On pole-swapping algorithms for the eigenvalue problem

Pole-swapping algorithms, which are generalizations of the QZ algorithm for the generalized eigenvalue problem, are studied. A new modular (and therefore more flexible) convergence theory that applies to all pole-swapping algorithms is developed. A key component of all such algorithms is a procedure that swaps two adjacent eigenvalues in a triangular pencil. An improved swapping routine is developed, and its superiority over existing methods is demonstrated by a backward error analysis and numerical tests. The modularity of the new convergence theory and the generality of the pole-swapping approach shed new light on bi-directional chasing algorithms, optimally packed shifts, and bulge pencils, and allow the design of novel algorithms

Yet another algorithm for the symmetric eigenvalue problem

In this paper we present a new algorithm for solving the symmetric matrix eigenvalue problem that works by first using a Cayley transformation to convert the symmetric matrix into a unitary one and then uses Gragg’s implicitly shifted unitary QR algorithm to solve the resulting unitary eigenvalue problem. We prove that under reasonable assumptions on the symmetric matrix this algorithm is backward stable and also demonstrate that this algorithm is comparable with other well known implementations in terms of both speed and accuracy

Fast and stable unitary QR algorithm

A fast Fortran implementation of a variant of Gragg's unitary Hessenberg QR algorithm is presented. It is proved, moreover, that all QR- And QZ-like algorithms for the unitary eigenvalue problems are equivalent. The algorithm is backward stable. Numerical experiments are presented that confirm the backward stability and compare the speed and accuracy of this algorithm with other methods

Fast and backward stable computation of roots of polynomials

A stable algorithm to compute the roots of polynomials is presented. The roots are found by computing the eigenvalues of the associated companion matrix by Francis's implicitly shifted QR algorithm. A companion matrix is an upper Hessenberg matrix that is unitary-plus-rankone, that is, it is the sum of a unitary matrix and a rank-one matrix. These properties are preserved by iterations of Francis's algorithm, and it is these properties that are exploited here. The matrix is represented as a product of 3n - 1 Givens rotators plus the rank-one part, so only O(n) storage space is required. In fact, the information about the rank-one part is also encoded in the rotators, so it is not necessary to store the rank-one part explicitly. Francis's algorithm implemented on this representation requires only O(n) flops per iteration and thus O(n2) flops overall. The algorithm is described, normwise backward stability is proved, and an extensive set of numerical experiments is presented. The algorithm is shown to be about as accurate as the (slow) Francis QR algorithm applied to the companion matrix without exploiting the structure. It is faster than other fast methods that have been proposed, and its accuracy is comparable or better

Fast and backward stable computation of eigenvalues and eigenvectors of matrix polynomials

In the last decade matrix polynomials have been investigated with the primary focus on adequate linearizations and good scaling techniques for computing their eigenvalues and eigenvectors. In this article we propose a new method for computing a factored Schur form of the associated companion pencil. The algorithm has a quadratic cost in the degree of the polynomial and a cubic one in the size of the coefficient matrices. Also the eigenvectors can be computed at the same cost. The algorithm is a variant of Francis's implicitly shifted QR algorithm applied on the companion pencil. A preprocessing unitary equivalence is executed on the matrix polynomial to simultaneously bring the leading matrix coefficient and the constant matrix term to triangular form before forming the companion pencil. The resulting structure allows us to stably factor each matrix of the pencil as a product of k matrices of unitary-plus-rank-one form, admitting cheap and numerically reliable storage. The problem is then solved as a product core chasing eigenvalue problem. A backward error analysis is included, implying normwise backward stability after a proper scaling. Computing the eigenvectors via reordering the Schur form is discussed as well. Numerical experiments illustrate stability and efficiency of the proposed methods

Roots of Polynomials: on twisted QR methods for companion matrices and pencils

Two generalizations of the companion QR algorithm by J.L. Aurentz, T. Mach, R. Vandebril, and D.S. Watkins, SIAM Journal on Matrix Analysis and Applications, 36(3): 942--973, 2015, to compute the roots of a polynomial are presented. First, we will show how the fast and backward stable QR algorithm for companion matrices can be generalized to a QZ algorithm for companion pencils. Companion pencils admit a greater flexibility in scaling the polynomial and distributing the matrix coefficients over both matrices in the pencil. This allows for an enhanced stability for polynomials with largely varying coefficients. Second, we will generalize the pencil approach further to a twisted QZ algorithm. Whereas in the classical QZ case Krylov spaces govern the convergence, the convergence of the twisted case is determined by a rational Krylov space. A backward error analysis to map the error back to the original pencil and to the polynomial coefficients shows that in both cases the error scales quadratically with the input. An extensive set of numerical experiments supports the theoretical backward error, confirms the numerical stability and shows that the computing time depends quadratically on the problem size

The Decay of Accreting Triple Systems as Brown Dwarf Formation Scenario

We investigate the dynamical decay of non-hierarchical accreting triple systems and its implications on the ejection model as Brown Dwarf formation scenario. A modified chain-regularization scheme is used to integrate the equations of motion, that also allows for mass changes over time as well as for momentum transfer from the accreted gas mass onto the bodies. We integrate an ensemble of triple systems within a certain volume with different accretion rates, assuming several prescriptions of how momentum is transferred onto the bodies. We follow their evolution until the systems have decayed. We analyze the end states and decay times of these systems and determine the fraction of Brown Dwarfs formed, their escape speeds as well as the semi-major axis distribution of the formed Brown Dwarf binaries. We find that the formation probability of Brown Dwarfs depends strongly on the assumed momentum transfer which is related to the motion of the gas. Due to ongoing accretion and consequent shrinkage of the systems, the median escape velocity is increased by a factor of 2 and the binary separations are decreased by a factor of 5 compared with non-accreting systems. Furthermore, the obtained semi-major axis distribution drops off sharply to either side of the median, which is also supported by observations. We conclude that accretion and momentum transfer of accreted gas during the dynamical decay of triple systems is able to produce the observed distribution of close binary Brown Dwarfs, making the ejection model a viable option as Brown Dwarf formation scenario.Comment: 31 pages, 8 figures, accepted for publication in Ap

Bubble, Bubble, Flow and Hubble: Large Scale Galaxy Flow from Cosmological Bubble Collisions

We study large scale structure in the cosmology of Coleman-de Luccia bubble collisions. Within a set of controlled approximations we calculate the effects on galaxy motion seen from inside a bubble which has undergone such a collision. We find that generically bubble collisions lead to a coherent bulk flow of galaxies on some part of our sky, the details of which depend on the initial conditions of the collision and redshift to the galaxy in question. With other parameters held fixed the effects weaken as the amount of inflation inside our bubble grows, but can produce measurable flows past the number of efolds required to solve the flatness and horizon problems.Comment: 30 pages, 8 figures, pdftex, minor corrections and references adde

First-principles calculation of intrinsic defect formation volumes in silicon

We present an extensive first-principles study of the pressure dependence of the formation enthalpies of all the know vacancy and self-interstitial configurations in silicon, in each charge state from -2 through +2. The neutral vacancy is found to have a formation volume that varies markedly with pressure, leading to a remarkably large negative value (-0.68 atomic volumes) for the zero-pressure formation volume of a Frenkel pair (V + I). The interaction of volume and charge was examined, leading to pressure--Fermi level stability diagrams of the defects. Finally, we quantify the anisotropic nature of the lattice relaxation around the neutral defects.Comment: 9 pages, 9 figure