Prototype

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

tgsen( const int_t ijob, const fortran_bool_t wantq,
        const fortran_bool_t wantz, const VectorSELECT& select, MatrixA& a,
        MatrixB& b, VectorALPHAR& alphar, VectorALPHAI& alphai,
        VectorBETA& beta, MatrixQ& q, MatrixZ& z, int_t& m,
        Scalar >, Scalar >, VectorDIF& dif );

tgsen( const int_t ijob, const fortran_bool_t wantq,
        const fortran_bool_t wantz, const VectorSELECT& select, MatrixA& a,
        MatrixB& b, VectorALPHA& alpha, VectorBETA& beta, MatrixQ& q,
        MatrixZ& z, int_t& m, Scalar >, Scalar >, VectorDIF& dif );

Description

tgsen (short for $FRIENDLY_NAME) provides a C++ interface to LAPACK routines STGSEN, DTGSEN, CTGSEN, and ZTGSEN. tgsen reorders the generalized Schur decomposition of a complex matrix pair (A, B) (in terms of an unitary equivalence trans- formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the pair (A,B). The leading columns of Q and Z form unitary bases of the corresponding left and right eigenspaces (deflating subspaces). (A, B) must be in generalized Schur canonical form, that is, A and B are both upper triangular.

tgsen also computes the generalized eigenvalues

w(j)= ALPHA(j) / BETA(j)

of the reordered matrix pair (A, B).

Optionally, the routine computes estimates of reciprocal condition numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) between the matrix pairs (A11, B11) and (A22,B22) that correspond to the selected cluster and the eigenvalues outside the cluster, resp., and norms of "projections" onto left and right eigenspaces w.r.t. the selected cluster in the (1,1)-block.

The selection of the LAPACK routine is done during compile-time, and is determined by the type of values contained in type VectorSELECT. The type of values is obtained through the value_type meta-function typename value_type<VectorSELECT>::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/tgsen.hpp.

Parameters or Requirements on Types

Complexity

Example

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

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

this will output

[5] 0 1 2 3 4 5

Notes

See Also