Data Types
type	exponential_t

interface	hamiltonian_operator_t

interface	operator_apply

type	operator_t

Functions/Subroutines
subroutine, public	exponential_init (te, namespace, full_batch)

subroutine, public	exponential_copy (teo, tei)

subroutine	exponential_apply_single (te, namespace, mesh, hm, zpsi, ist, ik, deltat, imag_time)
	Wrapper to batchified routine for applying exponential to an array. More...

subroutine	exponential_apply_batch (te, namespace, mesh, hm, psib, deltat, psib2, deltat2, imag_time, inh_psib, op)
	This routine performs the operation: More...

subroutine	exponential_taylor_series_batch (te, namespace, mesh, hm, psib, deltat, op, psib2, deltat2, inh_psib, phik_shift)

subroutine	exponential_lanczos_batch (te, namespace, mesh, hm, psib, deltat, op, inh_psib, psib2, deltat2)
	Lanczos/Arnoldi procedure for the exponential of an operator. More...

subroutine, public	exponential_lanczos_function_batch (te, namespace, mesh, hm, psib, deltat, fun, op, psib2, deltat2)
	Compute fun(H) psib, i.e. the application of a function of the Hamiltonian to a batch. More...

subroutine	exponential_cheby_batch (te, namespace, mesh, hm, psib, chebyshev_function, op)
	Calculates the exponential of the Hamiltonian through an expansion in Chebyshev polynomials. More...

subroutine, public	exponential_apply_all (te, namespace, gr, hm, st, deltat, order)
	Note that this routine not only computes the exponential, but also an extra term if there is a inhomogeneous term in the Hamiltonian hm. More...

subroutine	exponential_apply_phi_batch (te, namespace, mesh, hm, psib, deltat, k, op)

type(hamiltonian_operator_t) function, pointer	hamiltonian_operator_constructor (namespace, mesh, hm)

subroutine	hamiltonian_operator_init (this, namespace, mesh, hm)

subroutine	hamiltonian_operator_apply (this, psib, hpsib)

Variables
integer, parameter, public	exp_lanczos = 2

integer, parameter, public	exp_taylor = 3

integer, parameter, public	exp_chebyshev = 4

Function/Subroutine Documentation

◆ exponential_init()

subroutine, public exponential_oct_m::exponential_init	(	type(exponential_t), intent(out)	te,
		type(namespace_t), intent(in)	namespace,
		logical, intent(in), optional	full_batch
	)

Definition at line 211 of file exponential.F90.

◆ exponential_copy()

subroutine, public exponential_oct_m::exponential_copy	(	type(exponential_t), intent(inout)	teo,
		type(exponential_t), intent(in)	tei
	)

Definition at line 353 of file exponential.F90.

◆ exponential_apply_single()

subroutine exponential_oct_m::exponential_apply_single	(	class(exponential_t), intent(inout)	te,
		type(namespace_t), intent(in)	namespace,
		class(mesh_t), intent(in)	mesh,
		type(hamiltonian_elec_t), intent(inout)	hm,
		complex(real64), dimension(:, :), intent(inout), contiguous	zpsi,
		integer, intent(in)	ist,
		integer, intent(in)	ik,
		real(real64), intent(in)	deltat,
		logical, intent(in), optional	imag_time
	)

private

Wrapper to batchified routine for applying exponential to an array.

Definition at line 369 of file exponential.F90.

◆ exponential_apply_batch()

subroutine exponential_oct_m::exponential_apply_batch	(	class(exponential_t), intent(inout)	te,
		type(namespace_t), intent(in)	namespace,
		class(mesh_t), intent(in)	mesh,
		class(hamiltonian_abst_t), intent(inout)	hm,
		class(batch_t), intent(inout)	psib,
		real(real64), intent(in)	deltat,
		class(batch_t), intent(inout), optional	psib2,
		real(real64), intent(in), optional	deltat2,
		logical, intent(in), optional	imag_time,
		class(batch_t), intent(inout), optional	inh_psib,
		class(operator_t), intent(in), optional, target	op
	)

private

This routine performs the operation:

\[ \exp{-i*\Delta t*hm(t)}|\psi> <-- |\psi> \]

If imag_time is present and is set to true, it performs instead:

\[ \exp{ \Delta t*hm(t)}|\psi> <-- |\psi> \]

If an inhomogeneous term is present, the operation is:

\[ \exp{-i*\Delta t*hm(t)}|\psi> + \Delta t*\phi_1{-i*\Delta t*hm(t)}|inh\psi> <-- |\psi> \]

where:

\[ \phi_1(x) = (e^x - 1)/x \]

Parameters

[in,out]	inh_psib	inhomogeneous term
[in]	op	operator to be applied in the exponential

Definition at line 433 of file exponential.F90.

◆ exponential_taylor_series_batch()

subroutine exponential_oct_m::exponential_taylor_series_batch	(	type(exponential_t), intent(inout)	te,
		type(namespace_t), intent(in)	namespace,
		class(mesh_t), intent(in)	mesh,
		class(hamiltonian_abst_t), intent(inout)	hm,
		class(batch_t), intent(inout)	psib,
		complex(real64), intent(in)	deltat,
		class(operator_t), intent(in)	op,
		class(batch_t), intent(inout), optional	psib2,
		complex(real64), intent(in), optional	deltat2,
		class(batch_t), intent(inout), optional	inh_psib,
		integer, intent(in), optional	phik_shift
	)

private

Parameters

[in]	op	operator to be applied in the exponential
[in,out]	inh_psib	inhomogeneous term
[in]	phik_shift	shift in the Taylor expansion coefficients for phi_k

Definition at line 559 of file exponential.F90.

◆ exponential_lanczos_batch()

subroutine exponential_oct_m::exponential_lanczos_batch	(	type(exponential_t), intent(inout)	te,
		type(namespace_t), intent(in)	namespace,
		class(mesh_t), intent(in)	mesh,
		class(hamiltonian_abst_t), intent(inout)	hm,
		class(batch_t), intent(inout)	psib,
		complex(real64), intent(in)	deltat,
		class(operator_t), intent(in)	op,
		class(batch_t), intent(in), optional	inh_psib,
		class(batch_t), intent(inout), optional	psib2,
		complex(real64), intent(in), optional	deltat2
	)

private

Lanczos/Arnoldi procedure for the exponential of an operator.

Support two time-steps, where the construction of the Krylov subspace is reused for the two time-steps

Parameters

[in]	op	operator to be applied in the exponential
[in]	inh_psib	inhomogeneous term
[in,out]	psib2	second output, sharing the Krylov basis (deltat2)

Definition at line 647 of file exponential.F90.

◆ exponential_lanczos_function_batch()

subroutine, public exponential_oct_m::exponential_lanczos_function_batch	(	type(exponential_t), intent(inout)	te,
		type(namespace_t), intent(in)	namespace,
		class(mesh_t), intent(in)	mesh,
		class(hamiltonian_abst_t), intent(inout)	hm,
		class(batch_t), intent(inout)	psib,
		complex(real64), intent(in)	deltat,
			fun,
		class(operator_t), intent(in)	op,
		class(batch_t), intent(inout), optional	psib2,
		complex(real64), intent(in), optional	deltat2
	)

Compute fun(H) psib, i.e. the application of a function of the Hamiltonian to a batch.

Usually, the function is an exponential, or a related function.

Some details of the implementation can be understood from Saad, Y. (1992). Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM Journal on Numerical Analysis, 29(1), 209-228.

A pdf can be accessed [here](https: or [via the DOI](https: Equation numbers below refer to this paper.

Support two time-steps, where the construction of the Krylov subspace is reused for the two time-steps

Parameters

[in]	op	operator to be applied in the exponential
[in,out]	psib2	second output fun(deltat2*H) psib, sharing the Krylov basis

Definition at line 710 of file exponential.F90.

◆ exponential_cheby_batch()

subroutine exponential_oct_m::exponential_cheby_batch	(	type(exponential_t), intent(inout)	te,
		type(namespace_t), intent(in)	namespace,
		class(mesh_t), intent(in)	mesh,
		class(hamiltonian_abst_t), intent(inout)	hm,
		class(batch_t), intent(inout)	psib,
		class(chebyshev_function_t), intent(in)	chebyshev_function,
		class(operator_t), intent(in)	op
	)

private

Calculates the exponential of the Hamiltonian through an expansion in Chebyshev polynomials.

It uses the algorithm as outlined in [Hochbruck, M. & Ostermann, A.: Exponential Runge–Kutta methods for parabolic problems. Applied Numerical Mathematics 53, 323–339 (2005)](https: Section 4.1. See also [Kosloff, J. Phys. Chem. (1988), 92, 2087-2100](http: Section III.1 and [Kosloff, Annu. Rev. Phys. Chem. (1994), 45, 145–178](https: equations 7.1ff and especially figure 2 for the convergence properties of the coefficients.

Parameters

[in] op operator to be applied in the exponential

Definition at line 906 of file exponential.F90.

◆ exponential_apply_all()

subroutine, public exponential_oct_m::exponential_apply_all	(	type(exponential_t), intent(inout)	te,
		type(namespace_t), intent(in)	namespace,
		type(grid_t), intent(inout)	gr,
		type(hamiltonian_elec_t), intent(inout)	hm,
		type(states_elec_t), intent(inout)	st,
		real(real64), intent(in)	deltat,
		integer, intent(inout), optional	order
	)

Note that this routine not only computes the exponential, but also an extra term if there is a inhomogeneous term in the Hamiltonian hm.

Definition at line 1010 of file exponential.F90.

◆ exponential_apply_phi_batch()

subroutine exponential_oct_m::exponential_apply_phi_batch	(	class(exponential_t), intent(inout)	te,
		type(namespace_t), intent(in)	namespace,
		class(mesh_t), intent(in)	mesh,
		class(hamiltonian_abst_t), intent(inout)	hm,
		class(batch_t), intent(inout)	psib,
		real(real64), intent(in)	deltat,
		integer, intent(in)	k,
		class(operator_t), intent(in), optional, target	op
	)

private

Definition at line 1097 of file exponential.F90.

◆ hamiltonian_operator_constructor()

type(hamiltonian_operator_t) function, pointer exponential_oct_m::hamiltonian_operator_constructor	(	type(namespace_t), intent(in), target	namespace,
		class(mesh_t), intent(in), target	mesh,
		class(hamiltonian_abst_t), intent(in), target	hm
	)

private

Definition at line 1166 of file exponential.F90.

◆ hamiltonian_operator_init()

subroutine exponential_oct_m::hamiltonian_operator_init	(	class(hamiltonian_operator_t), intent(inout)	this,
		type(namespace_t), intent(in), target	namespace,
		class(mesh_t), intent(in), target	mesh,
		class(hamiltonian_abst_t), intent(in), target	hm
	)

private

Definition at line 1180 of file exponential.F90.

◆ hamiltonian_operator_apply()

subroutine exponential_oct_m::hamiltonian_operator_apply	(	class(hamiltonian_operator_t), intent(in)	this,
		class(batch_t), intent(inout)	psib,
		class(batch_t), intent(inout)	hpsib
	)

private

Definition at line 1195 of file exponential.F90.

Variable Documentation

◆ exp_lanczos

integer, parameter, public exponential_oct_m::exp_lanczos = 2

Definition at line 159 of file exponential.F90.

◆ exp_taylor

integer, parameter, public exponential_oct_m::exp_taylor = 3

Definition at line 159 of file exponential.F90.

◆ exp_chebyshev

integer, parameter, public exponential_oct_m::exp_chebyshev = 4

Definition at line 159 of file exponential.F90.

Data Types

Functions/Subroutines

Variables

Function/Subroutine Documentation

◆ exponential_init()

◆ exponential_copy()

◆ exponential_apply_single()

◆ exponential_apply_batch()

◆ exponential_taylor_series_batch()

◆ exponential_lanczos_batch()

◆ exponential_lanczos_function_batch()

◆ exponential_cheby_batch()

◆ exponential_apply_all()

◆ exponential_apply_phi_batch()

◆ hamiltonian_operator_constructor()

◆ hamiltonian_operator_init()

◆ hamiltonian_operator_apply()

Variable Documentation

◆ exp_lanczos

◆ exp_taylor

◆ exp_chebyshev