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Projection-based model reduction has been successfully employed in prominent applications ranging from inverse problems to uncertainty quantification to optimal control. The projection-based methods require access to internal dynamics, i.e., a state-space representation, which is not always accessible. Rather, in these cases, one might have access to input/output measurements in the form of transfer function evaluations. We will discuss how one can employ this available data to construct high-fidelity (in some cases optimal) data-driven dynamical reduced models, both for non-parametric and parametric systems. Finally, we will connect the data-driven modeling framework to nonlinear eigenvalue problems and discuss how the classical realization theory gives further insights into certain classes of methods for nonlinear eigenvalue computations.

Vector Fitting (VF) is a powerful and widely used method of constructing rational approximants that provides a least squares fit to frequency response measurements. The original method is based on a least-squares fit to the measurements by a rational function, using an iterative reallocation of the poles of the approximant. We show that one can improve the performance of VF significantly, by using a particular choice of frequency sampling points and properly weighting their contribution based on quadrature rules that connect the least squares objective with an $H_2$ error measure. In this way, the original transfer function is approximated with better global fidelity (as measured with respect to the $H_2$ norm), than the localized least squares approximation implicit in VF. This framework can be extended also to incorporate derivative information, leading to rational approximants that minimize system error with respect to a discrete Sobolev norm. One important issue that arises for VF on matrix-valued functions (transfer functions associated with finite-dimensional multi-input/multi-output (MIMO) dynamical systems), that has remained largely unaddressed, is the control of the McMillan degree of the resulting rational approximant; the McMillan degree can grow very high in the case of large input/output dimensions. We discuss two mechanisms for controlling the McMillan degree of the final approximant, one based on alternating least-squares minimization and one based on ancillary system-theoretic model order reduction methods, such as balanced truncation and realization independent IRKA. Further, we discuss subtle numerical details related to notoriously ill-conditioned Cauchy-type matrices that naturally appear in the barycentric form of the rational approximants.

Let $\mathcal H(s, p)$ denote the transfer function of a single-input/single-output parameterized linear dynamical system; thus we assume linear dynamics, but the parameter dependency could be non-linear. We also assume that we do not have access to the internal dynamics, but have access only to the input/output measurements in the form of the transfer function evaluations. Let $ \{ \xi_i,\mu_j, \mathcal H(\xi_i, \mu_j) \}_{i=1,j=1}^{i=m_s,j=m_p}\subset \C\times\C\times C $ be the measurement data set where $\{\xi_i\}$ and $\{\mu_j\}$ denote the samples in the frequency (s) and parameter (p) domain, respectively; and $m_s$ and $m_p$ are, respectively, the number of samples in s and p. Then, given this measurement data, we consider the problem of finding a parametric dynamical system, represented by its transfer function $\mathcal {\hat H} (s, p)$, that solves the least-squares problem (1) $ \sum^{m_s}_{i=1}\sum^{m_p}_{j=1} |\mathcal {\hat H} (\xi_i, \mu_j)- \mathcal H (\xi_i, \mu_j)|^2 \to min. $ Let $\mathcal{\hat {H_k}}(s)$ denote the local reduced model obtained via, for example, Vector Fitting for the parameter sample $\mu_k$. We combine the local reduced models $\mathcal{\hat {H_k}}(s)$ via coefficient functions $\alpha_k(p)$ so that the parametric reduced model (2) $\mathcal{\hat H}(s,p):=\sum^{m_p}_{k=1}\alpha_k(p)\mathcal{\hat {H_k}}(s)$ solves the joint least-squares problem (1) in frequency and parameter. For the coefficients $\alpha_k(p)$, we consider both the polynomial and rational parametrizations. In the latter case, we use an iterative algorithm based on the variable projection method4 to automatically adapt the poles of the rational functions $\alpha_k(p)$. We compare our approach to various existing methods. In the case of many parameter samples, the order of the approximant $\mathcal H (s, p)$ could be higher than desired. For those cases, we use a post-processing step that employs the optimal H2 model reduction techniques for special parameterizations. We demonstrate our algorithm on several benchmark examples with scalar and higher dimensional parameter space.

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Filtered subspace iteration with Rayleigh-Ritz eigenvalue extraction is a recently reviewed method in the form of the FEAST algorithm. In this talk we present a method motivated by the FEAST iteration and apply it directly on the operator level. The core of the algorithm is a numerical resolvent calculus based on contour quadratures coupled with discontinuous Petrov-Galerkin discretization of the resolvent. We show the contraction property for the iterative process under the assumption that the error in the approximation of the resolvent is asymptotically balanced with the best approximation rate for the target eigenfunctions from a given finite element space.

Port-Hamiltonian systems are a type of passive systems, appearing in many domains of physics, e.g. electric circuits and mechanical systems. Since such systems can consist of many ordinary differential equations, one approach to enable faster analysis, simulation, and control is to apply a model reduction method. Additionally, a method preserving the port-Hamiltonian structure is preferred, such that the reduced model has the same physical interpretation. For unstructured linear time-invariant systems, the Iterative Rational Krylov Algorithm (IRKA), an H2-optimal model reduction method based on interpolatory necessary optimality conditions, was introduced by Antoulas, Beattie, and Gugercin [2]. Later, Gugercin et al. [3] proposed an interpolatory model order reduction method for port-Hamiltonian systems inspired by IRKA. The method shows its effectiveness in numerical examples, but it does not necessarily find a (locally) H2-optimal reduced model. Beattie and Benner [1] derived interpolatory necessary optimality conditions, but did not propose a method to find a reduced model which would satisfy them. In this talk, we will derive Gramian-based necessary optimality conditions, which take the form of a system of matrix equations, and discuss possible solution methods for this system of equations.

Model order reduction (MOR) is a commonly used procedure that aims to replace large and complex models of dynamical processes with much simpler and lower dimensional models. This is done usually to ease and speed up certain tasks such as control, design, and simulation. Dynamical systems that are described by an interaction between continuous and discrete dynamics are usually called hybrid systems. We are interested in the study of continuous-time systems with isolated discrete switching events, which are known as switched systems, and constitute a subclass of hybrid systems. More specifically, we analyze linear switched systems (LSS) with time-dependent switching. For this family of systems, there exists a piecewise-constant function $\sigma$ (referred to as switching signal) that specifies, at each time instant t, the linear subsystem (or mode) which is activated. Additionally, we consider the LSS to have coupling matrices that scale the state variable after each switching instant. In a sense, linear switched systems have some similarities with bilinear systems. For example, the LSS time-domain kernels have similar structure to the bilinear ones (as stated in [1]). Moreover, in some special cases, it is possible to formulate an LSS as a bilinear system with fixed control inputs. We start by using an appropriate extension of the H2 norm definition (from bilinear systems [2, 3] to linear switched systems [4]). It can be computed by summing up inner products of specific time- domain kernels, or equivalently, by using infinite controllability and observability Gramians (which were originally introduced in [1]). Furthermore, we propose an iterative rational Krylov algorithm that can be viewed as an extension of B-IRKA [2] to the class of linear switched systems. It is based on repeatedly applying a two-sided projection (constructed in the Petrov-Galerkin framework) to the original LSS. The projection matrices Vi and Wi are computed as solutions of generalized Sylvester equations and correspond to the mode i of the LSS. The procedure is repeated until a stopping criterion is met (the pole deviation does not exceed a certain tolerance value). Finally, we can also show that it is possible to derive similar optimality conditions as the ones previously presented in in [2, 3]. The validity and practical applicability of the proposed methodology is shown by means of several numerical examples.

Model order reduction (MOR) is a tool used to replace large and complex models of dynamical processes with smaller and simpler dimensional models, which can be easily used for complete analysis such as control, design, and simulation. The methods considered in this study are interpolation based and aim to construct reduced-order models for which the corresponding rational transfer function matches that of the original model at selected interpolation points. Moreover, the approaches we present are data-driven, in the sense that the required information is extracted from data (data points or nodes and data measurements or function values) - no original model or function is required. The primary method of this study is the Loewner framework, as introduced in see [1]. It produces models directly from measurements in a straightforward manner. The main feature is that it provides a trade-off between accuracy of fit and complexity of model, by means of the singular value decay of the Loewner matrix. The original Loewner framework is based on compression of the (usually large) data set in order to extract the dominant features and, in the same time, eliminate the inherent redundancies. The procedure introduced in [1] relies on a full SVD (singular value decomposition) to compress the raw data model. This might be costly to perform, especially for very large data sets. In order to surpass this possible shortcoming, we propose a more ecient approach which replaces the SVD by a CUR decomposition. In general, a CUR approximation of a given matrix M is written in terms of three matrices C, U, and R such that C and R are composed of columns and, respectively rows, selected from the ones of the M matrix. Finally, the matrix U is chosen such that product CUR closely approximates M. We propose a modified Loewner framework, in the sense that the interpolation points are selected from the original ones, and not by means of compression. The indexing of the chosen points coincides to the indexing of the selected rows and columns of the Loewner matrix. The selection algorithm is based on the CUR-DEIM approach in [4]. We also studied the AAA algorithm, as introduced in [2]. It can be viewed as an adaptive and iterative version of the Loewner framework and aims to optimize the approximation error by means of a least squares approach. The order of the rational approximant is increased after each step for better accuracy. We managed to construct low order models for physical problems such as Euler-Bernoulli clamped beam model and 1D heat diffusion model. Additionally, we tested the methods for artificial examples which commonly appear in approximation theory, such as the Bessel function, the hyperbolic sine function, and the Heaviside step function. Although a direct method, the original Loewner framework in [1] proved to be comparable (in terms of approximation quality) to the other methods which rely on optimization tools. Moreover, in the case of the Heaviside function, we were able to find rational functions via the Loewner framework, which turned out to be very close to the optimal minimax rational function (which is the solution of the fourth Zolotarev problem [3]).

Enjoy!

Venue: Hotel Osijek, Šamačka 4, 31000 Osijek

The benefit of compliant robotic design is becoming more and more evident with increasing tendency to involve robots navigating on rough or human tailored terrain especially when locomotion is executed using legs. When dealing with legged robot locomotion, these concerns are primarily related to ensuring robustness and energy efficiency of locomotion which are both compromised due to nondeterministic nature of the outdoor environment. This lecture deals with design and implementation of a hybrid compliance control system for a stiff-by-nature quadruped robot comprising active and variable passive compliance control parts. We show how this new joint effort of both complaince types is beneficial to the preformance of the legged robotic system under modernate external disturbances by preforming mathematical analysis and real world experiments.

There exists a widely recognised need to better understand and manage complex large-scale dynamical networks, such as “smart” energy grids, biological networks or automated highways. Complexity, large-scale, interconnection/communication constraints and often non-existence of a suitable global model (e.g, due to confidentiality issues), require development of analytical and computational methods for tractable analysis and controller synthesis under structural constraints (e.g., plug-and-play requirements; structured stability/performance conditions; decentralized or distributed controllers). The key challenge is to develop a suitable framework which enables us to efficiently cope with inherent efficiency-robustness-scalability trade-offs in both analysis and control synthesis. In this talk we will address the above challenges in parallel with presenting several recent results regarding structured stability and performance conditions for dynamical networks. The presented approach is based on the dissipativity theory and closely related integral quadratic constraints (IQC). For dissipativity conditions defined with static supply rates, we will relate dissipativity properties of individual systems in a network with existence of a structured Lyapunov function and with certain robustness properties. Furthermore, we will introduce the notion of differential supply rates and derive novel structured stability conditions based on storage functions with higher order derivatives. Relations with stability/performance conditions based on more standard IQCs will be presented and discussed. Several real-life examples will be presented in parallel to the theoretical results.

We formulate the eigenvalue problem underlying the mathematical model of a linear vibrational system as an eigenvalue problem of a diagonal-plus-rank-one matrix $A$. The eigenvector matrix of $A$ has Cauchy-like structure. We compute the trace of the solution of the Lyapunov equation $AX+XA^*=-GG^*$ using fast computations with Cauchy-like and Toeplitz-like matrices. Here $G$ is a low-rank matrix which depends on the damped eigenfrequencies. This is joint work with Nevena Jakovčević Stor.

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Balanced truncation is one of the most established methods for model reduction of linear time-invariant systems. However, this method may give reduced models that result in large errors if the initial value is not equal to zero. We propose a new way of reformulating the system by shifting the state by an L2-function and extending the input vector. Than classical balanced truncation can be applied and a parameter-dependent error bound is obtained. We show how reduced-order models can be practically constructed and how to eficiently determine an optimal error bound. We conclude the talk with numerical experiments and a comparison to other approaches. This is joint work with Christian Schröder.

Quadratic eigenvalue problem (QEP) is often solved by linearizing and then deploying well known techniques to solve the resulting linear (generalized) eigenproblem. However, a generic linear eigensolver is unaware of the underlying structure of the quadratic problem, which may cause loss of important structural spectral properties of the original problem. Bai and Su (2005.) first realized that in the case of iterative Arnoldi-type methods, it is advantageous to apply the Rayleigh-Ritz projection directly to the initial QEP. To that end, they introduced second order Krylov subspaces, and the corresponding second order Arnoldi procedure for generating orthonormal bases. The resulting method, Second Order Arnoldi (SOAR), is further modified yielding TOAR (Lu, Su and Bai, 2016). SOAR procedure is also used for dimension reduction of large scale second order dynamical systems. The key feature of this approach is perseverance of the structure of the dynamical system. In this talk, we will present implicit restarting in SOAR(TOAR) in context of computing the prescribed number of eigenvalues of QEP. The wanted number of eigenvalues is much smaller than the dimension of the original problem. The emphasize of the talk will be the issues of better choices of starting vectors, choosing shifts to construct polynomial filters during the restart process, and extracting the wanted eigenvalues and eigenvectors, with particular attention to the peculiarities of the quadratic problem.

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We discuss the feedback problem for dynamical systems described by a (system of) unsteady partial differential equations, such as nonlinear heat conduction and (reactive) flow problems. Using the linearization principle, such problems can be steered towards equilibrium states using linear-quadratic regulators defined via the solution of algebraic Riccati equations (AREs). Due to the semi-discretization in space, usually achieved by finite element methods, the computational bottleneck then is the solution of the resulting large-scale AREs. We discuss recent developments to solve such problems, and compare their performance using several numerical examples.

Balanced truncation is one of the most common model order reduction techniques. For linear dynamical systems, it relies on the computation of reachability and observability Gramians, which are the solution $X$ of algebraic Lyapunov equations as $AX+XA^T+Q = 0.$ Later, the concept of algebraic Gramians was extended to bilinear systems. In this context, they satisfy generalized Lyapunov equations as $ AX+XA^T+ \sum_{j=1}^M D_jXD_j^T +Q = 0. $ As a result, these algebraic Gramians allow us to compute reduced-order systems for bilinear systems. In this talk, an extension of balanced truncation for model reduction of continuous-time Linear Switched Systems (LSS) is proposed. First, we recast a linear switched system as a bilinear system. Then, inspired by the bilinear theory, we propose algebraic Gramians for LSS, which satisfy generalized Lyapunov equations. Also, we prove that these Gramians encode the reachability and observability sets. This allows us to find those states that are hard to reach and hard to observe via an appropriate transformation. Truncating such states yields reduced-order systems. The efficiency of these approximations is, then, demonstrated in some numerical examples.

In this talk we deal with the approximation of optimal controls for parameter-dependent elliptic and parabolic equations. We adapt well-known results on greedy algorithms to approximate in an efficient way the optimal controls for parametrized elliptic control problems. Our results yield an optimal approximation procedure that, in particular, performs better than simply sampling the parameter- space to compute controls for each parameter value. Generalisation of the method to parabolic control problems would lead to greedy selections of the parameters that depend on the initial datum under consideration. To avoid this dificulty we employ the turnpike property for time evolution control problems that ensures the asymptotic simplification of optimal control problems for evolution equations towards the elliptic steady-state ones in long time horizons [0, T]. The combination of the turnpike property and greedy methods allows us to develop eficient methods for the approximation of the parameter-dependent parabolic optimals too. We present various numerical experiments discussing the efficiency of our methodology and its application to turnpike control problems.

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In optimal design problems the goal is to find the arrangement of given materials within the body which optimizes its properties with respect to some optimality criteria. The performance of the mixture is usually measured by an integral functional, while optimality of the mixture is achieved through minimization or maximization of this functional, under constraints on amount of materials and PDE constraints that underlay involved physics. We consider multiple-state optimal design problems from conductivity point of view, where thermal (or electrical) conductivity is modeled with stationary diffusion equation and restrict ourselves to domains filled with two isotropic materials. Since the classical solution usually does not exist, we use relaxation by the homogenization method [2] in order to get a proper relaxation of the original problem. One of numerical methods used for solving these problems is the optimality criteria method, an iterative method based on optimality conditions of the relaxed formulation. In the case of a single-state problem this method is described in [1], where it is also proved that it converges in the case of a self-adjoint optimization problems. Based on the optimality conditions derived in [1], a variant of optimality criteria method for multiple-state problems was introduced in [3] . It appears that this variant works properly for maximization of conic sum of energies, but fails for the minimization of the same functional. We rewrite optimality conditions for relaxed problem and develop a variant of optimality criteria method suitable for energy minimization problems. We also prove convergence of this method in a special case when a number of states is less then the space dimension and in the spherically symmetric case. Presented method can be adapted to similar problems in the context of linearized elasticity.

General, non-periodic homogenization theory is well developed for second order elliptic partial differ- ential equations, where the key role plays the notion of H-convergence. It was introduced by Spagnolo through the concept of G-convergence (1968) for the symmetric case, and further generalized by Tartar (1975) and Murat and Tartar (1978) for non-symmetric coeficients under the name H-convergence. Some aspects for higher order elliptic problems were also considered by Zhikov, Kozlov, Oleinik and Ngoan (1979). Homogenization theory is probably the most successful approach for dealing with optimal design problems (in conductivity or linearized elasticity), that consists in arranging given materials such that obtained body satisfies some optimality criteria, which is mathematically usually expressed as minimization of some (integral) functional under some (PDE) constrains. Motivated by a possible application of the homogenization theory in optimal design problems for elastic plates, we adapt the general homogenization theory for Kirchoff-Love elastic plate equation, which is a fourth order elliptic equation. In addition to the compactness result, we prove a number of properties of H-convergence, such as locality, irrelevance of the boundary conditions, corrector results, etc. Using this newly developed theory, we derive expressions for elastic coeficients of composite plate obtained by mixing two materials in thin layers (known as laminated materials), and for mixing two materials in low-contrast regime. Moreover, we also derive optimal bounds on the effective energy of a composite material, known as Hashin-Shtrikman bounds.

We analyse an optimal design problem for the stationary diffusion equation. The aim is to find an optimal arrangement of two materials minimising an objective function whereby the proportion of the materials is fixed and predetermined. Optimal design problem is ill-posed in the sense that it doesn't admit a minimizer in general. Therefore it is relaxed by using the assumption of small amplitude and H-measures. Generalisation of the method to higher order asymptotic and time dependent problems will be discussed. Numerical examples will be presented as well.

The modeling of physical systems should take into account conservation laws, but this can be a significant challenge when models are derived directly from system response data. Observational noise and unmodeled sources can further complicate the enterprise. I will discuss a few (equivalent) characterizations of energy conservation for dynamical systems and describe how one can ascertain whether an observed response profile could have been generated by a passive system. This leads naturally to a data-driven modeling framework that yields a convex family of passive models all of which are consistent with the observed response profile. Evidently, this sets the stage for selecting a model that is consistent with the observed response profile that also optimizes a separate performance goal or perhaps satisfies other ancillary conditions. I will discuss a particular approach that can provide a data-driven model having maximal passivity margins.

We consider a linear vibrational system $ M\ddot{x}+D\dot{x}+Kx=0, \qquad x(0)=x_0, \qquad \dot x(0)=\dot x_0, $ where, matrices $M$ and $K$ represent mass and stiffness, respectively, which are either positive definite or semidefinite of order $n$. The damping matrix is $D=C_u+C$, where $C_u$ represents internal damping and $C$ is an external damping which is usually low rank and semidefinite. The problem of damping optimization of systems like this is very demanding and can be approached form a several different aspects. Thus, if $A$ a matrix obtained using appropriate linearization of the corresponding QEP $(\lambda^2 M + \lambda (C_u+C) + K )x=0$, as the one approach we can considers a behaviour of the spectrum of the matrix $A$. For example one can use \emph{spectral abscissa criterion}, that is $ \min\limits_i {\rm Re}(\lambda_i) $, where $\lambda_i$ are eigenvalues of the matrix $A$. Another approach considers the solution $X$ of the corresponding Lyapunov equation $A X + X A^T = -I$. Within this approach one can minimize the trace of $X$ or the norm of $X$ as the function of the external damping $C$. Since, this kind of optimization processes are computationally very demanding it is useful to approximate original systems with smaller one to accelerate optimization processes. This can be obtained using a dimension reduction, for example, one can use reduction using the $r$ columns of the matrix $\Phi$ which simultaneously diagonalize pair $(M,K)$. The quality of obtained approximations strongly depend on chosen subspaces, thus one of our aims is to show how one can bound the norm of sines of canonical angles between subspaces of interest. We will present a several upper bounds as well as lower bounds of this kind. It turns out the both bounds can be used in the damping optimization process. The upper bounds of the norm of the matrix of sines of canonical angles determine the quality of the approximations and can be used to control the error bound for the splitting the corresponding GEP in two (or more) independent parts (usually this procedure is called decoupling). On the other hand the lower bounds can be used to measure the influence of the external damping on the part of the spectrum which is of interest. This bound can be used if one wants to assure that the damping will influence on the one particular part of the spectrum, or if one wants to remove some eigenvalues from the certain region.

We will present new performance measures for vibrational systems based on the H2 norm of linear control systems. Examples, both theoretical and concrete, will be given showing how these perfor- mance measures stack up against standard ones when used as an optimization criterion for the optimal damping of vibrational systems.

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We consider a parametric linear time invariant dynamical systems represented as \begin{align*} E \dot x(t) &= A(p)x(t) + Bu(t),\\ y(t) &= Cx(t), \end{align*} where $ E, A(p) \in \mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m} $ and $C\in \mathbb{R}^{l\times n}$. We assume that $A(p)$ depends on $k\ll n$ parameters $p=(p_1, p_2, \ldots, p_k)$ such that $A(p)=A_0+U\, \rm{diag} (p_1, p_2, \ldots, p_k)V^T,$ where $U, V \in \mathbb{R}^{n\times k}$ are given fixed matrices. We propose an approach for approximating the full-order transfer function with a reduced-order model that retains the structure of parametric dependence and (typically) offers uniformly high fidelity across the full parameter range. Remarkably, the proposed reduction process removes the need for parameter sampling and thus does not depend on identifying particular parameter values of interest. In our approach the Sherman-Morrison-Woodbury formula allows us to construct a parameterized reduced order model from transfer functions of four subsystems that do not depend on parameters. In this form one can apply well-established model reduction techniques for non-parametric systems. The overall process is well suited for computationally efficient parameter optimization and the study of important system properties. One of the main applications of our approach is for damping optimization where parameters represent viscosities. The main problem is to determine the best damping matrix that is able to minimize influence of the disturbances, on the output of the system. We use a minimization criteria based on the $\mathcal{H}_2$ system norm. In realistic settings, damping optimization is a very demanding problem. We find that the parametric model reduction approach described here offers a new tool with significant advantages for the efficient optimization of damping in such problems. This is joint work with Christopher Beattie and Serkan Gugercin from Virginia Tech, Blacksburg, USA.

We consider a quadratic eigenvalue problem (QEP): $$ (\lambda^2(v)M(v)+\lambda(v) D(v) + K(v))x(v)=0, $$ where matrices $M$ and $K$ are Hermitian semidefinite and at least one of them is positive definite, and also $M, D$ and $K$ depend on $v=[v_1,\dots, v_s]\in \mathbb{R}^s$. The most widely used approach for solving the polynomial (which includes QEP) eigenvalue problem is to linearize in order to produce a larger order pencil, whose eigensystem can be found by any method for generalized eigenproblems. To avoid lineatization, we propose two different types of bounds, the first is a simple first order approximation of function of several variables while the second one considers structured perturbation. The first bound is the upper bound for the first order approximation, based on Taylor's theorem, for the eigenvalues and the corresponding left and right eigenvectors of the QEP. The second bound contains bound for relative gaps that arise in perturbation theory. This is joint work with Ninoslav Truhar and Zoran Tomljanović.

We present new relative perturbation bounds for the eigenvalues and eigensubspaces for quadratic eigenvalue problem $\lambda^2 M x + \lambda C x + Kx=0$, where $M$ and $K$ are nonsingular Hermitian and $C$ is any Hermitian matrix. First, we derive relative perturbation bounds for the eigenvalues and the $\sin \Theta$ type theorems for the eigensubspaces of the regular matrix pairs $(A,B)$, where both $A$ and $B$ are Hermitian matrices. Using a proper linearization and new relative perturbation bounds for regular 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 regular quadratic eigenvalue problem. Our bound can be applied to the gyroscopic systems which will be also shown. The obtained bounds will be illustrated by numerical examples.