# Pole Expansion for Solving a Type of Parametrized Linear Systems in Electronic Structure Calculations

@article{Damle2014PoleEF, title={Pole Expansion for Solving a Type of Parametrized Linear Systems in Electronic Structure Calculations}, author={Anil Damle and Lin Lin and Lexing Ying}, journal={SIAM J. Sci. Comput.}, year={2014}, volume={36} }

We present a new method for solving parametrized linear systems. Under certain assumptions on the parametrization, solutions to the linear systems for all parameters can be accurately approximated by linear combinations of solutions to linear systems for a small set of fixed parameters. Combined with either direct solvers or preconditioned iterative solvers for each linear system with a fixed parameter, the method is particularly suitable for situations when solutions to a large number of… Expand

#### Figures, Tables, and Topics from this paper

#### One Citation

Preconditioning Orbital Minimization Method for Planewave Discretization

- Mathematics, Computer Science
- Multiscale Model. Simul.
- 2017

Numerical results validate the performance of the new preconditioner for the orbital minimization method, in particular, the iteration number is reduced to $\mathcal{O}(1)$ and often only a few iterations are enough for convergence. Expand

#### References

SHOWING 1-10 OF 53 REFERENCES

A fast direct solver for boundary integral equations in two dimensions

- Mathematics
- 2003

We describe an algorithm for the direct solution of systems of linear algebraic equations associated with the discretization of boundary integral equations with non-oscillatory kernels in two… Expand

A partial Padé-via-Lanczos method for reduced-order modeling

- Mathematics
- 2001

Abstract The classical Lanczos process can be used to efficiently generate Pade approximants of the transfer function of a given single-input single-output time-invariant linear dynamical system.… Expand

The Solution of Parametrized Symmetric Linear Systems

- Mathematics, Computer Science
- SIAM J. Matrix Anal. Appl.
- 2003

The Lanczos and MINRES methods for the solution of symmetric linear systems with a linear parameter arising from structural engineering are compared and an error estimation for the problem in finite precision arithmetic is proposed. Expand

Self-Consistent Equations Including Exchange and Correlation Effects

- Physics
- 1965

From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed. These methods are exact for systems of slowly varying or high… Expand

Restarted GMRES for Shifted Linear Systems

- Mathematics, Computer Science
- SIAM J. Sci. Comput.
- 1998

This work develops a variant of the restarted GMRES method exhibiting the same advantage and investigates its convergence for positive real matrices in some detail and applies it to speed up "multiple masses" calculations arising in lattice gauge computations in quantum chromodynamics, one of the most time-consuming supercomputer applications. Expand

A rational Lanczos algorithm for model reduction

- Mathematics, Computer Science
- Numerical Algorithms
- 2005

A variant of the nonsymmetric Lanczos method, rational Lanczos, is shown to yield a rational interpolant (multi-point Padé approximant) for the large-scale system. Expand

KSSOLV—a MATLAB toolbox for solving the Kohn-Sham equations

- Computer Science
- TOMS
- 2009

KSSOLV, a MATLAB toolbox for solving a class of nonlinear eigenvalue problems known as the Kohn-Sham equations, is described, designed to enable researchers in computational and applied mathematics to investigate the convergence properties of the existing algorithms. Expand

An iteration method for the solution of the eigenvalue problem of linear differential and integral operators

- Mathematics
- 1950

The present investigation designs a systematic method for finding the latent roots and the principal axes of a matrix, without reducing the order of the matrix. It is characterized by a wide field of… Expand

Efficient linear circuit analysis by Pade´ approximation via the Lanczos process

- Computer Science, Mathematics
- EURO-DAC '94
- 1994

PVL, an algorithm for computing the Pad6 approximation of Laplace-domain transfer functions of large linear networks via a Lanczos process, has significantly superior numerical stability and renders unnecessary many of the heuristics that AWE and its derivatives had to employ. Expand

The SIESTA method for ab initio order-N materials simulation

- Materials Science, Physics
- 2001

We have developed and implemented a selfconsistent density functional method using standard norm-conserving pseudopotentials and a flexible, numerical linear combination of atomic orbitals basis set,… Expand