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zhseqr.f

      SUBROUTINE ZHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ,
     $                   WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 2.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     September 30, 1994
*
*     .. Scalar Arguments ..
      CHARACTER          COMPZ, JOB
      INTEGER            IHI, ILO, INFO, LDH, LDZ, LWORK, N
*     ..
*     .. Array Arguments ..
      COMPLEX*16         H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  ZHSEQR computes the eigenvalues of a complex upper Hessenberg
*  matrix H, and, optionally, the matrices T and Z from the Schur
*  decomposition H = Z T Z**H, where T is an upper triangular matrix
*  (the Schur form), and Z is the unitary matrix of Schur vectors.
*
*  Optionally Z may be postmultiplied into an input unitary matrix Q,
*  so that this routine can give the Schur factorization of a matrix A
*  which has been reduced to the Hessenberg form H by the unitary
*  matrix Q:  A = Q*H*Q**H = (QZ)*T*(QZ)**H.
*
*  Arguments
*  =========
*
*  JOB     (input) CHARACTER*1
*          = 'E': compute eigenvalues only;
*          = 'S': compute eigenvalues and the Schur form T.
*
*  COMPZ   (input) CHARACTER*1
*          = 'N': no Schur vectors are computed;
*          = 'I': Z is initialized to the unit matrix and the matrix Z
*                 of Schur vectors of H is returned;
*          = 'V': Z must contain an unitary matrix Q on entry, and
*                 the product Q*Z is returned.
*
*  N       (input) INTEGER
*          The order of the matrix H.  N >= 0.
*
*  ILO     (input) INTEGER
*  IHI     (input) INTEGER
*          It is assumed that H is already upper triangular in rows
*          and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
*          set by a previous call to ZGEBAL, and then passed to CGEHRD
*          when the matrix output by ZGEBAL is reduced to Hessenberg
*          form. Otherwise ILO and IHI should be set to 1 and N
*          respectively.
*          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*
*  H       (input/output) COMPLEX*16 array, dimension (LDH,N)
*          On entry, the upper Hessenberg matrix H.
*          On exit, if JOB = 'S', H contains the upper triangular matrix
*          T from the Schur decomposition (the Schur form). If
*          JOB = 'E', the contents of H are unspecified on exit.
*
*  LDH     (input) INTEGER
*          The leading dimension of the array H. LDH >= max(1,N).
*
*  W       (output) COMPLEX*16 array, dimension (N)
*          The computed eigenvalues. If JOB = 'S', the eigenvalues are
*          stored in the same order as on the diagonal of the Schur form
*          returned in H, with W(i) = H(i,i).
*
*  Z       (input/output) COMPLEX*16 array, dimension (LDZ,N)
*          If COMPZ = 'N': Z is not referenced.
*          If COMPZ = 'I': on entry, Z need not be set, and on exit, Z
*          contains the unitary matrix Z of the Schur vectors of H.
*          If COMPZ = 'V': on entry Z must contain an N-by-N matrix Q,
*          which is assumed to be equal to the unit matrix except for
*          the submatrix Z(ILO:IHI,ILO:IHI); on exit Z contains Q*Z.
*          Normally Q is the unitary matrix generated by ZUNGHR after
*          the call to ZGEHRD which formed the Hessenberg matrix H.
*
*  LDZ     (input) INTEGER
*          The leading dimension of the array Z.
*          LDZ >= max(1,N) if COMPZ = 'I' or 'V'; LDZ >= 1 otherwise.
*
*  WORK    (workspace) COMPLEX*16 array, dimension (N)
*
*  LWORK   (input) INTEGER
*          This argument is currently redundant.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  if INFO = i, ZHSEQR failed to compute all the
*                eigenvalues in a total of 30*(IHI-ILO+1) iterations;
*                elements 1:ilo-1 and i+1:n of W contain those
*                eigenvalues which have been successfully computed.
*
*  =====================================================================
*
*     .. Parameters ..
      COMPLEX*16         ZERO, ONE
      PARAMETER          ( ZERO = ( 0.0D+0, 0.0D+0 ),
     $                   ONE = ( 1.0D+0, 0.0D+0 ) )
      DOUBLE PRECISION   RZERO, RONE, CONST
      PARAMETER          ( RZERO = 0.0D+0, RONE = 1.0D+0,
     $                   CONST = 1.5D+0 )
      INTEGER            NSMAX, LDS
      PARAMETER          ( NSMAX = 15, LDS = NSMAX )
*     ..
*     .. Local Scalars ..
      LOGICAL            INITZ, WANTT, WANTZ
      INTEGER            I, I1, I2, IERR, II, ITEMP, ITN, ITS, J, K, L,
     $                   MAXB, NH, NR, NS, NV
      DOUBLE PRECISION   OVFL, RTEMP, SMLNUM, TST1, ULP, UNFL
      COMPLEX*16         CDUM, TAU, TEMP
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   RWORK( 1 )
      COMPLEX*16         S( LDS, NSMAX ), V( NSMAX+1 ), VV( NSMAX+1 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV, IZAMAX
      DOUBLE PRECISION   DLAMCH, DLAPY2, ZLANHS
      EXTERNAL           LSAME, ILAENV, IZAMAX, DLAMCH, DLAPY2, ZLANHS
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLABAD, XERBLA, ZCOPY, ZDSCAL, ZGEMV, ZLACPY,
     $                   ZLAHQR, ZLARFG, ZLARFX, ZLASET, ZSCAL
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, DCONJG, DIMAG, MAX, MIN
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   CABS1
*     ..
*     .. Statement Function definitions ..
      CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters
*
      WANTT = LSAME( JOB, 'S' )
      INITZ = LSAME( COMPZ, 'I' )
      WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
*
      INFO = 0
      IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN
         INFO = -1
      ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
         INFO = -5
      ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDZ.LT.1 .OR. WANTZ .AND. LDZ.LT.MAX( 1, N ) ) THEN
         INFO = -10
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'ZHSEQR', -INFO )
         RETURN
      END IF
*
*     Initialize Z, if necessary
*
      IF( INITZ )
     $   CALL ZLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
*     Store the eigenvalues isolated by ZGEBAL.
*
      DO 10 I = 1, ILO - 1
         W( I ) = H( I, I )
   10 CONTINUE
      DO 20 I = IHI + 1, N
         W( I ) = H( I, I )
   20 CONTINUE
*
*     Quick return if possible.
*
      IF( N.EQ.0 )
     $   RETURN
      IF( ILO.EQ.IHI ) THEN
         W( ILO ) = H( ILO, ILO )
         RETURN
      END IF
*
*     Set rows and columns ILO to IHI to zero below the first
*     subdiagonal.
*
      DO 40 J = ILO, IHI - 2
         DO 30 I = J + 2, N
            H( I, J ) = ZERO
   30    CONTINUE
   40 CONTINUE
      NH = IHI - ILO + 1
*
*     I1 and I2 are the indices of the first row and last column of H
*     to which transformations must be applied. If eigenvalues only are
*     being computed, I1 and I2 are re-set inside the main loop.
*
      IF( WANTT ) THEN
         I1 = 1
         I2 = N
      ELSE
         I1 = ILO
         I2 = IHI
      END IF
*
*     Ensure that the subdiagonal elements are real.
*
      DO 50 I = ILO + 1, IHI
         TEMP = H( I, I-1 )
         IF( DIMAG( TEMP ).NE.RZERO ) THEN
            RTEMP = DLAPY2( DBLE( TEMP ), DIMAG( TEMP ) )
            H( I, I-1 ) = RTEMP
            TEMP = TEMP / RTEMP
            IF( I2.GT.I )
     $         CALL ZSCAL( I2-I, DCONJG( TEMP ), H( I, I+1 ), LDH )
            CALL ZSCAL( I-I1, TEMP, H( I1, I ), 1 )
            IF( I.LT.IHI )
     $         H( I+1, I ) = TEMP*H( I+1, I )
            IF( WANTZ )
     $         CALL ZSCAL( NH, TEMP, Z( ILO, I ), 1 )
         END IF
   50 CONTINUE
*
*     Determine the order of the multi-shift QR algorithm to be used.
*
      NS = ILAENV( 4, 'ZHSEQR', JOB // COMPZ, N, ILO, IHI, -1 )
      MAXB = ILAENV( 8, 'ZHSEQR', JOB // COMPZ, N, ILO, IHI, -1 )
      IF( NS.LE.1 .OR. NS.GT.NH .OR. MAXB.GE.NH ) THEN
*
*        Use the standard double-shift algorithm
*
         CALL ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI, Z,
     $                LDZ, INFO )
         RETURN
      END IF
      MAXB = MAX( 2, MAXB )
      NS = MIN( NS, MAXB, NSMAX )
*
*     Now 1 < NS <= MAXB < NH.
*
*     Set machine-dependent constants for the stopping criterion.
*     If norm(H) <= sqrt(OVFL), overflow should not occur.
*
      UNFL = DLAMCH( 'Safe minimum' )
      OVFL = RONE / UNFL
      CALL DLABAD( UNFL, OVFL )
      ULP = DLAMCH( 'Precision' )
      SMLNUM = UNFL*( NH / ULP )
*
*     ITN is the total number of multiple-shift QR iterations allowed.
*
      ITN = 30*NH
*
*     The main loop begins here. I is the loop index and decreases from
*     IHI to ILO in steps of at most MAXB. Each iteration of the loop
*     works with the active submatrix in rows and columns L to I.
*     Eigenvalues I+1 to IHI have already converged. Either L = ILO, or
*     H(L,L-1) is negligible so that the matrix splits.
*
      I = IHI
   60 CONTINUE
      IF( I.LT.ILO )
     $   GO TO 180
*
*     Perform multiple-shift QR iterations on rows and columns ILO to I
*     until a submatrix of order at most MAXB splits off at the bottom
*     because a subdiagonal element has become negligible.
*
      L = ILO
      DO 160 ITS = 0, ITN
*
*        Look for a single small subdiagonal element.
*
         DO 70 K = I, L + 1, -1
            TST1 = CABS1( H( K-1, K-1 ) ) + CABS1( H( K, K ) )
            IF( TST1.EQ.RZERO )
     $         TST1 = ZLANHS( '1', I-L+1, H( L, L ), LDH, RWORK )
            IF( ABS( DBLE( H( K, K-1 ) ) ).LE.MAX( ULP*TST1, SMLNUM ) )
     $         GO TO 80
   70    CONTINUE
   80    CONTINUE
         L = K
         IF( L.GT.ILO ) THEN
*
*           H(L,L-1) is negligible.
*
            H( L, L-1 ) = ZERO
         END IF
*
*        Exit from loop if a submatrix of order <= MAXB has split off.
*
         IF( L.GE.I-MAXB+1 )
     $      GO TO 170
*
*        Now the active submatrix is in rows and columns L to I. If
*        eigenvalues only are being computed, only the active submatrix
*        need be transformed.
*
         IF( .NOT.WANTT ) THEN
            I1 = L
            I2 = I
         END IF
*
         IF( ITS.EQ.20 .OR. ITS.EQ.30 ) THEN
*
*           Exceptional shifts.
*
            DO 90 II = I - NS + 1, I
               W( II ) = CONST*( ABS( DBLE( H( II, II-1 ) ) )+
     $                   ABS( DBLE( H( II, II ) ) ) )
   90       CONTINUE
         ELSE
*
*           Use eigenvalues of trailing submatrix of order NS as shifts.
*
            CALL ZLACPY( 'Full', NS, NS, H( I-NS+1, I-NS+1 ), LDH, S,
     $                   LDS )
            CALL ZLAHQR( .FALSE., .FALSE., NS, 1, NS, S, LDS,
     $                   W( I-NS+1 ), 1, NS, Z, LDZ, IERR )
            IF( IERR.GT.0 ) THEN
*
*              If ZLAHQR failed to compute all NS eigenvalues, use the
*              unconverged diagonal elements as the remaining shifts.
*
               DO 100 II = 1, IERR
                  W( I-NS+II ) = S( II, II )
  100          CONTINUE
            END IF
         END IF
*
*        Form the first column of (G-w(1)) (G-w(2)) . . . (G-w(ns))
*        where G is the Hessenberg submatrix H(L:I,L:I) and w is
*        the vector of shifts (stored in W). The result is
*        stored in the local array V.
*
         V( 1 ) = ONE
         DO 110 II = 2, NS + 1
            V( II ) = ZERO
  110    CONTINUE
         NV = 1
         DO 130 J = I - NS + 1, I
            CALL ZCOPY( NV+1, V, 1, VV, 1 )
            CALL ZGEMV( 'No transpose', NV+1, NV, ONE, H( L, L ), LDH,
     $                  VV, 1, -W( J ), V, 1 )
            NV = NV + 1
*
*           Scale V(1:NV) so that max(abs(V(i))) = 1. If V is zero,
*           reset it to the unit vector.
*
            ITEMP = IZAMAX( NV, V, 1 )
            RTEMP = CABS1( V( ITEMP ) )
            IF( RTEMP.EQ.RZERO ) THEN
               V( 1 ) = ONE
               DO 120 II = 2, NV
                  V( II ) = ZERO
  120          CONTINUE
            ELSE
               RTEMP = MAX( RTEMP, SMLNUM )
               CALL ZDSCAL( NV, RONE / RTEMP, V, 1 )
            END IF
  130    CONTINUE
*
*        Multiple-shift QR step
*
         DO 150 K = L, I - 1
*
*           The first iteration of this loop determines a reflection G
*           from the vector V and applies it from left and right to H,
*           thus creating a nonzero bulge below the subdiagonal.
*
*           Each subsequent iteration determines a reflection G to
*           restore the Hessenberg form in the (K-1)th column, and thus
*           chases the bulge one step toward the bottom of the active
*           submatrix. NR is the order of G.
*
            NR = MIN( NS+1, I-K+1 )
            IF( K.GT.L )
     $         CALL ZCOPY( NR, H( K, K-1 ), 1, V, 1 )
            CALL ZLARFG( NR, V( 1 ), V( 2 ), 1, TAU )
            IF( K.GT.L ) THEN
               H( K, K-1 ) = V( 1 )
               DO 140 II = K + 1, I
                  H( II, K-1 ) = ZERO
  140          CONTINUE
            END IF
            V( 1 ) = ONE
*
*           Apply G


' from the left to transform the rows of the matrix*           in columns K to I2.*            CALL ZLARFX( 'Left





', NR, I2-K+1, V, DCONJG( TAU ),     $                   H( K, K ), LDH, WORK )**           Apply G from the right to transform the columns of the*           matrix in rows I1 to min(K+NR,I).*            CALL ZLARFX( 'Right






', MIN( K+NR, I )-I1+1, NR, V, TAU,     $                   H( I1, K ), LDH, WORK )*            IF( WANTZ ) THEN**              Accumulate transformations in the matrix Z*               CALL ZLARFX( 'Right


















































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