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EigenSymmetric2Bisect

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. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. This method uses bisection algorithm (lapack functions SYGVX, HEGVX).

Computing for type matrix<double>

bool  matrix::EigenSymmetric2Bisect(
   ENUM_EIGS2_TYPE       itype,              // the problem type to be solved
   ENUM_EIG_VALUES       jobv,               // compute eigenvectors or not
   ENUM_BLAS_RANGE       range,              // subset of eigenvalues to compute
   double                lower,              // lower bound of the subset
   double                upper,              // Upper bound of the subset
   double                abstol,             // absolute error tolerance
   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::EigenSymmetric2Bisect(
   ENUM_EIGS2_TYPE       itype,              // the problem type to be solved
   ENUM_EIG_VALUES       jobv,               // compute eigenvectors or not
   ENUM_BLAS_RANGE       range,              // subset of eigenvalues to compute
   float                 lower,              // lower bound of the subset
   float                 upper,              // upper bound of the subset
   float                 abstol,             // absolute error tolerance
   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::EigenSymmetric2Bisect(
   ENUM_EIGS2_TYPE       itype,              // the problem type to be solved
   ENUM_EIG_VALUES       jobv,               // compute eigenvectors or not
   ENUM_BLAS_RANGE       range,              // subset of eigenvalues to compute
   double                lower,              // lower bound of the subset
   double                upper,              // Upper bound of the subset
   double                abstol,             // absolute error tolerance
   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::EigenSymmetric2Bisect(
   ENUM_EIGS2_TYPE       itype,              // the problem type to be solved
   ENUM_EIG_VALUES       jobv,               // compute eigenvectors or not
   ENUM_BLAS_RANGE       range,              // subset of eigenvalues to compute
   float                 lower,              // lower bound of the subset
   float                 upper,              // Upper bound of the subset
   float                 abstol,             // absolute error tolerance
   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.

range

[in] ENUM_BLAS_RANGE enumeration value that defines a subset of computable eigenvalues and vectors.

lower

[in] The lower bound of eigenvalues subset; it is specified depending on the value of the 'range' parameter.

upper

[in] The upper bound of eigenvalues subset; it is specified depending on the value of the 'range' parameter.

abstol

[in] Absolute error tolerance.

The absolute error tolerance to which each eigenvalue/eigenvector is required.

If jobv = 'V', the eigenvalues and eigenvectors output have residual norms bounded by abstol, and the dot products between different eigenvectors are bounded by abstol.

If abstol < n *eps*|T|, then n *eps*|T| is used instead, where eps is the machine precision, and |T| is the 1-norm of the matrix T. The eigenvalues are computed to an accuracy of eps*|T| irrespective of abstol.

If high relative precision is important, 'abstol' should be set to a safe minimum value X such that 1.0/X does not overflow.

[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

Computation depends on the values of the jobv and range parameters.

When BLASRANGE_A is set, all eigenvalues are computed, and the lower and upper parameters are ignored.

With the BLASRANGE_V value, only those eigenvalues (and their vectors) are computed, which fall within the range of real values specified by the 'lower' and 'upper' parameters.

With the BLASRANGE_I value, only those eigenvalues (and their vectors) are computed, which fall within the range of integer indices specified by the 'lower' and 'upper' parameters. For example, with lower=0 and upper=2, only the first three eigenvalues are computed.

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: Ax = (lambda)B*x
EIGS2TYPE_2	2: ABx = (lambda)*x
EIGS2TYPE_3	3: BAx = (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.

ENUM_BLAS_RANGE

An enumeration defining how right singular vectors should be computed.

ID	Description
BLASRANGE_A	All singular or eigenvalues will be found.
BLASRANGE_V	All singular or eigenvalues in the half-open interval (VL,VU] will be found.
BLASRANGE_I	The IL-th through IU-th singular or eigenvalues will be found.

EigenSymmetric2QR

Tridiagonal Matrices