Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices
SIAM Journal on Scientific and Statistical Computing
On the limited memory BFGS method for large scale optimization
Mathematical Programming: Series A and B
Efficient generalized conjugate gradient algorithms, Part 1: theory
Journal of Optimization Theory and Applications
Multilevel filtering preconditioners: extensions to more general elliptic problems
SIAM Journal on Scientific and Statistical Computing - Special issue on iterative methods in numerical linear algebra
TNPACK—A truncated Newton minimization package for large-scale problems: I. Algorithm and usage
ACM Transactions on Mathematical Software (TOMS)
TNPACK—a truncated Newton minimization package for large-scale problems: II. Implementation examples
ACM Transactions on Mathematical Software (TOMS)
Techniques for geometry optimization: a comparison of Cartesian and natural internal coordinates
Journal of Computational Chemistry
Updated Hessian matrix and the restricted step method for locating transition structures
Journal of Computational Chemistry
A truncated Newton minimizer adapted for CHARMM and biomolecular applications
Journal of Computational Chemistry
Line search algorithms with guaranteed sufficient decrease
ACM Transactions on Mathematical Software (TOMS)
A Sparse Approximate Inverse Preconditioner for the Conjugate Gradient Method
SIAM Journal on Scientific Computing
Matrix computations (3rd ed.)
The symmetric eigenvalue problem
The symmetric eigenvalue problem
The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
Unconstrained energy functionals for electronic structure calculations
Journal of Computational Physics
New advances in chemistry and materials science with CPMD and parallel computing
Parallel Computing - computational chemistry
Trust-region methods
A survey of truncated-Newton methods
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. IV: optimization and nonlinear equations
A Nonlinear Conjugate Gradient Method with a Strong Global Convergence Property
SIAM Journal on Optimization
Enriched Methods for Large-Scale Unconstrained Optimization
Computational Optimization and Applications
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
Improved algorithm for geometry optimisation using damped molecular dynamics
Journal of Computational Physics
A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search
SIAM Journal on Optimization
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent
ACM Transactions on Mathematical Software (TOMS)
Computer simulations for organic light-emitting diodes
IBM Journal of Research and Development
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The goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems. The paper is intended for a diverse scientific computing audience. For this reason, we assume the reader does not have an extensive background in the related physics. Our overview focuses on the nature of the numerical problems to be solved, their origin, and the methods used to solve the resulting linear algebra or nonlinear optimization problems. It is common knowledge that the behavior of matter at the nanoscale is, in principle, entirely determined by the Schrödinger equation. In practice, this equation in its original form is not tractable. Successful but approximate versions of this equation, which allow one to study nontrivial systems, took about five or six decades to develop. In particular, the last two decades saw a flurry of activity in developing effective software. One of the main practical variants of the Schrödinger equation is based on what is referred to as density functional theory (DFT). The combination of DFT with pseudopotentials allows one to obtain in an efficient way the ground state configuration for many materials. This article will emphasize pseudopotential-density functional theory, but other techniques will be discussed as well.