chebyshev_filter_oracle_oct_m Module Reference

Data Types
type	three_term_relation

Functions/Subroutines
subroutine	three_term_relation_init (this, n)
	Initialize an instance three_term_relation More...
subroutine	finalize (this)
	Finalize an instance three_term_relation More...
subroutine, public	optimal_chebyshev_polynomial_degree (namespace, st, ik, residual, residual_tol, bounds, max_degree, max_degree_estimate)
	Approximate an optimal Chebyshev polynomial filter degree. Batchified. More...
pure integer function	get_max_state_index (st, bounds, ik)
	Detect the highest meaningful eigenstate. More...
integer function	optimal_polynomial_degree (prior_residual, eigenvalues, bounds, T, target_residual, max_degree, first_eigenval)
	Approximate an optimal Chebyshev polynomial filter degree. More...

Function/Subroutine Documentation

three_term_relation_init()

subroutine chebyshev_filter_oracle_oct_m::three_term_relation_init ( class(three_term_relation)  this, integer, intent(in)  n  )

private

Initialize an instance three_term_relation

Definition at line 136 of file chebyshev_filter_oracle.F90.

finalize()

subroutine chebyshev_filter_oracle_oct_m::finalize ( type(three_term_relation), intent(inout)  this )

private

Finalize an instance three_term_relation

Definition at line 149 of file chebyshev_filter_oracle.F90.

optimal_chebyshev_polynomial_degree()

subroutine, public chebyshev_filter_oracle_oct_m::optimal_chebyshev_polynomial_degree	(	type(namespace_t), intent(in)	namespace,
		type(states_elec_t), intent(in)	st,
		integer, intent(in)	ik,
		real(real64), dimension(:), intent(in)	residual,
		real(real64), intent(in)	residual_tol,
		type(chebyshev_filter_bounds_t), intent(in)	bounds,
		integer, intent(in)	max_degree,
		integer, dimension(:), intent(out)	max_degree_estimate
	)

Approximate an optimal Chebyshev polynomial filter degree. Batchified.

Batch/block wrapper for approximating an optimal Chebyshev polynomial filter degree, based on how much the magnitude of the Chebyshev polynomial scales the corresponding residuals.

Parameters

[in]	st	KS states containing eigenvalues
[in]	ik	k-point index
[in]	residual	Residual vector from prior SCF iteration
[in]	residual_tol	Threshold at which the residuals are
[in]	bounds	Polynomial filter bounds.
[in]	max_degree	Maximum degree to consider.
[out]	max_degree_estimate	Maximum polynomial degree estimate,

Definition at line 164 of file chebyshev_filter_oracle.F90.

get_max_state_index()

pure integer function chebyshev_filter_oracle_oct_m::get_max_state_index ( type(states_elec_t), intent(in)  st, type(chebyshev_filter_bounds_t), intent(in)  bounds, integer, intent(in)  ik  )

private

Detect the highest meaningful eigenstate.

The topmost state is a fixed point, with a shifted eigenenergy of -1 or very close to it. We exclude this state at it will always make the last batch be very slow to converge

Parameters

[in]	st	KS states containing eigenvalues
[in]	bounds	Polynomial filter bounds.
[in]	ik	k-point index

Definition at line 242 of file chebyshev_filter_oracle.F90.

optimal_polynomial_degree()

integer function chebyshev_filter_oracle_oct_m::optimal_polynomial_degree ( real(real64), dimension(:), intent(in)  prior_residual, real(real64), dimension(:), intent(in)  eigenvalues, type(chebyshev_filter_bounds_t), intent(in)  bounds, type(three_term_relation), intent(inout)  T, real(real64), intent(in)  target_residual, integer, intent(in)  max_degree, real(real64), intent(in)  first_eigenval  )

private

Approximate an optimal Chebyshev polynomial filter degree.

Approximate an optimal Chebyshev polynomial filter degree based on how much the magnitude of the Chebyshev polynomial, evaluated for a set of eignevalues, scales the corresponding residuals. Based on the ABINIT implementation documented in [Parallel eigensolvers in plane-wave Density Functional Theory](https:

Selection Criteria

The criteria are implemented such that for early SCF cycles, when the absolute residual values are large, the degree returned should not result in a reduction in the smallest residual by more than an order of magnitude. Once the absolute values of residuals are the same order of magnitude as target_residual, the absolute criterion should take over.

\(\epsilon * |T_n(E_1)| \ll 1\) is implemented to prevent obtaining large polynomial degrees, as this will result in the Gram matrix being ill-conditioned. Like Abinit, we apply this condition to the lowest eigenvalue. In general, we do not expect this criterion to be evaluated unless \(|T_n|\) increases exponentially from \(n\) to \(n+1\).

If max_degree == 1, filter_degree = 1 is returned.

Note

• This function implements the simple Chebyshev polynomial filter, consistent with Algorithm 3.1 in [Chebyshev-filtered subspace iteration method free of sparse diagonalization for solving the Kohn–Sham equation](https: The scaled polynomial was tested but the magnitude of the polynomial did not appear to consistently increase as a function of degree. That implementation was assumed to be erroneous and removed, but should be revisited.
• The Chebyshev polynomial increases in magnitude as the degree is increased, but the sign switches per iteration. One therefore should implement \(|T_n|\), which is a subtle detail missing from the ABINIT paper.

Parameters

[in]	prior_residual	Residual vector from prior SCF iteration.
[in]	eigenvalues	Eigenvalues (or a subset) of eigenvalues
[in]	bounds	Polynomial filter bounds.
[in,out]	t	Filter work arrays: Polynomial filter
[in]	target_residual	Absolute value for target residual.
[in]	max_degree	Maximum polynomial degree to consider.
[in]	first_eigenval	First eigenvalue

Returns

Optimal maximum polynomial order.

Definition at line 292 of file chebyshev_filter_oracle.F90.