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EigenSolver2

Compute generalized eigenvalues and eigenvectors for a pair of ordinary square matrices (lapack function GGEV). Both matrices must be the same size.

Computing for type matrix<double>

bool  matrix::EigenSolver2(
   matrix&               B,                       // second matrix in the pair
   ENUM_EIG_VECTORS      jobv,                    // method to compute right and left vectors
   vectorc&              alpha,                   // vector of computed eigenvalues
   vector&               beta,                    // vector of eigenvalue divisors
   matrix&               left_eigenvectors,       // matrix of computed left vectors
   matrix&               right_eigenvectors       // matrix of computed right vectors  
   );

Computing for type matrix<float>

bool  matrixf::EigenSolver2(
   matrix&               B,                       // second matrix in the pair
   ENUM_EIG_VECTORS      jobv,                    // method to compute right and left vectors
   vectorcf&             alpha,                   // vector of computed eigenvalues
   vectorf&              beta,                    // vector of eigenvalue divisors
   matrixf&              left_eigenvectors,       // matrix of computed left vectors
   matrixf&              right_eigenvectors       // matrix of computed right vectors  
   );

Computing for type matrix<complex>

bool  matrixc::EigenSolver2(
   matrixc&              B,                       // second matrix in the pair
   ENUM_EIG_VECTORS      jobv,                    // method to compute right and left vectors
   vectorc&              alpha,                   // vector of computed eigenvalues
   vectorc&              beta,                    // vector of eigenvalue divisors
   matrixc&              left_eigenvectors,       // matrix of computed left vectors
   matrixc&              right_eigenvectors       // matrix of computed right vectors  
   );

Computing for type matrix<complexf>

bool  matrixcf::EigenSolver2(
   matrixcf&             B,                       // second matrix in the pair
   ENUM_EIG_VECTORS      jobv,                    // method to compute right and left vectors
   vectorcf&             alpha,                   // vector of computed eigenvalues
   vectorcf&             beta,                    // vector of eigenvalue divisors
   matrixcf&             left_eigenvectors,       // matrix of computed left vectors
   matrixcf&             right_eigenvectors       // matrix of computed right vectors  
   );

Parameters

B

[in]  The second matrix in the pair.

jobv

[in] ENUM_EIG_VECTORS enumeration value which determines the method for computing left and right eigenvectors.

alpha

[out] Vector of eigenvalues.

beta

[out]  Vector of eigen value divisors.

left_eigenvectors

[out] Matrix of left eigenvectors.

right_eigenvectors

[out] Matrix of right eigenvectors.

Return Value

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

Note

Computation depends on the value of the jobv parameter.

ENUM_EIG_VECTORS

An enumeration defining the need to compute eigenvectors.

ID

Description

EIGVECTORS_N

Only eigenvalues are computed, without vectors.

EIGVECTORS_L

Only left eigenvectors are computed.

EIGVECTORS_R

Only right eigenvectors are computed.

EIGVECTORS_LR

Left and right eigenvectors are computed, eigenvalues are always computed.

A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A - lambda*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both being zero.

The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

                 A * v(j) = lambda(j) * B * v(j).

The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

                 u(j)**H * A  = lambda(j) * u(j)**H * B .

where u(j)**H is the conjugate-transpose of u(j).

Real (non-complex) matrices can have a complex solution. Therefore, the input vector of eigenvalues must be complex. In case of a complex solution, the error code is set to 4019 (ERR_MATH_OVERFLOW). Otherwise, only the real parts of the complex values of the eigenvalue vector should be used.