svds Interface

public interface svds

Description

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

The subspaces and are constructed via Lanczos factorization, resulting in a bidiagonal matrix . The singular values of are approximated by those of 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(in) 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_sp or tolerance = rtol_dp.


Module Procedures

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

Arguments

Type IntentOptional Attributes Name
class(abstract_linop_rsp), intent(in) :: 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.
integer, intent(in), optional :: kdim

Desired number of eigenpairs.
real(kind=sp), intent(in), optional :: tolerance

Tolerance.

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

Arguments

Type IntentOptional Attributes Name
class(abstract_linop_rdp), intent(in) :: 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.
integer, intent(in), optional :: kdim

Desired number of eigenpairs.
real(kind=dp), intent(in), optional :: tolerance

Tolerance.

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

Arguments

Type IntentOptional Attributes Name
class(abstract_linop_csp), intent(in) :: 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.
integer, intent(in), optional :: kdim

Desired number of eigenpairs.
real(kind=sp), intent(in), optional :: tolerance

Tolerance.

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

Arguments

Type IntentOptional Attributes Name
class(abstract_linop_cdp), intent(in) :: 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.
integer, intent(in), optional :: kdim

Desired number of eigenpairs.
real(kind=dp), intent(in), optional :: tolerance

Tolerance.