Prototype

There are two prototypes of hgeqz available, please see below.

hgeqz( const char job, const char compq, const char compz,
        const int_t ilo, MatrixH& h, MatrixT& t,
        VectorALPHAR& alphar, VectorALPHAI& alphai, VectorBETA& beta,
        MatrixQ& q, MatrixZ& z );

hgeqz( const char job, const char compq, const char compz,
        const int_t ilo, MatrixH& h, MatrixT& t,
        VectorALPHA& alpha, VectorBETA& beta, MatrixQ& q, MatrixZ& z );

Description

hgeqz (short for $FRIENDLY_NAME) provides a C++ interface to LAPACK routines SHGEQZ, DHGEQZ, CHGEQZ, and ZHGEQZ. hgeqz computes the eigenvalues of a complex matrix pair (H,T), where H is an upper Hessenberg matrix and T is upper triangular, using the single-shift QZ method. Matrix pairs of this type are produced by the reduction to generalized upper Hessenberg form of a complex matrix pair (A,B):

A = Q1H*Z1*H, B = Q1T*Z1*H,

as computed by ZGGHRD.

If JOB='S', then the Hessenberg-triangular pair (H,T) is also reduced to generalized Schur form,

H = QS*Z*H, T = QP*Z*H,

where Q and Z are unitary matrices and S and P are upper triangular.

Optionally, the unitary matrix Q from the generalized Schur factorization may be postmultiplied into an input matrix Q1, and the unitary matrix Z may be postmultiplied into an input matrix Z1. If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced the matrix pair (A,B) to generalized Hessenberg form, then the output matrices Q1*Q and Z1*Z are the unitary factors from the generalized Schur factorization of (A,B):

A = (Q1*Q)S(Z1Z)*H, B = (Q1*Q)P(Z1Z)*H.

To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, of (A,B)) are computed as a pair of complex values (alpha,beta). If beta is nonzero, lambda = alpha / beta is an eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP) A*x = lambda*B*x and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the alternate form of the GNEP mu*A*y = B*y. The values of alpha and beta for the i-th eigenvalue can be read directly from the generalized Schur form: alpha = S(i,i), beta = P(i,i).

Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973), pp. 241--256.

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

Definition

Defined in header boost/numeric/bindings/lapack/computational/hgeqz.hpp.

Parameters or Requirements on Types

Complexity

Example

#include <boost/numeric/bindings/lapack/computational/hgeqz.hpp>
using namespace boost::numeric::bindings;

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

this will output

[5] 0 1 2 3 4 5

Notes

See Also