In this manuscript a new type of QR-iteration will be presented. Each step of this new iteration consists of two substeps. In the explicit version, first an RQ-factorization of the initial matrix A−κI = RQ will be computed, followed by a QR-factorization of the matrix (A−σI)QH . Applying the unitary similarity transformation defined by the QR-factorization of the transformed matrix (A−σI)QH , w...

Abstract. This paper presents a modification of Krylov subspace spectral (KSS) methods, which build on the work of Golub, Meurant and others, pertaining to moments and Gaussian quadrature to produce high-order accurate approximate solutions to variable-coefficient time-dependent PDEs. Whereas KSS methods currently use Lanczos iteration to compute the needed quadrature rules, our modification us...

We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigenspaces of a symmetric positive definite operator A defined on a finite dimensional real Hilbert space V . In our applications, the dimension of V is large and the cost of inverting A is prohibitive. In this paper, we shall develop an effective parallelizable te...

A semidefinite program (SDP) is an optimization problem over n × n symmetric matrices where a linear function of the entries is to be minimized subject to linear equality constraints, and the condition that the unknown matrix is positive semidefinite. Standard techniques for solving SDP’s require O(n) operations per iteration. We introduce subspace algorithms that greatly reduce the cost os sol...

We consider the numerical solution of the continuous algebraic Riccati equation AX +XA−XFX +G = 0, with F = F , G = G of low rank and A large and sparse. We develop an algorithm for the low rank approximation of X by means of an invariant subspace iteration on a function of the associated Hamiltonian matrix. We show that the sought after approximation can be obtained by a low rank update, in th...

The calculation of a few interior eigenvalues of a matrix has not received much attention in the past, most methods being some spin-off of either the complete eigenvalue calculation or a subspace method designed for the extremal part of the spectrum. The reason for this could be the rather chaotic behaviour of most methods tried. Only 'shift and invert' and polynomial iteration seemed to have a...

A new direct constrained optimization algorithm for minimizing the Kohn-Sham (KS) total energy functional is presented in this paper. The key ingredients of this algorithm involve projecting the total energy functional into a sequences of subspaces of small dimensions and seeking the minimizer of total energy functional within each subspace. The minimizer of a subspace energy functional not onl...

We compare two approaches to compute a fraction of the spectrum of dense symmetric definite generalized eigenproblems: one is based on the reduction to tridiagonal form, and the other on the Krylov-subspace iteration. Two large-scale applications, arising in molecular dynamics and material science, are employed to investigate the contributions of the application, architecture, and parallelism o...