qs_scf_subspace_math Module Reference

Small, matrix-free mathematical kernels used by ADIIS. More...

Functions/Subroutines
subroutine, public	simplex_quadratic_minimize (hessian, linear, coeff, objective, status, preferred_index)
	Minimize a quadratic model over the probability simplex.
subroutine, public	qs_scf_subspace_build_adiis_model (pf_metric, newest, hessian, linear, valid)
	Construct the canonical ADIIS model relative to the newest history entry.
pure integer function, public	qs_scf_subspace_fifo_slot (generation)
	Select the next physical slot for a strict FIFO history.

Variables
integer, parameter, public	simplex_qp_success = 0
integer, parameter, public	simplex_qp_invalid_shape = 1
integer, parameter, public	simplex_qp_nonfinite_input = 2
integer, parameter, public	simplex_qp_iteration_limit = 3
integer, parameter, public	simplex_qp_pairwise_stationary = 4

Detailed Description

Small, matrix-free mathematical kernels used by ADIIS.

The quadratic model uses the convention

  f(c) = 1/2 c^T hessian c + linear^T c,

with c on the probability simplex.  Only the symmetric part of the
supplied Hessian contributes to the objective.

Function/Subroutine Documentation

simplex_quadratic_minimize()

subroutine, public qs_scf_subspace_math::simplex_quadratic_minimize	(	real(kind=dp), dimension(:, :), intent(in)	hessian,
		real(kind=dp), dimension(:), intent(in)	linear,
		real(kind=dp), dimension(:), intent(out)	coeff,
		real(kind=dp), intent(out), optional	objective,
		integer, intent(out), optional	status,
		integer, intent(in), optional	preferred_index )

Minimize a quadratic model over the probability simplex.

Parameters

hessian	Hessian in f(c) = 1/2 c^T hessian c + linear^T c.
linear	Linear part of the quadratic model.
coeff	Resulting non-negative coefficients, normalized to sum to one.
objective	Objective value at coeff.
status	Completion status. Invalid/nonfinite inputs use the preferred-point fallback; iteration limits return the best legal point found.
preferred_index	History entry preferred for fallbacks and numerical ties; defaults to n. SCF histories are normally very small. Up to max_exact_dimension, all active faces are enumerated. A global minimizer of a quadratic over a simplex is a stationary point in the relative interior of one of these faces (a singular stationary face always contains an equivalent boundary solution). Larger problems use multistart, simplex-preserving pairwise line searches and return simplex_qp_pairwise_stationary. This only proves that no improving two-coordinate transfer was found; for an indefinite quadratic it is a heuristic result, not a general local/global optimum.

Definition at line 60 of file qs_scf_subspace_math.F.

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qs_scf_subspace_build_adiis_model()

subroutine, public qs_scf_subspace_math::qs_scf_subspace_build_adiis_model	(	real(kind=dp), dimension(:, :), intent(in)	pf_metric,
		integer, intent(in)	newest,
		real(kind=dp), dimension(:, :), intent(out)	hessian,
		real(kind=dp), dimension(:), intent(out)	linear,
		logical, intent(out), optional	valid )

Construct the canonical ADIIS model relative to the newest history entry.

Parameters

pf_metric	Matrix T_ij=Tr(P_i F_j), already summed over spin/k-points as needed.
newest	Index of the reference history entry.
hessian	Hessian of the canonical simplex quadratic model.
linear	Linear term of the canonical simplex quadratic model.
valid	Whether all dimensions and input values were valid.

Definition at line 157 of file qs_scf_subspace_math.F.

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qs_scf_subspace_fifo_slot()

pure integer function, public qs_scf_subspace_math::qs_scf_subspace_fifo_slot

(

integer, dimension(:), intent(in)

generation

)

Select the next physical slot for a strict FIFO history.

Parameters

generation

Insertion generation for each physical slot; zero denotes an empty slot.

Returns

First empty slot, otherwise the oldest occupied slot; zero for invalid input.

Definition at line 206 of file qs_scf_subspace_math.F.

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Variable Documentation

simplex_qp_success

integer, parameter, public qs_scf_subspace_math::simplex_qp_success = 0

Definition at line 27 of file qs_scf_subspace_math.F.

simplex_qp_invalid_shape

integer, parameter, public qs_scf_subspace_math::simplex_qp_invalid_shape = 1

Definition at line 28 of file qs_scf_subspace_math.F.

simplex_qp_nonfinite_input

integer, parameter, public qs_scf_subspace_math::simplex_qp_nonfinite_input = 2

Definition at line 29 of file qs_scf_subspace_math.F.

simplex_qp_iteration_limit

integer, parameter, public qs_scf_subspace_math::simplex_qp_iteration_limit = 3

Definition at line 30 of file qs_scf_subspace_math.F.

simplex_qp_pairwise_stationary

integer, parameter, public qs_scf_subspace_math::simplex_qp_pairwise_stationary = 4

Definition at line 31 of file qs_scf_subspace_math.F.