Odjel za matematiku

Ninoslav Truhar (Google Scholar Profile) 


Truhar Full Professor
Department of Mathematics
Josip Juraj Strossmayer University of Osijek
Trg Ljudevita Gaja 6
Osijek, HR-31000, Croatia¸
phone: +385-31-224-817
fax: +385-31-224-801
email: ntruhar @ mathos.hr
office: 1st  floor


Research Interests

  • Numerical Linear Algebra
  • Systems and Control  Theory
  • Applied Mathematics
Linear Matrix Equations, Linear Vibrating Systems, Damping Optimization, Matrix Perturbation Theory, Perturbation Theory of Invariant Subspaces


  • B. S. in Mathematics and Physics 1987, University of Osijek
  • M. S. In Mathematics 1995, University of Zagreb
  • Ph.D. in Mathematics 2000, University of Zagreb


Study Visits Abroad and Professional Improvement

  • 1997 10-12, visiting researher at The Pennsylvania State University,  State College, PA, USA
  • 1999–2001 post-Ph. D. research at FernUniversitat in Hagen, Germany
  • 2003 guest professor at FernUniversitat in Hagen, Germany (one month)
  • 2004 guest professor at FernUniversitat in Hagen, Germany (one month)
  • 2006 visiting researher at Department of Mathematics, University of Kentucky,
    Lexington, Kentucky, USA
  • 2007 visiting professor at Department of Mathematics at the University of Texas
    at Arlington, Arlington, Texas, USA (one semester)
  • 2013 visiting professor at Department of Mathematics at the University of Texas
    at Arlington, Arlington, Texas, USA (one semester)



Journal Publications

  1. J. Moro, S. Miodragović, F. de Teran, N. Truhar, Frequency isolation for gyroscopic systems via hyperbolic quadratic eigenvalue problems, Mechanical Systems and Signal Processing 184/109688 (2023), 1-19
    The solutions of a forced gyroscopic system of ODEs may undergo large oscillations whenever some eigenvalues of the corresponding quadratic eigenvalue problem (QEP) $(lambda^2 M + lambda G+K)v=0,quad 0 neq v in mathbb{C}^n,$ are close to the frequency of the external force (both $M,K$ are symmetric, $M$ is positive definite, $K$ is definite and $G$ is skew-symmetric). This is the phenomenon of {colr the} so-called resonance. One way to avoid resonance is to modify some (or all) of the coefficient matrices, $M$, $G$, and $K in mathbb{R}^{ntimes n}$ in such a way that the new system has no eigenvalues close to these frequencies. This is known as the frequency isolation problem. In this paper we present frequency isolation algorithms for tridiagonal systems in which only the gyroscopic term $G$ is modified. To derive these algorithms, the real gyroscopic QEP is first transformed into a complex hyperbolic one, which allows to translate many of the ideas in textcolor{red}{[Mech. Syst. Signal Process., 75:11-26, 2016]} for undamped systems into the full quadratic framework. Some numerical experiments are presented.
  2. N. Truhar, M. Petrač, Damping Optimization of Linear Vibrational Systems with a Singular Mass Matrix, Mathematics 10 (2022), 1-21
    We present two novel results for small damped oscillations described by the vector differential equation $M ddot{x} + C dot{x} + K x = 0$, where the mass matrix $M$ can be singular, but standard deflation techniques cannot be applied. The first result is a novel formula for the solution ${X}$ of the Lyapunov equation {${A}^T {X} + {X} {A} = -I$,} where ${A}={A}(v)$ is obtained from $M, C(v) in mathbb{R}^{n times n}$, and $K in mathbb{R}^{n times n} $, which are the so-called mass, damping, and stiffness matrices, respectively, and $rank(M)=n-1$. Here, $C(v)$ is positive semidefinite with $rank({C}(v))=1$. Using the obtained formula, we propose a very efficient way to compute the optimal damping matrix. The second result was obtained for a different structure, where we assume that $dim(mathcal{N}(M))geq 1$ and internal damping exists (usually a small percentage of the critical damping). For this structure, we introduce a novel linearization, i.e., a novel construction of the matrix $A$ in the Lyapunov equation {$A^T{X} + {X}{A} = - {I}$,} and a novel optimization process. The proposed optimization process computes the optimal damping $C(v)$ that minimizes a function $vmapsto{rm trace}({Z}{X})$ (where ${Z}$ is a chosen symmetric positive semidefinite matrix) using the approximation function $g(v) = c_v + frac{a}{v} + bv$, for the trace function $f(v) doteq {rm trace}({Z}{X}(v))$. Both results are illustrated with several corresponding numerical examples.
  3. N. Truhar, R. Li, On an Eigenvector-Dependent Nonlinear Eigenvalue Problem from the Perspective of Relative Perturbation Theory, Journal of Computational and Applied Mathematics, 395 (2021)
    We are concerned with the eigenvector-dependent nonlinear eigenvalue problem (NEPv) $H(V)V = V Lambda$, where $H(V) in bbC^{n times n}$ is a Hermitian matrix-valued function of $V in bbC^{n times k}$ with orthonormal columns, i.e., $V^{HH} V = I_k$, $k leq n$ (usually $k ll n$). Sufficient conditions on the solvability and solution uniqueness of NEPv are obtained, based on the well-known results from the relative perturbation theory. These results are complementary to recent ones in [Cai, Zhang, Bai, and Li, {em SIAM J. Matrix Anal. Appl.}, 39:2 (2018), pp.1360--1382], where, among others, one can find conditions for the solvability and solution uniqueness of NEPv, based on the well-known results from the absolute perturbation theory. Although the absolute perturbation theory is more versatile in applications, there are cases where the relative perturbation theory produces better results.
  4. P. Benner, X. Liang, S. Miodragović, N. Truhar, Relative Perturbation Theory for Quadratic Hermitian Eigenvalue Problem, Linear algebra and its applications 618 (2021), 97-128
    In this paper, we derive new relative perturbation bounds for eigenvectors and eigenvalues for regular quadratic eigenvalue problems of the form $(lambda^2 M + lambda C + K)x = 0$, where $M$ and $K$ are nonsingular Hermitian matrices and $C$ is a general Hermitian matrix. These results are based on new relative perturbation bounds for an equivalent regular Hermitian matrix pair $A-lambda B$. The new bounds can be applied to quadratic eigenvalue problems appearing in many relevant applications, such as mechanical models with indefinite damping. The quality of our bounds is demonstrated by several numerical experiments.
  5. N. Truhar, Z. Tomljanović, R. Li, Perturbation Theory for Hermitian Quadratic Eigenvalue Problem -- Damped and Simultaneously Diagonalizable Systems, Applied mathematics and computation 371 (2020)
    The main contribution of this paper is a novel approach to the perturbation theory of a structured Hermitian quadratic eigenvalue problems $(lambda^2 M + lambda D + K) x=0$. We propose a new concept without linearization, considering two structures: general quadratic eigenvalue problems (QEP) and simultaneously diagonalizable quadratic eigenvalue problems (SDQEP). Our first two results are upper bounds for the difference $left| | X_2^* M widetilde{;X};_1 |_F^2 - | X_2^* M {;X};_1 |_F^2 right|$, and for $| X_2^* M widetilde X_1 - X_2^* M X_1|_F$, where the columns of $X_1=[x_1, ldots, x_k]$ and $X_2=[x_{;k+1};, ldots, x_n]$ are linearly independent right eigenvectors and $M$ is positive definite Hermitian matrix. As an application of these results we present an eigenvalue perturbation bound for SDQEP. The third result is a lower and an upper bound for $|sin{;Theta(mathcal{;X};_1, widetilde{;mathcal{;X};};_1)}; |_F$, where $Theta$ is a matrix of canonical angles between the eigensubspaces $mathcal{;X};_1 $ and $widetilde{;mathcal{;X};};_1$, $mathcal{;X};_1 $ is spanned by the set of linearly independent right eigenvectors of SDQEP and $widetilde{;mathcal{;X};};_1$ is spanned by the corresponding perturbed eigenvectors. The quality of the mentioned results have been illustrated by numerical examples.




  • Mixed Integer Nonlinear Programming (MINLP) for damper optimization--scientific project; supported by the DAAD for period 2015--2016 (Project director); partner institution: Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg
  • European Model Reduction Network (EU-MORNET). Funded by: COST (European Cooperation in Science and Technology).

         Partner: researchers in model order reduction from 17 countries.

Project run 01/01/2013 - 12/31/2014 founded by DAAD in collaboration between Max Planck Institute for Dynamics Complex Technical Systems Magdeburg, Computational Methods in Systems and Control Theory, Magdeburg, Germany and Department of Mathematics, University of Osijek, Osijek, Croatia   

  • Solution of large-scale Lyapunov Differential Equations,  

    Funded by: FWF Austrian Science Fundation,  FWF project id: P27926
    Researchers: Dr. Hermann Mena (project director, University of Innsbruck, Innsbruck, Austria); Prof. Dr. Alexander Ostermann (University of Innsbruck, Innsbruck, Austria)
    Partners: Universidad Jaime I, Castellon (Spain), University of Tuebingen, (Germany),
    Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg (Germany), Department of Mathematics, University of Osijek (Croatia)

Editorial Service

Member of Editorial Board:

Forthcoming Meetings

  • ????
Committee Memberships
  • 11th Conference on Applied Mathematics and Scientific Computing   5-9 September 2022, Brijuni, Croatia 
  • 10th Conference on Applied Mathematics and Scientific Computing       14-18 September 2020, Brijuni, Croatia
  • Ninth Conference on Applied Mathematics and Scientific Computing    17-20 September 2018, Solaris, Šibenik, Croatia
  • Workshop on Model Reduction Methods and Optimization,      20-21 September 2016, in Opatija, Croatia, http://www.mathos.unios.hr/index.php/443
  •  Member of the Scientific Committee of the 6th Croatian Congress of Mathematics (Zagreb, 2016)   





  • SIAM Journal on Matrix Analysis and Applications (SIMAX)
  • SIAM Journal on Scientific Computing (SISC)
  • Linear Algebra and its Applications (LAA)
  • Numerische Mathematik
  • BIT Numerical Mathematics
  • Mathematical and Computer Modelling (MCM)
  • Applied Mathematics and Computation (AMC)
  • International Journal of Computer Mathematics
  • Journal of Applied Mathematics and Computing (JACM)
  • Journal of Sound and Vibration 
  • International Journal of Systems Science
  • International Journal of Computer Mathematics
  • Numerical Algorithms
  • Central European Journal of Mathematics
  • Bulletin of the Iranian Mathematical Society 
  • Glasnik matematički
  • Mathematical Communications


  • AMS Mathematical Review   (since 2006)
  • Zentralblatt MATH


Service Activities


  • Chairman of Osijek Mathematical Society, 2003--2013 
  • Chairman of the Mathematical Colloquium, 2005-2017





Konzultacije (Office Hours): Utorak (Tue)  11:30am-12:15pm, Srijeda (Wed) 11:30am-12:15pm. Konzultacije su moguće i po dogovoru.


Dodiplomska nastava:


 Diplomska nastava:


Teme za diplomske radove: popis tema







  • Birthdate: May 4, 1963
  • Birthplace: Osijek, Croatia
  • Citizenship: Croatian
  • Family: Married



I am a fan and supporter of basketball club KK Vrijednosnice Osijek