Hermitian to tridiagonal

The standard approach for computing eigenpairs of dense Hermitian matrices begins by performing a unitary similarity transformation which reduces the matrix to real symmetric tridiagonal form (usually through Householder transformations). The following routines perform said reduction on a Hermitian matrix and store the scaled Householder vectors in place of the introduced zeroes.

Note

While so-called Successive Band Reduction approaches, which reduce the matrix to tridiagonal form using a two-stage process, are sometimes preferred, they are not yet supported within Elemental. Please see ELPA for such an implementation.

Implementation

Subroutine implementations

Test driver

Reduction

C++ API

void HermitianTridiag(UpperOrLower uplo, Matrix<F> &A, Matrix<F> &t)

void HermitianTridiag(UpperOrLower uplo, AbstractDistMatrix<F> &A, AbstractDistMatrix<F> &t, const HermitianTridiagCtrl &ctrl = HermitianTridiagCtrl())

Overwrites the main and sub (super) diagonal of the real matrix A with its similar symmetric tridiagonal matrix and stores the scaled Householder vectors below (above) its tridiagonal entries. Complex Hermitian reductions have the added complication of needing to also store the scalings for the Householder vectors (the scaling can be inferred since the Householder vectors must be unit length) if they are to be applied (in the column vector t).

void herm_tridiag::ExplicitCondensed(UpperOrLower uplo, Matrix<F> &A)

void herm_tridiag::ExplicitCondensed(UpperOrLower uplo, AbstractDistMatrix<F> &A, const HermitianTridiagCtrl &ctrl = HermitianTridiagCtrl())

Returns just the (appropriate triangle of the) resulting tridiagonal matrix.

type HermitianTridiagCtrl

HermitianTridiagApproach approach

GridOrder order

HermitianTridiagCtrl()

Sets approach to HERMITIAN_TRIDIAG_SQUARE and order to ROW_MAJOR.

Note

Please see the Tuning parameters section for extensive information on maximizing the performance of Householder tridiagonalization.

C API

ElHermitianTridiagCtrl

ElHermitianTridiagApproach approach

ElGridOrder order

ElHermitianTridiagCtrlDefault(ElHermitianTridiagCtrl* ctrl)

Sets approach to HERMITIAN_TRIDIAG_SQUARE and order to ROW_MAJOR.

ElError ElHermitianTridiag_s(ElUpperOrLower uplo, ElMatrix_s A, ElMatrix_s t)

ElError ElHermitianTridiag_d(ElUpperOrLower uplo, ElMatrix_d A, ElMatrix_d t)

ElError ElHermitianTridiag_c(ElUpperOrLower uplo, ElMatrix_c A, ElMatrix_c t)

ElError ElHermitianTridiag_z(ElUpperOrLower uplo, ElMatrix_z A, ElMatrix_z t)

ElError ElHermitianTridiagDist_s(ElUpperOrLower uplo, ElDistMatrix_s A, ElDistMatrix_s t)

ElError ElHermitianTridiagDist_d(ElUpperOrLower uplo, ElDistMatrix_d A, ElDistMatrix_d t)

ElError ElHermitianTridiagDist_c(ElUpperOrLower uplo, ElDistMatrix_c A, ElDistMatrix_c t)

ElError ElHermitianTridiagDist_z(ElUpperOrLower uplo, ElDistMatrix_z A, ElDistMatrix_z t)

ElError ElHermitianTridiagXDist_s(ElUpperOrLower uplo, ElDistMatrix_s A, ElDistMatrix_s t, ElHermitianTridiagCtrl ctrl)

ElError ElHermitianTridiagXDist_d(ElUpperOrLower uplo, ElDistMatrix_d A, ElDistMatrix_d t, ElHermitianTridiagCtrl ctrl)

ElError ElHermitianTridiagXDist_c(ElUpperOrLower uplo, ElDistMatrix_c A, ElDistMatrix_c t, ElHermitianTridiagCtrl ctrl)

ElError ElHermitianTridiagXDist_z(ElUpperOrLower uplo, ElDistMatrix_z A, ElDistMatrix_z t, ElHermitianTridiagCtrl ctrl)

ElError ElHermitianTridiagExplicitCondensed_s(ElUpperOrLower uplo, ElMatrix_s A)

ElError ElHermitianTridiagExplicitCondensed_d(ElUpperOrLower uplo, ElMatrix_d A)

ElError ElHermitianTridiagExplicitCondensed_c(ElUpperOrLower uplo, ElMatrix_c A)

ElError ElHermitianTridiagExplicitCondensed_z(ElUpperOrLower uplo, ElMatrix_z A)

ElError ElHermitianTridiagExplicitCondensedDist_s(ElUpperOrLower uplo, ElDistMatrix_s A)

ElError ElHermitianTridiagExplicitCondensedDist_d(ElUpperOrLower uplo, ElDistMatrix_d A)

ElError ElHermitianTridiagExplicitCondensedDist_c(ElUpperOrLower uplo, ElDistMatrix_c A)

ElError ElHermitianTridiagExplicitCondensedDist_z(ElUpperOrLower uplo, ElDistMatrix_z A)

ElError ElHermitianTridiagExplicitCondensedXDist_s(ElUpperOrLower uplo, ElDistMatrix_s A, ElHermitianTridiag ctrl)

ElError ElHermitianTridiagExplicitCondensedXDist_d(ElUpperOrLower uplo, ElDistMatrix_d A, ElHermitianTridiag ctrl)

ElError ElHermitianTridiagExplicitCondensedXDist_c(ElUpperOrLower uplo, ElDistMatrix_c A, ElHermitianTridiag ctrl)

ElError ElHermitianTridiagExplicitCondensedXDist_z(ElUpperOrLower uplo, ElDistMatrix_z A, ElHermitianTridiag ctrl)

Applying the change of basis

Apply (from the left or right) the implicitly defined unitary matrix (or its adjoint) represented by the Householder transformations stored within the specified triangle of A and their scalings are stored in the vector t.

Implementation

C++ API

void herm_tridiag::ApplyQ(LeftOrRight side, UpperOrLower uplo, Orientation orientation, const Matrix<F> &A, const Matrix<F> &t, Matrix<F> &B)

void herm_tridiag::ApplyQ(LeftOrRight side, UpperOrLower uplo, Orientation orientation, const AbstractDistMatrix<F> &A, const AbstractDistMatrix<F> &t, AbstractDistMatrix<F> &B)

C API

ElError ElApplyQAfterHermitianTridiag_s(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstMatrix_s A, ElConstMatrix_s t, ElMatrix_s B)

ElError ElApplyQAfterHermitianTridiag_d(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstMatrix_d A, ElConstMatrix_d t, ElMatrix_d B)

ElError ElApplyQAfterHermitianTridiag_c(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstMatrix_c A, ElConstMatrix_c t, ElMatrix_c B)

ElError ElApplyQAfterHermitianTridiag_z(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstMatrix_z A, ElConstMatrix_z t, ElMatrix_z B)

ElError ElApplyQAfterHermitianTridiagDist_s(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstDistMatrix_s A, ElConstDistMatrix_s t, ElDistMatrix_s B)

ElError ElApplyQAfterHermitianTridiagDist_d(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstDistMatrix_d A, ElConstDistMatrix_d t, ElDistMatrix_d B)

ElError ElApplyQAfterHermitianTridiagDist_c(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstDistMatrix_c A, ElConstDistMatrix_c t, ElDistMatrix_c B)

ElError ElApplyQAfterHermitianTridiagDist_z(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstDistMatrix_z A, ElConstDistMatrix_z t, ElDistMatrix_z B)