SHGEQZ(3)      LAPACK routine of NEC Numeric Library Collection      SHGEQZ(3)



NAME
       SHGEQZ

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



PURPOSE
            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).





ARGUMENTS
           JOB       (input)
                     JOB is CHARACTER*1
                     = 'E': Compute eigenvalues only;
                     = 'S': Compute eigenvalues and the Schur form.

           COMPQ     (input)
                     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     (input)
                     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         (input)
                     N is INTEGER
                     The order of the matrices H, T, Q, and Z.  N >= 0.

           ILO       (input)
                     ILO is INTEGER

           IHI       (input)
                     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         (input/output)
                     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       (input)
                     LDH is INTEGER
                     The leading dimension of the array H.  LDH >= max( 1, N ).

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

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

           ALPHAI    (output)
                     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      (output)
                     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         (input/output)
                     Q is REAL 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)
                     LDQ is INTEGER
                     The leading dimension of the array Q.  LDQ >= 1.
                     If COMPQ='V' or 'I', then LDQ >= N.

           Z         (input/output)
                     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       (input)
                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1.
                     If COMPZ='V' or 'I', then LDZ >= N.

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

           LWORK     (input)
                     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      (output)
                     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.






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.



LAPACK routine                  31 October 2017                      SHGEQZ(3)