119 lines
5.4 KiB
Markdown
119 lines
5.4 KiB
Markdown
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# EigenSolver2
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Compute generalized eigenvalues and eigenvectors for a pair of ordinary square matrices (LAPACK function [GGEV](https://www.intel.com/content/www/us/en/docs/onemkl/developer-reference-fortran/2025-2/ggev.html)). Both matrices must be the same size.
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Computing for type matrix<double>
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```
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bool matrix::EigenSolver2(
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matrix& B, // second matrix in the pair
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ENUM_EIG_VECTORS jobv, // method to compute right and left vectors
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vectorc& alpha, // vector of computed eigenvalues
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vector& beta, // vector of eigenvalue divisors
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matrix& left_eigenvectors, // matrix of computed left vectors
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matrix& right_eigenvectors // matrix of computed right vectors
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);
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```
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Computing for type matrix<float>
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```
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bool matrixf::EigenSolver2(
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matrix& B, // second matrix in the pair
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ENUM_EIG_VECTORS jobv, // method to compute right and left vectors
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vectorcf& alpha, // vector of computed eigenvalues
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vectorf& beta, // vector of eigenvalue divisors
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matrixf& left_eigenvectors, // matrix of computed left vectors
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matrixf& right_eigenvectors // matrix of computed right vectors
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);
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```
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Computing for type matrix<complex>
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```
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bool matrixc::EigenSolver2(
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matrixc& B, // second matrix in the pair
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ENUM_EIG_VECTORS jobv, // method to compute right and left vectors
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vectorc& alpha, // vector of computed eigenvalues
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vectorc& beta, // vector of eigenvalue divisors
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matrixc& left_eigenvectors, // matrix of computed left vectors
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matrixc& right_eigenvectors // matrix of computed right vectors
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);
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```
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Computing for type matrix<complexf>
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```
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bool matrixcf::EigenSolver2(
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matrixcf& B, // second matrix in the pair
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ENUM_EIG_VECTORS jobv, // method to compute right and left vectors
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vectorcf& alpha, // vector of computed eigenvalues
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vectorcf& beta, // vector of eigenvalue divisors
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matrixcf& left_eigenvectors, // matrix of computed left vectors
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matrixcf& right_eigenvectors // matrix of computed right vectors
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);
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```
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Parameters
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B
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[in] The second matrix in the pair.
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jobv
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[in] [ENUM_EIG_VECTORS](/en/docs/matrix/openblas/eigen_values/general_matrices/eigensolver#enum_eig_vectors) enumeration value which determines the method for computing left and right eigenvectors.
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alpha
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[out] Vector of eigenvalues.
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beta
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[out] Vector of eigen value divisors.
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left_eigenvectors
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[out] Matrix of left eigenvectors.
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right_eigenvectors
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[out] Matrix of right eigenvectors.
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Return Value
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Return true if successful, otherwise false in case of an [error](/en/docs/constants/errorswarnings/errorcodes).
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Note
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Computation depends on the value of the jobv parameter.
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ENUM_EIG_VECTORS
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An enumeration defining the need to compute eigenvectors.
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| ID | Description |
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| --- | --- |
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| EIGVECTORS_N | Only eigenvalues are computed, without vectors. |
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| EIGVECTORS_L | Only left eigenvectors are computed. |
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| EIGVECTORS_R | Only right eigenvectors are computed. |
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| EIGVECTORS_LR | Left and right eigenvectors are computed, eigenvalues are always computed. |
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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.
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The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies
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A * v(j) = lambda(j) * B * v(j).
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The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies
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u(j)**H * A = lambda(j) * u(j)**H * B .
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where u(j)**H is the conjugate-transpose of u(j).
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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)](/en/docs/constants/errorswarnings/errorcodes). Otherwise, only the real parts of the complex values of the eigenvalue vector should be used.
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