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EigenSymmetric2QR

Compute all the eigenvalues, and optionally, the eigenvectors of a generalized symmetric-definite eigenproblem, of the form

A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.

Here A and B are assumed to be symmetric (Hermitian) and B is also positive definite. This method uses QR algorithm (lapack functions SYGV, HEGV).

Computing for type matrix<double>

bool  matrix::EigenSymmetric2QR(
   ENUM_EIGS2_TYPE       itype,              // the problem type to be solved
   ENUM_EIG_VALUES       jobv,               // compute eigenvectors or not
   matrix&               B,                  // second matrix
   vector&               eigen_values,       // vector of computed eigenvalues
   matrix&               eigen_vectors,      // matrix of computed eigenvectors
   matrix&               triangular_factor   // triangular factor from the Cholesky factorization B
   );

Computing for type matrix<float>

bool  matrixf::EigenSymmetric2QR(
   ENUM_EIGS2_TYPE       itype,              // the problem type to be solved
   ENUM_EIG_VALUES       jobv,               // compute eigenvectors or not
   matrixf&              B,                  // second matrix
   vectorf&              eigen_values,       // vector of computed eigenvalues
   matrixf&              eigen_vectors,      // matrix of computed eigenvectors
   matrixf&              triangular_factor   // triangular factor from the Cholesky factorization B
   );

Computing for type matrix<complex>

bool  matrixc::EigenSymmetric2QR(
   ENUM_EIGS2_TYPE       itype,              // the problem type to be solved
   ENUM_EIG_VALUES       jobv,               // compute eigenvectors or not
   matrixc&              B,                  // second matrix
   vector&               eigen_values,       // vector of computed eigenvalues
   matrixc&              eigen_vectors,      // matrix of computed eigenvectors
   matrixc&              triangular_factor   // triangular factor from the Cholesky factorization B
   );

Computing for type matrix<complexf>

bool  matrixcf::EigenSymmetric2QR(
   ENUM_EIGS2_TYPE       itype,              // the problem type to be solved
   ENUM_EIG_VALUES       jobv,               // compute eigenvectors or not
   matrixcf&             B,                  // second matrix
   vectorf&              eigen_values,       // vector of computed eigenvalues
   matrixcf&             eigen_vectors,      // matrix of computed eigenvectors
   matrixcf&             triangular_factor   // triangular factor from the Cholesky factorization B
   );

Parameters

itype

[in]  ENUM_EIGS2_TYPE enumeration value which specified the problem type to be solved : A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.

jobv

[in]  ENUM_EIG_VALUES enumeration value which determines the method for computing eigenvectors.

B

[in]  Second matrix B. Must be positive definite symmetric (or Hermitian conjugated) matrix.

eigen_values

[out] Vector of eigenvalues.

eigen_vectors

[out] Matrix of eigenvectors.

triangular_factor

[out] The triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T.

Return Value

Return true if successful, otherwise false in case of an error.

Note

The input can be a symmetric (Hermitian), upper triangular or lower triangular matrix. Triangular matrices are assumed to be symmetric (Hermitian conjugated). Second matrix B must be positive definite symmetric. If the input matrix and second matrix B are triangular, then both must be the same triangular, upper or lower.

ENUM_EIGS2_TYPE

An enumeration that specifies the problem type to be solved.

ID

Description

EIGS2TYPE_1

1:  A*x = (lambda)*B*x

EIGS2TYPE_2

2:  A*B*x = (lambda)*x

EIGS2TYPE_3

3:  B*A*x = (lambda)*x

ENUM_EIG_VALUES

An enumeration that specifies whether to calculate eigenvectors.

ID

Description

EIGVALUES_V

Eigenvectors and eigenvalues are calculated.

EIGVALUES_N

Only eigenvalues are calculated, without vectors.