negf_integr_simpson Module Reference

Adaptive Simpson's rule algorithm to integrate a complex-valued function in a complex plane. More...

Data Types
type	simpsonrule_type
	A structure to store data needed for adaptive Simpson's rule algorithm. More...

Functions/Subroutines
subroutine, public	simpsonrule_init (sr_env, xnodes, nnodes, a, b, shape_id, conv, weights, tnodes_restart)
	Initialise a Simpson's rule environment variable.
subroutine, public	simpsonrule_release (sr_env)
	Release a Simpson's rule environment variable.
subroutine, public	simpsonrule_get_next_nodes (sr_env, xnodes_next, nnodes)
	Get the next set of nodes where to compute integrand.
subroutine, public	simpsonrule_refine_integral (sr_env, zdata_next)
	Compute integral using the simpson's rules.

Variables
integer, parameter, public	sr_shape_linear = contour_shape_linear
integer, parameter, public	sr_shape_arc = contour_shape_arc

Detailed Description

Adaptive Simpson's rule algorithm to integrate a complex-valued function in a complex plane.

Function/Subroutine Documentation

simpsonrule_init()

subroutine, public negf_integr_simpson::simpsonrule_init	(	type(simpsonrule_type), intent(out)	sr_env,
		complex(kind=dp), dimension(nnodes), intent(out)	xnodes,
		integer, intent(inout)	nnodes,
		complex(kind=dp), intent(in)	a,
		complex(kind=dp), intent(in)	b,
		integer, intent(in)	shape_id,
		real(kind=dp), intent(in)	conv,
		type(cp_fm_type), intent(in)	weights,
		real(kind=dp), dimension(nnodes), intent(in), optional	tnodes_restart
	)

Initialise a Simpson's rule environment variable.

Parameters

sr_env	Simpson's rule environment (initialised on exit)
xnodes	points at which an integrand needs to be computed (initialised on exit)
nnodes	initial number of points to compute (initialised on exit)
a	integral lower boundary
b	integral upper boundary
shape_id	shape of a curve along which the integral will be evaluated
conv	convergence threshold
weights	weights associated with matrix elements; used to compute cumulative error
tnodes_restart	list of nodes over the interval [-1 .. 1] from a previous integral evaluation. If present, the same set of 'xnodes' will be used to compute this integral.

History

• 05.2017 created [Sergey Chulkov]

Note

When we integrate the retarded Green's function times the Fermi function over the energy domain and pass the overlap matrix (S) as the 'weights' matrix, the convergence threshold ('conv') becomes the maximum error in the total number of electrons multiplied by pi.

Definition at line 121 of file negf_integr_simpson.F.

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simpsonrule_release()

subroutine, public negf_integr_simpson::simpsonrule_release

(

type(simpsonrule_type), intent(inout)

sr_env

)

Release a Simpson's rule environment variable.

Parameters

sr_env

Simpson's rule environment (modified on exit)

History

• 05.2017 created [Sergey Chulkov]

Definition at line 188 of file negf_integr_simpson.F.

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simpsonrule_get_next_nodes()

subroutine, public negf_integr_simpson::simpsonrule_get_next_nodes	(	type(simpsonrule_type), intent(inout)	sr_env,
		complex(kind=dp), dimension(nnodes), intent(out)	xnodes_next,
		integer, intent(inout)	nnodes
	)

Get the next set of nodes where to compute integrand.

Parameters

sr_env	Simpson's rule environment (modified on exit)
xnodes_next	list of additional points (initialised on exit)
nnodes	actual number of points to compute (modified on exit)

History

• 05.2017 created [Sergey Chulkov]

Note

The number of nodes returned is limited by the initial value of the nnodes variable; un exit nnodes == 0 means that the target accuracy has been achieved.

Definition at line 259 of file negf_integr_simpson.F.

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simpsonrule_refine_integral()

subroutine, public negf_integr_simpson::simpsonrule_refine_integral	(	type(simpsonrule_type), intent(inout)	sr_env,
		type(cp_cfm_type), dimension(:), intent(inout)	zdata_next
	)

Compute integral using the simpson's rules.

Parameters

sr_env	Simpson's rule environment
zdata_next	precomputed integrand values at points xnodes_next (nullified on exit)

History

• 05.2017 created [Sergey Chulkov]

Definition at line 340 of file negf_integr_simpson.F.

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Variable Documentation

sr_shape_linear

integer, parameter, public negf_integr_simpson::sr_shape_linear = contour_shape_linear

Definition at line 46 of file negf_integr_simpson.F.

sr_shape_arc

integer, parameter, public negf_integr_simpson::sr_shape_arc = contour_shape_arc

Definition at line 46 of file negf_integr_simpson.F.