Hermitian generalized-definite eigensolvers

The following Hermitian generalized-definite eigenvalue problems frequently appear, where both \(A\) and \(B\) are Hermitian, and \(B\) is additionally positive-definite:

\[ABx = \lambda x,\]

which is denoted with the value ABX via the Pencil enum,

\[BAx = \lambda x,\]

which uses the BAX value, and finally

\[Ax = \lambda B x,\]

which uses the AXBX enum value.

type Pencil

An enum for specifying either the ABX, BAX, or AXBX generalized eigenvalue problems (described above).

Implementation

Test driver

Computing eigenvalues

Compute the set of eigenvalues of the Hermitian-definite matrix pencil determined by the subset structure (the default is to compute all eigenvalues). By default, Elemental is used to transform the problem to and from real symmetric tridiagonal form and PMRRR is used to solve the tridiagonal EVP. Alternatively, the ctrl structure may be altered to request the usage of a (prototype) Spectral Divide and Conquer algorithm.

Note

Strict subset computation is not currently supported with Spectral Divide and Conquer.

C++ API

void HermitianGenDefEig(Pencil pencil, UpperOrLower uplo, Matrix<F> &A, Matrix<F> &B, Matrix<Base<F>> &w, SortType sort = ASCENDING, const HermitianEigSubset<Base<F>> subset = HermitianEigSubset<Base<F>>(), const HermitianEigCtrl<Base<F>> ctrl = HermitianEigCtrl<Base<F>>())

void HermitianGenDefEig(Pencil pencil, UpperOrLower uplo, AbstractDistMatrix<F> &A, AbstractDistMatrix<F> &B, AbstractDistMatrix<Base<F>> &w, SortType sort = ASCENDING, const HermitianEigSubset<Base<F>> subset = HermitianEigSubset<Base<F>>(), const HermitianEigCtrl<Base<F>> ctrl = HermitianEigCtrl<Base<F>>())

C API

ElError ElHermitianGenDefEig_s(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_s A, ElMatrix_s B, ElMatrix_s w, ElSortType sort)

ElError ElHermitianGenDefEig_d(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_d A, ElMatrix_d B, ElMatrix_d w, ElSortType sort)

ElError ElHermitianGenDefEig_c(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_c A, ElMatrix_c B, ElMatrix_s w, ElSortType sort)

ElError ElHermitianGenDefEig_z(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_z A, ElMatrix_z B, ElMatrix_d w, ElSortType sort)

ElError ElHermitianGenDefEigDist_s(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_s A, ElDistMatrix_s B, ElDistMatrix_s w, ElSortType sort)

ElError ElHermitianGenDefEigDist_d(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_d A, ElDistMatrix_d B, ElDistMatrix_d w, ElSortType sort)

ElError ElHermitianGenDefEigDist_c(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_c A, ElDistMatrix_c B, ElDistMatrix_s w, ElSortType sort)

ElError ElHermitianGenDefEigDist_z(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_z A, ElDistMatrix_z B, ElDistMatrix_d w, ElSortType sort)

ElError ElHermitianGenDefEigPartial_s(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_s A, ElMatrix_s B, ElMatrix_s w, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElHermitianGenDefEigPartial_d(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_d A, ElMatrix_d B, ElMatrix_d w, ElSortType sort, ElHermitianEigSubset_d subset)

ElError ElHermitianGenDefEigPartial_c(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_c A, ElMatrix_c B, ElMatrix_s w, ElMatrix_c Z, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElHermitianGenDefEigPartial_z(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_z A, ElMatrix_z B, ElMatrix_d w, SortType sort, ElHermitianEigSubset_d subset)

ElError ElHermitianGenDefEigPartialDist_s(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_s A, ElDistMatrix_s B, ElDistMatrix_s w, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElHermitianGenDefEigPartialDist_d(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_d A, ElDistMatrix_d B, ElDistMatrix_d w, ElSortType sort, ElHermitianEigSubset_d subset)

ElError ElHermitianGenDefEigPartialDist_c(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_c A, ElDistMatrix_c B, ElDistMatrix_s w, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElHermitianGenDefEigPartialDist_z(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_z A, ElDistMatrix_z B, ElDistMatrix_d w, ElSortType sort, ElHermitianEigSubset_d subset)

Computing eigenpairs

Compute the set of eigenpairs of the Hermitian-definite matrix pencil determined by the subset structure (the default is to compute all eigenpairs). By default, Elemental is used to transform the problem to and from real symmetric tridiagonal form and PMRRR is used to solve the tridiagonal EVP. Alternatively, the ctrl structure may be altered to request the usage of a (prototype) Spectral Divide and Conquer algorithm.

Note

Strict subset computation is not currently supported with Spectral Divide and Conquer.

C++ API

void HermitianGenDefEig(Pencil pencil, UpperOrLower uplo, Matrix<F> &A, Matrix<F> &B, Matrix<Base<F>> &w, Matrix<F> &Z, SortType sort = ASCENDING, const HermitianEigSubset<Base<F>> subset = HermitianEigSubset<Base<F>>(), const HermitianEigCtrl<Base<F>> ctrl = HermitianEigCtrl<Base<F>>())

void HermitianGenDefEig(Pencil pencil, UpperOrLower uplo, AbstractDistMatrix<F> &A, AbstractDistMatrix<F> &B, AbstractDistMatrix<Base<F>> &w, AbstractDistMatrix<F> &Z, SortType sort = ASCENDING, const HermitianEigSubset<Base<F>> subset = HermitianEigSubset<Base<F>>(), const HermitianEigCtrl<Base<F>> ctrl = HermitianEigCtrl<Base<F>>())

C API

ElError ElHermitianGenDefEigPair_s(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_s A, ElMatrix_s B, ElMatrix_s w, ElMatrix_s Z, ElSortType sort)

ElError ElHermitianGenDefEigPair_d(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_d A, ElMatrix_d B, ElMatrix_d w, ElMatrix_d Z, ElSortType sort)

ElError ElHermitianGenDefEigPair_c(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_c A, ElMatrix_c B, ElMatrix_s w, ElMatrix_c Z, ElSortType sort)

ElError ElHermitianGenDefEigPair_z(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_z A, ElMatrix_z B, ElMatrix_d w, ElMatrix_z Z, ElSortType sort)

ElError ElHermitianGenDefEigPairDist_s(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_s A, ElDistMatrix_s B, ElDistMatrix_s w, ElDistMatrix_s Z, ElSortType sort)

ElError ElHermitianGenDefEigPairDist_d(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_d A, ElDistMatrix_d B, ElDistMatrix_d w, ElDistMatrix_d Z, ElSortType sort)

ElError ElHermitianGenDefEigPairDist_c(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_c A, ElDistMatrix_c B, ElDistMatrix_s w, ElDistMatrix_c Z, ElSortType sort)

ElError ElHermitianGenDefEigPairDist_z(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_z A, ElDistMatrix_z B, ElDistMatrix_d w, ElDistMatrix_z Z, ElSortType sort)

ElError ElHermitianGenDefEigPairPartial_s(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_s A, ElMatrix_s B, ElMatrix_s w, ElMatrix_s Z, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElHermitianGenDefEigPairPartial_d(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_d A, ElMatrix_d B, ElMatrix_d w, ElMatrix_d Z, ElSortType sort, ElHermitianEigSubset_d subset)

ElError ElHermitianGenDefEigPairPartial_c(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_c A, ElMatrix_c B, ElMatrix_s w, ElMatrix_c Z, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElHermitianGenDefEigPairPartial_z(ElPencil pencil, ElUpperOrLower uplo, ElMatrix_z A, ElMatrix_z B, ElMatrix_d w, ElMatrix_z Z, ElSortType sort, ElHermitianEigSubset_d subset)

ElError ElHermitianGenDefEigPairPartialDist_s(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_s A, ElDistMatrix_s B, ElDistMatrix_s w, ElDistMatrix_s Z, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElHermitianGenDefEigPairPartialDist_d(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_d A, ElDistMatrix_d B, ElDistMatrix_d w, ElDistMatrix_d Z, ElSortType sort, ElHermitianEigSubset_d subset)

ElError ElHermitianGenDefEigPairPartialDist_c(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_c A, ElDistMatrix_c B, ElDistMatrix_s w, ElDistMatrix_c Z, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElHermitianGenDefEigPairPartialDist_z(ElPencil pencil, ElUpperOrLower uplo, ElDistMatrix_z A, ElDistMatrix_z B, ElDistMatrix_d w, ElDistMatrix_z Z, ElSortType sort, ElHermitianEigSubset_d subset)