eigs – LightKrylov

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(in) 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.

Module Procedures

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

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_linop_rsp),	intent(in)			::	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.
integer,	intent(in),	optional		::	kdim
procedure(eigvals_select_sp),		optional		::	select	Desired number of eigenpairs.
real(kind=sp),	intent(in),	optional		::	tolerance	Tolerance.
logical,	intent(in),	optional		::	transpose	Determine whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.

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

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_linop_rdp),	intent(in)			::	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.
integer,	intent(in),	optional		::	kdim
procedure(eigvals_select_dp),		optional		::	select	Desired number of eigenpairs.
real(kind=dp),	intent(in),	optional		::	tolerance	Tolerance.
logical,	intent(in),	optional		::	transpose	Determine whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.

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

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_linop_csp),	intent(in)			::	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.
integer,	intent(in),	optional		::	kdim
procedure(eigvals_select_sp),		optional		::	select	Desired number of eigenpairs.
real(kind=sp),	intent(in),	optional		::	tolerance	Tolerance.
logical,	intent(in),	optional		::	transpose	Determine whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.

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

Arguments

Type	Intent	Optional	Attributes		Name
class(abstract_linop_cdp),	intent(in)			::	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.
integer,	intent(in),	optional		::	kdim
procedure(eigvals_select_dp),		optional		::	select	Desired number of eigenpairs.
real(kind=dp),	intent(in),	optional		::	tolerance	Tolerance.
logical,	intent(in),	optional		::	transpose	Determine whether $\mathbf{A}$ or $\mathbf{A}^H$ is being used.

eigs Interface

Contents

Module Procedures

public interface eigs

Description

Syntax

Arguments

Module Procedures

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

Arguments

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

Arguments

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

Arguments

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

Arguments