Toward a High Performance Tile Divide and Conquer Algorithm for the Dense Symmetric Eigenvalue Problem

TL;DR: The Eigenvalue soLvers for Petascale Applications (ELPA) library is focused on, which facilitates the efficient algebraic solution of symmetric and Hermitian eigenvalue problems for dense matrices that have real-valued and complex-valued matrix entries, respectively, on parallel computer platforms.

TL;DR: A data-sparse approximation based on hierarchical matrices can be used to overcome the presence of small relative spectral gaps in the approximate computation of spectral projectors for symmetric banded matrices and is proved a priori bounds on the approximation error.

TL;DR: A data-sparse approximation based on hierarchical matrices can be used to overcome the presence of small relative spectral gaps in the approximate computation of spectral projectors for symmetric banded matrices and is proved a priori bounds on the approximation error.

TL;DR: This work aims to compare different direct linear solvers that can exploit the block-tridiagonal structure and develops a parallel implementation based on MPI in the context of the SLEPc library.

TL;DR: The third edition of LAPACK provided a guide to troubleshooting and installation of Routines, as well as providing examples of how to convert from LINPACK or EISPACK to BLAS.

Abstract: Exercise 1 (4 points) (a) Let unitary matrices Q 1 ,. .. , Q k ∈ C m×m be fixed and consider the problem of computing, for A ∈ C m×n , the product B = Q k · · · Q 1 A. Let the computation be carried out on a computer satisfying axioms (A1) and (A2). Show that this algorithm is backward stable. (Here A is thought of as data that can be perturbed; the matrices Q j are fixed and not to be perturbed.) (b) Give an example to show that this result no longer holds if the unitary matrices Q j are replaced with arbitrary matrices X j ∈ C m×m. Recall the axioms from exercise sheet 9, exercise 2: (A1) For all x ∈ R, there exists with || ≤ machine such that fl(x) = x(1 +). (A2) For all x, y ∈ F, there exists with || ≤ machine such that x y = (x * y)(1 +). Here * denotes an arithmetic operation and the floating point equivalent. Exercise 2 (4 points) Consider the example A =   1 1 1 1.0001 1 1.0001   , b =   2 0.0001 4.0001  . Do the computations in part (a) and (b) by hand. Show intermediate results where appropriate. Give exact results. From part (c) on, numerical results computed by MATLAB are acceptable. (a) What are the matrices A + and P = AA + for this example? Hint: Compute A + by solving the linear system (A * A)A + = A *. (b) Find the exact solutions x and y = Ax for the least squares problem Ax ≈ b. (c) What are κ(A), θ, and η? (d) What are the four relative condition numbers? (e) Give examples of perturbations δb and δA that approximately attain these four condition numbers.

TL;DR: In this paper, the Kohn-Sham ansatz is used to solve the problem of determining the electronic structure of atoms, and the three basic methods for determining electronic structure are presented.