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We investigate the numerical solution to a low rank perturbed Lyapunov equation A T X + XA = W via the sign function method (SFM). The sign function method has been proposed to solve Lyapunov equations, see e.g. [1], but here we focus on a framework where the matrix A has a special structure, i.e. A = B + U CV T , where B is a blockdiagonal matrix and U CV T is a low rank perturbation. We show that this structure can be kept throughout the sign function iteration but the rank of the perturbation doubles per iteration. Therefore, we apply a low rank approximation to the perturbation in order to keep its numerical rank small. We compare the standard SFM with its structure preserving variant presented in this paper by means of numerical examples from viscously damped mechanical systems. The algebraic Lyapunov equationwhere A, W = W T ∈ R n×n with its solution X ∈ R n×n plays a fundamental role e.g. in the stability analysis of linear systems. We focus on a structured algebraic Lyapunov equation, where , c k being some scaling for k = 0, 1, . . . with the solution X = 1 2 lim k→∞ W k [1]. The given structure can be exploited by this iteration as the following theorem shows.r×r and r n. Then the next sign function iterate A k+1 can be expressed as, whereHence, A k . In the k-th step of the sign function method, the complexity of computing A −1 k can be reduced from O(n 3 ) to O(r 3 ) by using its structure. Nevertheless, the complexity grows per sign function iteration since the rank r of the perturbation doubles per iteration (cf. Theorem 1.1). In order to save workspace and reduce the complexity, we propose to perform rank revealing QR factorizations (RR-QR): Q U R U = U k+1 and Q V R V = V k+1 and a truncated SVDÛΣV T ≈ R U C k+1 R T V w.r.t. some predefined truncation tolerance ε > 0, i.e.,Σ = diag(σ 1 , . . . , σ p ), where σ i ≥ ε for i = 1, . . . , p. This results in Algorithm 1.

Linear time-periodic systems have been an active area of research in the last decades. They arise in various applications such as anisotropic rotor-bearing systems and nonlinear systems linearized about a periodic trajectory. Rigorous bounds support the transient analysis of these systems. Optimal constants are determined by the differential calculus for norms of matrix functions. Bounds based on trigonometric spline approximations of the solution are introduced and convergence results for the approximations are stated. Bounds are illustrated by means of an anisotropic rotor-bearing system. Linear time-periodic systemsWe are considering a set of linear ordinary differential equations (ODEs) with time-periodic coefficients with periodicity T and a given initial condition as a linear time-periodic systeṁwhere x ∈ R n and A : R → R n×n . Under well known assumptions a unique solution to (1) is guaranteed. The solution can then be represented by the following theorem due to Floquet [1], but in a different form.Theorem 1.1 A fundamental matrix Φ(t) of (1) can be represented aswhere R, Z(t) ∈ C n×n and Z(t) = Z(t + T ) is nonsingular ∀t ∈ R.The solution x(t) is a linear combination of n linearly independent solutions w.r.t. the initial condition x 0 : x(t) = Φ(t)x 0 = Z(t)e Rt x 0 with Φ(0) = I. Determining Φ(t) in the interval [0, T ] is sufficient due to the semigroup property of the solution. For the sake of simplicity we consider equidistant nodes t i = ih for i = −1, 0, 1, . . . , r with h = T r and we use quadratic trigonometric splines T i (t) [2] to approximate the solution. . .whereT denotes a vector of the i-th coefficients α (j)i of the spline approximation s j (t) for j = 1, . . . , n. Quadratic trigonometric splines with compact support are chosen in order to simplify the computation of the coefficientswhere θ = 1 sin h sin h 2 . Solving ODEs by spline approximations has been investigated by F. R. Loscalzo and T. D. Talbot [3] and many others, e.g. A. Nikolis [4]. The focus on trigonometric splines is due to the periodicity of (1) and furthermore, we will equip the computation with rigorous upper and lower bounds on the solution. Demanding that the approximation s fulfills (1) at the nodes t i , yields a series of r + 1 linear systems A

Linear time-periodic systems arise whenever a nonlinear system is linearized about a periodic trajectory. Examples include anisotropic rotor-bearing systems and parametrically excited systems. The structure of the solution to linear time-periodic systems is known due to Floquet's Theorem. We use this information to derive a new norm which yields two-sided bounds on the solution and in this norm vibrations of the solution are suppressed. The obtained results are a generalization for linear time-invariant systems. Since Floquet's Theorem is non-constructive, the applicability of the aforementioned results suffers in general from an unknown Floquet normal form. Hence, we discuss trigonometric splines and spectral methods that are both equipped with rigorous bounds on the solution. The methodology differs systematically for the two methods. While in the first method the solution is approximated by trigonometric splines and the upper bound depends on the approximation quality, in the second method the linear time-periodic system is approximated and its solution is represented as an infinite series. Depending on the smoothness of the time-periodic system, we formulate two upper bounds which incorporate the approximation error of the linear time-periodic system and the truncation error of the series representation. Rigorous bounds on the solution are necessary whenever reliable results are needed, and hence they can support the analysis and, e.g., stability or robustness of the solution may be proven or falsified. The theoretical results are illustrated and compared to trigonometric spline bounds and spectral bounds by means of three examples that include an anisotropic rotor-bearing system and a parametrically excited Cantilever beam.