Square to Hessenberg¶

Reduction¶

Elemental’s algorithms for reducing square matrices to Hessenberg form via products of Householder similarity transformations are straight-forward parallelizations (and extensions to the complex field) of an algorithm described within [QvdG2006], which can be viewed as a modification of the corresponding algorithm from [DSH1989] which shifts some of the work from level 2 to level 3 BLAS.

Python API¶

Hessenberg(uplo, A[, hessOnly=False])¶

C++ API¶

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

void Hessenberg(UpperOrLower uplo, AbstractDistMatrix<F> &A, AbstractDistMatrix<F> &t)¶: Return the in-place reduction of the matrix A to lower-/upper-Hessenberg form. The vector t contains the scalings for the Householder reflectors, which are stored in the locations of the zeros that they introduced.

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

void hessenberg::ExplicitCondensed(UpperOrLower uplo, AbstractDistMatrix<F> &A)¶: Return just the Hessenberg matrix.

C API¶

Single-precision¶

ElError ElHessenberg_s(ElUpperOrLower uplo, ElMatrix_s A, ElMatrix_s t)¶

ElError ElHessenbergDist_s(ElUpperOrLower uplo, ElDistMatrix_s A, ElDistMatrix_s t)¶: Return the in-place reduction of the matrix A to lower-/upper-Hessenberg form. The vector t contains the scalings for the Householder reflectors, which are stored in the locations of the zeros that they introduced.

ElError ElHessenbergExplicitCondensed_s(ElUpperOrLower uplo, ElMatrix_s A)¶

ElError ElHessenbergExplicitCondensedDist_s(ElUpperOrLower uplo, ElDistMatrix_s A)¶: Return just the Hessenberg matrix.

Double-precision¶

ElError ElHessenberg_d(ElUpperOrLower uplo, ElMatrix_d A, ElMatrix_d t)¶

ElError ElHessenbergDist_d(ElUpperOrLower uplo, ElDistMatrix_d A, ElDistMatrix_d t)¶: Return the in-place reduction of the matrix A to lower-/upper-Hessenberg form. The vector t contains the scalings for the Householder reflectors, which are stored in the locations of the zeros that they introduced.

ElError ElHessenbergExplicitCondensed_d(ElUpperOrLower uplo, ElMatrix_d A)¶

ElError ElHessenbergExplicitCondensedDist_d(ElUpperOrLower uplo, ElDistMatrix_d A)¶: Return just the Hessenberg matrix.

Single-precision complex¶

ElError ElHessenberg_c(ElUpperOrLower uplo, ElMatrix_c A, ElMatrix_c t)¶

ElError ElHessenbergDist_c(ElUpperOrLower uplo, ElDistMatrix_c A, ElDistMatrix_c t)¶: Return the in-place reduction of the matrix A to lower-/upper-Hessenberg form. The vector t contains the scalings for the Householder reflectors, which are stored in the locations of the zeros that they introduced.

ElError ElHessenbergExplicitCondensed_c(ElUpperOrLower uplo, ElMatrix_c A)¶

ElError ElHessenbergExplicitCondensedDist_c(ElUpperOrLower uplo, ElDistMatrix_c A)¶: Return just the Hessenberg matrix.

Double-precision complex¶

ElError ElHessenberg_z(ElUpperOrLower uplo, ElMatrix_z A, ElMatrix_z t)¶

ElError ElHessenbergDist_z(ElUpperOrLower uplo, ElDistMatrix_z A, ElDistMatrix_z t)¶: Return the in-place reduction of the matrix A to lower-/upper-Hessenberg form. The vector t contains the scalings for the Householder reflectors, which are stored in the locations of the zeros that they introduced.

ElError ElHessenbergExplicitCondensed_z(ElUpperOrLower uplo, ElMatrix_z A)¶

ElError ElHessenbergExplicitCondensedDist_z(ElUpperOrLower uplo, ElDistMatrix_z A)¶: Return just the Hessenberg matrix.

Applying the change of basis¶

Python API¶

ApplyQAfterHessenberg(side, uplo, orient, A, t, B)¶

C++ API¶

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

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

C API¶

Single-precision¶

ElError ElApplyQAfterHessenberg_s(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstMatrix_s A, ElConstMatrix_s t, ElMatrix_s H)¶

ElError ElApplyQAfterHessenbergDist_s(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstDistMatrix_s A, ElConstDistMatrix_s t, ElDistMatrix_s H)¶

Double-precision¶

ElError ElApplyQAfterHessenberg_d(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstMatrix_d A, ElConstMatrix_d t, ElMatrix_d H)¶

ElError ElApplyQAfterHessenbergDist_d(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstDistMatrix_d A, ElConstDistMatrix_d t, ElDistMatrix_d H)¶

Single-precision complex¶

ElError ElApplyQAfterHessenberg_c(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstMatrix_c A, ElConstMatrix_c t, ElMatrix_c H)¶

ElError ElApplyQAfterHessenbergDist_c(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstDistMatrix_c A, ElConstDistMatrix_c t, ElDistMatrix_c H)¶

Double-precision complex¶

ElError ElApplyQAfterHessenberg_z(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstMatrix_z A, ElConstMatrix_z t, ElMatrix_z H)¶

ElError ElApplyQAfterHessenbergDist_z(ElLeftOrRight side, ElUpperOrLower uplo, ElOrientation orientation, ElConstDistMatrix_z A, ElConstDistMatrix_z t, ElDistMatrix_z H)¶

References¶

Implementation

Subroutine header files

Test driver

DSH1989: Jack J. Dongarra, Danny C. Sorensen, and Sven J. Hammarling, Block reduction of matrices to condensed forms for eigenvalue computations, Journal of Computational and Applied Mathematics, Vol. 27, Issues 1–2, pp. 215–227, 1989. DOI: http://dx.doi.org/10.1016/0377-0427(89)90367-1 <http://dx.doi.org/10.1016/0377-0427(89)90367-1>`__
QvdG2006: Gregorio Quintana-Orti and Robert van de Geijn, Improving the performance of reduction to Hessenberg form, ACM Transactions on Mathematical Software, Vol. 32, Issue 2, pp. 180–194, 2006. DOI: http://dx.doi.org/10.1145/1141885.1141887