# COMPLEX*16 or DOUBLE COMPLEX routines for upper Hessenberg matrix, generalized problem (i.e a Hessenberg and a triangular matrix) matrix

## zhgeqz

```USAGE:
alpha, beta, work, info, h, t, q, z = NumRu::Lapack.zhgeqz( job, compq, compz, ilo, ihi, h, t, q, z, [:lwork => lwork, :usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO )

*  Purpose
*  =======
*
*  ZHGEQZ 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 = Q1*H*Z1**H,  B = Q1*T*Z1**H,
*
*  as computed by ZGGHRD.
*
*  If JOB='S', then the Hessenberg-triangular pair (H,T) is
*  also reduced to generalized Schur form,
*
*     H = Q*S*Z**H,  T = Q*P*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*(Z1*Z)**H,  B = (Q1*Q)*P*(Z1*Z)**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.
*

*  Arguments
*  =========
*
*  JOB     (input) CHARACTER*1
*          = 'E': Compute eigenvalues only;
*          = 'S': Computer eigenvalues and the Schur form.
*
*  COMPQ   (input) CHARACTER*1
*          = 'N': Left Schur vectors (Q) are not computed;
*          = 'I': Q is initialized to the unit matrix and the matrix Q
*                 of left Schur vectors of (H,T) is returned;
*          = 'V': Q must contain a unitary matrix Q1 on entry and
*                 the product Q1*Q is returned.
*
*  COMPZ   (input) CHARACTER*1
*          = 'N': Right Schur vectors (Z) are not computed;
*          = 'I': Q is initialized to the unit matrix and the matrix Z
*                 of right Schur vectors of (H,T) is returned;
*          = 'V': Z must contain a unitary matrix Z1 on entry and
*                 the product Z1*Z is returned.
*
*  N       (input) INTEGER
*          The order of the matrices H, T, Q, and Z.  N >= 0.
*
*  ILO     (input) INTEGER
*  IHI     (input) INTEGER
*          ILO and IHI mark the rows and columns of H which are in
*          Hessenberg form.  It is assumed that A is already upper
*          triangular in rows and columns 1:ILO-1 and IHI+1:N.
*          If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
*
*  H       (input/output) COMPLEX*16 array, dimension (LDH, N)
*          On entry, the N-by-N upper Hessenberg matrix H.
*          On exit, if JOB = 'S', H contains the upper triangular
*          matrix S from the generalized Schur factorization.
*          If JOB = 'E', the diagonal of H matches that of S, but
*          the rest of H is unspecified.
*
*  LDH     (input) INTEGER
*          The leading dimension of the array H.  LDH >= max( 1, N ).
*
*  T       (input/output) COMPLEX*16 array, dimension (LDT, N)
*          On entry, the N-by-N upper triangular matrix T.
*          On exit, if JOB = 'S', T contains the upper triangular
*          matrix P from the generalized Schur factorization.
*          If JOB = 'E', the diagonal of T matches that of P, but
*          the rest of T is unspecified.
*
*  LDT     (input) INTEGER
*          The leading dimension of the array T.  LDT >= max( 1, N ).
*
*  ALPHA   (output) COMPLEX*16 array, dimension (N)
*          The complex scalars alpha that define the eigenvalues of
*          GNEP.  ALPHA(i) = S(i,i) in the generalized Schur
*          factorization.
*
*  BETA    (output) COMPLEX*16 array, dimension (N)
*          The real non-negative scalars beta that define the
*          eigenvalues of GNEP.  BETA(i) = P(i,i) in the generalized
*          Schur factorization.
*
*          Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
*          represent the j-th eigenvalue of the matrix pair (A,B), in
*          one of the forms lambda = alpha/beta or mu = beta/alpha.
*          Since either lambda or mu may overflow, they should not,
*          in general, be computed.
*
*  Q       (input/output) COMPLEX*16 array, dimension (LDQ, N)
*          On entry, if COMPZ = 'V', the unitary matrix Q1 used in the
*          reduction of (A,B) to generalized Hessenberg form.
*          On exit, if COMPZ = 'I', the unitary matrix of left Schur
*          vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
*          left Schur vectors of (A,B).
*          Not referenced if COMPZ = 'N'.
*
*  LDQ     (input) INTEGER
*          The leading dimension of the array Q.  LDQ >= 1.
*          If COMPQ='V' or 'I', then LDQ >= N.
*
*  Z       (input/output) COMPLEX*16 array, dimension (LDZ, N)
*          On entry, if COMPZ = 'V', the unitary matrix Z1 used in the
*          reduction of (A,B) to generalized Hessenberg form.
*          On exit, if COMPZ = 'I', the unitary matrix of right Schur
*          vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
*          right Schur vectors of (A,B).
*          Not referenced if COMPZ = 'N'.
*
*  LDZ     (input) INTEGER
*          The leading dimension of the array Z.  LDZ >= 1.
*          If COMPZ='V' or 'I', then LDZ >= N.
*
*  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
*          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) INTEGER
*          The dimension of the array WORK.  LWORK >= max(1,N).
*
*          If LWORK = -1, then a workspace query is assumed; the routine
*          only calculates the optimal size of the WORK array, returns
*          this value as the first entry of the WORK array, and no error
*          message related to LWORK is issued by XERBLA.
*
*  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*          = 1,...,N: the QZ iteration did not converge.  (H,T) is not
*                     in Schur form, but ALPHA(i) and BETA(i),
*                     i=INFO+1,...,N should be correct.
*          = N+1,...,2*N: the shift calculation failed.  (H,T) is not
*                     in Schur form, but ALPHA(i) and BETA(i),
*                     i=INFO-N+1,...,N should be correct.
*

*  Further Details
*  ===============
*
*  We assume that complex ABS works as long as its value is less than
*  overflow.
*
*  =====================================================================
*

```
go to the page top
back to matrix types
back to data types