!>
!> SHGEQZ computes the eigenvalues of a real matrix pair (H,T),
!> where H is an upper Hessenberg matrix and T is upper triangular,
!> using the double-shift QZ method.
!> Matrix pairs of this type are produced by the reduction to
!> generalized upper Hessenberg form of a real matrix pair (A,B):
!>
!>    A = Q1*H*Z1**T,  B = Q1*T*Z1**T,
!>
!> as computed by SGGHRD.
!>
!> If JOB='S', then the Hessenberg-triangular pair (H,T) is
!> also reduced to generalized Schur form,
!>
!>    H = Q*S*Z**T,  T = Q*P*Z**T,
!>
!> where Q and Z are orthogonal matrices, P is an upper triangular
!> matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
!> diagonal blocks.
!>
!> The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
!> (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
!> eigenvalues.
!>
!> Additionally, the 2-by-2 upper triangular diagonal blocks of P
!> corresponding to 2-by-2 blocks of S are reduced to positive diagonal
!> form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
!> P(j,j) > 0, and P(j+1,j+1) > 0.
!>
!> Optionally, the orthogonal matrix Q from the generalized Schur
!> factorization may be postmultiplied into an input matrix Q1, and the
!> orthogonal matrix Z may be postmultiplied into an input matrix Z1.
!> If Q1 and Z1 are the orthogonal matrices from SGGHRD that reduced
!> the matrix pair (A,B) to generalized upper Hessenberg form, then the
!> output matrices Q1*Q and Z1*Z are the orthogonal factors from the
!> generalized Schur factorization of (A,B):
!>
!>    A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T.
!>
!> To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
!> of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
!> complex and beta real.
!> 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.
!> Real eigenvalues can be read directly from the generalized Schur
!> form:
!>   alpha = S(i,i), beta = P(i,i).
!>
!> Ref: C.B. Moler & G.W. Stewart, , SIAM J. Numer. Anal., 10(1973),
!>      pp. 241--256.
!> 

JOB

!>          JOB is CHARACTER*1
!>          = 'E': Compute eigenvalues only;
!>          = 'S': Compute eigenvalues and the Schur form.
!> 

COMPQ

!>          COMPQ is 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 an orthogonal matrix Q1 on entry and
!>                 the product Q1*Q is returned.
!> 

COMPZ

!>          COMPZ is CHARACTER*1
!>          = 'N': Right Schur vectors (Z) are not computed;
!>          = 'I': Z is initialized to the unit matrix and the matrix Z
!>                 of right Schur vectors of (H,T) is returned;
!>          = 'V': Z must contain an orthogonal matrix Z1 on entry and
!>                 the product Z1*Z is returned.
!> 

N

!>          N is INTEGER
!>          The order of the matrices H, T, Q, and Z.  N >= 0.
!> 

ILO

!>          ILO is INTEGER
!> 

IHI

!>          IHI is 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

!>          H is REAL array, dimension (LDH, N)
!>          On entry, the N-by-N upper Hessenberg matrix H.
!>          On exit, if JOB = 'S', H contains the upper quasi-triangular
!>          matrix S from the generalized Schur factorization.
!>          If JOB = 'E', the diagonal blocks of H match those of S, but
!>          the rest of H is unspecified.
!> 

LDH

!>          LDH is INTEGER
!>          The leading dimension of the array H.  LDH >= max( 1, N ).
!> 

T

!>          T is REAL 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;
!>          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
!>          are reduced to positive diagonal form, i.e., if H(j+1,j) is
!>          non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
!>          T(j+1,j+1) > 0.
!>          If JOB = 'E', the diagonal blocks of T match those of P, but
!>          the rest of T is unspecified.
!> 

LDT

!>          LDT is INTEGER
!>          The leading dimension of the array T.  LDT >= max( 1, N ).
!> 

ALPHAR

!>          ALPHAR is REAL array, dimension (N)
!>          The real parts of each scalar alpha defining an eigenvalue
!>          of GNEP.
!> 

ALPHAI

!>          ALPHAI is REAL array, dimension (N)
!>          The imaginary parts of each scalar alpha defining an
!>          eigenvalue of GNEP.
!>          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
!>          positive, then the j-th and (j+1)-st eigenvalues are a
!>          complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
!> 

BETA

!>          BETA is REAL array, dimension (N)
!>          The scalars beta that define the eigenvalues of GNEP.
!>          Together, the quantities alpha = (ALPHAR(j),ALPHAI(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

!>          Q is REAL array, dimension (LDQ, N)
!>          On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
!>          the reduction of (A,B) to generalized Hessenberg form.
!>          On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
!>          vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix
!>          of left Schur vectors of (A,B).
!>          Not referenced if COMPQ = 'N'.
!> 

LDQ

!>          LDQ is INTEGER
!>          The leading dimension of the array Q.  LDQ >= 1.
!>          If COMPQ='V' or 'I', then LDQ >= N.
!> 

Z

!>          Z is REAL array, dimension (LDZ, N)
!>          On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
!>          the reduction of (A,B) to generalized Hessenberg form.
!>          On exit, if COMPZ = 'I', the orthogonal matrix of
!>          right Schur vectors of (H,T), and if COMPZ = 'V', the
!>          orthogonal matrix of right Schur vectors of (A,B).
!>          Not referenced if COMPZ = 'N'.
!> 

LDZ

!>          LDZ is INTEGER
!>          The leading dimension of the array Z.  LDZ >= 1.
!>          If COMPZ='V' or 'I', then LDZ >= N.
!> 

WORK

!>          WORK is REAL array, dimension (MAX(1,LWORK))
!>          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
!> 

LWORK

!>          LWORK is 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.
!> 

INFO

!>          INFO is 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 ALPHAR(i), ALPHAI(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 ALPHAR(i), ALPHAI(i), and
!>                     BETA(i), i=INFO-N+1,...,N should be correct.
!> 

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!>
!>  Iteration counters:
!>
!>  JITER  -- counts iterations.
!>  IITER  -- counts iterations run since ILAST was last
!>            changed.  This is therefore reset only when a 1-by-1 or
!>            2-by-2 block deflates off the bottom.
!>