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, "An Algorithm for Generalized Matrix


      Eigenvalue Problems", 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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

June 2016

  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.