Valeria Simoncini

Available software



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  • EVEN_IRA

    This Matlab code is available here

    Related article:
    Volker Mehrmann , Christian Schroeder and V. Simoncini
    An Implicitly-restarted Krylov Method for Real Symmetric/Skew-Symmetric Eigenproblems
    Linear Algebra and Appl., 436 (10), (2012), pp. 4070-4087.

  • KPIK

    This Matlab code approximately solves a large-scale Lyapunov matrix equation by means of the
    Extended Krylov method with Galerkin condition. Version 1.0. Download kpik.m
    Here is an example of how to call the function kpik.m. Download example_kpik.m
    The code does not include any checking of the input data.

    Related article:
    V. Simoncini,
    A new iterative method for solving large-scale Lyapunov matrix equations
    SIAM J. Scient. Computing, v.29, n.3 (2007), pp. 1268-1288.

  • KPIK_SYLV

    This Matlab code approximately solves a large-scale Sylvester matrix equation by means of the
    Extended Krylov method with Galerkin condition. Version 1.0. Download kpik_sylv.m
    Here is an example of how to call the function kpik_sylv.m. Download example_kpik_sylv.m
    The code does not include any checking of the input data.
    Deflated version to be included.

    Related article:
    T. Breiten , V. Simoncini and M. Stoll
    Fast iterative solvers for fractional differential equations
    pp.1-32, January 2014. To appear in ETNA.

  • RKSM for Lyapunov equations

    This Matlab code approximately solves a large-scale Lyapunov matrix equation by means of the
    adaptive Rational Krylov method with Galerkin condition.
    Version 1.0, June 2014. Download rksm.m
    Here is an example of how to call the function rksm.m. Download example_rksm.m
    The code does not include any checking of the input data.
    Version 2.0. All-real-basis version (real basis also with complex shifts), Aug 2019. Thanks to Gerhard Kirsten, PhD Student @UniBO. Download RKSM_Lyap_real.m
    Here is an example of how to call the function RKSM_Lyap_real.m. Download example_lyap_real.m
    The code does not include any checking of the input data.

    Related article:
    V. Druskin and V. Simoncini,
    Adaptive rational Krylov subspaces for large-scale dynamical systems
    Systems & Control Letters, 60 (2011), pp. 546-560.


  • Tangential RKSM for Lyapunov equations

    This Matlab code approximately solves a large-scale Lyapunov matrix equation by means of the
    adaptive Rational Krylov method with Galerkin condition.
    A tangential approach is used to deal with large rank of the known ``rhs'' matrix BB'.
    Version 1.0. Download rksm_tan_lyap_amg_Mdir.m
    Here is an example of how to call the function rksm.m. Download example_rksm_tan.m
    The code does not include any checking of the input data.

    Related article:
    V. Druskin, V. Simoncini and M. Zaslavsky,
    Adaptive tangential interpolation in rational Krylov subspaces for MIMO model reduction data
    SIAM J. Matrix Analysis and Appl., v.35, n.2 (2014), 476-498.


  • Iterative solver for generalized Lyapunov equations

    This Matlab code approximately solves a generalized (multiterm) large-scale Lyapunov matrix equation by means
    of a splitting iterative method. At each iteration, the algorithm solves
    a Lyapunov equation with the extended Krylov method with Galerkin condition.
    Version 1.0. Download efficientGenLyap.m , Download EKS.m , Download invertA.m ,
    Here is an example of how to call the function efficientGenLyap.m. Download example_Glyap.m
    The code does not include any checking of the input data.

    Related article:
    Stephen D. Shank , V. Simoncini and Daniel B. Szyld
    Efficient low-rank solutions of Generalized Lyapunov equations
    Numerische Mathematik, 134(2), (2016), 327-342. DOI 10.1007/s00211-015-0777-7



  • SKSM_cTri: Enhanced standard Krylov subspace method for symmetric Lyapunov equations

    This Matlab code iteratively solves a large-scale Lyapunov matrix equation with symmetric coeff. matrix by means of the
    standard Krylov method with Galerkin condition (low CPU and memory requirements). Version 1.0.
    Download: SKSM_cTri.m , cTri.mexa64
    Here is an example of how to call the function SKSM_cTri.m. Download example_SKSM_cTri.m
    The code does not include any checking of the input data.

    Related article:
    Davide Palitta and V. Simoncini,
    Computationally enhanced projection methods for symmetric Sylvester and Lyapunov matrix equations
    J. Computational and Applied Mathematics, 330 (2018) 648-659.



  • RKSM for Riccati equations

    This Matlab code approximately solves a large-scale Riccati matrix equation by means of the
    adaptive Rational Krylov method with Galerkin condition. Version 1.0. Download ARKSM_Riccati.m
    Here is an example of how to call the function rksm.m. Download test_ARKSM_Riccati.m
    The code does not include any checking of the input data.

    Related article:
    V. Simoncini,
    Analysis of the rational Krylov subspace projection method for large-scale algebraic Riccati equations
    SIAM J. Matrix Analysis and Appl. 37-4 (2016), pp. 1655-1674. http://dx.doi.org/10.1137/16M1059382

  • RKSM for Differential Algebraic Riccati equations

    This Matlab code approximately solves a large-scale differential algebraic Riccati matrix equation by means of the
    adaptive Rational Krylov method with Galerkin condition (MOR style: projecting and then time-integrating) Version 1.2.
    Download DRE.tar.gz
    The code does not include any checking of the input data. See also here for the GitHub version.

    Related article:
    Gerhard Kirsten and V. Simoncini,
    Order reduction methods for solving large-scale differential matrix Riccati equations
    To appear in SIAM J. Scient. Comput. arXiv: 1905.12119. https://arxiv.org/abs/1905.12119

  • LYAP_BANDED for Lyapunov equations

    This Matlab code approximately solves a large-scale Lyapunov matrix equation with banded data. Version 1.0.
    Download here .
    The code does not include any checking of the input data.

    Related article:
    Davide Palitta and V. Simoncini
    Numerical methods for large-scale Lyapunov equations with symmetric banded data
    SIAM J. Scientific Computing. v.40 (5), pp.A3581--A3608 (2018). https://doi.org/10.1137/17M1156575



  • MultiRB: Galerkin method for solving multiterm matrix equations from SGFEM

    This package contains matlab codes to iteratively solving linear multiterm matrix equations stemming from Stochastic Galerkin diffusion problems.
    Version 1.0.
    Download: MultiRB.zip .
    The package contains the driver SGFEM_matdriver.m to replicate the examples in the related article.
    The code does not include any checking of the input data.

    Related article:
    C.E. Powell, D. Silvester, V. Simoncini,
    An Efficient Reduced Basis Solver for Stochastic Galerkin Matrix Equations,
    SIAM J. Sci. Comput., Vol 39, No. 1, pp. A141--A163, (2017)





  • MotCG: Matrix-Oriented Truncated CG

    This Matlab code approximately solves a large-scale linear matrix equation by means of the
    Preconditioned CG method with truncation, in matrix form. Version 1.0. Download tcg.tar.gz
    The tar file contains a driver with an example calling the function MotCG.m.
    The code does not include any checking of the input data.

    Related article:
    V. Simoncini and Yue Hao
    Analysis of the truncated conjugate gradient method for linear matrix equations
    pp. 1-24, Dipartimento di Matematica, Universita' di Bologna, February 2022. HAL archive hal-03579267. To appear in SIAM J. Matrix Analysis and Appl.
    (see also references inthere)

  • Sketched and truncated polynomial Krylov for Sylvester equations

    This Matlab code approximately solves a large-scale Sylvester matrix equation by means of Sketched and truncated polynomial Krylov methods. Codes available here

    Related article:
    Davide Palitta , Marcel Schweitzer , and V. Simoncini
    Sketched and truncated polynomial Krylov methods: Sylvester matrix equations
    arXiv: 2311.16019, pp. 1-28, Dipartimento di Matematica, Universita' di Bologna, Nov 2023 (and later versions)
    To appear in Math. Comp.


  • MR_RKSM

    (to be included)