# DOUBLE PRECISION routines for upper Hessenberg matrix, generalized problem (i.e a Hessenberg and a triangular matrix) matrix

## dhgeqz

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

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

*  Purpose
*  =======
*
*  DHGEQZ 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 DGGHRD.
*
*  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 DGGHRD 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, "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': Compute 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 an orthogonal matrix Q1 on entry and
*                 the product Q1*Q is returned.
*
*  COMPZ   (input) 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       (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) DOUBLE PRECISION 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;
*          2-by-2 diagonal blocks (corresponding to complex conjugate
*          pairs of eigenvalues) are returned in standard form, with
*          H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0.
*          If JOB = 'E', the diagonal blocks of H match those 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) DOUBLE PRECISION 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     (input) INTEGER
*          The leading dimension of the array T.  LDT >= max( 1, N ).
*
*  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
*          The real parts of each scalar alpha defining an eigenvalue
*          of GNEP.
*
*  ALPHAI  (output) DOUBLE PRECISION 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    (output) DOUBLE PRECISION 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       (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
*          On entry, if COMPZ = 'V', the orthogonal matrix Q1 used in
*          the reduction of (A,B) to generalized Hessenberg form.
*          On exit, if COMPZ = 'I', the orthogonal matrix of left Schur
*          vectors of (H,T), and if COMPZ = 'V', the orthogonal 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) DOUBLE PRECISION 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     (input) INTEGER
*          The leading dimension of the array Z.  LDZ >= 1.
*          If COMPZ='V' or 'I', then LDZ >= N.
*
*  WORK    (workspace/output) DOUBLE PRECISION 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.
*
*  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 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.
*

*  Further Details
*  ===============
*
*  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.
*
*  =====================================================================
*

```
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