General to bidiagonal

Reduces a general fully-populated \(m \times n\) matrix to bidiagonal form through two-sided Householder transformations; when the \(m \ge n\), the result is upper bidiagonal, otherwise it is lower bidiagonal. This routine is most commonly used as a preprocessing step in computing the SVD of a general matrix.

Implementation

Subroutine implementations

Test driver

Reduction

C++ API

void Bidiag(Matrix<F> &A, Matrix<F> &tP, Matrix<F> &tQ)

void Bidiag(AbstractDistMatrix<F> &A, AbstractDistMatrix<F> &tP, AbstractDistMatrix<F> &tQ)

Overwrites the main and sub (or super) diagonal of the real matrix A with the resulting bidiagonal matrix and stores the scaled Householder vectors in the remainder of the matrix. The complex case must also store the scalings of the Householder transformations (in tP and tQ) if they are to be applied.

void bidiag::Explicit(Matrix<F> &A, Matrix<F> &P, Matrix<F> &Q)

void bidiag::Explicit(AbstractDistMatrix<F> &A, AbstractDistMatrix<F> &P, AbstractDistMatrix<F> &Q)

Overwrite \(A\) with the bidiagonal matrix, \(B = Q^H A P\), and also return \(P\) and \(Q\).

void bidiag::ExplicitCondensed(Matrix<F> &A)

void bidiag::ExplicitCondensed(AbstractDistMatrix<F> &A)

Returns just the resulting bidiagonal matrix, \(B = Q^H A P\).

C API

ElError ElBidiag_s(ElMatrix_s A, ElMatrix_s tP, ElMatrix_s tQ)

ElError ElBidiag_d(ElMatrix_d A, ElMatrix_d tP, ElMatrix_d tQ)

ElError ElBidiag_c(ElMatrix_c A, ElMatrix_c tP, ElMatrix_c tQ)

ElError ElBidiag_z(ElMatrix_z A, ElMatrix_z tP, ElMatrix_z tQ)

ElError ElBidiagDist_s(ElDistMatrix_s A, ElDistMatrix_s tP, ElDistMatrix_s tQ)

ElError ElBidiagDist_d(ElDistMatrix_d A, ElDistMatrix_d tP, ElDistMatrix_d tQ)

ElError ElBidiagDist_c(ElDistMatrix_c A, ElDistMatrix_c tP, ElDistMatrix_c tQ)

ElError ElBidiagDist_z(ElDistMatrix_z A, ElDistMatrix_z tP, ElDistMatrix_z tQ)

Overwrites the main and sub (or super) diagonal of the real matrix A with the resulting bidiagonal matrix and stores the scaled Householder vectors in the remainder of the matrix. The complex case must also store the scalings of the Householder transformations (in tP and tQ) if they are to be applied.

ElError ElBidiagExplicit_s(ElMatrix_s A, ElMatrix_s P, ElMatrix_s Q)

ElError ElBidiagExplicit_d(ElMatrix_d A, ElMatrix_d P, ElMatrix_d Q)

ElError ElBidiagExplicit_c(ElMatrix_c A, ElMatrix_c P, ElMatrix_c Q)

ElError ElBidiagExplicit_z(ElMatrix_z A, ElMatrix_z P, ElMatrix_z Q)

ElError ElBidiagExplicitDist_s(ElDistMatrix_s A, ElDistMatrix_s P, ElDistMatrix_s Q)

ElError ElBidiagExplicitDist_d(ElDistMatrix_d A, ElDistMatrix_d P, ElDistMatrix_d Q)

ElError ElBidiagExplicitDist_c(ElDistMatrix_c A, ElDistMatrix_c P, ElDistMatrix_c Q)

ElError ElBidiagExplicitDist_z(ElDistMatrix_z A, ElDistMatrix_z P, ElDistMatrix_z Q)

Overwrite \(A\) with the bidiagonal matrix, \(B = Q^H A P\), and also return \(P\) and \(Q\).

ElError ElBidiagExplicitCondensed_s(ElMatrix_s A)

ElError ElBidiagExplicitCondensed_d(ElMatrix_d A)

ElError ElBidiagExplicitCondensed_c(ElMatrix_c A)

ElError ElBidiagExplicitCondensed_z(ElMatrix_z A)

ElError ElBidiagExplicitCondensedDist_s(ElDistMatrix_s A)

ElError ElBidiagExplicitCondensedDist_d(ElDistMatrix_d A)

ElError ElBidiagExplicitCondensedDist_c(ElDistMatrix_c A)

ElError ElBidiagExplicitCondensedDist_z(ElDistMatrix_z A)

Returns just the resulting bidiagonal matrix, \(B = Q^H A P\).

Applying the changes of basis

C++ API

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

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

void bidiag::ApplyP(LeftOrRight side, Orientation orientation, const Matrix<F> &A, const Matrix<F> &t, Matrix<F> &B)

void bidiag::ApplyP(LeftOrRight side, Orientation orientation, const AbstractDistMatrix<F> &A, const AbstractDistMatrix<F> &t, AbstractDistMatrix<F> &B)

C API

ElError ElApplyQAfterBidiag_s(ElLeftOrRight side, ElOrientation orientation, ElConstMatrix_s A, ElConstMatrix_s t, ElMatrix_s B)

ElError ElApplyQAfterBidiag_d(ElLeftOrRight side, ElOrientation orientation, ElConstMatrix_d A, ElConstMatrix_d t, ElMatrix_d B)

ElError ElApplyQAfterBidiag_c(ElLeftOrRight side, ElOrientation orientation, ElConstMatrix_c A, ElConstMatrix_c t, ElMatrix_c B)

ElError ElApplyQAfterBidiag_z(ElLeftOrRight side, ElOrientation orientation, ElConstMatrix_z A, ElConstMatrix_z t, ElMatrix_z B)

ElError ElApplyQAfterBidiagDist_s(ElLeftOrRight side, ElOrientation orientation, ElConstDistMatrix_s A, ElConstDistMatrix_s t, ElDistMatrix_s B)

ElError ElApplyQAfterBidiagDist_d(ElLeftOrRight side, ElOrientation orientation, ElConstDistMatrix_d A, ElConstDistMatrix_d t, ElDistMatrix_d B)

ElError ElApplyQAfterBidiagDist_c(ElLeftOrRight side, ElOrientation orientation, ElConstDistMatrix_c A, ElConstDistMatrix_c t, ElDistMatrix_c B)

ElError ElApplyQAfterBidiagDist_z(ElLeftOrRight side, ElOrientation orientation, ElConstDistMatrix_z A, ElConstDistMatrix_z t, ElDistMatrix_z B)

ElError ElApplyPAfterBidiag_s(ElLeftOrRight side, ElOrientation orientation, ElConstMatrix_s A, ElConstMatrix_s t, ElMatrix_s B)

ElError ElApplyPAfterBidiag_d(ElLeftOrRight side, ElOrientation orientation, ElConstMatrix_d A, ElConstMatrix_d t, ElMatrix_d B)

ElError ElApplyPAfterBidiag_c(ElLeftOrRight side, ElOrientation orientation, ElConstMatrix_c A, ElConstMatrix_c t, ElMatrix_c B)

ElError ElApplyPAfterBidiag_z(ElLeftOrRight side, ElOrientation orientation, ElConstMatrix_z A, ElConstMatrix_z t, ElMatrix_z B)

ElError ElApplyPAfterBidiagDist_s(ElLeftOrRight side, ElOrientation orientation, ElConstDistMatrix_s A, ElConstDistMatrix_s t, ElDistMatrix_s B)

ElError ElApplyPAfterBidiagDist_d(ElLeftOrRight side, ElOrientation orientation, ElConstDistMatrix_d A, ElConstDistMatrix_d t, ElDistMatrix_d B)

ElError ElApplyPAfterBidiagDist_c(ElLeftOrRight side, ElOrientation orientation, ElConstDistMatrix_c A, ElConstDistMatrix_c t, ElDistMatrix_c B)

ElError ElApplyPAfterBidiagDist_z(ElLeftOrRight side, ElOrientation orientation, ElConstDistMatrix_z A, ElConstDistMatrix_z t, ElDistMatrix_z B)