LightKrylov_IterativeSolvers

This module provides some of the most important computational routines provided by LightKrylov. These include:

eigs : Compute the leading eigenpairs of a square linear operator $A$ .
eighs : Compute the leading eigenpairs of a symmetric positive definite operator $A$ .
svds : Compute the leading singular triplets of a linear operator $A$ .
gmres : Solve the linear system $Ax = b$ using the generalized minimum residual method.
cg : Solve the linear system $Ax = b$ where $A$ is symmetric positive definite using the Conjugate Gradient method.

It also provides abstract interfaces to pass user-defined solvers and preconditioners to LightKrylov. Note that these features are still experimental however.

Uses

- stdlib_stats
- LightKrylov_Logger
- LightKrylov_Constants
- LightKrylov_Timing
- iso_fortran_env
- stdlib_optval
- LightKrylov_Utils
- LightKrylov_AbstractLinops
- stdlib_sorting
- LightKrylov_BaseKrylov
- stdlib_io_npy
- LightKrylov_AbstractVectors

Used by

- Descendants:
- cg_solver
- fgmres_solver
- gmres_solver
- hermitian_eigensolvers
- svds_solver

Interfaces

public interface cg

Description

Given a symmetric (positive definite) matrix $A$ , solves the linear system

$Ax = b$

using the Conjugate Gradient method.

References

Hestenes, M. R., and Stiefel, E. (1952). "Methods of Conjugate Gradients for Solving Linear Systems," Journal of Research of the National Bureau of Standards, 49(6), 409–436.

Syntax

    call cg(A, b, x, info [, rtol] [, atol] [, preconditioner] [, options])

Arguments

A : Linear operator derived from one of the abstract_sym_linop or abstract_hermitian_linop types provided
b : Right-hand side vector derived from one the abstract_vector types provided by the AbstractVectors module. It needs to have the same type and kind as A. It is an intent(in) argument.
x : On entry, initial guess for the solution. On exit, the solution computed by cg. It is a vector derived from one the abstract_vector types provided by the AbstractVectors module. It needs to have the same type and kind as A. It is an intent(inout) argument.
info : integer information flag.
rtol (optional) : real relative tolerance for the solver.
atol (optional) : real absolute tolerance for the solver.
preconditioner (optional) : Right preconditioner used to solve the system. It needs to be consistent with the abstract_preconditioner interface. It is an intent(in) argument.
options (optional) : Container for the gmres options given by the cg_opts type. It is an intent(in) argument.

private module subroutine cg_cdp(A, b, x, info, rtol, atol, preconditioner, options, meta)

Arguments

Type	Intent	Optional		Name
class(abstract_hermitian_linop_cdp),	intent(inout)		::	A	Linear operator to be inverted.
class(abstract_vector_cdp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_cdp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=dp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=dp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_cdp),	intent(inout),	optional	::	preconditioner	Preconditioner (not yet supported).
type(cg_dp_opts),	intent(in),	optional	::	options	Options for the conjugate gradient solver.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine cg_csp(A, b, x, info, rtol, atol, preconditioner, options, meta)

Arguments

Type	Intent	Optional		Name
class(abstract_hermitian_linop_csp),	intent(inout)		::	A	Linear operator to be inverted.
class(abstract_vector_csp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_csp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=sp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=sp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_csp),	intent(inout),	optional	::	preconditioner	Preconditioner (not yet supported).
type(cg_sp_opts),	intent(in),	optional	::	options	Options for the conjugate gradient solver.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine cg_rdp(A, b, x, info, rtol, atol, preconditioner, options, meta)

Arguments

Type	Intent	Optional		Name
class(abstract_sym_linop_rdp),	intent(inout)		::	A	Linear operator to be inverted.
class(abstract_vector_rdp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_rdp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=dp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=dp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_rdp),	intent(inout),	optional	::	preconditioner	Preconditioner (not yet supported).
type(cg_dp_opts),	intent(in),	optional	::	options	Options for the conjugate gradient solver.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine cg_rsp(A, b, x, info, rtol, atol, preconditioner, options, meta)

Arguments

Type	Intent	Optional		Name
class(abstract_sym_linop_rsp),	intent(inout)		::	A	Linear operator to be inverted.
class(abstract_vector_rsp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_rsp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=sp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=sp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_rsp),	intent(inout),	optional	::	preconditioner	Preconditioner (not yet supported).
type(cg_sp_opts),	intent(in),	optional	::	options	Options for the conjugate gradient solver.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

public interface eighs

Description

Computes the leading eigenpairs of a symmetric operator $A$ using the Lanczos iterative process. Given a square linear operator $A$ , it finds the leading eigvalues and eigvectors such that:

$Ax = \lambda x$

The subspace $X$ is constructed via Lanczos factorization, resulting in a symmetric tridiagonal matrix $T$ . The eigenvalues of $A$ are approximated by those of $T$ and the eigenvectors are computed accordingly.

References

Lanczos, C. (1950). "An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators". United States Governm. Press Office.

Syntax

    call eighs(A, X, eigvals, residuals, info [, kdim] [,tolerance])

Arguments

A : Linear operator derived from abstract_sym_linop_rsp, abstract_sym_linop_rdp, abstract_hermitian_linop_csp or abstract_hermitian_linop_cdp whose leading eigenpairs need to be computed. It is an intent(inout) argument.
X : Array of abstract_vectors with the same type and kind as A. On exit, it contains the leading eigenvectors of A. Note that the dimension of X fixes the number of eigenpairs computed.
eigvals : Rank-1 array of real numbers. On exit, it contains the leading eigenvalues of A. It is an intent(out) argument.
residuals : Rank-1 array of real numbers. On exit, it contains the residuals associated with each eigenpairs. It is an intent(out) argument.
info : integer Information flag.
kdim (optional) : integer, maximum dimension of the Krylov subspace used to approximate the leading eigenpairs. It is an intent(in) argument. By default, kdim = 4*size(X).
tolerance (optional) : real tolerance below which an eigenpair is considered as being converged. It is an intent(in) agument. By default, tolerance = rtol_sp or tolerance = rtol_dp.

Note

This implementation does not currently include an automatic restarting procedure such as krylov_schur for eigs. This is work in progress.

private module subroutine eighs_cdp(A, X, eigvals, residuals, info, x0, kdim, tolerance, write_intermediate)

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_hermitian_linop_cdp),	intent(inout)			::	A	Linear operator whose leading eigenpairs need to be computed.
class(abstract_vector_cdp),	intent(out)			::	X(:)	Leading eigevectors of $\mathbf{A}$ .
real(kind=dp),	intent(out),		allocatable	::	eigvals(:)	Leading eigenvalues of $\mathbf{A}$ .
real(kind=dp),	intent(out),		allocatable	::	residuals(:)	Residuals associated to each Ritz eigenpairs.
integer,	intent(out)			::	info	Information flag.
class(abstract_vector_cdp),	intent(in),	optional		::	x0	Optional starting vector to generate the Krylov subspace.
integer,	intent(in),	optional		::	kdim	Desired number of eigenpairs.
real(kind=dp),	intent(in),	optional		::	tolerance	Tolerance
logical,	intent(in),	optional		::	write_intermediate	Write intermediate eigenvalues to file during iteration?

private module subroutine eighs_csp(A, X, eigvals, residuals, info, x0, kdim, tolerance, write_intermediate)

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_hermitian_linop_csp),	intent(inout)			::	A	Linear operator whose leading eigenpairs need to be computed.
class(abstract_vector_csp),	intent(out)			::	X(:)	Leading eigevectors of $\mathbf{A}$ .
real(kind=sp),	intent(out),		allocatable	::	eigvals(:)	Leading eigenvalues of $\mathbf{A}$ .
real(kind=sp),	intent(out),		allocatable	::	residuals(:)	Residuals associated to each Ritz eigenpairs.
integer,	intent(out)			::	info	Information flag.
class(abstract_vector_csp),	intent(in),	optional		::	x0	Optional starting vector to generate the Krylov subspace.
integer,	intent(in),	optional		::	kdim	Desired number of eigenpairs.
real(kind=sp),	intent(in),	optional		::	tolerance	Tolerance
logical,	intent(in),	optional		::	write_intermediate	Write intermediate eigenvalues to file during iteration?

private module subroutine eighs_rdp(A, X, eigvals, residuals, info, x0, kdim, tolerance, write_intermediate)

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_sym_linop_rdp),	intent(inout)			::	A	Linear operator whose leading eigenpairs need to be computed.
class(abstract_vector_rdp),	intent(out)			::	X(:)	Leading eigevectors of $\mathbf{A}$ .
real(kind=dp),	intent(out),		allocatable	::	eigvals(:)	Leading eigenvalues of $\mathbf{A}$ .
real(kind=dp),	intent(out),		allocatable	::	residuals(:)	Residuals associated to each Ritz eigenpairs.
integer,	intent(out)			::	info	Information flag.
class(abstract_vector_rdp),	intent(in),	optional		::	x0	Optional starting vector to generate the Krylov subspace.
integer,	intent(in),	optional		::	kdim	Desired number of eigenpairs.
real(kind=dp),	intent(in),	optional		::	tolerance	Tolerance
logical,	intent(in),	optional		::	write_intermediate	Write intermediate eigenvalues to file during iteration?

private module subroutine eighs_rsp(A, X, eigvals, residuals, info, x0, kdim, tolerance, write_intermediate)

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_sym_linop_rsp),	intent(inout)			::	A	Linear operator whose leading eigenpairs need to be computed.
class(abstract_vector_rsp),	intent(out)			::	X(:)	Leading eigevectors of $\mathbf{A}$ .
real(kind=sp),	intent(out),		allocatable	::	eigvals(:)	Leading eigenvalues of $\mathbf{A}$ .
real(kind=sp),	intent(out),		allocatable	::	residuals(:)	Residuals associated to each Ritz eigenpairs.
integer,	intent(out)			::	info	Information flag.
class(abstract_vector_rsp),	intent(in),	optional		::	x0	Optional starting vector to generate the Krylov subspace.
integer,	intent(in),	optional		::	kdim	Desired number of eigenpairs.
real(kind=sp),	intent(in),	optional		::	tolerance	Tolerance
logical,	intent(in),	optional		::	write_intermediate	Write intermediate eigenvalues to file during iteration?

public interface eigs

Description

Computes the leading eigenpairs of a square linear operator $A$ using the Arnoldi iterative process. Given a square linear operator $A$ , it finds the leading eigvalues and eigvectorss such that:

$Ax = \lambda x$

$A^H x = \lambda x.$

The subspace $X$ is constructed via Arnoldi factorization, resulting in an upper Hessenberg matrix $H$ . The eigenvalues of $A$ are approximated by those of $H$ and the eigenvectors are computed accordingly.

References

Arnoldi, W. E. (1951). "The Principle of Minimized Iterations in the Solution of the Matrix Eigenvalue Problem." Quarterly of Applied Mathematics, 9(1), 17–29.

Syntax

    call eigs(A, X, eigvals, residuals, info [, kdim] [, select] [,tolerance] [, transpose])

Arguments

A : Linear operator derived from abstract_sym_linop_rsp, abstract_sym_linop_rdp, abstract_hermitian_linop_csp or abstract_hermitian_linop_cdp whose leading eigenpairs need to be computed. It is an intent(inout) argument.
X : Array of abstract_vectors with the same type and kind as A. On exit, it contains the leading eigenvectors of A. Note that the dimension of X fixes the number of eigenpairs computed.
eigvals : Rank-1 array of real numbers. On exit, it contains the leading eigenvalues of A. It is an intent(out) argument.
residuals : Rank-1 array of real numbers. On exit, it contains the residuals associated with each eigenpairs. It is an intent(out) argument.
info : integer Information flag.
kdim (optional) : integer, maximum dimension of the Krylov subspace used to approximate the leading eigenpairs. It is an intent(in) argument. By default, kdim = 4*size(X).
select (optional) : Function to select which eigenvalues to compute.
tolerance (optional) : real tolerance below which an eigenpair is considered as being converged. It is an intent(in) agument. By default, tolerance = rtol_sp or tolerance = rtol_dp.
transpose (optional) : logical flag determining whether the eigenvalues of $A$ or $A^H$ need to be computed.

private subroutine eigs_rsp(A, X, eigvals, residuals, info, x0, kdim, tolerance, transpose, write_intermediate)

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_linop_rsp),	intent(inout)			::	A	Linear operator whose leading eigenpairs need to be computed.
class(abstract_vector_rsp),	intent(out)			::	X(:)	Leading eigenvectors of $\mathbf{A}$ .
complex(kind=sp),	intent(out),		allocatable	::	eigvals(:)	Leading eigenvalues of $\mathbf{A}$ .
real(kind=sp),	intent(out),		allocatable	::	residuals(:)	Residuals associated to each Ritz eigenpair.
integer,	intent(out)			::	info	Information flag.
class(abstract_vector_rsp),	intent(in),	optional		::	x0	Optional starting vector for generating the Krylov subspace.
integer,	intent(in),	optional		::	kdim	Maximum dimension of the Krylov subspace (optional).
real(kind=sp),	intent(in),	optional		::	tolerance	Tolerance.
logical,	intent(in),	optional		::	transpose	Determine whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
logical,	intent(in),	optional		::	write_intermediate	Write intermediate eigenvalues to file during iteration?

private subroutine eigs_rdp(A, X, eigvals, residuals, info, x0, kdim, tolerance, transpose, write_intermediate)

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_linop_rdp),	intent(inout)			::	A	Linear operator whose leading eigenpairs need to be computed.
class(abstract_vector_rdp),	intent(out)			::	X(:)	Leading eigenvectors of $\mathbf{A}$ .
complex(kind=dp),	intent(out),		allocatable	::	eigvals(:)	Leading eigenvalues of $\mathbf{A}$ .
real(kind=dp),	intent(out),		allocatable	::	residuals(:)	Residuals associated to each Ritz eigenpair.
integer,	intent(out)			::	info	Information flag.
class(abstract_vector_rdp),	intent(in),	optional		::	x0	Optional starting vector for generating the Krylov subspace.
integer,	intent(in),	optional		::	kdim	Maximum dimension of the Krylov subspace (optional).
real(kind=dp),	intent(in),	optional		::	tolerance	Tolerance.
logical,	intent(in),	optional		::	transpose	Determine whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
logical,	intent(in),	optional		::	write_intermediate	Write intermediate eigenvalues to file during iteration?

private subroutine eigs_csp(A, X, eigvals, residuals, info, x0, kdim, tolerance, transpose, write_intermediate)

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_linop_csp),	intent(inout)			::	A	Linear operator whose leading eigenpairs need to be computed.
class(abstract_vector_csp),	intent(out)			::	X(:)	Leading eigenvectors of $\mathbf{A}$ .
complex(kind=sp),	intent(out),		allocatable	::	eigvals(:)	Leading eigenvalues of $\mathbf{A}$ .
real(kind=sp),	intent(out),		allocatable	::	residuals(:)	Residuals associated to each Ritz eigenpair.
integer,	intent(out)			::	info	Information flag.
class(abstract_vector_csp),	intent(in),	optional		::	x0	Optional starting vector for generating the Krylov subspace.
integer,	intent(in),	optional		::	kdim	Maximum dimension of the Krylov subspace (optional).
real(kind=sp),	intent(in),	optional		::	tolerance	Tolerance.
logical,	intent(in),	optional		::	transpose	Determine whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
logical,	intent(in),	optional		::	write_intermediate	Write intermediate eigenvalues to file during iteration?

private subroutine eigs_cdp(A, X, eigvals, residuals, info, x0, kdim, tolerance, transpose, write_intermediate)

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_linop_cdp),	intent(inout)			::	A	Linear operator whose leading eigenpairs need to be computed.
class(abstract_vector_cdp),	intent(out)			::	X(:)	Leading eigenvectors of $\mathbf{A}$ .
complex(kind=dp),	intent(out),		allocatable	::	eigvals(:)	Leading eigenvalues of $\mathbf{A}$ .
real(kind=dp),	intent(out),		allocatable	::	residuals(:)	Residuals associated to each Ritz eigenpair.
integer,	intent(out)			::	info	Information flag.
class(abstract_vector_cdp),	intent(in),	optional		::	x0	Optional starting vector for generating the Krylov subspace.
integer,	intent(in),	optional		::	kdim	Maximum dimension of the Krylov subspace (optional).
real(kind=dp),	intent(in),	optional		::	tolerance	Tolerance.
logical,	intent(in),	optional		::	transpose	Determine whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
logical,	intent(in),	optional		::	write_intermediate	Write intermediate eigenvalues to file during iteration?

public interface fgmres

Description

Solve a square linear system of equations

$Ax = b$

using the Flexible Generalized Minimum RESidual (FGMRES) method.

References

Saad Y. and Schultz M. H. "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems." SIAM Journal on Scientific and Statistical Computing, 7(3), 1986.

Syntax

    call fgmres(A, b, x, info [, rtol] [, atol] [, preconditioner] [, options] [, transpose])

Arguments

A : Linear operator derived from one of the abstract_linop provided by the AbstractLinops module. It is an intent(inout) argument.
b : Right-hand side vector derived from one the abstract_vector types provided by the AbstractVectors module. It needs to have the same type and kind as A. It is an intent(in) argument.
x : On entry, initial guess for the solution. On exit, the solution computed by gmres. It is a vector derived from one the abstract_vector types provided by the AbstractVectors module. It needs to have the same type and kind as A. It is an intent(inout) argument.
info : integer information flag.
rtol (optional) : real relative tolerance for the solver.
atol (optional) : real absolute tolerance for the solver.
preconditioner (optional) : Right preconditioner used to solve the system. It needs to be consistent with the abstract_preconditioner interface. It is an intent(in) argument.
options (optional) : Container for the gmres options given by the gmres_opts type. It is an intent(in) argument.
transpose (optional): logical flag controlling whether $Ax = b$ or $A^H x = b$ is being solved.

private module subroutine dense_fgmres_cdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
complex(kind=dp),	intent(in)		::	A(:,:)	Linear operator to be inverted.
complex(kind=dp),	intent(in)		::	b(:)	Right-hand side vector.
complex(kind=dp),	intent(inout)		::	x(:)	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=dp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=dp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_cdp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine dense_fgmres_csp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
complex(kind=sp),	intent(in)		::	A(:,:)	Linear operator to be inverted.
complex(kind=sp),	intent(in)		::	b(:)	Right-hand side vector.
complex(kind=sp),	intent(inout)		::	x(:)	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=sp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=sp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_csp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine dense_fgmres_rdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
real(kind=dp),	intent(in)		::	A(:,:)	Linear operator to be inverted.
real(kind=dp),	intent(in)		::	b(:)	Right-hand side vector.
real(kind=dp),	intent(inout)		::	x(:)	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=dp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=dp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_rdp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine dense_fgmres_rsp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
real(kind=sp),	intent(in)		::	A(:,:)	Linear operator to be inverted.
real(kind=sp),	intent(in)		::	b(:)	Right-hand side vector.
real(kind=sp),	intent(inout)		::	x(:)	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=sp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=sp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_rsp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine fgmres_cdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
class(abstract_linop_cdp),	intent(inout)		::	A	Linear operator to be inverted.
class(abstract_vector_cdp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_cdp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=dp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=dp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_cdp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine fgmres_csp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
class(abstract_linop_csp),	intent(inout)		::	A	Linear operator to be inverted.
class(abstract_vector_csp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_csp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=sp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=sp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_csp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine fgmres_rdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
class(abstract_linop_rdp),	intent(inout)		::	A	Linear operator to be inverted.
class(abstract_vector_rdp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_rdp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=dp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=dp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_rdp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine fgmres_rsp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
class(abstract_linop_rsp),	intent(inout)		::	A	Linear operator to be inverted.
class(abstract_vector_rsp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_rsp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=sp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=sp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_rsp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

public interface gmres

Description

Solve a square linear system of equations

$Ax = b$

using the Generalized Minimum RESidual (GMRES) method.

References

Saad Y. and Schultz M. H. "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems." SIAM Journal on Scientific and Statistical Computing, 7(3), 1986.

Syntax

    call gmres(A, b, x, info [, rtol] [, atol] [, preconditioner] [, options] [, transpose])

Arguments

A : Linear operator derived from one of the abstract_linop provided by the AbstractLinops module. It is an intent(inout) argument.
b : Right-hand side vector derived from one the abstract_vector types provided by the AbstractVectors module. It needs to have the same type and kind as A. It is an intent(in) argument.
x : On entry, initial guess for the solution. On exit, the solution computed by gmres. It is a vector derived from one the abstract_vector types provided by the AbstractVectors module. It needs to have the same type and kind as A. It is an intent(inout) argument.
info : integer information flag.
rtol (optional) : real relative tolerance for the solver.
atol (optional) : real absolute tolerance for the solver.
preconditioner (optional) : Right preconditioner used to solve the system. It needs to be consistent with the abstract_preconditioner interface. It is an intent(in) argument.
options (optional) : Container for the gmres options given by the gmres_opts type. It is an intent(in) argument.
transpose (optional): logical flag controlling whether $Ax = b$ or $A^H x = b$ is being solver.
meta (optional) : Container for the gmres metada. It needs to be of type gmres_medata.

private module subroutine dense_gmres_cdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
complex(kind=dp),	intent(in)		::	A(:,:)	Linear operator to be inverted.
complex(kind=dp),	intent(in)		::	b(:)	Right-hand side vector.
complex(kind=dp),	intent(inout)		::	x(:)	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=dp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=dp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_cdp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine dense_gmres_csp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
complex(kind=sp),	intent(in)		::	A(:,:)	Linear operator to be inverted.
complex(kind=sp),	intent(in)		::	b(:)	Right-hand side vector.
complex(kind=sp),	intent(inout)		::	x(:)	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=sp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=sp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_csp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine dense_gmres_rdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
real(kind=dp),	intent(in)		::	A(:,:)	Linear operator to be inverted.
real(kind=dp),	intent(in)		::	b(:)	Right-hand side vector.
real(kind=dp),	intent(inout)		::	x(:)	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=dp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=dp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_rdp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine dense_gmres_rsp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
real(kind=sp),	intent(in)		::	A(:,:)	Linear operator to be inverted.
real(kind=sp),	intent(in)		::	b(:)	Right-hand side vector.
real(kind=sp),	intent(inout)		::	x(:)	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=sp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=sp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_rsp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine gmres_cdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
class(abstract_linop_cdp),	intent(inout)		::	A	Linear operator to be inverted.
class(abstract_vector_cdp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_cdp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=dp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=dp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_cdp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine gmres_csp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
class(abstract_linop_csp),	intent(inout)		::	A	Linear operator to be inverted.
class(abstract_vector_csp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_csp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=sp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=sp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_csp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine gmres_rdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
class(abstract_linop_rdp),	intent(inout)		::	A	Linear operator to be inverted.
class(abstract_vector_rdp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_rdp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=dp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=dp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_rdp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

private module subroutine gmres_rsp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

Type	Intent	Optional		Name
class(abstract_linop_rsp),	intent(inout)		::	A	Linear operator to be inverted.
class(abstract_vector_rsp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_rsp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag.
real(kind=sp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=sp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_rsp),	intent(inout),	optional	::	preconditioner	Preconditioner (optional).
class(abstract_opts),	intent(in),	optional	::	options	GMRES options.
logical,	intent(in),	optional	::	transpose	Whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

public interface save_eigenspectrum

Description

Utility function to save the eigenspectrum computed from the Arnoldi factorization. It outpost a .npy file.

Syntax

    call save_eigenspectrum(eigvals, residuals, fname)

Arguments

eigvals : complex rank-1 array containing the eigenvalues.
residuals : real rank-1 array containing the residuals associated to each eigenvalues.

fname : Name of the file to save the eigenspectrum.

private module subroutine save_eigenspectrum_cdp(lambda, residuals, fname)

Saves the eigenspectrum and corresponding residuals to disk use the npy binary format.

Arguments

Type	Intent		Name
complex(kind=dp),	intent(in)	::	lambda(:)	Eigenalues.
real(kind=dp),	intent(in)	::	residuals(:)	Residual of the corresponding Ritz eigenpairs.
character(len=*),	intent(in)	::	fname	Name of the output file.

private module subroutine save_eigenspectrum_csp(lambda, residuals, fname)

Saves the eigenspectrum and corresponding residuals to disk use the npy binary format.

Arguments

Type	Intent		Name
complex(kind=sp),	intent(in)	::	lambda(:)	Eigenalues.
real(kind=sp),	intent(in)	::	residuals(:)	Residual of the corresponding Ritz eigenpairs.
character(len=*),	intent(in)	::	fname	Name of the output file.

private module subroutine save_eigenspectrum_rdp(lambda, residuals, fname)

Saves the eigenspectrum and corresponding residuals to disk use the npy binary format.

Arguments

Type	Intent		Name
real(kind=dp),	intent(in)	::	lambda(:)	Eigenalues.
real(kind=dp),	intent(in)	::	residuals(:)	Residual of the corresponding Ritz eigenpairs.
character(len=*),	intent(in)	::	fname	Name of the output file.

private module subroutine save_eigenspectrum_rsp(lambda, residuals, fname)

Saves the eigenspectrum and corresponding residuals to disk use the npy binary format.

Arguments

Type	Intent		Name
real(kind=sp),	intent(in)	::	lambda(:)	Eigenalues.
real(kind=sp),	intent(in)	::	residuals(:)	Residual of the corresponding Ritz eigenpairs.
character(len=*),	intent(in)	::	fname	Name of the output file.

public interface svds

Description

Computes the leading singular triplets of an arbitrary linear operator $A$ using the Lanczos iterative process. Given a linear operator $A$ , it finds the leading singular values and singular vectors such that:

$\begin{aligned} Av & = \sigma u \\ A^H u & = \sigma v. \end{aligned}$

The subspaces $U$ and $V$ are constructed via Lanczos factorization, resulting in a bidiagonal matrix $B$ . The singular values of $A$ are approximated by those of $B$ and the singular vectors are computed accordingly.

References

Golub, G. H., & Kahan, W. (1965). "Calculating the Singular Values and Pseudo-Inverse of a Matrix."
Baglama, J., & Reichel, L. (2005). "Augmented implicitly restarted Lanczos bidiagonalization methods."
R. M. Larsen. "Lanczos bidiagonalization with partial reorthogonalization." Technical Report, 1998.

Syntax

    call svds(A, U, S, V, residuals, info [, kdim] [,tolerance])

Arguments

A : Linear operator derived from abstract_sym_linop_rsp, abstract_sym_linop_rdp, abstract_hermitian_linop_csp or abstract_hermitian_linop_cdp whose leading eigenpairs need to be computed. It is an intent(inout) argument.
U : Array of abstract_vectors with the same type and kind as A. On exit, it contains the left singular vectors of A. Note that the dimension of U fixes the number of eigenpairs computed.
S : Rank-1 array of real numbers. On exit, it contains the leading singular values of A. It is an intent(out) argument.
V : Array of abstract_vectors with the same type and kind as A. On exit, it contains the left singular vectors of A. Note that the dimension of U fixes the number of eigenpairs computed.
residuals : Rank-1 array of real numbers. On exit, it contains the residuals associated with each singular triplet. It is an intent(out) argument.
info : integer Information flag.
kdim (optional) : integer, maximum dimension of the Krylov subspace used to approximate the leading singular triplets. It is an intent(in) argument. By default, kdim = 4*size(X).
tolerance (optional) : real tolerance below which a triplet is considered as being converged. It is an intent(in) agument. By default, tolerance = rtol_sportolerance = rtol_dp`.

Note

This implementation does not currently include an automatic restarting procedure such as krylov_schur for eigs. This is work in progress.

private module subroutine svds_cdp(A, U, S, V, residuals, info, u0, kdim, tolerance, write_intermediate)

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_linop_cdp),	intent(inout)			::	A	Linear operator whose leading singular triplets need to be computed.
class(abstract_vector_cdp),	intent(out)			::	U(:)	Leading left singular vectors.
real(kind=dp),	intent(out),		allocatable	::	S(:)	Leading singular values.
class(abstract_vector_cdp),	intent(out)			::	V(:)	Leading right singular vectors.
real(kind=dp),	intent(out),		allocatable	::	residuals(:)	Residuals associated to each Ritz eigenpair.
integer,	intent(out)			::	info	Information flag.
class(abstract_vector_cdp),	intent(in),	optional		::	u0
integer,	intent(in),	optional		::	kdim	Desired number of eigenpairs.
real(kind=dp),	intent(in),	optional		::	tolerance	Tolerance.
logical,	intent(in),	optional		::	write_intermediate	Write intermediate eigenvalues to file during iteration?

private module subroutine svds_csp(A, U, S, V, residuals, info, u0, kdim, tolerance, write_intermediate)

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_linop_csp),	intent(inout)			::	A	Linear operator whose leading singular triplets need to be computed.
class(abstract_vector_csp),	intent(out)			::	U(:)	Leading left singular vectors.
real(kind=sp),	intent(out),		allocatable	::	S(:)	Leading singular values.
class(abstract_vector_csp),	intent(out)			::	V(:)	Leading right singular vectors.
real(kind=sp),	intent(out),		allocatable	::	residuals(:)	Residuals associated to each Ritz eigenpair.
integer,	intent(out)			::	info	Information flag.
class(abstract_vector_csp),	intent(in),	optional		::	u0
integer,	intent(in),	optional		::	kdim	Desired number of eigenpairs.
real(kind=sp),	intent(in),	optional		::	tolerance	Tolerance.
logical,	intent(in),	optional		::	write_intermediate	Write intermediate eigenvalues to file during iteration?

private module subroutine svds_rdp(A, U, S, V, residuals, info, u0, kdim, tolerance, write_intermediate)

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_linop_rdp),	intent(inout)			::	A	Linear operator whose leading singular triplets need to be computed.
class(abstract_vector_rdp),	intent(out)			::	U(:)	Leading left singular vectors.
real(kind=dp),	intent(out),		allocatable	::	S(:)	Leading singular values.
class(abstract_vector_rdp),	intent(out)			::	V(:)	Leading right singular vectors.
real(kind=dp),	intent(out),		allocatable	::	residuals(:)	Residuals associated to each Ritz eigenpair.
integer,	intent(out)			::	info	Information flag.
class(abstract_vector_rdp),	intent(in),	optional		::	u0
integer,	intent(in),	optional		::	kdim	Desired number of eigenpairs.
real(kind=dp),	intent(in),	optional		::	tolerance	Tolerance.
logical,	intent(in),	optional		::	write_intermediate	Write intermediate eigenvalues to file during iteration?

private module subroutine svds_rsp(A, U, S, V, residuals, info, u0, kdim, tolerance, write_intermediate)

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_linop_rsp),	intent(inout)			::	A	Linear operator whose leading singular triplets need to be computed.
class(abstract_vector_rsp),	intent(out)			::	U(:)	Leading left singular vectors.
real(kind=sp),	intent(out),		allocatable	::	S(:)	Leading singular values.
class(abstract_vector_rsp),	intent(out)			::	V(:)	Leading right singular vectors.
real(kind=sp),	intent(out),		allocatable	::	residuals(:)	Residuals associated to each Ritz eigenpair.
integer,	intent(out)			::	info	Information flag.
class(abstract_vector_rsp),	intent(in),	optional		::	u0
integer,	intent(in),	optional		::	kdim	Desired number of eigenpairs.
real(kind=sp),	intent(in),	optional		::	tolerance	Tolerance.
logical,	intent(in),	optional		::	write_intermediate	Write intermediate eigenvalues to file during iteration?

Abstract Interfaces

abstract interface

public subroutine abstract_linear_solver_cdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Abstract interface to use a user-defined linear solver in LightKrylov.

Arguments

Type	Intent	Optional		Name
class(abstract_linop_cdp),	intent(inout)		::	A	Linear operator to invert.
class(abstract_vector_cdp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_cdp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag. In case of successful exit, the flag should return the number of iterations required for convergence.
real(kind=dp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=dp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_cdp),	intent(inout),	optional	::	preconditioner	Preconditioner.
class(abstract_opts),	intent(in),	optional	::	options	Options passed to the linear solver.
logical,	intent(in),	optional	::	transpose	Determine whether $\mathbf{A}$ (`.false.`) or $\mathbf{A}^T$ (`.true.`) is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

abstract interface

public subroutine abstract_linear_solver_csp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Abstract interface to use a user-defined linear solver in LightKrylov.

Arguments

Type	Intent	Optional		Name
class(abstract_linop_csp),	intent(inout)		::	A	Linear operator to invert.
class(abstract_vector_csp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_csp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag. In case of successful exit, the flag should return the number of iterations required for convergence.
real(kind=sp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=sp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_csp),	intent(inout),	optional	::	preconditioner	Preconditioner.
class(abstract_opts),	intent(in),	optional	::	options	Options passed to the linear solver.
logical,	intent(in),	optional	::	transpose	Determine whether $\mathbf{A}$ (`.false.`) or $\mathbf{A}^T$ (`.true.`) is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

abstract interface

public subroutine abstract_linear_solver_rdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Abstract interface to use a user-defined linear solver in LightKrylov.

Arguments

Type	Intent	Optional		Name
class(abstract_linop_rdp),	intent(inout)		::	A	Linear operator to invert.
class(abstract_vector_rdp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_rdp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag. In case of successful exit, the flag should return the number of iterations required for convergence.
real(kind=dp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=dp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_rdp),	intent(inout),	optional	::	preconditioner	Preconditioner.
class(abstract_opts),	intent(in),	optional	::	options	Options passed to the linear solver.
logical,	intent(in),	optional	::	transpose	Determine whether $\mathbf{A}$ (`.false.`) or $\mathbf{A}^T$ (`.true.`) is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

abstract interface

public subroutine abstract_linear_solver_rsp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Abstract interface to use a user-defined linear solver in LightKrylov.

Arguments

Type	Intent	Optional		Name
class(abstract_linop_rsp),	intent(inout)		::	A	Linear operator to invert.
class(abstract_vector_rsp),	intent(in)		::	b	Right-hand side vector.
class(abstract_vector_rsp),	intent(inout)		::	x	Solution vector.
integer,	intent(out)		::	info	Information flag. In case of successful exit, the flag should return the number of iterations required for convergence.
real(kind=sp),	intent(in),	optional	::	rtol	Relative solver tolerance
real(kind=sp),	intent(in),	optional	::	atol	Absolute solver tolerance
class(abstract_precond_rsp),	intent(inout),	optional	::	preconditioner	Preconditioner.
class(abstract_opts),	intent(in),	optional	::	options	Options passed to the linear solver.
logical,	intent(in),	optional	::	transpose	Determine whether $\mathbf{A}$ (`.false.`) or $\mathbf{A}^T$ (`.true.`) is being used.
class(abstract_metadata),	intent(out),	optional	::	meta	Metadata.

Derived Types

type, public, abstract :: abstract_precond_cdp

Type-Bound Procedures

procedure(abstract_apply_cdp), public, deferred, pass(self) :: apply

type, public, abstract :: abstract_precond_csp

Type-Bound Procedures

procedure(abstract_apply_csp), public, deferred, pass(self) :: apply

type, public, abstract :: abstract_precond_rdp

Type-Bound Procedures

procedure(abstract_apply_rdp), public, deferred, pass(self) :: apply

type, public, abstract :: abstract_precond_rsp

Type-Bound Procedures

procedure(abstract_apply_rsp), public, deferred, pass(self) :: apply

type, public, extends(abstract_metadata) :: cg_dp_metadata

Conjugate gradient metadata.

Components

Type	Visibility	Attributes		Name		Initial
logical,	public		::	converged	=	.false.	Convergence flag
integer,	public		::	info	=	0	Copy of the information flag for completeness
integer,	public		::	n_iter	=	0	Iteration counter
real(kind=dp),	public,	dimension(:), allocatable	::	res			Residual history

Type-Bound Procedures

procedure, public, pass(self) :: print => print_cg_dp
procedure, public, pass(self) :: reset => reset_cg_dp

type, public, extends(abstract_opts) :: cg_dp_opts

Conjugate gradient options.

Components

Type	Visibility	Attributes		Name		Initial
logical,	public		::	if_print_metadata	=	.false.	Print interation metadata on exit (default = .false.)
integer,	public		::	maxiter	=	100	Maximum number of `cg` iterations (default: 100).

type, public, extends(abstract_metadata) :: cg_sp_metadata

Conjugate gradient metadata.

Components

Type	Visibility	Attributes		Name		Initial
logical,	public		::	converged	=	.false.	Convergence flag
integer,	public		::	info	=	0	Copy of the information flag for completeness
integer,	public		::	n_iter	=	0	Iteration counter
real(kind=sp),	public,	dimension(:), allocatable	::	res			Residual history

Type-Bound Procedures

procedure, public, pass(self) :: print => print_cg_sp
procedure, public, pass(self) :: reset => reset_cg_sp

type, public, extends(abstract_opts) :: cg_sp_opts

Conjugate gradient options.

Components

Type	Visibility	Attributes		Name		Initial
logical,	public		::	if_print_metadata	=	.false.	Print interation metadata on exit (default = .false.)
integer,	public		::	maxiter	=	100	Maximum number of `cg` iterations (default: 100).

type, public, extends(abstract_metadata) :: fgmres_dp_metadata

FGMRES metadata.

Components

Type	Visibility	Attributes		Name		Initial
logical,	public		::	converged	=	.false.	Convergence flag.
integer,	public		::	info	=	0	Copy of the information flag for completeness.
integer,	public		::	n_inner	=	0	Number of inner iterations.
integer,	public		::	n_iter	=	0	Total iteration counter.
integer,	public		::	n_outer	=	0	Number of outer iterations (i.e. restarts)
real(kind=dp),	public,	dimension(:), allocatable	::	res			Residual history.

Type-Bound Procedures

procedure, public, pass(self) :: print => print_fgmres_dp
procedure, public, pass(self) :: reset => reset_fgmres_dp

type, public, extends(abstract_opts) :: fgmres_dp_opts

FGMRES options.

Components

Type	Visibility		Name		Initial
logical,	public	::	if_print_metadata	=	.false.	Print iteration metadata on exit (default: .false.).
integer,	public	::	kdim	=	30	Dimension of the Krylov subspace (default: 30).
integer,	public	::	maxiter	=	10	Maximum number of `fgmres` restarts (default: 10).
logical,	public	::	sanity_check	=	.true.	Performs extra matrix-vector product for sanity check.

type, public, extends(abstract_metadata) :: fgmres_sp_metadata

FGMRES metadata.

Components

Type	Visibility	Attributes		Name		Initial
logical,	public		::	converged	=	.false.	Convergence flag.
integer,	public		::	info	=	0	Copy of the information flag for completeness.
integer,	public		::	n_inner	=	0	Number of inner iterations.
integer,	public		::	n_iter	=	0	Total iteration counter.
integer,	public		::	n_outer	=	0	Number of outer iterations (i.e. restarts)
real(kind=sp),	public,	dimension(:), allocatable	::	res			Residual history.

Type-Bound Procedures

procedure, public, pass(self) :: print => print_fgmres_sp
procedure, public, pass(self) :: reset => reset_fgmres_sp

type, public, extends(abstract_opts) :: fgmres_sp_opts

FGMRES options.

Components

Type	Visibility		Name		Initial
logical,	public	::	if_print_metadata	=	.false.	Print iteration metadata on exit (default: .false.).
integer,	public	::	kdim	=	30	Dimension of the Krylov subspace (default: 30).
integer,	public	::	maxiter	=	10	Maximum number of `fgmres` restarts (default: 10).
logical,	public	::	sanity_check	=	.true.	Performs extra matrix-vector product for sanity check.

type, public, extends(abstract_metadata) :: gmres_dp_metadata

GMRES metadata.

Components

Type	Visibility	Attributes		Name		Initial
logical,	public		::	converged	=	.false.	Convergence flag.
integer,	public		::	info	=	0	Copy of the information flag for completeness.
integer,	public		::	n_inner	=	0	Number of inner iterations.
integer,	public		::	n_iter	=	0	Total iteration counter.
integer,	public		::	n_outer	=	0	Number of outer iterations (i.e. restarts)
real(kind=dp),	public,	dimension(:), allocatable	::	res			Residual history.

Type-Bound Procedures

procedure, public, pass(self) :: print => print_gmres_dp
procedure, public, pass(self) :: reset => reset_gmres_dp

type, public, extends(abstract_opts) :: gmres_dp_opts

GMRES options.

Components

Type	Visibility		Name		Initial
logical,	public	::	if_print_metadata	=	.false.	Print iteration metadata on exit (default: .false.).
integer,	public	::	kdim	=	30	Dimension of the Krylov subspace (default: 30).
integer,	public	::	maxiter	=	10	Maximum number of `gmres` restarts (default: 10).
logical,	public	::	sanity_check	=	.true.	Performs extra matrix-vector product for sanity check.

type, public, extends(abstract_metadata) :: gmres_sp_metadata

GMRES metadata.

Components

Type	Visibility	Attributes		Name		Initial
logical,	public		::	converged	=	.false.	Convergence flag.
integer,	public		::	info	=	0	Copy of the information flag for completeness.
integer,	public		::	n_inner	=	0	Number of inner iterations.
integer,	public		::	n_iter	=	0	Total iteration counter.
integer,	public		::	n_outer	=	0	Number of outer iterations (i.e. restarts)
real(kind=sp),	public,	dimension(:), allocatable	::	res			Residual history.

Type-Bound Procedures

procedure, public, pass(self) :: print => print_gmres_sp
procedure, public, pass(self) :: reset => reset_gmres_sp

type, public, extends(abstract_opts) :: gmres_sp_opts

GMRES options.

Components

Type	Visibility		Name		Initial
logical,	public	::	if_print_metadata	=	.false.	Print iteration metadata on exit (default: .false.).
integer,	public	::	kdim	=	30	Dimension of the Krylov subspace (default: 30).
integer,	public	::	maxiter	=	10	Maximum number of `gmres` restarts (default: 10).
logical,	public	::	sanity_check	=	.true.	Performs extra matrix-vector product for sanity check.

Subroutines

public subroutine write_results_cdp(filename, vals, res, tol)

Prints the intermediate results of iterative eigenvalue/singular value decompositions

Arguments

Type	Intent		Name
character(len=*),	intent(in)	::	filename	Output filename. This file will be overwritten
complex(kind=dp),	intent(in)	::	vals(:)	Intermediate values
real(kind=dp),	intent(inout)	::	res(:)	Residuals
real(kind=dp),	intent(in)	::	tol	Convergence tolerance

public subroutine write_results_csp(filename, vals, res, tol)

Prints the intermediate results of iterative eigenvalue/singular value decompositions

Arguments

Type	Intent		Name
character(len=*),	intent(in)	::	filename	Output filename. This file will be overwritten
complex(kind=sp),	intent(in)	::	vals(:)	Intermediate values
real(kind=sp),	intent(inout)	::	res(:)	Residuals
real(kind=sp),	intent(in)	::	tol	Convergence tolerance

public subroutine write_results_rdp(filename, vals, res, tol)

Prints the intermediate results of iterative eigenvalue/singular value decompositions

Arguments

Type	Intent		Name
character(len=*),	intent(in)	::	filename	Output filename. This file will be overwritten
real(kind=dp),	intent(in)	::	vals(:)	Intermediate values
real(kind=dp),	intent(inout)	::	res(:)	Residuals
real(kind=dp),	intent(in)	::	tol	Convergence tolerance

public subroutine write_results_rsp(filename, vals, res, tol)

Prints the intermediate results of iterative eigenvalue/singular value decompositions

Arguments

Type	Intent		Name
character(len=*),	intent(in)	::	filename	Output filename. This file will be overwritten
real(kind=sp),	intent(in)	::	vals(:)	Intermediate values
real(kind=sp),	intent(inout)	::	res(:)	Residuals
real(kind=sp),	intent(in)	::	tol	Convergence tolerance

Module Procedures

module procedure /github/workspace/API-doc/module/lightkrylov_iterativesolvers.html save_eigenspectrum_cdp private module subroutine save_eigenspectrum_cdp(lambda, residuals, fname)

Arguments

Type	Intent		Name
complex(kind=dp),	intent(in)	::	lambda(:)	Eigenalues.
real(kind=dp),	intent(in)	::	residuals(:)	Residual of the corresponding Ritz eigenpairs.
character(len=*),	intent(in)	::	fname	Name of the output file.

module procedure /github/workspace/API-doc/module/lightkrylov_iterativesolvers.html save_eigenspectrum_csp private module subroutine save_eigenspectrum_csp(lambda, residuals, fname)

Arguments

Type	Intent		Name
complex(kind=sp),	intent(in)	::	lambda(:)	Eigenalues.
real(kind=sp),	intent(in)	::	residuals(:)	Residual of the corresponding Ritz eigenpairs.
character(len=*),	intent(in)	::	fname	Name of the output file.

module procedure /github/workspace/API-doc/module/lightkrylov_iterativesolvers.html save_eigenspectrum_rdp private module subroutine save_eigenspectrum_rdp(lambda, residuals, fname)

Arguments

Type	Intent		Name
real(kind=dp),	intent(in)	::	lambda(:)	Eigenalues.
real(kind=dp),	intent(in)	::	residuals(:)	Residual of the corresponding Ritz eigenpairs.
character(len=*),	intent(in)	::	fname	Name of the output file.

module procedure /github/workspace/API-doc/module/lightkrylov_iterativesolvers.html save_eigenspectrum_rsp private module subroutine save_eigenspectrum_rsp(lambda, residuals, fname)

Arguments

Type	Intent		Name
real(kind=sp),	intent(in)	::	lambda(:)	Eigenalues.
real(kind=sp),	intent(in)	::	residuals(:)	Residual of the corresponding Ritz eigenpairs.
character(len=*),	intent(in)	::	fname	Name of the output file.

LightKrylov_IterativeSolvers Module

Contents

Interfaces

Abstract Interfaces

Derived Types

Subroutines

Module Procedures

Uses

Used by

Descendants:

Interfaces

public interface cg

Description

Syntax

Arguments

private module subroutine cg_cdp(A, b, x, info, rtol, atol, preconditioner, options, meta)

Arguments

private module subroutine cg_csp(A, b, x, info, rtol, atol, preconditioner, options, meta)

Arguments

private module subroutine cg_rdp(A, b, x, info, rtol, atol, preconditioner, options, meta)

Arguments

private module subroutine cg_rsp(A, b, x, info, rtol, atol, preconditioner, options, meta)

Arguments

public interface eighs

Description

Syntax

Arguments

private module subroutine eighs_cdp(A, X, eigvals, residuals, info, x0, kdim, tolerance, write_intermediate)

Arguments

private module subroutine eighs_csp(A, X, eigvals, residuals, info, x0, kdim, tolerance, write_intermediate)

Arguments

private module subroutine eighs_rdp(A, X, eigvals, residuals, info, x0, kdim, tolerance, write_intermediate)

Arguments

private module subroutine eighs_rsp(A, X, eigvals, residuals, info, x0, kdim, tolerance, write_intermediate)

Arguments

public interface eigs

Description

Syntax

Arguments

private subroutine eigs_rsp(A, X, eigvals, residuals, info, x0, kdim, tolerance, transpose, write_intermediate)

Arguments

private subroutine eigs_rdp(A, X, eigvals, residuals, info, x0, kdim, tolerance, transpose, write_intermediate)

Arguments

private subroutine eigs_csp(A, X, eigvals, residuals, info, x0, kdim, tolerance, transpose, write_intermediate)

Arguments

private subroutine eigs_cdp(A, X, eigvals, residuals, info, x0, kdim, tolerance, transpose, write_intermediate)

Arguments

public interface fgmres

Description

Syntax

Arguments

private module subroutine dense_fgmres_cdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

private module subroutine dense_fgmres_csp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

private module subroutine dense_fgmres_rdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

private module subroutine dense_fgmres_rsp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

private module subroutine fgmres_cdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

private module subroutine fgmres_csp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

private module subroutine fgmres_rdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

private module subroutine fgmres_rsp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

public interface gmres

Description

Syntax

Arguments

private module subroutine dense_gmres_cdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

private module subroutine dense_gmres_csp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

private module subroutine dense_gmres_rdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

private module subroutine dense_gmres_rsp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

Arguments

private module subroutine gmres_cdp(A, b, x, info, rtol, atol, preconditioner, options, transpose, meta)

private module subroutine save_eigenspectrum_cdp(lambda, residuals, fname)

private module subroutine save_eigenspectrum_csp(lambda, residuals, fname)

private module subroutine save_eigenspectrum_rdp(lambda, residuals, fname)

private module subroutine save_eigenspectrum_rsp(lambda, residuals, fname)