22 (No. 3) (1988), 236 – 237.
    • V. Olshevsky. Semi-stability of dimension of linear span of eigenvectors and generalized eigenvectors of holomorphic operator-function, (in Russian), Matematicheskie Issledovania,   93 (1988), 119-128.