Skew-Hermitian eigensolvers

Implementation

Essentially identical to the Hermitian eigensolver, HermitianEig(); for any skew-Hermitian matrix \(G\), \(iG\) is Hermitian, as

\[(iG)^H = -iG^H = iG.\]

This fact implies a fast method for solving skew-Hermitian eigenvalue problems:

1. Form \(iG\) in \(O(n^2)\) work (switching to complex arithmetic in the real case)

2. Run a Hermitian eigensolve on \(iG\), yielding \(iG=Z \Lambda Z^H\).

3. Recognize that \(G=Z (-i \Lambda) Z^H\) provides an EVD of \(G\).

Please see the HermitianEig() documentation for more details.

Computing eigenvalues

The following routines compute the imaginary components of the eigenvalues of the skew-Hermitian matrix G 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 SkewHermitianEig(UpperOrLower uplo, Matrix<F> &G, Matrix<Base<F>> &w, SortType sort = ASCENDING, const HermitianEigSubset<Base<F>> &subset = HermitianEigSubset<Base<F>>(), const HermitianEigCtrl<Base<F>> &ctrl = HermitianEigCtrl<Base<F>>())

void SkewHermitianEig(UpperOrLower uplo, AbstractDistMatrix<F> &G, 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 ElSkewHermitianEig_s(ElUpperOrLower uplo, ElMatrix_s G, ElMatrix_s w, ElSortType sort)

ElError ElSkewHermitianEig_d(ElUpperOrLower uplo, ElMatrix_d G, ElMatrix_d w, ElSortType sort)

ElError ElSkewHermitianEig_c(ElUpperOrLower uplo, ElMatrix_c G, ElMatrix_s w, ElSortType sort)

ElError ElSkewHermitianEig_z(ElUpperOrLower uplo, ElMatrix_z G, ElMatrix_d w, ElSortType sort)

ElError ElSkewHermitianEigDist_s(ElUpperOrLower uplo, ElDistMatrix_s G, ElDistMatrix_s w, ElSortType sort)

ElError ElSkewHermitianEigDist_d(ElUpperOrLower uplo, ElDistMatrix_d G, ElDistMatrix_d w, ElSortType sort)

ElError ElSkewHermitianEigDist_c(ElUpperOrLower uplo, ElDistMatrix_c G, ElDistMatrix_s w, ElSortType sort)

ElError ElSkewHermitianEigDist_z(ElUpperOrLower uplo, ElDistMatrix_z G, ElDistMatrix_d w, ElSortType sort)

ElError ElSkewHermitianEigPartial_s(ElUpperOrLower uplo, ElMatrix_s G, ElMatrix_s w, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElSkewHermitianEigPartial_d(ElUpperOrLower uplo, ElMatrix_d G, ElMatrix_d w, ElSortType sort, ElHermitianEigSubset_d subset)

ElError ElSkewHermitianEigPartial_c(ElUpperOrLower uplo, ElMatrix_c G, ElMatrix_s w, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElSkewHermitianEigPartial_z(ElUpperOrLower uplo, ElMatrix_z G, ElMatrix_d w, ElSortType sort, ElHermitianEigSubset_d subset)

ElError ElSkewHermitianEigPartialDist_s(ElUpperOrLower uplo, ElDistMatrix_s G, ElDistMatrix_s w, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElSkewHermitianEigPartialDist_d(ElUpperOrLower uplo, ElDistMatrix_d G, ElDistMatrix_d w, ElSortType sort, ElHermitianEigSubset_d subset)

ElError ElSkewHermitianEigPartialDist_c(ElUpperOrLower uplo, ElDistMatrix_c G, ElDistMatrix_s w, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElSkewHermitianEigPartialDist_z(ElUpperOrLower uplo, ElDistMatrix_z G, ElDistMatrix_d w, ElSortType sort, ElHermitianEigSubset_d subset)

Computing eigenpairs

The following routines compute the set of eigenpairs (note that only the imaginary components of the eigenvalues are returned) of the skew-Hermitian matrix G 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 SkewHermitianEig(UpperOrLower uplo, Matrix<F> &G, Matrix<Base<F>> &w, Matrix<Complex<Base<F>>> &Z, SortType sort = ASCENDING, const HermitianEigSubset<Base<F>> &subset = HermitianEigSubset<Base<F>>(), const HermitianEigCtrl<Base<F>> &ctrl = HermitianEigCtrl<Base<F>>())

void SkewHermitianEig(UpperOrLower uplo, AbstractDistMatrix<F> &G, AbstractDistMatrix<Base<F>> &w, AbstractDistMatrix<Complex<Base<F>>> &Z, SortType sort = ASCENDING, const HermitianEigSubset<Base<F>> &subset = HermitianEigSubset<Base<F>>(), const HermitianEigCtrl<Base<F>> &ctrl = HermitianEigCtrl<Base<F>>())

C API

ElError ElSkewHermitianEigPair_s(ElUpperOrLower uplo, ElMatrix_s G, ElMatrix_s w, ElMatrix_c Z, ElSortType sort)

ElError ElSkewHermitianEigPair_d(ElUpperOrLower uplo, ElMatrix_d G, ElMatrix_d w, ElMatrix_z Z, ElSortType sort)

ElError ElSkewHermitianEigPair_c(ElUpperOrLower uplo, ElMatrix_c G, ElMatrix_s w, ElMatrix_c Z, ElSortType sort)

ElError ElSkewHermitianEigPair_z(ElUpperOrLower uplo, ElMatrix_z G, ElMatrix_d w, ElMatrix_z Z, ElSortType sort)

ElError ElSkewHermitianEigPairDist_s(ElUpperOrLower uplo, ElDistMatrix_s G, ElDistMatrix_s w, ElDistMatrix_c Z, ElSortType sort)

ElError ElSkewHermitianEigPairDist_d(ElUpperOrLower uplo, ElDistMatrix_d G, ElDistMatrix_d w, ElDistMatrix_z Z, ElSortType sort)

ElError ElSkewHermitianEigPairDist_c(ElUpperOrLower uplo, ElDistMatrix_c G, ElDistMatrix_s w, ElDistMatrix_c Z, ElSortType sort)

ElError ElSkewHermitianEigPairDist_z(ElUpperOrLower uplo, ElDistMatrix_z G, ElDistMatrix_d w, ElDistMatrix_z Z, ElSortType sort)

ElError ElSkewHermitianEigPairPartial_s(ElUpperOrLower uplo, ElMatrix_s G, ElMatrix_s w, ElMatrix_c Z, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElSkewHermitianEigPairPartial_d(ElUpperOrLower uplo, ElMatrix_d G, ElMatrix_d w, ElMatrix_z Z, ElSortType sort, ElHermitianEigSubset_d subset)

ElError ElSkewHermitianEigPairPartial_c(ElUpperOrLower uplo, ElMatrix_c G, ElMatrix_s w, ElMatrix_c Z, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElSkewHermitianEigPairPartial_z(ElUpperOrLower uplo, ElMatrix_z G, ElMatrix_d w, ElMatrix_z Z, ElSortType sort, ElHermitianEigSubset_d subset)

ElError ElSkewHermitianEigPairPartialDist_s(ElUpperOrLower uplo, ElDistMatrix_s G, ElDistMatrix_s w, ElDistMatrix_c Z, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElSkewHermitianEigPairPartialDist_d(ElUpperOrLower uplo, ElDistMatrix_d G, ElDistMatrix_d w, ElDistMatrix_z Z, ElSortType sort, ElHermitianEigSubset_d subset)

ElError ElSkewHermitianEigPairPartialDist_c(ElUpperOrLower uplo, ElDistMatrix_c G, ElDistMatrix_s w, ElDistMatrix_c Z, ElSortType sort, ElHermitianEigSubset_s subset)

ElError ElSkewHermitianEigPairPartialDist_z(ElUpperOrLower uplo, ElDistMatrix_z G, ElDistMatrix_d w, ElDistMatrix_z Z, ElSortType sort, ElHermitianEigSubset_d subset)