Non-Hermitian band theory distinguishes between line gaps and point gaps. While can give rise to intrinsic non-Hermitian topology without Hermitian counterparts, line-gapped systems always be adiabatically deformed a limit. Here we show that line-gap point-gap intricately connected: topological in $d$ dimensions induce nontrivial on their $(d-1)$-dimensional boundaries when suitable internal sp...

This paper explores the early implementation of highperformance routines for the solution of multiple large Hermitian eigenvector and eigenvalue systems on a Graphics Processing Unit (GPU). We report a performance increase of up to two orders of magnitude over the original EISPACK routines with a NVIDIA Tesla C2050 GPU, potentially allowing an order of magnitude increase in the complexity or re...

Algebraic multigrid methods solve sparse linear systems Ax = b by automatic construction of a multilevel hierarchy. This hierarchy is defined by grid transfer operators that must accurately capture algebraically smooth error relative to the relaxation method. We propose a methodology to improve grid transfers through energy minimization. The proposed strategy is applicable to Hermitian, non-Her...

We study inexact Rayleigh quotient iteration (IRQI) for computing a simple interior eigenpair of the generalized eigenvalue problem Av = λBv, providing new insights into a special type of preconditioners with “tuning” for the efficient iterative solution of the shifted linear systems that arise in this algorithm. We first give a new convergence analysis of IRQI, showing that locally cubic and q...

Conventional convergence acceleration techniques for the SCF procedure like the direct inversion in the iterative subspace and the level-shifting can not be applied in the general case of non-Hermitian Fock operators. The commutator of the Fock matrix and the density matrix do not vanish upon convergence of nonHermitian SCF equations and hence the usual error vector choice is not applicable. In...

The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobi...

We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields over Q( √ −m) for all m. For each imaginary quadratic field Q( √ −m), we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice L represents 1, 2, 3, 5, 6, 7, 10, 13, 14 and 15, then L is universal. We call this the fifteen theorem for universal Hermitian lattices. Note t...

The loop equations for a chain of hermitian random matrices are computed explicitely, including the 1/N2 corrections. To leading order, the master loop equation reduces to an algebraic equation, whose solution can be written in terms of geometric properties of the underlying algebraic curve. In particular we compute the free energy, the resolvents, the 2-loop functions and some mixed one loop f...

The loop equations for a chain of hermitian random matrices are computed explicitely, including the 1/N 2 corrections. To leading order, the master loop equation reduces to an algebraic equation, whose solution can be written in terms of geometric properties of the underlying algebraic curve. In particular we compute the free energy, the resolvents, the 2-loop functions and some mixed one loop ...

We discuss how concepts such as geodesic length and the volume of space-time can appear in 2d topological gravity. We then construct a detailed mapping between the reduced Hermitian matrix model and 2d topological gravity at genus zero. This leads to a complete solution of the counting problem for planar graphs with vertices of even coordination number. The connection between multi-critical mat...