zhgeqz.f

famous linear algebra library (LAPACK) ports to windows

      SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
     $                   ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
     $                   RWORK, INFO )
*
*  -- LAPACK routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
      CHARACTER          COMPQ, COMPZ, JOB
      INTEGER            IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   RWORK( * )
      COMPLEX*16         ALPHA( * ), BETA( * ), H( LDH, * ),
     $                   Q( LDQ, * ), T( LDT, * ), WORK( * ),
     $                   Z( LDZ, * )
*     ..
*
*  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.
*
*  =====================================================================
*
*     .. Parameters ..
      COMPLEX*16         CZERO, CONE
      PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ),
     $                   CONE = ( 1.0D+0, 0.0D+0 ) )
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
      DOUBLE PRECISION   HALF
      PARAMETER          ( HALF = 0.5D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ILAZR2, ILAZRO, ILQ, ILSCHR, ILZ, LQUERY
      INTEGER            ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
     $                   ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
     $                   JR, MAXIT
      DOUBLE PRECISION   ABSB, ANORM, ASCALE, ATOL, BNORM, BSCALE, BTOL,
     $                   C, SAFMIN, TEMP, TEMP2, TEMPR, ULP
      COMPLEX*16         ABI22, AD11, AD12, AD21, AD22, CTEMP, CTEMP2,
     $                   CTEMP3, ESHIFT, RTDISC, S, SHIFT, SIGNBC, T1,
     $                   U12, X
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, ZLANHS
      EXTERNAL           LSAME, DLAMCH, ZLANHS
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA, ZLARTG, ZLASET, ZROT, ZSCAL
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN,
     $                   SQRT
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   ABS1
*     ..
*     .. Statement Function definitions ..
      ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
*     ..
*     .. Executable Statements ..
*
*     Decode JOB, COMPQ, COMPZ
*
      IF( LSAME( JOB, 'E' ) ) THEN
         ILSCHR = .FALSE.
         ISCHUR = 1
      ELSE IF( LSAME( JOB, 'S' ) ) THEN
         ILSCHR = .TRUE.
         ISCHUR = 2
      ELSE
         ISCHUR = 0
      END IF
*
      IF( LSAME( COMPQ, 'N' ) ) THEN
         ILQ = .FALSE.
         ICOMPQ = 1
      ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
         ILQ = .TRUE.
         ICOMPQ = 2
      ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
         ILQ = .TRUE.
         ICOMPQ = 3
      ELSE
         ICOMPQ = 0
      END IF
*
      IF( LSAME( COMPZ, 'N' ) ) THEN
         ILZ = .FALSE.
         ICOMPZ = 1
      ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
         ILZ = .TRUE.
         ICOMPZ = 2
      ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
         ILZ = .TRUE.
         ICOMPZ = 3
      ELSE
         ICOMPZ = 0
      END IF
*
*     Check Argument Values
*
      INFO = 0
      WORK( 1 ) = MAX( 1, N )
      LQUERY = ( LWORK.EQ.-1 )
      IF( ISCHUR.EQ.0 ) THEN
         INFO = -1
      ELSE IF( ICOMPQ.EQ.0 ) THEN
         INFO = -2
      ELSE IF( ICOMPZ.EQ.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( ILO.LT.1 ) THEN
         INFO = -5
      ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
         INFO = -6
      ELSE IF( LDH.LT.N ) THEN
         INFO = -8
      ELSE IF( LDT.LT.N ) THEN
         INFO = -10
      ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
         INFO = -14
      ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
         INFO = -16
      ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
         INFO = -18
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'ZHGEQZ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
*     WORK( 1 ) = CMPLX( 1 )
      IF( N.LE.0 ) THEN
         WORK( 1 ) = DCMPLX( 1 )
         RETURN
      END IF
*
*     Initialize Q and Z
*
      IF( ICOMPQ.EQ.3 )
     $   CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
      IF( ICOMPZ.EQ.3 )
     $   CALL ZLASET( 'Full', N, N, CZERO, CONE, Z, LDZ )
*
*     Machine Constants
*
      IN = IHI + 1 - ILO
      SAFMIN = DLAMCH( 'S' )
      ULP = DLAMCH( 'E' )*DLAMCH( 'B' )
      ANORM = ZLANHS( 'F', IN, H( ILO, ILO ), LDH, RWORK )
      BNORM = ZLANHS( 'F', IN, T( ILO, ILO ), LDT, RWORK )
      ATOL = MAX( SAFMIN, ULP*ANORM )
      BTOL = MAX( SAFMIN, ULP*BNORM )
      ASCALE = ONE / MAX( SAFMIN, ANORM )
      BSCALE = ONE / MAX( SAFMIN, BNORM )
*
*
*     Set Eigenvalues IHI+1:N
*
      DO 10 J = IHI + 1, N
         ABSB = ABS( T( J, J ) )
         IF( ABSB.GT.SAFMIN ) THEN
            SIGNBC = DCONJG( T( J, J ) / ABSB )
            T( J, J ) = ABSB
            IF( ILSCHR ) THEN
               CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 )
               CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 )
            ELSE
               H( J, J ) = H( J, J )*SIGNBC
            END IF
            IF( ILZ )
     $         CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 )
         ELSE
            T( J, J ) = CZERO
         END IF
         ALPHA( J ) = H( J, J )
         BETA( J ) = T( J, J )
   10 CONTINUE
*
*     If IHI < ILO, skip QZ steps
*
      IF( IHI.LT.ILO )
     $   GO TO 190
*
*     MAIN QZ ITERATION LOOP
*
*     Initialize dynamic indices
*
*     Eigenvalues ILAST+1:N have been found.
*        Column operations modify rows IFRSTM:whatever
*        Row operations modify columns whatever:ILASTM
*
*     If only eigenvalues are being computed, then
*        IFRSTM is the row of the last splitting row above row ILAST;
*        this is always at least ILO.
*     IITER counts iterations since the last eigenvalue was found,
*        to tell when to use an extraordinary shift.
*     MAXIT is the maximum number of QZ sweeps allowed.
*
      ILAST = IHI
      IF( ILSCHR ) THEN
         IFRSTM = 1
         ILASTM = N
      ELSE
         IFRSTM = ILO
         ILASTM = IHI
      END IF
      IITER = 0
      ESHIFT = CZERO