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       1            1 : // Distributed under the MIT License.
       2              : // See LICENSE.txt for details.
       3              : 
       4              : /// \file
       5              : /// Defines function find_generalized_eigenvalues.
       6              : 
       7              : #pragma once
       8              : 
       9              : /// \cond
      10              : class DataVector;
      11              : class Matrix;
      12              : namespace gsl {
      13              : template <typename T>
      14              : class not_null;
      15              : }  // namespace gsl
      16              : /// \endcond
      17              : 
      18              : /*!
      19              :  * \ingroup NumericalAlgorithmsGroup
      20              :  * \brief Solve the generalized eigenvalue problem for two matrices.
      21              :  *
      22              :  * This function uses the lapack routine dggev
      23              :  * (http://www.netlib.org/lapack/explore-3.1.1-html/dggev.f.html)
      24              :  * to solve the
      25              :  * generalized eigenvalue problem \f$A v_a =\lambda_a B v_a \f$
      26              :  * for the generalized eigenvalues \f$\lambda_a\f$ and corresponding
      27              :  * eigenvectors \f$v_a\f$.
      28              :  * `matrix_a` and `matrix_b` are each a `Matrix`; they correspond to square
      29              :  * matrices \f$A\f$ and \f$B\f$ that are the same dimension \f$N\f$.
      30              :  * `eigenvalues_real_part` is a `DataVector` of size \f$N\f$ that
      31              :  * will store the real parts of the eigenvalues,
      32              :  * `eigenvalues_imaginary_part` is a `DataVector` of size \f$N\f$
      33              :  * that will store the imaginary parts of the eigenvalues.
      34              :  * Complex eigenvalues always form complex conjugate pairs, and
      35              :  * the \f$j\f$ and \f$j+1\f$ eigenvalues will have the forms
      36              :  * \f$a+ib\f$ and \f$a-ib\f$, respectively. The eigenvectors
      37              :  * are returned as the columns of a square `Matrix` of dimension \f$N\f$
      38              :  * called `eigenvectors`. If eigenvalue \f$j\f$ is real, then column \f$j\f$ of
      39              :  * `eigenvectors` is
      40              :  * the corresponding eigenvector. If eigenvalue \f$j\f$ and \f$j+1\f$ are
      41              :  * complex-conjugate pairs, then the eigenvector for
      42              :  * eigenvalue \f$j\f$ is (column j) + \f$i\f$ (column j+1), and the
      43              :  * eigenvector for eigenvalue \f$j+1\f$ is (column j) - \f$i\f$ (column j+1).
      44              :  *
      45              :  */
      46            1 : void find_generalized_eigenvalues(
      47              :     gsl::not_null<DataVector*> eigenvalues_real_part,
      48              :     gsl::not_null<DataVector*> eigenvalues_imaginary_part,
      49              :     gsl::not_null<Matrix*> eigenvectors, Matrix matrix_a, Matrix matrix_b);