We are concerned with the computation of the L∞-norm for an L∞-function of the form H(s) = C(s)D(s) −1 B(s), where the middle factor is the inverse of a meromorphic matrix-valued function, and C(s), B(s) are meromorphic functions mapping to shortand-fat and tall-and-skinny matrices, respectively. For instance, transfer functions of descriptor systems and delay systems fall into this family. We focus on the case where the middle factor is large-scale. We propose a subspace projection method to obtain approximations of the function H where the middle factor is of much smaller dimension. The L∞-norms are computed for the resulting reduced functions, then the subspaces are refined by means of the optimal points on the imaginary axis where the L∞-norm of the reduced function is attained. The subspace method is designed so that certain Hermite interpolation properties hold between the largest singular values of the original and reduced functions. This leads to a locally superlinearly convergent algorithm with respect to the subspace dimension, which we prove and illustrate on various numerical examples.

We deal with the minimization of the H∞-norm of the transfer function of a parameterdependent descriptor system over the set of admissible parameter values. Subspace frameworks are proposed for such minimization problems where the involved systems are of large order. The proposed algorithms are greedy interpolatary approaches inspired by our recent work [

A linear time-invariant dissipative Hamiltonian (DH) systemẋ = (J − R)Qx, with a skew-Hermitian J, an Hermitian positive semi-definite R, and an Hermitian positive definite Q, is always Lyapunov stable and under weak further conditions even asymptotically stable. In various applications there is uncertainty on the system matrices J, R, Q, and it is desirable to know whether the system remains asymptotically stable uniformly against all possible uncertainties within a given perturbation set. Such robust stability considerations motivate the concept of stability radius for DH systems, i.e., what is the maximal perturbation permissible to the coefficients J, R, Q, while preserving the asymptotic stability. We consider two stability radii, the unstructured one where J, R, Q are subject to unstructured perturbation, and the structured one where the perturbations preserve the DH structure. We employ characterizations for these radii that have been derived recently in [SIAM J. Matrix Anal. Appl., 37, pp. 2016 ] and propose new algorithms to compute these stability radii for large scale problems by tailoring subspace frameworks that are interpolatory and guaranteed to converge at a super-linear rate in theory. At every iteration, they first solve a reduced problem and then expand the subspaces in order to attain certain Hermite interpolation properties between the full and reduced problems. The reduced problems are solved by means of the adaptations of existing level-set algorithms for H∞-norm computation in the unstructured case, while, for the structured radii, we benefit from algorithms that approximate the objective eigenvalue function with a piece-wise quadratic global underestimator. The performance of the new approaches is illustrated with several examples including a system that arises from a finite-element modeling of an industrial disk brake.

We consider the computation of the L∞-norm for a general class of L∞-functions and focus on the case where the function is represented in terms of large-scale matrix-valued factors. We propose a subspace projection method to obtain reduced approximations of this function by interpolation techniques. The L∞-norms are computed for the resulting reduced functions, then the subspaces are refined by means of the optimizer of the L∞-norm of the reduced function. In this way we obtain much better performance compared to existing methods. Problem StatementIn this paper we consider functions of the formwhere Ω ⊆ C is assumed to be an open domain enclosing the imaginary axis iR. Moreover, we assume that the matrix-valuedfor given matrices B 1 , . . . ,and given functions f 1 , . . . , f κ B , g 1 , . . . , g κ C , h 1 , . . . , h κ D : Ω → C that are meromorphic in Ω.In this paper we suppose the function H (more precisely, its restriction to the imaginary axis) to be an element of the normed spacewhere σ max (·) denotes the maximum singular value of its matrix argument. In this paper we address the computation of the L ∞ -norm with an emphasis on the case where n is large with n m, p and thus the evaluation of H(s) is expensive. The MethodOur approach makes use of ideas from the field of model order reduction, i. e., instead of computing the L ∞ -norm of H directly, we construct reduced functions of the form