Schur¶

template<Number T, DataLocation L, template<typename, typename> typename S> internal::Schur<T, L> schur(const Matrix<T, L, S> &A)¶

Computes the Schur form of a square matrix A: \( \mathbf{A} = \mathbf{UZU}^*\), where \( \mathbf{U} \) is a unitary matrix and \( \mathbf{Z} \) is an upper triangular matrix. This routine can also calculate the eigenvalues of A.

This operation can be done in-place, e.g., A = schur(A). In this case, A will be replaced by the matrix Z.

Usage:

Matrix<double> A;
Vector<std::complex<double>> eigenvalues;
Matrix<double> U;
Matrix<double> Z;
Z = schur(A);                           // Calculates only the upper triangular matrix Z
tie(Z, U) = schur(A);                   // Calculates Z and the unitary matrix U
tie(Z, U, eigenvalues) = schur(A);      // Calculates Z, U and the eigenvalues

Parameters:: A – [in] input matrix
Returns:: An object representing the factorization.

template<Number T, DataLocation L, template<typename, typename> typename S> internal::HessenbergSchur<T, L> hessenberg_schur(const Matrix<T, L, S> &H)¶

Computes the Schur form of an upper Hessenberg matrix H: \( \mathbf{H} = \mathbf{UZU}^*\), where \( \mathbf{U} \) is a unitary matrix and \( \mathbf{Z} \) is an upper triangular matrix. This routine can also calculate the eigenvalues of H.

This operation can be done in-place, e.g., H = schur(H). In this case, H will be replaced by the matrix Z.

Usage:

Matrix<double> H;
Vector<std::complex<double>> eigenvalues;
Matrix<double> U;
Matrix<double> Z;
Z = hessenberg_schur(A);                           // Calculates only the upper triangular matrix Z
tie(Z, U) = hessenberg_schur(A);                   // Calculates Z and the unitary matrix U
tie(Z, U, eigenvalues) = hessenberg_schur(A);      // Calculates Z, U and the eigenvalues

Parameters:: H – [in] input matrix
Returns:: An object representing the factorization.

template<Number EigenT, Number T, DataLocation L, template<typename, typename> typename S1, template<typename, typename> typename S2, template<typename, typename> typename S3> void reorder_schur(Matrix<T, L, S1> &Z, Matrix<T, L, S2> &U, Vector<EigenT, L, S3> &eigenvalues, SortOrder order, index_t &ell)¶

Reorder the Schur form such that the ell smallest or largest eigenvalues are located at the begining of Z. The eigenvalues are also reorder.

Warning

If the ell-th and ell + 1-th eigenvalues are complex conjugate from each other, than the ell + 1 is also included in the reorder form and ell is increased by 1.

Parameters:

Z – [inout] quasi-upper triangular matrix from the Schur form
U – [inout] unitary matrix from the Schur form
eigenvalues – [inout] eigenvalues of Z
order – [in] either kAscending or kDescending
ell – [inout] the number of eigenvalues selectecd