Hermitian SVD

Suppose that $A$ is Hermitian, with a spectral decomposition

$$A = V \Lambda V^H.$$

Then an SVD of $A$ can easily be computed as

$$A = U |\Lambda| V^H,$$

where the columns of $U$ equal the columns of $V$, modulo sign flips introduced by negative eigenvalues.

Computing singular values

The following routines return the singular values of $A$ in s. Note that the appropriate triangle of A is overwritten during computation.

C++ API

void HermitianSVD(UpperOrLower uplo, Matrix<F> &A, Matrix<Base<F>> &s)

void HermitianSVD(UpperOrLower uplo, AbstractDistMatrix<F> &A, AbstractDistMatrix<Base<F>> &s)

C API

ElError ElHermitianSingularValues_s(ElUpperOrLower uplo, ElMatrix_s A, ElMatrix_s s)

ElError ElHermitianSingularValues_d(ElUpperOrLower uplo, ElMatrix_d A, ElMatrix_d s)

ElError ElHermitianSingularValues_c(ElUpperOrLower uplo, ElMatrix_c A, ElMatrix_s s)

ElError ElHermitianSingularValues_z(ElUpperOrLower uplo, ElMatrix_z A, ElMatrix_d s)

ElError ElHermitianSingularValuesDist_s(ElUpperOrLower uplo, ElDistMatrix_s A, ElDistMatrix_s s)

ElError ElHermitianSingularValuesDist_d(ElUpperOrLower uplo, ElDistMatrix_d A, ElDistMatrix_d s)

ElError ElHermitianSingularValuesDist_c(ElUpperOrLower uplo, ElDistMatrix_c A, ElDistMatrix_s s)

ElError ElHermitianSingularValuesDist_z(ElUpperOrLower uplo, ElDistMatrix_z A, ElDistMatrix_d s)

Computing Singular Value Decompositions

The following routines return a vector of singular values, $s$, and the left and right singular vector matrices, $U$ and $V$, such that $A=U \mathrm{diag}(s) V^H$.

C++ API

void HermitianSVD(UpperOrLower uplo, Matrix<F> &A, Matrix<Base<F>> &s, Matrix<F> &U, Matrix<F> &V)

void HermitianSVD(UpperOrLower uplo, AbstractDistMatrix<F> &A, AbstractDistMatrix<Base<F>> &s, AbstractDistMatrix<F> &U, AbstractDistMatrix<F> &V)

C API

ElError ElHermitianSVD_s(ElUpperOrLower uplo, ElMatrix_s A, ElMatrix_s s, ElMatrix_s U, ElMatrix_s V)

ElError ElHermitianSVD_d(ElUpperOrLower uplo, ElMatrix_d A, ElMatrix_d s, ElMatrix_d U, ElMatrix_d V)

ElError ElHermitianSVD_c(ElUpperOrLower uplo, ElMatrix_c A, ElMatrix_s s, ElMatrix_c U, ElMatrix_c V)

ElError ElHermitianSVD_z(ElUpperOrLower uplo, ElMatrix_z A, ElMatrix_d s, ElMatrix_z U, ElMatrix_z V)

ElError ElHermitianSVDDist_s(ElUpperOrLower uplo, ElDistMatrix_s A, ElDistMatrix_s s, ElDistMatrix_s U, ElDistMatrix_s V)

ElError ElHermitianSVDDist_d(ElUpperOrLower uplo, ElDistMatrix_d A, ElDistMatrix_d s, ElDistMatrix_d U, ElDistMatrix_d V)

ElError ElHermitianSVDDist_c(ElUpperOrLower uplo, ElDistMatrix_c A, ElDistMatrix_s s, ElDistMatrix_c U, ElDistMatrix_c V)

ElError ElHermitianSVDDist_z(ElUpperOrLower uplo, ElDistMatrix_z A, ElDistMatrix_d s, ElDistMatrix_z U, ElDistMatrix_z V)