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 @
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. N. Truhar, A Note on an Upper and a Lower Bound on Sines between Eigenspaces for regular Hermitian Matrix Pairs, Journal of Computational and Applied Mathematics, (2019), prihvaćen za objavljivanje
    The main results of the paper are un upper and a lower bound for the Frobenius norm of the matrix $sin Theta$, of the sines of the canonical angles between unperturbed and perturbed eigenspaces of a regular generalized Hermitian eigenvalue problem $A x = lambda B x$ where $A$ and $B$ are Hermitian $n times n$ matrices, under a feasible non-Hermitian perturbation. As one application of the obtained bounds we present the corresponding upper and the lower bounds for eigenspaces of a matrix pair $(A,B)$ obtained by a linearization of regular quadratic eigenvalue problem $left( lambda^2 M + lambda D + K right) u = 0 $, where $M$ is positive definite and $D$ and $K$ are semidefinite. We also apply obtained upper and lower bounds to the important problem which considers the influence of adding a damping on mechanical systems. The new results show that for certain additional damping the upper bound can be too pessimistic, but the lower bound can reflect a behaviour of considered eigenspaces properly. The obtained results have been illustrated with several numerical examples.
  2. N. Truhar, Z. Tomljanović, M. Puvača, Approximation of damped quadratic eigenvalue problem by dimension reduction, Applied mathematics and computation 347 (2019), 40-53
    This paper presents an approach to the efficient calculation of all or just one important part of the eigenvalues of the parameter dependent quadratic eigenvalue problem $(lambda^2(mathbf{;v};) M + lambda(mathbf{;v};) D(mathbf{;v};) + K) x(mathbf{;v};) = 0$, where $M, K$ are positive definite Hermitian $ntimes n$ matrices and $D(mathbf{;v};)$ is an $ntimes n$ Hermitian semidefinite matrix which depends on a damping parameter vector $mathbf{;v};= begin{;bmatrix}; v_1 & ldots & v_k end{;bmatrix};in mathbb{;R};_+^k$. With the new approach one can efficiently (and accurately enough) calculate all (or just part of the) eigenvalues even for the case when the parameters $v_i$, which in this paper represent damping viscosities, are of the modest magnitude. Moreover, we derive two types of approximations with corresponding error bounds. The quality of error bounds as well as the performance of the achieved eigenvalue tracking are illustrated in several numerical experiments.
  3. N. Truhar, Z. Tomljanović, M. Puvača, An Efficient Approximation For Optimal Damping In Mechanical Systems, International journal of numerical analysis and modeling 14/2 (2017), 201-217
    This paper is concerned with an efficient algorithm for damping optimization in mechanical systems with a prescribed structure. Our approach is based on the minimization of the total energy of the system which is equivalent to the minimization of the trace of the corresponding Lyapunov equation. Thus, the prescribed structure in our case means that a mechanical system is close to a modally damped system. Although our approach is very efficient (as expected) for mechanical systems close to modally damped system, our experiments show that for some cases when systems are not modally damped, the proposed approach provides efficient approximation of optimal damping.
  4. I. Kuzmanović, Z. Tomljanović, N. Truhar, Damping optimization over the arbitrary time of the excited mechanical system, Journal of Computational and Applied Mathematics, 304 (2016), 120-129
    In this paper we consider damping optimization in mechanical system excited by an external force. We use optimization criteria based on minimizing average energy amplitude and average displacement amplitude over the arbitrary time. As the main result we derive explicit formulas for objective functions. These formulas can be implemented efficiently and accelerate optimization process significantly, which is illustrated in a numerical example.
  5. L. Grubišić, S. Miodragović, N. Truhar, Double angle theorems for definite matrix pairs, Electronic Transactions on Numerical Analysis 45 (2016), 33-57
    In this paper we present new double angle theorems for the rotation of the eigenspaces for Hermitian matrix pairs $(H,M)$, where $H$ is a non-singular matrix which can be factorized as $H = G J G^*$, $J = diag(pm 1)$, and $M$ is non-singular. The rotation of the eigenspaces is measured in the matrix dependent scalar product and the bounds belong to the relative perturbation theory. The quality of the new bounds are illustrated in the 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)

Professional Activities


  • Mathematical Communications (since 2007)
  • Osječki matematički list (since 2003)

Forthcoming Meetings

Committee Memberships
  •  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): Srijeda (Wed) 11:00am-12:15pm, Četvrtak (Thu)  9:00am-10:00pm. Konzultacije su moguće i po dogovoru.


Dodiplomska nastava:


 Diplomska nastava:


Teme za diplomske radove: popis tema



Građevinski fakultet Osijek, Razlikovna godina 2018-2019:

Rezultati 1. Kolkvija iz Matematike su ovdje

Rezultati 2. Kolkvija sa ocjenama iz Matematike su ovdje.

2. kolkvij iz Matematike se može pogledati u utorak 4.12.2018  četvrtak 6.12. 2019 u 12 sati na Odjelu za matematiku, u sobi prof. Truhara.



Građevinski fakultet Osijek, Razlikovna godina 2018-2019:

Rezultati psimenog ispita 2.rok (18.2.2019)

MJ000   ocjena 2

MC711  ocjena 1

Ispite možete pogledati u pon. 25.2.2019 u 10.30 na Odjelu za matematiku.




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



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