negf_integr_cc Module Reference

Adaptive Clenshaw-Curtis quadrature algorithm to integrate a complex-valued function in a complex plane. More...

Functions/Subroutines
subroutine, public	ccquad_init (cc_env, xnodes, nnodes, a, b, interval_id, shape_id, weights, tnodes_restart)
	Initialise a Clenshaw-Curtis quadrature environment variable. More...
subroutine, public	ccquad_release (cc_env)
	Release a Clenshaw-Curtis quadrature environment variable. More...
subroutine, public	ccquad_double_number_of_points (cc_env, xnodes_next)
	Get the next set of points at which the integrand needs to be computed. These points are then can be used to refine the integral approximation. More...
subroutine, public	ccquad_reduce_and_append_zdata (cc_env, zdata_next)
	Prepare Clenshaw-Curtis environment for the subsequent refinement of the integral. More...
subroutine, public	ccquad_refine_integral (cc_env)
	Refine approximated integral. More...

Variables
integer, parameter, public	cc_interval_full = 0
integer, parameter, public	cc_interval_half = 1
integer, parameter, public	cc_shape_linear = contour_shape_linear
integer, parameter, public	cc_shape_arc = contour_shape_arc

Detailed Description

Adaptive Clenshaw-Curtis quadrature algorithm to integrate a complex-valued function in a complex plane.

History

• 05.2017 created [Sergey Chulkov]

Function/Subroutine Documentation

ccquad_init()

subroutine, public negf_integr_cc::ccquad_init	(	type(ccquad_type), intent(out)	cc_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)	interval_id,
		integer, intent(in)	shape_id,
		type(cp_fm_type), intent(in)	weights,
		real(kind=dp), dimension(nnodes), intent(in), optional	tnodes_restart
	)

Initialise a Clenshaw-Curtis quadrature environment variable.

Parameters

cc_env	environment variable to initialise
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 bound
b	integral upper bound
interval_id	full [-1 .. 1] or half [-1 .. 0] interval
shape_id	shape of a curve along which the integral will be evaluated
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

Clenshaw-Curtis quadratures are defined on the interval [-1 .. 1] and have non-uniforms node distribution which is symmetric and much sparse about 0. When the half-interval [-1 .. 0] is requested, the integrand value on another subinterval (0 .. 1] is assumed to be zero. Half interval mode is typically useful for rapidly decaying integrands (e.g. multiplied by Fermi function), so we do not actually need a fine grid spacing on this tail.

Definition at line 112 of file negf_integr_cc.F.

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ccquad_release()

subroutine, public negf_integr_cc::ccquad_release

(

type(ccquad_type), intent(inout)

cc_env

)

Release a Clenshaw-Curtis quadrature environment variable.

Parameters

cc_env

environment variable to release (modified on exit)

History

• 05.2017 created [Sergey Chulkov]

Definition at line 201 of file negf_integr_cc.F.

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ccquad_double_number_of_points()

subroutine, public negf_integr_cc::ccquad_double_number_of_points	(	type(ccquad_type), intent(inout)	cc_env,
		complex(kind=dp), dimension(:), intent(inout), allocatable	xnodes_next
	)

Get the next set of points at which the integrand needs to be computed. These points are then can be used to refine the integral approximation.

Parameters

cc_env	environment variable (modified on exit)
xnodes_next	set of additional points (allocated and initialised on exit)

History

• 05.2017 created [Sergey Chulkov]

Definition at line 249 of file negf_integr_cc.F.

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ccquad_reduce_and_append_zdata()

subroutine, public negf_integr_cc::ccquad_reduce_and_append_zdata	(	type(ccquad_type), intent(inout)	cc_env,
		type(cp_cfm_type), dimension(:), intent(inout)	zdata_next
	)

Prepare Clenshaw-Curtis environment for the subsequent refinement of the integral.

Parameters

cc_env	environment variable (modified on exit)
zdata_next	additional integrand value at additional points (modified on exit)

History

• 05.2017 created [Sergey Chulkov]

Note

Due to symmetry of Clenshaw-Curtis quadratures (weight(x) == weight(-x)), we do not need to keep all the matrices from 'zdata_next', only 'zdata_next(x) + zdata_next(-x)' is needed. In order to reduce the number of matrix allocations, we move some of the matrices from the end of the 'zdata_new' array to the 'cc_envzdata_cache' array, and nullify the corresponding pointers at 'zdata_next' array. So the calling subroutine need to release the remained matrices or reuse them but taking into account the missed ones.

Definition at line 332 of file negf_integr_cc.F.

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ccquad_refine_integral()

subroutine, public negf_integr_cc::ccquad_refine_integral

(

type(ccquad_type), intent(inout)

cc_env

)

Refine approximated integral.

Parameters

cc_env

environment variable (modified on exit)

History

• 05.2017 created [Sergey Chulkov]

Definition at line 427 of file negf_integr_cc.F.

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Variable Documentation

cc_interval_full

integer, parameter, public negf_integr_cc::cc_interval_full = 0

Definition at line 48 of file negf_integr_cc.F.

cc_interval_half

integer, parameter, public negf_integr_cc::cc_interval_half = 1

Definition at line 48 of file negf_integr_cc.F.

cc_shape_linear

integer, parameter, public negf_integr_cc::cc_shape_linear = contour_shape_linear

Definition at line 51 of file negf_integr_cc.F.

cc_shape_arc

integer, parameter, public negf_integr_cc::cc_shape_arc = contour_shape_arc

Definition at line 51 of file negf_integr_cc.F.