Davidson Eigensolver

This package contains a Modern Fortran implementation of the Davidson diagonalization algorithms. 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 kinds of symmetric matrices.

Usage

The following program call the eigensolver subroutine from the davidson module and computes the lowest 3 eigenvalues and corresponding eigenvectors, using the GJD method with a tolerance of 1e-8 and 100 maximum iteration.

program main
  use numeric_kinds, only: dp
  use davidson, only: generalized_eigensolver, generate_diagonal_dominant
 
  implicit none

  integer, parameter :: dim = 50
  integer, parameter :: lowest = 3
  real(dp), dimension(dim, dim) :: matrix, second_matrix
  real(dp), dimension(lowest) :: eigenvalues
  real(dp), dimension(dim, lowest) :: eigenvectors
  real(dp) :: tolerance
  integer:: max_dim_subspace, max_iterations, lowest

  matrix = generate_diagonal_dominant(dim, 1d-4)
  second_matrix = generate_diagonal_dominant(dim, 1d-4, 1.0_dp)
  max_iterations = 1000
  max_dim_subspace = 20
  tolerance = 1d-8
  call generalized_eigensolver(matrix, eigenvalues, eigenvectors, lowest, "GJD", max_iterations, &
       tolerance, final_iterations, max_dim_subspace, second_matrix)
  print *, eigenvalues
  print *, eigenvectors

end program main

The helper generate_diagonal_dominant function generates a diagonal dominant matrix with entries to the diagonal close to row number (i=1, number_of_rows) and random number of the order 1e-4 on the off-diagonal entries.

Variables:

matrix (in) matrix to diagonalize
eigenvalues (out) resulting eigenvalues
eigenvectors (out) resulting eigenvectors
lowest(in) number of eigenvalues to compute
method(in) Either "DPR" or "GJD"
max_iterations(in) maximum number of iterations
tolerance(in) Numerical tolerance for convergence
final_iterations(output) returns the number of iterations that were needed to converge
max_dim_subspace(in, optional) Dimension of the subspace of search
second_matrix(in, optional) Optional matrix to compute the generalized eigenvalue problem

References:

Installation and Testing

To compile execute:

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

To use another compiler (e.g. ifort):

cmake -H. -Bbuild -DCMAKE_Fortran_COMPILER=ifort && cmake --build build

To run the test:

cmake -H. -Bbuild -DENABLE_TEST=ON && cmake --build build
cd build && ctest -V

To Debug compile as:

cmake -H. -Bbuild  -DCMAKE_BUILD_TYPE=Debug && cmake --build build

Dependencies

This packages assumes that you have installed the following packages:

A Fortran compiler >= version 6.0
CMake
Lapack