### 15.6 Real Generalized Nonsymmetric Eigensystems

Given two square matrices (A, B), the generalized nonsymmetric eigenvalue problem is to find eigenvalues \lambda and eigenvectors x such that

    A x = \lambda B x

We may also define the problem as finding eigenvalues \mu and eigenvectors y such that

    \mu A y = B y

Note that these two problems are equivalent (with \lambda = 1/\mu) if neither \lambda nor \mu is zero. If say, \lambda is zero, then it is still a well defined eigenproblem, but its alternate problem involving \mu is not. Therefore, to allow for zero (and infinite) eigenvalues, the problem which is actually solved is

    \beta A x = \alpha B x

The eigensolver routines below will return two values \alpha and \beta and leave it to the user to perform the divisions \lambda = \alpha / \beta and \mu = \beta / \alpha.

If the determinant of the matrix pencil A - \lambda B is zero for all \lambda, the problem is said to be singular; otherwise it is called regular. Singularity normally leads to some \alpha = \beta = 0 which means the eigenproblem is ill-conditioned and generally does not have well defined eigenvalue solutions. The routines below are intended for regular matrix pencils and could yield unpredictable results when applied to singular pencils.

The solution of the real generalized nonsymmetric eigensystem problem for a matrix pair (A, B) involves computing the generalized Schur decomposition

    A = Q S Z^T
    B = Q T Z^T

where Q and Z are orthogonal matrices of left and right Schur vectors respectively, and (S, T) is the generalized Schur form whose diagonal elements give the \alpha and \beta values. The algorithm used is the QZ method due to Moler and Stewart (see references).

Function: gsl_eigen_gen_workspace * gsl_eigen_gen_alloc (const size_t n)

  This function allocates a workspace for computing eigenvalues of n-by-n real generalized nonsymmetric eigensystems. The size of the workspace is O(n).

Function: void gsl_eigen_gen_free (gsl_eigen_gen_workspace * w)

  This function frees the memory associated with the workspace w.

Function: void gsl_eigen_gen_params (const int compute_s, const int compute_t, const int balance, gsl_eigen_gen_workspace * w)

  This function sets some parameters which determine how the eigenvalue problem is solved in subsequent calls to `gsl_eigen_gen`.

  If compute_s is set to 1, the full Schur form S will be computed by `gsl_eigen_gen`. If it is set to 0, S will not be computed (this is the default setting). S is a quasi upper triangular matrix with 1-by-1 and 2-by-2 blocks on its diagonal. 1-by-1 blocks correspond to real eigenvalues, and 2-by-2 blocks correspond to complex eigenvalues.

  If compute_t is set to 1, the full Schur form T will be computed by `gsl_eigen_gen`. If it is set to 0, T will not be computed (this is the default setting). T is an upper triangular matrix with non-negative elements on its diagonal. Any 2-by-2 blocks in S will correspond to a 2-by-2 diagonal block in T.

  The balance parameter is currently ignored, since generalized balancing is not yet implemented.

Function: int gsl_eigen_gen (gsl_matrix * A, gsl_matrix * B, gsl_vector_complex * alpha, gsl_vector * beta, gsl_eigen_gen_workspace * w)

  This function computes the eigenvalues of the real generalized nonsymmetric matrix pair (A, B), and stores them as pairs in (alpha, beta), where alpha is complex and beta is real. If \beta_i is non-zero, then \lambda = \alpha_i / \beta_i is an eigenvalue. Likewise, if \alpha_i is non-zero, then \mu = \beta_i / \alpha_i is an eigenvalue of the alternate problem \mu A y = B y. The elements of beta are normalized to be non-negative.

  If S is desired, it is stored in A on output. If T is desired, it is stored in B on output. The ordering of eigenvalues in (alpha, beta) follows the ordering of the diagonal blocks in the Schur forms S and T. In rare cases, this function may fail to find all eigenvalues. If this occurs, an error code is returned.

Function: int gsl_eigen_gen_QZ (gsl_matrix * A, gsl_matrix * B, gsl_vector_complex * alpha, gsl_vector * beta, gsl_matrix * Q, gsl_matrix * Z, gsl_eigen_gen_workspace * w)

  This function is identical to `gsl_eigen_gen` except that it also computes the left and right Schur vectors and stores them into Q and Z respectively.

Function: gsl_eigen_genv_workspace * gsl_eigen_genv_alloc (const size_t n)

  This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real generalized nonsymmetric eigensystems. The size of the workspace is O(7n).

Function: void gsl_eigen_genv_free (gsl_eigen_genv_workspace * w)

  This function frees the memory associated with the workspace w.

Function: int gsl_eigen_genv (gsl_matrix * A, gsl_matrix * B, gsl_vector_complex * alpha, gsl_vector * beta, gsl_matrix_complex * evec, gsl_eigen_genv_workspace * w)

  This function computes eigenvalues and right eigenvectors of the n-by-n real generalized nonsymmetric matrix pair (A, B). The eigenvalues are stored in (alpha, beta) and the eigenvectors are stored in evec. It first calls `gsl_eigen_gen` to compute the eigenvalues, Schur forms, and Schur vectors. Then it finds eigenvectors of the Schur forms and backtransforms them using the Schur vectors. The Schur vectors are destroyed in the process, but can be saved by using `gsl_eigen_genv_QZ`. The computed eigenvectors are normalized to have unit magnitude. On output, (A, B) contains the generalized Schur form (S, T). If `gsl_eigen_gen` fails, no eigenvectors are computed, and an error code is returned.

Function: int gsl_eigen_genv_QZ (gsl_matrix * A, gsl_matrix * B, gsl_vector_complex * alpha, gsl_vector * beta, gsl_matrix_complex * evec, gsl_matrix * Q, gsl_matrix * Z, gsl_eigen_genv_workspace * w)

  This function is identical to `gsl_eigen_genv` except that it also computes the left and right Schur vectors and stores them into Q and Z respectively.