Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set

CHARMM (Chemistry at HARvard Macromolecular Mechanics) is a highly flexible computer program which uses empirical energy functions to model macromolecular systems. The program can read or model build structures, energy minimize them by first- or second-derivative techniques, perform a normal mode or molecular dynamics simulation, and analyze the structural, equilibrium, and dynamic properties determined in these calculations. The operations that CHARMM can perform are described, and some implementation details are given. A set of parameters for the empirical energy function and a sample run are included.

CHARMM: A program for macromolecular energy, minimization, and dynamics calculations

https://www2.econ.iastate.edu/tesfatsi/DeepLearningInNeuralNetworksOverview.JSchmidhuber2015.pdf

Deep learning in neural networks

Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

/pdf/iterative-methods-for-sparse-linear-systems-34ozhpwq89.pdf

Iterative Methods for Sparse Linear Systems

A Scaling Method for Priorities in Hierarchical Structures

Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography Index.

The algebraic eigenvalue problem

According to Parlett, 'Vibrations are everywhere, and so too are the eigenvalues associated with them. As mathematical models invade more and more disciplines, we can anticipate a demand for eigenvalue calculations in an ever richer variety of contexts.' Anyone who performs these calculations will welcome the reprinting of Parlett's book (originally published in 1980). In this unabridged, amended version, Parlett covers aspects of the problem that are not easily found elsewhere. The chapter titles convey the scope of the material succinctly. The aim of the book is to present mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few. The author explains why the selected information really matters and he is not shy about making judgments. The commentary is lively but the proofs are terse.

The Symmetric Eigenvalue Problem.

The Algebraic Eigenvalue Problem

/pdf/rounding-errors-in-algebraic-processes-3vx7l0y3ad.pdf

Rounding Errors in Algebraic Processes

1. In order to assess the relative effectiveness of methods of inverting a matrix it is useful to have a priori bounds for the errors in the computed inverses. In this paper we determine such error bounds for a number of the most effective direct methods. To illustrate fully the techniques we have used, some of the analysis has been done for floating-point computat ion and some for fixed-point. In all cases it has been assumed tha t the computat ion has been performed using a precision of t binary places, though it should be appreciated tha t on a computer which has both fixed and floating-point facilities the number of permissible digits in a fixed-point number is greater than the number of digits in the mantissa of a floating-point number. The techniques used for analyzing floating-point computat ion are essentially those of [8], and a familiarity with tha t paper is assumed. 2. The error bounds are most conveniently expressed in terms of vector and matr ix norms, and throughout we have used the Euclidean vector norm and the spectral matrix norm except when explicit reference is made to the contrary. For convenience the main properties of these norms are given in Section 9. In a recent paper [7] we analyzed the effect of the rounding errors made in the solution of the equations

James Hardy Wilkinson

Papers

The algebraic eigenvalue problem

The Symmetric Eigenvalue Problem.

The Algebraic Eigenvalue Problem

Rounding Errors in Algebraic Processes

Error Analysis of Direct Methods of Matrix Inversion