LightKrylov_ExpmLib – LightKrylov

This module implements the evaluation of the "matrix-exponential times vector" procedure using Krylov methods.

Uses

- LightKrylov_BaseKrylov
- LightKrylov_Constants
- LightKrylov_Utils
- LightKrylov_AbstractVectors
- LightKrylov_AbstractLinops
- stdlib_linalg
- iso_fortran_env
- LightKrylov_Logger
- stdlib_optval

Interfaces

public interface kexpm

Description

This interface provides methods to evaluate the matrix-vector product $c = \exp(\tau A) b$ based on the Arnoldi method.

Syntax

    call kexpm(c, A, b, tau, tol, info [, trans] [, kdim])

Arguments

c : Output vector (or vectors). It is an intent(out) argument.
A : Linear operator to be exponentiated. It is an intent(inout) argument.
b : Vector to be multiplied by $\exp(\tau A)$ . It is an intent(in) argument.
tau : real (singe or double) time over which the matrix exponential needs to be computed. It is an intent(in) argument.
info : integer Information flag.
trans (optional) : Whether $A$ or $A^H$ is being used. (default trans=.false.)
kdim (optional) : Dimension of the Krylov subspace used in the Arnoldi method.

private subroutine kexpm_vec_rsp(c, A, b, tau, tol, info, trans, kdim)

Best approximation of $\exp(\tau \mathbf{A}) \mathbf{b}$ in the computed Krylov subspace

Arguments

Type	Intent	Optional		Name
class(abstract_vector_rsp),	intent(out)		::	c
class(abstract_linop_rsp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_rsp),	intent(in)		::	b	Input vector on which to apply $\exp(\tau \mathbf{A})$ .
real(kind=sp),	intent(in)		::	tau	Time horizon for the exponentiation.
real(kind=sp),	intent(in)		::	tol	Solution tolerance based on error estimates.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use transpose?
integer,	intent(in),	optional	::	kdim	Maximum size of the Krylov subspace.

private subroutine kexpm_mat_rsp(C, A, B, tau, tol, info, trans, kdim)

Best Krylov approximation of $\mathbf{C} = \exp(\tau \mathbf{A}) \mathbf{B}$ in the computed Krylov subspace.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_rsp),	intent(out)		::	C(:)
class(abstract_linop_rsp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_rsp),	intent(in)		::	B(:)	Input matrix on which to apply $\exp(\tau \mathbf{A})$ .
real(kind=sp),	intent(in)		::	tau	Time horizon for the exponentiation.
real(kind=sp),	intent(in)		::	tol	Solution toleance based on error estimates.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use transpose ?
integer,	intent(in),	optional	::	kdim	Maximum size of the Krylov subspace.

private subroutine kexpm_vec_rdp(c, A, b, tau, tol, info, trans, kdim)

Best approximation of $\exp(\tau \mathbf{A}) \mathbf{b}$ in the computed Krylov subspace

Arguments

Type	Intent	Optional		Name
class(abstract_vector_rdp),	intent(out)		::	c
class(abstract_linop_rdp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_rdp),	intent(in)		::	b	Input vector on which to apply $\exp(\tau \mathbf{A})$ .
real(kind=dp),	intent(in)		::	tau	Time horizon for the exponentiation.
real(kind=dp),	intent(in)		::	tol	Solution tolerance based on error estimates.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use transpose?
integer,	intent(in),	optional	::	kdim	Maximum size of the Krylov subspace.

private subroutine kexpm_mat_rdp(C, A, B, tau, tol, info, trans, kdim)

Best Krylov approximation of $\mathbf{C} = \exp(\tau \mathbf{A}) \mathbf{B}$ in the computed Krylov subspace.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_rdp),	intent(out)		::	C(:)
class(abstract_linop_rdp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_rdp),	intent(in)		::	B(:)	Input matrix on which to apply $\exp(\tau \mathbf{A})$ .
real(kind=dp),	intent(in)		::	tau	Time horizon for the exponentiation.
real(kind=dp),	intent(in)		::	tol	Solution toleance based on error estimates.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use transpose ?
integer,	intent(in),	optional	::	kdim	Maximum size of the Krylov subspace.

private subroutine kexpm_vec_csp(c, A, b, tau, tol, info, trans, kdim)

Best approximation of $\exp(\tau \mathbf{A}) \mathbf{b}$ in the computed Krylov subspace

Arguments

Type	Intent	Optional		Name
class(abstract_vector_csp),	intent(out)		::	c
class(abstract_linop_csp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_csp),	intent(in)		::	b	Input vector on which to apply $\exp(\tau \mathbf{A})$ .
real(kind=sp),	intent(in)		::	tau	Time horizon for the exponentiation.
real(kind=sp),	intent(in)		::	tol	Solution tolerance based on error estimates.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use transpose?
integer,	intent(in),	optional	::	kdim	Maximum size of the Krylov subspace.

private subroutine kexpm_mat_csp(C, A, B, tau, tol, info, trans, kdim)

Best Krylov approximation of $\mathbf{C} = \exp(\tau \mathbf{A}) \mathbf{B}$ in the computed Krylov subspace.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_csp),	intent(out)		::	C(:)
class(abstract_linop_csp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_csp),	intent(in)		::	B(:)	Input matrix on which to apply $\exp(\tau \mathbf{A})$ .
real(kind=sp),	intent(in)		::	tau	Time horizon for the exponentiation.
real(kind=sp),	intent(in)		::	tol	Solution toleance based on error estimates.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use transpose ?
integer,	intent(in),	optional	::	kdim	Maximum size of the Krylov subspace.

private subroutine kexpm_vec_cdp(c, A, b, tau, tol, info, trans, kdim)

Best approximation of $\exp(\tau \mathbf{A}) \mathbf{b}$ in the computed Krylov subspace

Arguments

Type	Intent	Optional		Name
class(abstract_vector_cdp),	intent(out)		::	c
class(abstract_linop_cdp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_cdp),	intent(in)		::	b	Input vector on which to apply $\exp(\tau \mathbf{A})$ .
real(kind=dp),	intent(in)		::	tau	Time horizon for the exponentiation.
real(kind=dp),	intent(in)		::	tol	Solution tolerance based on error estimates.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use transpose?
integer,	intent(in),	optional	::	kdim	Maximum size of the Krylov subspace.

private subroutine kexpm_mat_cdp(C, A, B, tau, tol, info, trans, kdim)

Best Krylov approximation of $\mathbf{C} = \exp(\tau \mathbf{A}) \mathbf{B}$ in the computed Krylov subspace.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_cdp),	intent(out)		::	C(:)
class(abstract_linop_cdp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_cdp),	intent(in)		::	B(:)	Input matrix on which to apply $\exp(\tau \mathbf{A})$ .
real(kind=dp),	intent(in)		::	tau	Time horizon for the exponentiation.
real(kind=dp),	intent(in)		::	tol	Solution toleance based on error estimates.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use transpose ?
integer,	intent(in),	optional	::	kdim	Maximum size of the Krylov subspace.

public interface krylov_exptA

Description

Utility function to evaluate the matrix-exponential times vector.

Syntax

    call k_exptA(vec_out, A, vec_in, tau, info, trans)

Arguments

vec_out : Output vector.
A : Matrix to be exponentiated.
vec_in : Input vector.
tau : Integration time.
info : Information flag.
trans : Whether $A$ or $A^H$ is being used.

public subroutine krylov_exptA_rsp(vec_out, A, vec_in, tau, info, trans)

Wrapper for the Krylov-based evaluation of the action of the matrix exponential operator on a vector that conforms to the abstract_exptA_rsp interface.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_rsp),	intent(out)		::	vec_out	Solution vector.
class(abstract_linop_rsp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_rsp),	intent(in)		::	vec_in	Input vector to be multiplied by $\exp(\tau \mathbf{A})$ .
real(kind=sp),	intent(in)		::	tau	Time horizon for the exponentiation.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use adjoint ?

public subroutine krylov_exptA_rdp(vec_out, A, vec_in, tau, info, trans)

Wrapper for the Krylov-based evaluation of the action of the matrix exponential operator on a vector that conforms to the abstract_exptA_rdp interface.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_rdp),	intent(out)		::	vec_out	Solution vector.
class(abstract_linop_rdp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_rdp),	intent(in)		::	vec_in	Input vector to be multiplied by $\exp(\tau \mathbf{A})$ .
real(kind=dp),	intent(in)		::	tau	Time horizon for the exponentiation.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use adjoint ?

public subroutine krylov_exptA_csp(vec_out, A, vec_in, tau, info, trans)

Wrapper for the Krylov-based evaluation of the action of the matrix exponential operator on a vector that conforms to the abstract_exptA_csp interface.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_csp),	intent(out)		::	vec_out	Solution vector.
class(abstract_linop_csp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_csp),	intent(in)		::	vec_in	Input vector to be multiplied by $\exp(\tau \mathbf{A})$ .
real(kind=sp),	intent(in)		::	tau	Time horizon for the exponentiation.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use adjoint ?

public subroutine krylov_exptA_cdp(vec_out, A, vec_in, tau, info, trans)

Wrapper for the Krylov-based evaluation of the action of the matrix exponential operator on a vector that conforms to the abstract_exptA_cdp interface.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_cdp),	intent(out)		::	vec_out	Solution vector.
class(abstract_linop_cdp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_cdp),	intent(in)		::	vec_in	Input vector to be multiplied by $\exp(\tau \mathbf{A})$ .
real(kind=dp),	intent(in)		::	tau	Time horizon for the exponentiation.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use adjoint ?

Abstract Interfaces

abstract interface

public subroutine abstract_exptA_cdp(vec_out, A, vec_in, tau, info, trans)

Abstract interface to define the matrix exponential-vector product.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_cdp),	intent(out)		::	vec_out	Solution vector.
class(abstract_linop_cdp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_cdp),	intent(in)		::	vec_in	Input vector.
real(kind=dp),	intent(in)		::	tau	Time horizon for integration.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use transpose ?

abstract interface

public subroutine abstract_exptA_csp(vec_out, A, vec_in, tau, info, trans)

Abstract interface to define the matrix exponential-vector product.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_csp),	intent(out)		::	vec_out	Solution vector.
class(abstract_linop_csp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_csp),	intent(in)		::	vec_in	Input vector.
real(kind=sp),	intent(in)		::	tau	Time horizon for integration.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use transpose ?

abstract interface

public subroutine abstract_exptA_rdp(vec_out, A, vec_in, tau, info, trans)

Abstract interface to define the matrix exponential-vector product.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_rdp),	intent(out)		::	vec_out	Solution vector.
class(abstract_linop_rdp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_rdp),	intent(in)		::	vec_in	Input vector.
real(kind=dp),	intent(in)		::	tau	Time horizon for integration.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use transpose ?

abstract interface

public subroutine abstract_exptA_rsp(vec_out, A, vec_in, tau, info, trans)

Abstract interface to define the matrix exponential-vector product.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_rsp),	intent(out)		::	vec_out	Solution vector.
class(abstract_linop_rsp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_rsp),	intent(in)		::	vec_in	Input vector.
real(kind=sp),	intent(in)		::	tau	Time horizon for integration.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use transpose ?

Subroutines

public subroutine krylov_exptA_cdp(vec_out, A, vec_in, tau, info, trans)

Wrapper for the Krylov-based evaluation of the action of the matrix exponential operator on a vector that conforms to the abstract_exptA_cdp interface.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_cdp),	intent(out)		::	vec_out	Solution vector.
class(abstract_linop_cdp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_cdp),	intent(in)		::	vec_in	Input vector to be multiplied by $\exp(\tau \mathbf{A})$ .
real(kind=dp),	intent(in)		::	tau	Time horizon for the exponentiation.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use adjoint ?

public subroutine krylov_exptA_csp(vec_out, A, vec_in, tau, info, trans)

Wrapper for the Krylov-based evaluation of the action of the matrix exponential operator on a vector that conforms to the abstract_exptA_csp interface.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_csp),	intent(out)		::	vec_out	Solution vector.
class(abstract_linop_csp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_csp),	intent(in)		::	vec_in	Input vector to be multiplied by $\exp(\tau \mathbf{A})$ .
real(kind=sp),	intent(in)		::	tau	Time horizon for the exponentiation.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use adjoint ?

public subroutine krylov_exptA_rdp(vec_out, A, vec_in, tau, info, trans)

Wrapper for the Krylov-based evaluation of the action of the matrix exponential operator on a vector that conforms to the abstract_exptA_rdp interface.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_rdp),	intent(out)		::	vec_out	Solution vector.
class(abstract_linop_rdp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_rdp),	intent(in)		::	vec_in	Input vector to be multiplied by $\exp(\tau \mathbf{A})$ .
real(kind=dp),	intent(in)		::	tau	Time horizon for the exponentiation.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use adjoint ?

public subroutine krylov_exptA_rsp(vec_out, A, vec_in, tau, info, trans)

Wrapper for the Krylov-based evaluation of the action of the matrix exponential operator on a vector that conforms to the abstract_exptA_rsp interface.

Arguments

Type	Intent	Optional		Name
class(abstract_vector_rsp),	intent(out)		::	vec_out	Solution vector.
class(abstract_linop_rsp),	intent(inout)		::	A	Linear operator to be exponentiated.
class(abstract_vector_rsp),	intent(in)		::	vec_in	Input vector to be multiplied by $\exp(\tau \mathbf{A})$ .
real(kind=sp),	intent(in)		::	tau	Time horizon for the exponentiation.
integer,	intent(out)		::	info	Information flag.
logical,	intent(in),	optional	::	trans	Use adjoint ?

LightKrylov_ExpmLib Module

Contents

Interfaces

Abstract Interfaces

Subroutines

Uses

Interfaces

public interface kexpm

Description

Syntax

Arguments

private subroutine kexpm_vec_rsp(c, A, b, tau, tol, info, trans, kdim)

Arguments

private subroutine kexpm_mat_rsp(C, A, B, tau, tol, info, trans, kdim)

Arguments

private subroutine kexpm_vec_rdp(c, A, b, tau, tol, info, trans, kdim)

Arguments

private subroutine kexpm_mat_rdp(C, A, B, tau, tol, info, trans, kdim)

Arguments

private subroutine kexpm_vec_csp(c, A, b, tau, tol, info, trans, kdim)

Arguments

private subroutine kexpm_mat_csp(C, A, B, tau, tol, info, trans, kdim)

Arguments

private subroutine kexpm_vec_cdp(c, A, b, tau, tol, info, trans, kdim)

Arguments

private subroutine kexpm_mat_cdp(C, A, B, tau, tol, info, trans, kdim)

Arguments

public interface krylov_exptA

Description

Syntax

Arguments

public subroutine krylov_exptA_rsp(vec_out, A, vec_in, tau, info, trans)

Arguments

public subroutine krylov_exptA_rdp(vec_out, A, vec_in, tau, info, trans)

Arguments

public subroutine krylov_exptA_csp(vec_out, A, vec_in, tau, info, trans)

Arguments

public subroutine krylov_exptA_cdp(vec_out, A, vec_in, tau, info, trans)

Arguments

Abstract Interfaces

abstract interface

public subroutine abstract_exptA_cdp(vec_out, A, vec_in, tau, info, trans)

Arguments

abstract interface

public subroutine abstract_exptA_csp(vec_out, A, vec_in, tau, info, trans)

Arguments

abstract interface

public subroutine abstract_exptA_rdp(vec_out, A, vec_in, tau, info, trans)

Arguments

abstract interface

public subroutine abstract_exptA_rsp(vec_out, A, vec_in, tau, info, trans)

Arguments

Subroutines

public subroutine krylov_exptA_cdp(vec_out, A, vec_in, tau, info, trans)

Arguments

public subroutine krylov_exptA_csp(vec_out, A, vec_in, tau, info, trans)

Arguments

public subroutine krylov_exptA_rdp(vec_out, A, vec_in, tau, info, trans)

Arguments

public subroutine krylov_exptA_rsp(vec_out, A, vec_in, tau, info, trans)

Arguments