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 hermitian matrices, that are quite common in certain scientific problems like electronic structure. The Davidson method could be not practical for other kinds of hermitian 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
  complex(dp), dimension(dim, dim) :: matrix, second_matrix
  real(dp), dimension(lowest) :: eigenvalues
  complex(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:

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

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 7.0
• CMake
• Lapack

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