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We propose a fast algorithm for computing optimal viscosities of dampers of a linear vibrational system. We are using a standard approach where the vibrational system is first modeled using the second-order structure. This structure yields a quadratic eigenvalue problem which is then linearized. Optimal viscosities are those for which the trace of the solution of the Lyapunov equation with the linearized matrix is minimal. Here, the free term of the Lyapunov equation is a low-rank matrix that depends on the eigenfrequencies that need to be damped. The optimization process in the standard approach requires O(n3) floating-point operations. In our approach, we transform the linearized matrix into an eigenvalue problem of a diagonal-plus-low-rank matrix whose eigenvectors have a Cauchy-like structure. Our algorithm is based on a new fast eigensolver for complex symmetric diagonal-plus-rank-one matrices and fast multiplication of linked Cauchy-like matrices, yielding computation of optimal viscosities for each choice of external dampers in O(kn2) operations, k being the number of dampers. The accuracy of our algorithm is compatible with the accuracy of the standard approach.

MSC: 65F15 65G50 15-04 15B99 Keywords: Eigenvalue decomposition Diagonal-plus-rank-one matrix Real symmetric matrix Arrowhead matrix High relative accuracy Forward stabilityWe present a new algorithm for solving an eigenvalue problem for a real symmetric matrix which is a rank-one modification of a diagonal matrix. The algorithm computes each eigenvalue and all components of the corresponding eigenvector with high relative accuracy in O(n) operations. The algorithm is based on a shift-and-invert approach. Only a single element of the inverse of the shifted matrix eventually needs to be computed with double the working precision. Each eigenvalue and the corresponding eigenvector can be computed separately, which makes the algorithm adaptable for parallel computing. Our results extend to the complex Hermitian case. The algorithm is similar to the algorithm for solving the eigenvalue problem for real symmetric arrowhead matrices from N. Jakovčević Stor et al. (2015) [16].

We consider frequency-weighted damping optimization for vibrating systems described by a second-order differential equation. The goal is to determine viscosity values such that eigenvalues are kept away from certain undesirable areas on the imaginary axis. To this end, we present two complementary techniques. First, we propose new frameworks using nonsmooth constrained optimization problems, whose solutions both damp undesirable frequency bands and maintain stability of the system. These frameworks also allow us to weight which frequency bands are the most important to damp. Second, we also propose a fast new eigensolver for the structured quadratic eigenvalue problems that appear in such vibrating systems. In order to be efficient, our new eigensolver exploits special properties of diagonal-plus-rank-one complex symmetric matrices, which we leverage by showing how each quadratic eigenvalue problem can be transformed into a short sequence of such linear eigenvalue problems. The result is an eigensolver that is substantially faster than standard techniques. By combining this new solver with our new optimization frameworks, we obtain our overall algorithm for fast computation of optimal viscosities. The efficiency and performance of our new methods are verified and illustrated on several numerical examples.

We discuss the eigenproblem for the symmetric arrowhead matrixwhere D ∈ R n×n is diagonal, z ∈ R n , and α ∈ R in order to examine criteria for when components of z may be set to zero. We show that whenever two eigenvalues of C are sufficiently close, some component of z may be deflated to zero, without significantly perturbing the eigenvalues of C, by either substituting zero for that component or performing a Givens rotation on each side of C. The strategy for this deflation requires O(n 2 ) comparisons. Although it is too costly for many applications, when we use it as a benchmark, we can analyze the effectiveness of O(n) heuristics that are more practical approaches to deflation. We show that one such O(n) heuristic finds all sets of three or more nearby eigenvalues, misses sets of two or more nearby eigenvalues under limited circumstances, and produces a reduced matrix whose eigenvalues are distinct in double the working precision. Using the O(n) heuristic, we develop a more aggressive method for finding converged eigenvalues in the symmetric Lanczos algorithm. It is shown that except for pathological exceptions, the O(n) heuristic finds nearly as much deflation as the O(n 2 ) algorithm that reduces an arrowhead matrix to one that cannot be deflated further. The deflation algorithms and their analysis are extended to the symmetric diagonal-plus-rank-one eigenvalue problem and lead to a better deflation strategy for the LAPACK routine dstedc.f.