DavidsonEigen

This package contains a C++ implementation of the Davidson diagonalization algorithms. The calculation can be performed matrix free, i.e. without having to ever store the entire matrix. Different schemas are available to compute the correction.

Available correction methods are:

• DPR: Diagonal-Preconditioned-Residue
• GJD: Generalized Jacobi Davidson

Note:

The Davidson method is suitable for diagonal-dominant symmetric matrices, that are quite common in certain scientific problems like electronic structure. The Davidson method could be not practical for other kind of symmetric matrice.

Usage

The following program instantiates the DavidsonOperator class and uses the DavidsonSolver to compute the lowest 5 eigenvalues and corresponding eigenvectors, using the GJD method.

#include <iostream>
#include "DavidsonSolver.hpp"
#include "DavidsonOperator.hpp"

int main (int argc, char *argv[]) {

    // Matrix Free Operator
    DavidsonOperator A(1000);

    // Davidson Solver
    DavidsonSolver DS;
    DS.set_correction("JACOBI");
    DS.set_jacobi_linsolve("CG");
    DS.solve(A,5);

    auto eigenvalues = DS.eigenvalues();
    auto eigenvectors = DS.eigenvectors();

    return;
}

References:

• Davidson diagonalization method and its applications to electronic structure calculations
• Numerical Methods for Large Eigenvalue Problem

Installation

To compile execute:

cmake -H. -Bbuild && cmake --build build

Dependencies

This packages assumes that you have installed the following packages:

• Eigen
• cxxopts
• Lapack

Optionally, If an MKL library is available the package will try to find it.

Binaries

The cmake compiles a binary in bin/ that can be directly used to test the routines :

./main --help
Eigen Davidson Iterative Solver
Usage:
  ./main [OPTION...]
      --size arg      dimension of the matrix (default: 100)
      --neigen arg    number of eigenvalues required (default: 5)
      --jocc          use Jacobi-Davidson
      --linsolve arg  method to solve the linear system of JOCC (0:CG,
                      1:GMRES, 2:LLT) (default: 0)
      --help          Print the help

./main --size 1000
Matrix size : 1000x1000
Num Threads : 1

===========================
= Matrix Free Method
= Davidson (DPR)
===========================

iter	Search Space	Norm
   0	          10	9.22e-01/1e-06
   1	          20	4.62e-06/1e-06
   2	          30	2.49e-06/1e-06
   3	          40	1.57e-07/1e-06

Davidson               : 27.6797 secs
Eigen                  : 53.4498 secs

      Davidson  	Eigen
   0 0.3994790 	0.3994790
   1 0.6749251 	0.6749251
   2 1.0814170 	1.0814170
   3 1.3745905 	1.3745903
   4 1.4875875 	1.4875875