Data Types
type	exponential_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, vmagnus, 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, vmagnus, imag_time, inh_psib)
	This routine performs the operation: More...

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

subroutine	exponential_lanczos_batch (te, namespace, mesh, hm, psib, deltat, vmagnus, inh_psib)

subroutine, public	exponential_lanczos_function_batch (te, namespace, mesh, hm, psib, deltat, fun, vmagnus)
	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, deltat, chebyshev_function, vmagnus)
	Calculates the exponential of the Hamiltonian through an expansion in Chebyshev polynomials. More...

subroutine	operate_batch (hm, namespace, mesh, psib, hpsib, vmagnus)

subroutine, public	exponential_apply_all (te, namespace, mesh, 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, vmagnus)

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 176 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 314 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,
		real(real64), dimension(mesh%np, hm%d%nspin, 2), intent(in), optional	vmagnus,
		logical, intent(in), optional	imag_time
	)

private

Wrapper to batchified routine for applying exponential to an array.

Definition at line 330 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,
		real(real64), dimension(:,:,:), intent(in), optional	vmagnus,
		logical, intent(in), optional	imag_time,
		class(batch_t), intent(inout), optional	inh_psib
	)

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

Definition at line 395 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(batch_t), intent(inout), optional	psib2,
		complex(real64), intent(in), optional	deltat2,
		real(real64), dimension(:,:,:), intent(in), optional	vmagnus,
		class(batch_t), intent(inout), optional	inh_psib,
		integer, intent(in), optional	phik_shift
	)

private

Parameters

[in,out]	inh_psib	inhomogeneous term
[in]	phik_shift	shift in the Taylor expansion coefficients for phi_k

Definition at line 511 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,
		real(real64), dimension(:,:,:), intent(in), optional	vmagnus,
		class(batch_t), intent(in), optional	inh_psib
	)

private

Parameters

[in] inh_psib inhomogeneous term

Definition at line 596 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,
		real(real64), dimension(:,:,:), intent(in), optional	vmagnus
	)

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.

Definition at line 635 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,
		real(real64), intent(in)	deltat,
		class(chebyshev_function_t), intent(in)	chebyshev_function,
		real(real64), dimension(:,:,:), intent(in), optional	vmagnus
	)

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.

Definition at line 792 of file exponential.F90.

◆ operate_batch()

subroutine exponential_oct_m::operate_batch	(	class(hamiltonian_abst_t), intent(inout)	hm,
		type(namespace_t), intent(in)	namespace,
		class(mesh_t), intent(in)	mesh,
		class(batch_t), intent(inout)	psib,
		class(batch_t), intent(inout)	hpsib,
		real(real64), dimension(:, :, :), intent(in), optional	vmagnus
	)

private

Definition at line 896 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,
		class(mesh_t), intent(inout)	mesh,
		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 919 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,
		real(real64), dimension(:,:,:), intent(in), optional	vmagnus
	)

private

Definition at line 1008 of file exponential.F90.

Variable Documentation

◆ exp_lanczos

integer, parameter, public exponential_oct_m::exp_lanczos = 2

Definition at line 154 of file exponential.F90.

◆ exp_taylor

integer, parameter, public exponential_oct_m::exp_taylor = 3

Definition at line 154 of file exponential.F90.

◆ exp_chebyshev

integer, parameter, public exponential_oct_m::exp_chebyshev = 4

Definition at line 154 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()

◆ operate_batch()

◆ exponential_apply_all()

◆ exponential_apply_phi_batch()

Variable Documentation

◆ exp_lanczos

◆ exp_taylor

◆ exp_chebyshev