##### ggesx

###### Prototype

There are two prototypes of `ggesx` available, please see below.

    ggesx( const char jobvsl, const char jobvsr, const char sort,
            external_fp selctg, const char sense, MatrixA& a, MatrixB& b,
            int_t& sdim, VectorALPHAR& alphar, VectorALPHAI& alphai,
            VectorBETA& beta, MatrixVSL& vsl, MatrixVSR& vsr,
            VectorRCONDE& rconde, VectorRCONDV& rcondv );

    ggesx( const char jobvsl, const char jobvsr, const char sort,
            external_fp selctg, const char sense, MatrixA& a, MatrixB& b,
            int_t& sdim, VectorALPHA& alpha, VectorBETA& beta,
            MatrixVSL& vsl, MatrixVSR& vsr, VectorRCONDE& rconde,
            VectorRCONDV& rcondv );

###### Description

`ggesx` (short for $FRIENDLY_NAME) provides a C++ interface to LAPACK routines SGGESX, DGGESX, CGGESX, and ZGGESX. `ggesx` computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, the complex Schur form (S,T), and, optionally, the left and/or right matrices of Schur vectors (VSL and VSR). This gives the generalized Schur factorization

(A,B) = ( (VSL) S (VSR)*H, (VSL) T (VSR)*H )

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

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper triangular matrix S and the upper triangular matrix T; computes a reciprocal condition number for the average of the selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigenvalues (RCONDV). The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces).

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

A pair of matrices (S,T) is in generalized complex Schur form if T is upper triangular with non-negative diagonal and S is upper triangular.

The selection of the LAPACK routine is done during compile-time, and is determined by the type of values contained in type `MatrixA`. The type of values is obtained through the `value_type` meta-function `typename value_type<MatrixA>::type`. The dispatching table below illustrates to which specific routine the code path will be generated.

Table 1.103. Dispatching of ggesx

Value type of MatrixA | LAPACK routine
--- | ---
`float` | SGGESX
`double` | DGGESX
`complex<float>` | CGGESX
`complex<double>` | ZGGESX

###### Definition

Defined in header `boost/numeric/bindings/lapack/driver/ggesx.hpp`.

###### Parameters or Requirements on Types

Parameters

MatrixA

  The definition of term 1

MatrixB

  The definition of term 2

MatrixC

  The definition of term 3.

  Definitions may contain paragraphs.

###### Complexity

###### Example

    #include <boost/numeric/bindings/lapack/driver/ggesx.hpp>
    using namespace boost::numeric::bindings;

    lapack::ggesx( x, y, z );

this will output

    [5] 0 1 2 3 4 5

###### Notes

###### See Also

- Originating Fortran source files sggesx.f, dggesx.f, cggesx.f, and zggesx.f at Netlib.