##### geevx

###### Prototype

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

    geevx( const char balanc, const char jobvl, const char jobvr,
            const char sense, MatrixA& a, VectorWR& wr, VectorWI& wi,
            MatrixVL& vl, MatrixVR& vr, int_t& ilo,
            int_t& ihi, VectorSCALE& scale, Scalar >,
            VectorRCONDE& rconde, VectorRCONDV& rcondv );

    geevx( const char balanc, const char jobvl, const char jobvr,
            const char sense, MatrixA& a, VectorW& w, MatrixVL& vl, MatrixVR& vr,
            int_t& ilo, int_t& ihi, VectorSCALE& scale,
            Scalar >, VectorRCONDE& rconde, VectorRCONDV& rcondv );

###### Description

`geevx` (short for $FRIENDLY_NAME) provides a C++ interface to LAPACK routines SGEEVX, DGEEVX, CGEEVX, and ZGEEVX. `geevx` computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors.

Optionally also, it computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ILO, IHI, SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and reciprocal condition numbers for the right eigenvectors (RCONDV).

The right eigenvector v(j) of A satisfies A * v(j) = lambda(j) * v(j) where lambda(j) is its eigenvalue. The left eigenvector u(j) of A satisfies u(j)*H * A = lambda(j) * u(j)H where u(j)*H denotes the conjugate transpose of u(j).

The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real.

Balancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and applying a diagonal similarity transformation D * A * D**(-1), where D is a diagonal matrix, to make its rows and columns closer in norm and the condition numbers of its eigenvalues and eigenvectors smaller. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see section 4.10.2 of the LAPACK Users' Guide.

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 1.154. Dispatching of geevx

Value type of MatrixA | LAPACK routine
--- | ---
`float` | SGEEVX
`double` | DGEEVX
`complex<float>` | CGEEVX
`complex<double>` | ZGEEVX

###### Definition

Defined in header `boost/numeric/bindings/lapack/driver/geevx.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/geevx.hpp>
    using namespace boost::numeric::bindings;

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

this will output

    [5] 0 1 2 3 4 5

###### Notes

###### See Also

- Originating Fortran source files sgeevx.f, dgeevx.f, cgeevx.f, and zgeevx.f at Netlib.