nag_zhgeqz (f08xsc) implements the  method for finding generalized eigenvalues of the complex matrix pair  of order , which is in the generalized upper Hessenberg form.

nag_zhgeqz (f08xsc) implements a single-shift version of the  method for finding the generalized eigenvalues of the complex matrix pair  which is in the generalized upper Hessenberg form. If the matrix pair  is not in the generalized upper Hessenberg form, then the function f08wsc (nag_zgghrd) should be called before invoking nag_zhgeqz (f08xsc).

This problem is mathematically equivalent to solving the matrix equation

Note that, to avoid underflow, overflow and other arithmetic problems, the generalized eigenvalues  are never computed explicitly by this function but defined as ratios between two computed values,  and :

The arguments , in general, are finite complex values and  are finite real non-negative values.

If desired, the matrix pair  may be reduced to generalized Schur form. That is, the transformed matrices  and  are upper triangular and the diagonal values of  and  provide  and .

The argument job specifies two options. If  then the matrix pair  is simultaneously reduced to Schur form by applying one unitary transformation (usually called ) on the left and another (usually called ) on the right. That is,

If  then at each iteration the same transformations are computed but they are only applied to those parts of  and  which are needed to compute  and . This option could be used if generalized eigenvalues are required but not generalized eigenvectors.

If  and   "Nag_AccumulateQ" or "Nag_InitQ" and   "Nag_AccumulateZ" or "Nag_InitZ" then the unitary transformations used to reduce the pair  are accumulated into the input arrays q and z. If generalized eigenvectors are required then job must be set to  and if left (right) generalized eigenvectors are to be computed then compq (compz) must be set to   "Nag_AccumulateQ" or "Nag_InitQ" rather than .

If , then eigenvectors are accumulated on the identity matrix and on exit the array q contains the left eigenvector matrix . However, if  then the transformations are accumulated in the user-supplied matrix  in array q on entry and thus on exit q contains the matrix product . A similar convention is used for compz.

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument  had an illegal value.

"NE_CONVERGENCE"

 An unexpected Library error has occurred.

 The computation of shifts failed and the matrix pair  is not inthe generalized Schur form.

 The computation of shifts failed and the matrix pair  is not inthe generalized Schur form.The computed  and  should be correctfor .

 The  iteration did not converge and the matrix pair  is notin the generalized Schur form.

 The  iteration did not converge and the matrix pair  is notin the generalized Schur form.The computed  and  should be correctfor .

"NE_ENUM_INT"

On entry, , . Constraint: .

On entry, , . Constraint: .

"NE_INT"

On entry, . Constraint: .

"NE_INT_3"

On entry, , , . Constraint: if ,  and .

On entry, , , . Constraint: if , .

"NE_INTERNAL_ERROR"

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance.

Please consult section 4.11 of the LAPACK Users' Guide (see Anderson et al. (1999)) and Chapter 6 of Stewart and Sun (1990), for more information.

nag_zhgeqz (f08xsc) is the fifth step in the solution of the complex generalized eigenvalue problem and is called after f08wsc (nag_zgghrd).

The number of floating-point operations taken by this function is proportional to .

The real analogue of this function is f08xec (nag_dhgeqz).