Büyük ölçekli kararlilik yariçapi hesaplamalari

The stability radius of a matrix is the distance from the matrix to a nearest matrix with an eigenvalue on the closed right half of the complex plane with respect to the matrix 2-norm. It also corresponds to a distance from the associated continuous-time linear control system to the set of systems that are not asymptotically stable, and can be posed as a singular value optimization problem.In this thesis, our primary interest is the large-scale complex stability-radii problems. In the first part, we propose a subspace framework for the complex stability radius of a large matrix. The subspace framework operates on the singular value optimization characterization. In particular, the state space of the control system is restricted to a small subspace at every iteration. The resulting singular value optimization problem is much smaller than the original one. As a result, it can be solved numerically by means of level-set methods that converge quickly at a quadratic rate. We describe how the existing Boyd-Balakrishnan and Bruinsma-Steinbuch algorithms can be modified for this purpose. After solving such a reduced singular value optimization problem at every iteration, we expand the restriction subspace by including certain singular vectors of the original problem so as to ensure the Hermite interpolation properties between the reduced and original problem at the optimizer of the reduced problem. We formally argue why this subspace framework converges at a super-linear rate. One important issue with the subspace framework proposed for estimating the stability radius is that it converges locally. In the second part, to avoid convergence to a local optimizer that is not a global optimizer, we propose to use the eigenvalues of the matrix closest to the imaginary axis as the initial interpolation points for the subspace framework introduced in the first part. This initialization is motivated by the observation that the global optimizers of the singular value optimization problem are very likely to be located near the imaginary parts of the eigenvalues closest to the imaginary axis. We estimate these eigenvalues by employing another derivative interpolating subspace framework recently tailored for large-scale nonlinear eigenvalue problems by Aziz et al. In this subspace approach for eigenvalue problems, two-sided Petrov-Galerkin projections are employed leading to a small generalized eigenvalue problem such that interpolation properties hold between the original large-scale and small generalized eigenvalue problems. All of the eigenvalues of the small problem can be computed at ease for instance by means of the QZ algorithm. Then, again the projection subspaces are expanded at every iteration so that Hermite interpolation properties hold between the original and the small problem at the eigenvalues of the small problem closest to the imaginary axis. This framework is guaranteed to converge at least quadratically, which means that the original large-scale eigenvalue problem is approximated by a small eigenvalue problem with nearly the same eigenvalues near the imaginary axis. Putting these ingredients together, we obtain an efficient approach that most often approximates the stability radius of large matrices well, which we confirm by numerical experiments on several sparse matrices.