```      SUBROUTINE CLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
\$                   IHIZ, Z, LDZ, INFO )
*
*  -- LAPACK auxiliary routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
LOGICAL            WANTT, WANTZ
*     ..
*     .. Array Arguments ..
COMPLEX            H( LDH, * ), W( * ), Z( LDZ, * )
*     ..
*
*     Purpose
*     =======
*
*     CLAHQR is an auxiliary routine called by CHSEQR to update the
*     eigenvalues and Schur decomposition already computed by CHSEQR, by
*     dealing with the Hessenberg submatrix in rows and columns ILO to
*     IHI.
*
*     Arguments
*     =========
*
*     WANTT   (input) LOGICAL
*          = .TRUE. : the full Schur form T is required;
*          = .FALSE.: only eigenvalues are required.
*
*     WANTZ   (input) LOGICAL
*          = .TRUE. : the matrix of Schur vectors Z is required;
*          = .FALSE.: Schur vectors are not required.
*
*     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 IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1).
*          CLAHQR works primarily with the Hessenberg submatrix in rows
*          and columns ILO to IHI, but applies transformations to all of
*          H if WANTT is .TRUE..
*          1 <= ILO <= max(1,IHI); IHI <= N.
*
*     H       (input/output) COMPLEX array, dimension (LDH,N)
*          On entry, the upper Hessenberg matrix H.
*          On exit, if INFO is zero and if WANTT is .TRUE., then H
*          is upper triangular in rows and columns ILO:IHI.  If INFO
*          is zero and if WANTT is .FALSE., then the contents of H
*          are unspecified on exit.  The output state of H in case
*          INF is positive is below under the description of INFO.
*
*     LDH     (input) INTEGER
*          The leading dimension of the array H. LDH >= max(1,N).
*
*     W       (output) COMPLEX array, dimension (N)
*          The computed eigenvalues ILO to IHI are stored in the
*          corresponding elements of W. If WANTT is .TRUE., 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).
*
*     ILOZ    (input) INTEGER
*     IHIZ    (input) INTEGER
*          Specify the rows of Z to which transformations must be
*          applied if WANTZ is .TRUE..
*          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*
*     Z       (input/output) COMPLEX array, dimension (LDZ,N)
*          If WANTZ is .TRUE., on entry Z must contain the current
*          matrix Z of transformations accumulated by CHSEQR, and on
*          exit Z has been updated; transformations are applied only to
*          the submatrix Z(ILOZ:IHIZ,ILO:IHI).
*          If WANTZ is .FALSE., Z is not referenced.
*
*     LDZ     (input) INTEGER
*          The leading dimension of the array Z. LDZ >= max(1,N).
*
*     INFO    (output) INTEGER
*           =   0: successful exit
*          .GT. 0: if INFO = i, CLAHQR failed to compute all the
*                  eigenvalues ILO to IHI in a total of 30 iterations
*                  per eigenvalue; elements i+1:ihi of W contain
*                  those eigenvalues which have been successfully
*                  computed.
*
*                  If INFO .GT. 0 and WANTT is .FALSE., then on exit,
*                  the remaining unconverged eigenvalues are the
*                  eigenvalues of the upper Hessenberg matrix
*                  rows and columns ILO thorugh INFO of the final,
*                  output value of H.
*
*                  If INFO .GT. 0 and WANTT is .TRUE., then on exit
*          (*)       (initial value of H)*U  = U*(final value of H)
*                  where U is an orthognal matrix.    The final
*                  value of H is upper Hessenberg and triangular in
*                  rows and columns INFO+1 through IHI.
*
*                  If INFO .GT. 0 and WANTZ is .TRUE., then on exit
*                      (final value of Z)  = (initial value of Z)*U
*                  where U is the orthogonal matrix in (*)
*                  (regardless of the value of WANTT.)
*
*     Further Details
*     ===============
*
*     02-96 Based on modifications by
*     David Day, Sandia National Laboratory, USA
*
*     12-04 Further modifications by
*     Ralph Byers, University of Kansas, USA
*
*       This is a modified version of CLAHQR from LAPACK version 3.0.
*       It is (1) more robust against overflow and underflow and
*       (2) adopts the more conservative Ahues & Tisseur stopping
*       criterion (LAWN 122, 1997).
*
*     =========================================================
*
*     .. Parameters ..
INTEGER            ITMAX
PARAMETER          ( ITMAX = 30 )
COMPLEX            ZERO, ONE
PARAMETER          ( ZERO = ( 0.0e0, 0.0e0 ),
\$                   ONE = ( 1.0e0, 0.0e0 ) )
REAL               RZERO, RONE, HALF
PARAMETER          ( RZERO = 0.0e0, RONE = 1.0e0, HALF = 0.5e0 )
REAL               DAT1
PARAMETER          ( DAT1 = 3.0e0 / 4.0e0 )
*     ..
*     .. Local Scalars ..
COMPLEX            CDUM, H11, H11S, H22, SC, SUM, T, T1, TEMP, U,
\$                   V2, X, Y
REAL               AA, AB, BA, BB, H10, H21, RTEMP, S, SAFMAX,
\$                   SAFMIN, SMLNUM, SX, T2, TST, ULP
INTEGER            I, I1, I2, ITS, J, JHI, JLO, K, L, M, NH, NZ
*     ..
*     .. Local Arrays ..
COMPLEX            V( 2 )
*     ..
*     .. External Functions ..
REAL               SLAMCH
*     ..
*     .. External Subroutines ..
*     ..
*     .. Statement Functions ..
REAL               CABS1
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          ABS, AIMAG, CONJG, MAX, MIN, REAL, SQRT
*     ..
*     .. Statement Function definitions ..
CABS1( CDUM ) = ABS( REAL( CDUM ) ) + ABS( AIMAG( CDUM ) )
*     ..
*     .. Executable Statements ..
*
INFO = 0
*
*     Quick return if possible
*
IF( N.EQ.0 )
\$   RETURN
IF( ILO.EQ.IHI ) THEN
W( ILO ) = H( ILO, ILO )
RETURN
END IF
*
*     ==== clear out the trash ====
DO 10 J = ILO, IHI - 3
H( J+2, J ) = ZERO
H( J+3, J ) = ZERO
10 CONTINUE
IF( ILO.LE.IHI-2 )
\$   H( IHI, IHI-2 ) = ZERO
*     ==== ensure that subdiagonal entries are real ====
DO 20 I = ILO + 1, IHI
IF( AIMAG( H( I, I-1 ) ).NE.RZERO ) THEN
*           ==== The following redundant normalization
*           .    avoids problems with both gradual and
*           .    sudden underflow in ABS(H(I,I-1)) ====
SC = H( I, I-1 ) / CABS1( H( I, I-1 ) )
SC = CONJG( SC ) / ABS( SC )
H( I, I-1 ) = ABS( H( I, I-1 ) )
IF( WANTT ) THEN
JLO = 1
JHI = N
ELSE
JLO = ILO
JHI = IHI
END IF
CALL CSCAL( JHI-I+1, SC, H( I, I ), LDH )
CALL CSCAL( MIN( JHI, I+1 )-JLO+1, CONJG( SC ), H( JLO, I ),
\$                  1 )
IF( WANTZ )
\$         CALL CSCAL( IHIZ-ILOZ+1, CONJG( SC ), Z( ILOZ, I ), 1 )
END IF
20 CONTINUE
*
NH = IHI - ILO + 1
NZ = IHIZ - ILOZ + 1
*
*     Set machine-dependent constants for the stopping criterion.
*
SAFMIN = SLAMCH( 'SAFE MINIMUM' )
SAFMAX = RONE / SAFMIN
ULP = SLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( REAL( NH ) / ULP )
*
*     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 set inside the main loop.
*
IF( WANTT ) THEN
I1 = 1
I2 = N
END IF
*
*     The main loop begins here. I is the loop index and decreases from
*     IHI to ILO in steps of 1. 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
30 CONTINUE
IF( I.LT.ILO )
\$   GO TO 150
*
*     Perform QR iterations on rows and columns ILO to I until a
*     submatrix of order 1 splits off at the bottom because a
*     subdiagonal element has become negligible.
*
L = ILO
DO 130 ITS = 0, ITMAX
*
*        Look for a single small subdiagonal element.
*
DO 40 K = I, L + 1, -1
IF( CABS1( H( K, K-1 ) ).LE.SMLNUM )
\$         GO TO 50
TST = CABS1( H( K-1, K-1 ) ) + CABS1( H( K, K ) )
IF( TST.EQ.ZERO ) THEN
IF( K-2.GE.ILO )
\$            TST = TST + ABS( REAL( H( K-1, K-2 ) ) )
IF( K+1.LE.IHI )
\$            TST = TST + ABS( REAL( H( K+1, K ) ) )
END IF
*           ==== The following is a conservative small subdiagonal
*           .    deflation criterion due to Ahues & Tisseur (LAWN 122,
*           .    1997). It has better mathematical foundation and
*           .    improves accuracy in some examples.  ====
IF( ABS( REAL( H( K, K-1 ) ) ).LE.ULP*TST ) THEN
AB = MAX( CABS1( H( K, K-1 ) ), CABS1( H( K-1, K ) ) )
BA = MIN( CABS1( H( K, K-1 ) ), CABS1( H( K-1, K ) ) )
AA = MAX( CABS1( H( K, K ) ),
\$              CABS1( H( K-1, K-1 )-H( K, K ) ) )
BB = MIN( CABS1( H( K, K ) ),
\$              CABS1( H( K-1, K-1 )-H( K, K ) ) )
S = AA + AB
IF( BA*( AB / S ).LE.MAX( SMLNUM,
\$             ULP*( BB*( AA / S ) ) ) )GO TO 50
END IF
40    CONTINUE
50    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 1 has split off.
*
IF( L.GE.I )
\$      GO TO 140
*
*        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.10 .OR. ITS.EQ.20 ) THEN
*
*           Exceptional shift.
*
S = DAT1*ABS( REAL( H( I, I-1 ) ) )
T = S + H( I, I )
ELSE
*
*           Wilkinson's shift.
*
T = H( I, I )
U = SQRT( H( I-1, I ) )*SQRT( H( I, I-1 ) )
S = CABS1( U )
IF( S.NE.RZERO ) THEN
X = HALF*( H( I-1, I-1 )-T )
SX = CABS1( X )
S = MAX( S, CABS1( X ) )
Y = S*SQRT( ( X / S )**2+( U / S )**2 )
IF( SX.GT.RZERO ) THEN
IF( REAL( X / SX )*REAL( Y )+AIMAG( X / SX )*
\$                AIMAG( Y ).LT.RZERO )Y = -Y
END IF
T = T - U*CLADIV( U, ( X+Y ) )
END IF
END IF
*
*        Look for two consecutive small subdiagonal elements.
*
DO 60 M = I - 1, L + 1, -1
*
*           Determine the effect of starting the single-shift QR
*           iteration at row M, and see if this would make H(M,M-1)
*           negligible.
*
H11 = H( M, M )
H22 = H( M+1, M+1 )
H11S = H11 - T
H21 = H( M+1, M )
S = CABS1( H11S ) + ABS( H21 )
H11S = H11S / S
H21 = H21 / S
V( 1 ) = H11S
V( 2 ) = H21
H10 = H( M, M-1 )
IF( ABS( H10 )*ABS( H21 ).LE.ULP*
\$          ( CABS1( H11S )*( CABS1( H11 )+CABS1( H22 ) ) ) )
\$          GO TO 70
60    CONTINUE
H11 = H( L, L )
H22 = H( L+1, L+1 )
H11S = H11 - T
H21 = H( L+1, L )
S = CABS1( H11S ) + ABS( H21 )
H11S = H11S / S
H21 = H21 / S
V( 1 ) = H11S
V( 2 ) = H21
70    CONTINUE
*
*        Single-shift QR step
*
DO 120 K = M, 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.
*
*           V(2) is always real before the call to CLARFG, and hence
*           after the call T2 ( = T1*V(2) ) is also real.
*
IF( K.GT.M )
\$         CALL CCOPY( 2, H( K, K-1 ), 1, V, 1 )
CALL CLARFG( 2, V( 1 ), V( 2 ), 1, T1 )
IF( K.GT.M ) THEN
H( K, K-1 ) = V( 1 )
H( K+1, K-1 ) = ZERO
END IF
V2 = V( 2 )
T2 = REAL( T1*V2 )
*
*           Apply G from the left to transform the rows of the matrix
*           in columns K to I2.
*
DO 80 J = K, I2
SUM = CONJG( T1 )*H( K, J ) + T2*H( K+1, J )
H( K, J ) = H( K, J ) - SUM
H( K+1, J ) = H( K+1, J ) - SUM*V2
80       CONTINUE
*
*           Apply G from the right to transform the columns of the
*           matrix in rows I1 to min(K+2,I).
*
DO 90 J = I1, MIN( K+2, I )
SUM = T1*H( J, K ) + T2*H( J, K+1 )
H( J, K ) = H( J, K ) - SUM
H( J, K+1 ) = H( J, K+1 ) - SUM*CONJG( V2 )
90       CONTINUE
*
IF( WANTZ ) THEN
*
*              Accumulate transformations in the matrix Z
*
DO 100 J = ILOZ, IHIZ
SUM = T1*Z( J, K ) + T2*Z( J, K+1 )
Z( J, K ) = Z( J, K ) - SUM
Z( J, K+1 ) = Z( J, K+1 ) - SUM*CONJG( V2 )
100          CONTINUE
END IF
*
IF( K.EQ.M .AND. M.GT.L ) THEN
*
*              If the QR step was started at row M > L because two
*              consecutive small subdiagonals were found, then extra
*              scaling must be performed to ensure that H(M,M-1) remains
*              real.
*
TEMP = ONE - T1
TEMP = TEMP / ABS( TEMP )
H( M+1, M ) = H( M+1, M )*CONJG( TEMP )
IF( M+2.LE.I )
\$            H( M+2, M+1 ) = H( M+2, M+1 )*TEMP
DO 110 J = M, I
IF( J.NE.M+1 ) THEN
IF( I2.GT.J )
\$                  CALL CSCAL( I2-J, TEMP, H( J, J+1 ), LDH )
CALL CSCAL( J-I1, CONJG( TEMP ), H( I1, J ), 1 )
IF( WANTZ ) THEN
CALL CSCAL( NZ, CONJG( TEMP ), Z( ILOZ, J ), 1 )
END IF
END IF
110          CONTINUE
END IF
120    CONTINUE
*
*        Ensure that H(I,I-1) is real.
*
TEMP = H( I, I-1 )
IF( AIMAG( TEMP ).NE.RZERO ) THEN
RTEMP = ABS( TEMP )
H( I, I-1 ) = RTEMP
TEMP = TEMP / RTEMP
IF( I2.GT.I )
\$         CALL CSCAL( I2-I, CONJG( TEMP ), H( I, I+1 ), LDH )
CALL CSCAL( I-I1, TEMP, H( I1, I ), 1 )
IF( WANTZ ) THEN
CALL CSCAL( NZ, TEMP, Z( ILOZ, I ), 1 )
END IF
END IF
*
130 CONTINUE
*
*     Failure to converge in remaining number of iterations
*
INFO = I
RETURN
*
140 CONTINUE
*
*     H(I,I-1) is negligible: one eigenvalue has converged.
*
W( I ) = H( I, I )
*
*     return to start of the main loop with new value of I.
*
I = L - 1
GO TO 30
*
150 CONTINUE
RETURN
*
*     End of CLAHQR
*
END

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