### Ninoslav Truhar

Head of Department of Applied Mathematics and Computer Science

School of Applied Mathematics and Informatics

Josip Juraj Strossmayer University of Osijek

Josip Juraj Strossmayer University of Osijek

### 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**

### Degrees

- 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

### Publications

Journal Publications

- N. Truhar, Z. Tomljanović, M. Ugrica, M. Karow, Efficient approximation of novel residual bounds for a parameter dependent quadratic eigenvalue problem, Numerical Algebra, Control and Optimization (2024), prihvaćen za objavljivanje
- R. Li, N. Truhar, L. Zhang, On Stewart's Perturbation Theorem for SVD, Annals of Mathematical Sciences and Applications (2024), 1-28This paper establishes a variant of Stewart's theorem [Theorem 6.4 of Stewart, SIAM Rev., 15:727{764, 1973] for the singular subspaces associated with the SVD of a matrix subject to perturbations. Stewart's original version uses both the Frobenius and spectral norms, whereas the new variant uses the spectral norm and any unitarily invariant norm that offer choices per convenience of particular applications and lead to sharper bounds than that straightforwardly derived from Stewart's original theorem with the help of the well-known equivalence inequalities between matrix norms. Of interest in their own right, bounds on the solution to two couple Sylvester equations are established for a few different circumstances.
- S. Miodragović, N. Truhar, I. Kuzmanović Ivičić, Relative perturbation $\tan \Theta$ theorems for definite matrix pairs, Electronic Transactions on Numerical Analysis
**60**(2024), 364-380 - 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), 1-19The 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. - N. Truhar, M. Petrač, Damping Optimization of Linear Vibrational Systems with a Singular Mass Matrix, Mathematics
**10**(2022), 1-21We 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. - 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. - P. Benner, X. Liang, S. Miodragović, N. Truhar, Relative Perturbation Theory for Quadratic Hermitian Eigenvalue Problem, Linear algebra and its applications
**618**(2021), 97-128In 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. - 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. - 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,
**358**(2019), 374-384The 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. - N. Truhar, Z. Tomljanović, M. Puvača, Approximation of damped quadratic eigenvalue problem by dimension reduction, Applied mathematics and computation
**347**(2019), 40-53This 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. - I. Ali, N. Truhar, Location of right eigenvalues of quaternionic matrix polynomials, Advances in Applied Clifford Algebras
**29**/4 (2019), 1-21In this paper, inclusion regions for the right eigenvalues of a quaternionic matrix polynomial are derived from Ostrowski's type theorem for quaternionic block companion matrices. Furthermore, a right spectral radius inequality and its applications for finding bounds for the right eigenvalues of a quaternionic matrix polynomial is presented. Consequently, these bounds give bounds for the zeros of quaternionic polynomials. Finally, bounds on the eigenvalues of complex matrix polynomials are derived. The comparison between the new bounds and some existing bounds have been illustrated with several examples. - I. Nakić, Z. Tomljanović, N. Truhar, Mixed control of vibrational systems, Journal of Applied Mathematics and Mechanics
**99**/9 (2019), 1-15We consider new performance measures for vibrational systems based on the $H_2$ norm of linear time invariant systems. New measures will be used as an optimization criterion for the optimal damping of vibrational systems. We consider both theoretical and concrete cases in order to show how new measures stack up against the standard measures. The quality and advantages of new measures as well as the behaviour of optimal damping positions and corresponding damping viscosities are illustrated in numerical experiments. - Y. Kanno, M. Puvača, Z. Tomljanović, N. Truhar, Optimization Of Damping Positions In A Mechanical System, Rad HAZU, Matematičke znanosti.
**23**(2019), 141-157This paper deals with damping optimization of the mechanical system based on the minimization of the so-called "average displacement amplitude". Further, we propose three different approaches to solving this minimization problems, and present their performance on two examples. - 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), 201-217This 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. - I. Kuzmanović Ivičić, Z. Tomljanović, N. Truhar, Damping optimization over the arbitrary time of the excited mechanical system, Journal of Computational and Applied Mathematics,
**304**(2016), 120-129In 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. - L. Grubišić, S. Miodragović, N. Truhar, Double angle theorems for definite matrix pairs, Electronic Transactions on Numerical Analysis
**45**(2016), 33-57In 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. - P. Benner, P. Kurschner, Z. Tomljanović, N. Truhar, Semi-active damping optimization of vibrational systems using the parametric dominant pole algorithm, Journal of Applied Mathematics and Mechanics
**96**/5 (2016), 604-619We consider the problem of determining an optimal semi-active damping of vibrating systems. For this damping optimization we use a minimization criterion based on the impulse response energy of the system. The optimization approach yields a large number of Lyapunov equations which have to be solved. In this work, we propose an optimization approach that works with reduced systems which are generated using the parametric dominant pole algorithm. This optimization process is accelerated with a modal approach while the initial parameters for the parametric dominant pole algorithm are chosen in advance using residual bounds. Our approach calculates a satisfactory approximation of the impulse response energy while providing a significant acceleration of the optimization process. Numerical results illustrate the effectiveness of the proposed algorithm. - N. Truhar, Z. Tomljanović, K. Veselić, Damping optimization in mechanical systems with external force, Applied mathematics and computation
**250**(2015), 270-279We consider a mechanical system excited by external force. Model of such a system is described by the system of ordinary differential equations: $M ddot x(t) + D dot x(t) + K x(t) = {hat f}(t)$, where matrices $M, K$ (mass and stiffness) are positive definite and the vector ${hat f} $ corresponds to an external force. The damping matrix D is assumed to be positive semidefinite and has a small rank. We introduce two criteria that allow damping optimization of mechanical system excited by an external force. Since in general a damping optimization is a very demanding problem, we provide a new formulas which have been used for efficient damping optimization. The efficiency of new formulas is illustrated with a numerical experiment. - N. Truhar, S. Miodragović, Relative perturbation theory for denite matrix pairs and hyperbolic eigenvalue problem, Applied Numerical Mathematics
**98**(2015), 106-121In this paper, new relative perturbation bounds for the eigenvalues as well as for the eigensubspaces are developed for definite Hermitian matrix pairs and the quadratic hyperbolic eigenvalue problem. First, we derive relative perturbation bounds for the eigenvalues and the $sin Theta$ type theorems for the eigensubspaces of the definite matrix pairs $(A,B)$, where both $A, B in mathbb{C}^{mtimes m}$ are Hermitian nonsingular matrices with particular emphasis, where $B$ is a diagonal of $pm 1$. Further, we consider the following quadratic hyperbolic eigenvalue problem $(mu^2 M + mu C + K) x =0$, where $M, C, K in mathbb{C}^{ntimes n}$ are given Hermitian matrices. Using proper linearization and new relative perturbation bounds for definite matrix pairs $(A,B)$, we develop corresponding relative perturbation bounds for the eigenvalues and the $sin Theta$ type theorems for the eigensubspaces for the considered quadratic hyperbolic eigenvalue problem. The new bounds are uniform and depend only on matrices $M$, $C$, $K$, perturbations $delta M$, $delta C$ and $delta K$ and standard relative gaps. The quality of new bounds is illustrated through numerical examples. - N. Truhar, L. Grubišić, S. Miodragović, The Rotation of Eigenspaces of Perturbed Matrix Pairs II, Linear and multilinear algebra
**68**/8 (2014), 1010-1031This paper studies the perturbation theory for spectral projections of Hermitian matrix pairs $(H, M)$, where $H$ is a non-singular Hermitian matrix which can be factorized as $H = G J G^*$, $J = diag(pm 1)$, and $M$ is positive definite. The class of allowed perturbations is so restricted that the corresponding perturbed pair $(wtd H, wtd M)= (H+delta H, M+ delta M)$ must have the form $wtd H = wtd G J wtd G^*$, $J = diag(pm 1)$ and $wtd M$ is positive definite. The main contribution of the paper is a $sinTheta$ theorem which generalizes the main result from the first part of the paper to this more general setting. Our estimate, in its most general form, depends on a uniform norm bound on a set of all $J$-unitary matrices which diagonalize $G^*G$. The second main contribution is a new sharp uniform estimate of a norm of all $J$-unitary matrices which diagonalize $G^*G$ such that $H=G^*JG$ is a quasi-definite matrix. The case of a quasi-definite pair is therefore the case where our bounds are most competitive. We present numerical experiments to corroborate the theory. - E. Mengi, D. Kressner, I. Nakić, N. Truhar, Generalized Eigenvalue Problems with Specified Eigenvalues, The IMA Journal of Numerical Analysis
**34**/2 (2014), 480-501We consider the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specified eigenvalues. We derive a singular value optimization characterization for this problem and illustrate its usefulness for two applications. First, the characterization yields a singular value formula for determining the nearest pencil whose eigenvalues lie in a specified region in the complex plane. For instance, this enables the numerical computation of the nearest stable descriptor system in control theory. Second, the characterization partially solves the problem posed in [Boutry et al. 2005] regarding the distance from a general rectangular pencil to the nearest pencil with a complete set of eigenvalues. The involved singular value optimization problems are solved by means of BFGS and Lipschitz-based global optimization algorithms. - P. Benner, Z. Tomljanović, N. Truhar, Optimal Damping of Selected Eigenfrequencies Using Dimension Reduction, Numerical Linear Algebra with Applications
**20**/1 (2013), 1-17We consider a mathematical model of a linear vibrational system described by the second-order differential equation $M ddot{x} + D dot{x} + Kx = 0$, where $M$ and $K$ are positive definite matrices, representing mass and stiffness, respectively. The damping matrix $D$ is positive semidefinite. We are interested in finding an optimal damping matrix which will damp a certain (critical) part of the undamped eigenfrequencies. For this we use an optimization criterion based on minimization of the average total energy of the system. This is equivalent to the minimization of the trace of the solution of the corresponding Lyapunov equation $A X+ X A^T =-GG^T$, where $A$ is the matrix obtained from linearizing the second-order differential equation and $G$ depends on the critical part of the eigenfrequencies to be damped. The main result is the efficient approximation and corresponding error bound for the trace of the solution of the Lyapunov equation obtained by dimension reduction, which includes the influence of the right-hand side $G G^T$ and allows us to control the accuracy of the trace approximation. This trace approximation yields a much accelerated optimization algorithm for determining the optimal damping. - I. Nakić, Z. Tomljanović, N. Truhar, Optimal Direct Velocity Feedback, Applied mathematics and computation
**225**(2013), 590-600We present a novel approach to the problem of Direct Velocity Feedback (DVF) optimization of vibrational structures, which treats simultaneously small as well as large gains. For that purpose, we use two different approaches. The first one is based on the gains optimization using the Lyapunov equation. In the scope of this approach we present a new formula for the optimal gain and we present a relative error for modal approximation. In addition, we present a new formula for the solution of the corresponding Lyapunov equation for the case with multiple undamped eigenfrequencies, which is a generalization of existing formulae. The second approach studies the behavior of the eigenvalues of the corresponding quadratic eigenvalue problem. Since this approach leads to the parametric eigenvalue problem we consider small and large gains separately. For the small gains, which are connected to a modal damping approximation, we present a standard approach based on Gerschgorin discs. For the large gains we present a new approach which allows us to approximate all eigenvalues very accurately and efficiently. - I. Kuzmanović Ivičić, N. Truhar, Optimization of the solution of the parameter-dependent Sylvester equation and applications, Journal of Computational and Applied Mathematics,
**237**/1 (2013), 136-144This paper deals with an efficient algorithm for optimization of the solution of the parameter-dependent Sylvester equation $(A_0-vC_1C_2^T)X(v)+X(v)(B_0-vD_1D_2^T)=E, $ where $A_0$ is $mtimes m$, $B_0$ is $ntimes n$, $C_1$ and $C_2$ are $mtimes r_1$, $D_1$ and $D_2$ are $ntimes r_2$ and $X$ and $E$ are $mtimes n$ matrices. For optimization we use the following two optimization criteria: $tr(X(v))rightarrowmin$ and $|X(v)|_Frightarrowmin$. We present an efficient algorithm which derives formulas for the trace and for the Frobenius norm of the solution $X$ as functions $vrightarrowtr(X(v))$ and $vrightarrow|X(v)|_F$ as well as derivatives of these functions in a small amount of operations. That ensures fast optimization of these functions via standard optimization methods like the Newton method. A special case of this problem is a very important problem of dampers’ viscosity optimization in mechanical systems. - I. Kuzmanović Ivičić, N. Truhar, Sherman-Morrison-Woodbury formula for Sylvester and $T$-Sylvester equation with applications, International Journal of Computer Mathematics
**90**/2 (2013), 306-324In this paper we present the Sherman-Morrison-Woodbury-type formula for the solution of the Sylvester equation of the form [(A_0+U_1V_1)X+X(B_0+U_2V_2)=E, ] as well as for the solution of the $T$-Sylvester equation of the form [ (A_0+U_1V_1)X+X^T(B_0+U_2V_2)=E, ] where $U_1, U_2, V_1, V_2$ are low-rank matrices. Although the matrix version of this formula for the Sylvester equation has been used in several different applications (but not for the case of a $T$-Sylvester equation), we present a novel approach using a proper operator representation. This novel approach allows us to derive a matrix version of the Sherman-Morrison-Woodbury-type formula for the Sylvester equation, as well as for the $T$-Sylvester equation which seems to be new. We also present algorithms for the efficient calculation of the solution of Sylvester and $T$-Sylvester equations by using these formulas and illustrate their application in several examples. - I. Kuzmanović Ivičić, Z. Tomljanović, N. Truhar, Optimization of material with modal damping, Applied mathematics and computation
**218**(2012), 7326-7338This paper considers optimal parameters for modal damping $D=Mf_1(M^{-1}K;alpha_1,dots,alpha_k)+Kf_2(K^{-1}M;alpha_1,dots,alpha_k)$ in mechanical systems described by the equation $Mddot{x}+Ddot{x}+Kx=0 $, where matrices $M$ and $K$ are mass and stiffness matrices, respectively. Different models of proportional and generalized proportional damping are considered and optimal parameters with respect to different optimization criteria related to the solution of the corresponding Lyapunov equation are given. Also, some specific example problems are compared with respect to the optimal and estimated parameters. - R. Li, Y. Nakatsukasa, N. Truhar, W. Wang, Perturbation of multipleeigenvalues of Hermitian matrices, Linear algebra and its applications
**437**/1 (2012), 202-213This paper is concerned with the perturbation of the multiple eigenvalue $mu$ of Hermitian matrices of the form $A=mbox{; ; ; ; diag}; ; ; ; (mu I, A_{; ; ; ; 22}; ; ; ; )$, when the matrix undergoes an off-diagonal perturbation $E$ whose columns have widely varying magnitudes. When some of $E$'s columns are much smaller than the others, some copies of $mu$ are much insensitive than any existing bound suggests. We explain this phenomenon by showing that when $A_{; ; ; ; 22}; ; ; ; -mu I$ is definite the $i$th bound scales quadratically with the norm of the $i$th column, and in the indefinite case the bound is necessarily proportional to the product of $E$'s $i$th column norm and $E$'s norm. An extension to generalized Hermitian eigenvalue problems is presented. - L. Grubišić, N. Truhar, K. Veselić, The Rotation of Eigenspaces of Perturbed Matrix Pairs, Linear algebra and its applications
**436**/11 (2012), 4161-4178We revisit the relative perturbation theory for invariant subspaces of positive definite matrix pairs. As a prototype example for our results we consider parameter dependent families of eigenvalue problems. We show that new estimates are a natural way to obtain sharp---as functions of the parameter indexing the family of matrix pairs---estimates for the rotation of spectral subspaces. - P. Benner, Z. Tomljanović, N. Truhar, Dimension reduction for damping optimization in linear vibrating system, Journal of Applied Mathematics and Mechanics
**91**/3 (2011), 179-191We consider a mathematical model of a linear vibrational system described by the second-order differential equation $M ddot{;x}; + D dot{;x}; + Kx = 0$, where $M$ and $K$ are positive definite matrices, called mass and stiffness, espectively. We consider the case where the damping matrix $D$ is positive semidefinite. The main problem considered in the paper is the construction of efficient algorithm for calculating an optimal damping. As optimization criterion we use the minimization of the average total energy of the system which is equivalent to the minimization of the trace of the solution of the corresponding Lyapunov equation $A X+ X A^T =-I$, where $A$ is the matrix obtained from linearizing the second-order differential equation. Finding the optimal $D$ such that the trace of $X$ is minimal is a very demanding problem, caused by the large number of trace calculations, which are required for bigger matrix dimensions. We propose a dimension reduction to accelerate the optimization process. We will present an approximation of the solution of the structured Lyapunov equation and a corresponding error bound for the approximation. Our algorithm for efficient approximation of the optimal damping is based on this approximation. - Z. Tomljanović, N. Truhar, K. Veselić, Optimizing a damped system - a case study, International Journal of Computer Mathematics
**88**/7 (2011), 1533-1545We consider a second order damped-vibrational system described by the equation $ M ddot{;x}; + C(v) dot{;x}; + K x = 0 $, where $M, C(v), K$ are real, symmetric matrices of order $n$. We assume that the undamped eigenfrequencies (eigenvalues of $(lambda^2 M + K) x = 0$) $omega_1, omega_2, ldots, omega_n , $, are multiple in the sense that $omega_1 = omega_2$, $omega_3 = omega_4$, ldots, $omega_{;n-1}; = omega_n$, or are given in close pairs $omega_1 approx omega_2$, $omega_3 approx omega_4$, ldots, $omega_{;n-1}; approx omega_n$. We present a formula which gives the solution of the corresponding phase space Lyapunov equation, which then allows us to calculate the first and second derivatives of the trace of the solution, with no extra cost. This one can serve for the efficient trace minimization. - R. Li, Y. Nakatsukasa, N. Truhar, S. Xu, Perturbation of Partitioned Hermitian Generalized Eigenvalue Problem, SIAM Journal on Matrix Analysis and Applications
**32**/2 (2011), 642-663This paper is concerned with Hermitian positive definite generalized eigenvalue problem $A-lambda B$ for partitioned $$ A=[A_{; ; ; ; 11}; ; ; ; 0 \ 0 & A_{; ; ; ; 22}; ; ; ; ] , B=[B_{; ; ; ; 11}; ; ; ; & 0 \ 0 & B_{; ; ; ; 22}; ; ; ; ], $$ where both $A$ and $B$ are Hermitian and $B$ is positive definite. Bounds on how its eigenvalues varies when $A$ and $B$ are perturbed by Hermitian matrices. These bounds are generally of linear order with respect to the perturbations in the diagonal blocks and of quadratic order with respect to the perturbations in the off-diagonal blocks. The results for the special case of no perturbations in the diagonal blocks can also be used to bound the changes of eigenvalues of a Hermitian positive definite generalized eigenvalue problem after its off-diagonal blocks are dropped, a situation occurs frequently in eigenvalue computations. Presented results extend those of Li and Li ({; ; ; ; em Linear Algebra Appl.}; ; ; ; , 395 (2005), pp.183--190). Another possible extension here is to derive quadratic eigenvalue perturbation bounds for diagonalizable matrix pencils with real spectra. Also established are perturbation bounds for a multiple eigenvalue to reflect the distinguished feature that its different copies may exhibit quite different sensitivities towards perturbations. - N. Truhar, Z. Tomljanović, R. Li, Analysis of the solution of the Sylvester equation using low-rank ADI with exact shifts, Systems and Control Letters
**59**(2010), 248-257The solution to a general Sylvester equation $AX+XB = GF^*$ with a low rank right- hand side is analyzed quantitatively through Low-rank Alternating-Directional- Implicit method (LR-ADI) with exact shifts. New bounds and perturbation bounds on X are obtained. A distinguished feature of these bounds is that they reflect the interplay between the eigenvalue decompositions of A and B and the right-hand side factors G and F. Numerical examples suggest that because of this inclusion of details, new perturbation bounds are much sharper than the existing ones. - N. Truhar, K. Veselić, An efficient method for estimating the optimal dampers' viscosity for linear vibrating systems using Lyapunov equation, SIAM Journal on Matrix Analysis and Applications
**31**/1 (2009), 18-39This paper deals with an efficient algorithm for dampers' viscosity optimization in mechanical systems. Our algorithm optimizes the trace of the solution of the corresponding Lyapunov equation using an iterative method which calculates a low rank Cholesky factor for the solution of the corresponding Lyapunov equation. We have shown that the new algorithm calculates the trace in $mathcal{; ; ; ; ; O}; ; ; ; ; (m)$ flops per iteration, where $m$ is a dimension of matrices in Lyapunov equation (our coefficient matrices are treated as dense). - N. Truhar, Z. Tomljanović, Estimation of optimal damping for mechanical vibrating systems, International Journal of Applied Mathematics and Mechanics
**5**/5 (2009), 14-26This paper is concerned with the efficient algorithm for dampers' and viscosity optimization in mechanical systems. Our algorithm optimize simultaneously the dampers' positions and their viscosities. For the criterion for optimization we use minimization of the average total energy of the system which can be done by the minimization of the trace of the solution of the corresponding Lyapunov equation. Efficiency of the algorithm is obtained by new heuristics for finding the optimal dampers' positions and for the approximation of the trace of the solution of the Lyapunov equation. - P. Benner, R. Li, N. Truhar, On ADI Method for Sylvester Equations, Journal of Computational and Applied Mathematics,
**233**/4 (2009), 1035-1045This paper is concerned with numerical solutions of large scale Sylvester equations AX − XB = C, Lyapunov equations as a special case in particular included, with C having very small rank. For stable Lyapunov equations, Penzl (2000) and Li and White (2002) demonstrated that the so called Cholesky factored ADI method with decent shift parameters can be very effective. In this paper we present a generalization of Cholesky factored ADI for Sylvester equations. We also demonstrate that often much more accurate solutions than ADI solutions can be gotten by performing Galerkin projection via the column space and row space of the computed approximate solutions. - K. Veselić, N. Truhar, On bounds for discrete semigroups, Operators and Matrices
**3**/3 (2009), 357-365This note studies the exponential decay of the powers $T^k$ of a Hilbert space operator $T$. The main result is extension on the infinite dimension of the following known result for finite matrices: while the spectral radius $spr(T)$ gives only asymptotic decay estimates the solution $X$ of the {; ; discrete Lyapunov equation}; ; $T^*XT-X=-BB^*$ yields rigorous bounds. We also present a new upper bound for the norm of the solution $X$ in the matrix case which depends on the structure of the right hand side. The new bound shows that the structure of $B$ can greatly influence $|X|$. - N. Truhar, K. Veselić, Bounds on the trace of a solution to the Lyapunov equation with a general stable matrix, Systems and Control Letters
**56**/7-8 (2007), 493-503Some new estimates for the eigenvalue decay rate of the Lyapunov equation (A X + X A^T = B) with a low rank right-hand side (B) are derived. The new bounds show that the right-hand side (B) can greatly influence the eigenvalue decay rate of the solution. This suggests a new choice of the ADI-parameters for the iterative solution. The advantage of these new parameters is illustrated on second order damped systems with a low rank damping matrix. - N. Truhar, The perturbation bound for the solution of the Lyapunov equation, Mathematical Communications
**1**/12 (2007), 83-94We present the first order error bound for the Lyapunov equation AX +XA* = − GG*, where A is perturbed to A+δ A. We use the structure of the solution of the Lyapunov equation X = sum_{;k=1};^m W_k W_k^{;*}; , where W_k is the k-th matrix obtained by the Low Rank Cholesky Factor ADI (LRCF-ADI) algorithm using the set of ADI parameters equal to exact eigenvalues of A, that is with ADI parameters {;p_1, ... , p_m}; = σ (A). Our bound depends on the structure of the right-hand side G of the Lyapunov equation, and sometimes it can be sharper than the classical error bounds - N. Truhar, Relative residual bounds for eigenvalues of Hermitian matrices, SIAM Journal on Matrix Analysis and Applications
**28**/4 (2006), 949-960This paper presents a linear and quadratic residual bound for eigenvalues of an indefinite possible singular Hermitian matrix. These bounds are a generalization of results on a semidefinite Hermitian matrix [Z. Drmač and V. Hari, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 21–29]. The bounds here contain an extra factor which depends on the norm of a J‐unitary matrix, where J is diagonal matrix with $pm 1$ on its diagonal. - N. Truhar, I. Slapničar, Relative residual bounds for indefinite Hermitian matrices, Linear algebra and its applications
**417**/2-3 (2006), 466-477We prove several residual bounds for relative perturbations of the eigenvalues of indefinite Hermitian matrix. The bounds fall into two categories– – the Weyl-type bounds and the Hofmann– Wielandt-type bounds. The bounds are expressed in terms of sines of acute principal angles between certain subspaces associated with the indefinite decomposition of the given matrix. The bounds are never worse than the classical residual bounds and can be much sharper in some cases. The bounds generalize the existing relative residual bounds for positive definite matrices to indefinite case. - N. Truhar, An efficient algorithm for damper optimization for linear vibrating systems using Lyapunov equation, Journal of Computational and Applied Mathematics,
**172**/1 (2004), 169-182We consider a second order damped-vibration equation $ M ddot{;x};+ D(epsilon) dot{;x};+ K x = 0 $, where $M, D(epsilon), K$ are real, symmetric matrices of order $n$. The damping matrix $D(epsilon)$ is defined by $D(epsilon)=C_u+C(epsilon)$, where $C_u$ presents internal damping and $rank(C(epsilon))=r$, where $epsilon$ is dampers' viscosity. We present an algorithm which derives a formula for the trace of the solution $mathbf{;X};$ of the Lyapunov equation $ mathbf{;A};^T mathbf{;X};+ mathbf{;X};mathbf{;A};= - mathbf{;B};, $ as a function $epsilon rightarrow Tr(mathbf{;ZX};(epsilon))$, where $mathbf{;A};=mathbf{;A};(epsilon)$ is a $2n times 2n$ matrix (obtained from $M, D(epsilon), K$) such that the eigenvalue problem $mathbf{;A};mathbf{;y};= {;lambda , mathbf{;y};};$ is equivalent with the quadratic eigenvalue problem $(lambda^2 M + lambda D(epsilon) + K) x = 0$ ($mathbf{;B};$ and $mathbf{;Z};$ are suitably chosen positive semidefinite matrices). Moreover, our algorithm provides the first and the second derivative of the function $epsilon rightarrow Tr(mathbf{;ZX};(epsilon))$ almost for free. The optimal dampers' viscosity is derived as $epsilon_{;opt};= {;rm argmin};, Tr(mathbf{;ZX};(epsilon))$. If $r$ is small, our algorithm allows a sensibly more efficient optimization, than standard methods based on the Bartels--Stewart's Lyapunov solver. - N. Truhar, K. Veselić, On some properties of the Lyapunov equation for damped systems, Mathematical Communications
**2**/9 (2004), 189-197We consider a damped linear vibrational system whose dampers depend linearly on the viscosity parameter $v$. We show that the trace of the corresponding Lyapunov solution can be represented as a rational function of $v$ whose poles are the eigenvalues of a certain skew symmetric matrix. This makes it possible to derive an asymptotic expansion of the solution in the neighborhood of zero (small damping). - N. Truhar, R. Li, A sin 2theta theorem for graded indefinite Hermitian matrices, Linear algebra and its applications
**359**/1-3 (2003), 263-276This paper gives double angle theorems that bound the change in an invariant subspace of an indefinite Hermitian matrix in the graded form H=D^*AD subject to a perturbation H -> tilde H=D^*(A+Delta A)D. These theorems extend recent results on a definite Hermitian matrix in the graded form (Linear Algebra Appl. 311 (2000) 45) but the bounds here are more complicated in that they depend on not only relative gaps and norms of Delta A as in the definite case but also norms of some J-unitary matrices, where J is diagonal with +1, -1 on its diagonal. For two special but interesting cases, bounds on these J-unitary matrices are obtained to show that their norms are of moderate magnitude. - I. Slapničar, N. Truhar, Relative perturbation theory for hyperbolic singular value problem, Linear algebra and its applications
**358**/1-3 (2003), 367-386We give relative perturbation bounds for singular values and perturbation bounds for singular subspaces of a hyperbolic singular value problem for the pair (G, J), where G is a full rank matrix and J is a diagonal matrix of signs. We consider two types of relative perturbations: G+delta G=(B+delta B)D and G+delta G=bar D(bar B+bar delta B), depending whether G has full column or full row rank, respectively. In both cases we also consider relative element-wise perturbations of G which typically occur in numerical computations. - N. Truhar, I. Slapničar, Relative perturbation bound for invariant subspaces of Hermitian matrix, Glasnik Matematički
**2**/35 (2000), 221-232We give a bound for the perturbations of invariant subspaces of a non-singular Hermitian matrix $H$ under relative additive perturbations of $H$. Such perturbations include the case when the elements of $H$ are known up to some relative tolerance. Our bound is, in appropriate cases, sharper than the classical bounds, and it generalizes some of the recent relative perturbation results. - I. Slapničar, N. Truhar, Relative perturbation theory for hyperbolic eigenvalue problem, Linear algebra and its applications
**309**(2000), 57-72We give relative perturbation bounds for eigenvalues and perturbation bounds for eigenspaces of a hyperbolic eigenvalue problem Hx=Jx, where H is a positive definite matrix and J is a diagonal matrix of signs. We consider two types of perturbations: when a graded matrix H=D*AD is perturbed in a graded sense to H+H=D*(A+A)D, and the multiplicative perturbations of the form H+H=(I+E)*H(I+E). Our bounds are simple to compute, compare well to the classical results, and can be used when analyzing numerical algorithms. - N. Truhar, I. Slapničar, Relative perturbation bounds for invariant subspaces of graded indefinite Hermitian matrices, Linear algebra and its applications
**301**(1999), 171-185We give a bound for invariant subspaces of graded indefinite Hermitian matrix H=D*AD which is perturbed into H+dH=D*(A+dA)D. Such relative perturbations include important case where H is given with an element-wise relative error.

Refereed Proceedings

- N. Truhar, W. Weber, A numerical approach for determining the optimal position of a single damper for damping a distinct eigenfrequency of multi-body systems, GAMM Annual meeting 2023, Dresden, 2023, 1-6In this paper the damping optimization of a multi-body oscillator with arbitrary $n$ degrees of freedom is considered. The main contribution is a theoretical result for the optimal position of a single damper in a structured optimization. For that purpose the optimization criterion based on the average total energy is considered. As the damping optimization by means of the chosen optimization criterion in general is not a straight-forward task, a numerical approach based on the formula of textsc{Veseli'{c}} is introduced in this contribution. This approach can be used for efficiently calculating optimal damping for medium-size problems with~$n leq 10^4$.
- I. Kuzmanović Ivičić, Z. Tomljanović, N. Truhar, Applications of Lyapunov and T-Lyapunov equations in mechanics, Fourth Mathematical Conference of the Republic of Srpska,, Trebinje, 2014, 83-92This paper considers Lyapunov and T -Lyapunov matrix equations. Lyapunov equation is a matrix equation of the form AX + XA^T = E which plays a vital role in a number of applications, while T -Lyapunov equation is a matrix equation of the form AX +X^TA^T = E. In this paper the relation between these equations will be exploit with purpose of applying obtained results in problems regarding damping optimization in mechanical systems.
- P. Benner, Z. Tomljanović, N. Truhar, Damping Optimization for Linear Vibrating Systems Using Dimension Reduction, The 10th International Conference on Vibration Problems ICOVP 2011, Prag, 2011, 297-305We consider a mathematical model of a linear vibrational system described by the second-order system of differential equations $M ddot{; ; x}; ; + D dot{; ; x}; ; + Kx = 0$, where M, K and D are positive deffinite matrices, called mass, stiffness and damping, respectively. We are interested in finding an optimal damping matrix which will damp a certain part of the undamped eigenfrequencies. For this we use a minimization criterion which minimizes the average total energy of the system. This is equivalent to the minimization of the trace of the solution of a corresponding Lyapunov equation. In this paper we consider an algorithm for the efficient optimization of the damping positions based on dimension reduction techniques. Numerical results illustrate the efficiency of our approach.
- D. Markulak, D. Varevac, N. Truhar, he dynamic response of a highway bridge under actual traffic load, Progress in Steel, Composite and Aluminium Structures , Rzesow, 2006, 264-265The problem of the dynamic response of the highway bridge under the actual traffic load has been analysed. The dynamic influence of the heavy lorry vehicles under usual circumstances does not affect on a bearing load capacity, but continuosly degrades serviceability of the bridge and thus its service life. Increasing number of the lorry vehicles and their weight, as well as more lighter and flexibile bridges, additionally empahasize this fact. Determination of the dynamic response of the bridge resulting from the passage of a heavy vehicles across the bridge is very complicated and depends on many parameters such as dynamic characteristics of the vehicle and the bridge, heavy vehicle geometry and the speed, roughness of the surface profile of the bridge, the natural frequencies of the vehicle and the etc. The classical calculation procedure includes simplified calculation of the dynamic bridge response based on increased static load. This procedure neglects the fact that degradation of the bridge appears not only by the extreme traffic load but continuously by the real heavy vehicles. Moreover, the dynamic interaction between the vehicle and the bridge and stochastic nature of the traffic load also is usually neglected. In this paper we present approach which is based on a stochastic characteristic of the actual traffic flow and numerical calculation of dynamic interaction system heavy vehicle-bridge. Vehicles are modelled as a group of axles (SDOF model) with the dynamic characteristic (stiffness and damping), and the weight is treated as stochastic variable. Dynamic vehicle-bridge model has been defined by the system of coupled partial differential equations.
- D. Varevac, D. Markulak, N. Truhar, Seismic Response Of The Long Countinuous RC Bridge, Concrete Structures for Traffic Network, Hradec Kralove, 2006Bridges are structures which are very sensitive to the transversal excitation. Some recent events have shown this fact in very drastic fashion (Loma Prieta 1989, Northridge 1994, Kobe 1995.). Dynamic analysis of such structures may be time consumable due to complicated response and large number of the degrees of freedom, but it is necessary step in the preliminary stage of the design process. This paper presents simple method for obtaining eigenvalues, eigenvectors and participation masses of the continuous girder bridge. It is shown that bridges with the same stiffness ratio between piers and the bridge deck have dynamic properties in the direct relationship, assuming that their span ratios are the same. Procedure shown in the paper presents simple method for obtaining these dynamic characteristics, using as input just simple geometry of the cross sections. This is useful in the preliminary stage of the design because all the important dynamic parametars are calculated very fast. By changing stiffness ratio one can customize bridge response and reduce seismic forces acting on the bridge in the transversal direction, according to microseismic location of the bridge.
- D. Varevac, D. Morić, N. Truhar, Mathematical model for calculating eigenvalues of continuous bridge ggirder in transversal direction, Durability and Maintenance of Concrete Structures, Dubrovnik, 2004, 147-154The article describes mathematical model of RC continuous bridge girder in transversal direction. Chosen disposition of the bridge is continuous beam over three spans (0, 7L+L+0, 7L) with two single-column-bent, and two different cases of end supports: pinned support at the ends of the bridge and free translation supports at the end of the bridge(console in the the transversal direction). Columns are pinned to bridge girder in transversal direction and have free rotation at the top. Vibration of the structure is described with partial differential equation of IVth order (wave equation) as well as with partial differential equations of the theory of elasticity. Columns are modeled in transversal direction as springs with stiffnesess which correspondent to the console bending stiffness. Shear stiffness, as well as rotation and translation of the column foundation are neglected, but it can be easily incorporated in the calculations through reduction of the stiffness. After introduction of the boundary condition, system of partial differential equations of IV order is reduced to system of 12 linearly independent equations with 12 unknowns. The solution of the system is found in the roots of matrix. Since determinant of the system matrix is very complex, it is not possible to find direct solution. One of the iterative method must be used. Newton s method in this case.
- N. Truhar, D. Markulak, D. Varevac, Optimizing stiffnesses and positions of dampers for cable on the steel high voltage towers, 7th Congress of the European Society for Research in Mathematics Education, Rzeszow, Poljska, 2002
- N. Truhar, B. Dukić, Low rank approximation using singular value decomposition, 6th International Conference on Operational Research KOI 1996, Rovinj, 1996, 75-80In many applications we have to deal with matrices which have a large amount of redundancy. The matrix of this kind can be expressed much more efficiently than simply by listing all the entries. In this paper we will show application of the low rank approximation using singular value decomposition in representation of matrices of such kind. We will illustrate this application in few examples from different fields of science.

Others

- Z. Tomljanović, N. Truhar, K. Veselić, Damping optimization in mechanical systems with external force (2015)
- N. Truhar, Dva dokaza Heronove formule, Osječki matematički list
**8**/2 (2009), 65-68Ovaj članak sadrži izuzetno zanimljiv i elegantan dokaz Heronove formule srenjoškolca Milesa Edwardsa objavljenog u "FA Proof of Heron's Formula", American Mathematics Monthly, Vol. 114, No. 10, 2007, p. 937. Ono što ga posebno čini zanimljivim jest upotreba kompleksnih brojeva. Željeli smo istaći da ovaj dokaz pokazuje da ponekad izlazak ili nepridržavanje standardnih klišea ili rigidnih pravila može proizvesti lijep i zanimljiv matematički rezultat. - D. Markulak, N. Truhar, Dinamička interakcija vozila i cestovnog grednog mosta, Tehnički vjesnik
**3**/3,4 (2004), 15-21Traffic dynamic interaction of the traffic bridge - N. Truhar, I. Slapničar, Relative perturbation of invariant subspaces, Mathematical Communications (1996), 169-174In this paper we consider the upper bound for the sine of the greatest canonical angle between the original invariant subspace and its perturbation. We present our recent results which generalize some of the results from the relative perturbation theory of indefinite Hermitian matrices.
- G. Bučar, N. Truhar, Utjecaj vremena građenja na cijenu radova, Građevinar
**48**(1996)

Books

- R. Scitovski, N. Truhar, Z. Tomljanović, Metode optimizacije, Svučilište Josipa Jurja Strossmayera u Osijeku, Odjel za matematiku, Osijek, 2014.Namjena ovog teksta je upoznati čitatelja s glavnim metodama jednodimenzionalne i višedimenzionalne minimizacije sa i bez ograničnja. Posebna je pozornost posvećena metodama minimizacije nediferencijabilnih funkcija. Pri tome je izbjegavano dokazivanje zahtjevnih teorema, osim u slučajevima konstruktivnih dokaza koji sami po sebi upućuju na izgradnju ideja ili metoda. Navedeni optimizacijski problemi imaju veliku primjenu u raznim dijelovima života. Na primjer, često se javljaju problemi poput optimalnog oblikovanja odredenih mehaničkih sustava (oblikovanje dijelova automobilskih motora, nosivih konstrukcija u gradjevinarstvu, . . .), problem modeliranja ponašsanja tržišta, problemi iz teorije upravljanja (smirivanje sustava, optimalno upravljanje, . . . ) i mnogi drugi. Upravo činjenica da se razni problemi optimizicije pojavljuju u raznim dijelovima ljudske djelatnosti osigurava ovom tekstu široku primjenu.
- D. Matijević, N. Truhar, Uvod u računarstvo, Odjel za matematiku, Sveučilište Josipa Jurja Strossmayera u Osijeku, Osijek, 2012.
- N. Truhar, Numerička linearna algebra, Odjel za matematiku, svučilišta J. J. Strosmayera u Osijeku, Osijek, 2010.Osnova za proučavanje numeričke linearne algebre je linearna algebra, to znači da je osnovni naglasak u okviru ovog teksta stavljen na proučavaje istih problemi kao i u linearnoj algebri, ali se ovdje ti problemi proučavaju s potpuno drugačijeg aspekta. Prije svega je proučavano rješavanje velikih problema pomoću računala, te proučavanje brzine, točnosti i stabilnosti pojedinih algoritama.

Technical Reports

- N. Truhar, Z. Tomljanović, M. Puvača, An efficient approximation for the optimal damping in mechanical systems (2016)This paper is concerned with the efficient algorithm for damping optimization in mechanical systems with prescribed structure. Our approach is based on the minimization of the total energy of the system which is equivalent with 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 the modally damped system. Even though our approach is very efficient (as it was expected) for the 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 the optimal damping.
- N. Truhar, Z. Tomljanović, Dimension reduction approach for the parameter dependent quadratic eigenvalue problem (2016)This paper presents the novel approach in efficient calculation of the 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 $ntimes n$ Hermitian semidefinite matrix which depends on a parameter $mathbf{v}= begin{bmatrix} v_1 & ldots & v_k end{bmatrix}in mathbb{R}_+^k$. With the new approach one can efficiently (and accurate enough) calculate the all (or just part of the) eigenvalues even for the case when $v_i$ are of the modest magnitude. Moreover, for the both cases of approximations we have derived the corresponding upper bounds. The quality of the error bounds as well as the performance of the achieved eigenvalue tracking was illustrated in several numerical experiments.
- N. Truhar, Z. Tomljanović, K. Veselić, Damping optimization in mechanical systems with external force (2014)We consider a mechanical system excited by external force. Model of such a system is described by the system of ordinary differential equations: $M ddot x(t) + D dot x(t) + K x(t) = {hat f}(t)$, where matrices $M, K$ (mass and stiffness) are positive definite and the vector ${hat f} $ corresponds to an external force. The damping matrix D is assumed to be positive semidefinite and has a small rank. We introduce two criteria that allow damping optimization of mechanical system excited by an external force. Since in general a damping optimization is a very demanding problem, we provide a new formulas which have been used for efficient damping optimization. The efficiency of new formulas is illustrated with a numerical experiment.

### Professional Activities

## Editorial Service

Member of Editorial Board:- Computational Mathematics and Computer Modeling with Applications (since 2022–)
- Numerical Algebra, Control and Optimization (since 2021–)
- Mathematics (MDPI) (since 2021–)
- Mathematical Communications (since 2007–)
- Osječki matematički list (since 2003–)