```      SUBROUTINE SHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI,
\$                   VL, LDVL, VR, LDVR, MM, M, WORK, IFAILL,
\$                   IFAILR, INFO )
*
*  -- LAPACK routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
CHARACTER          EIGSRC, INITV, SIDE
INTEGER            INFO, LDH, LDVL, LDVR, M, MM, N
*     ..
*     .. Array Arguments ..
LOGICAL            SELECT( * )
INTEGER            IFAILL( * ), IFAILR( * )
REAL               H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
\$                   WI( * ), WORK( * ), WR( * )
*     ..
*
*  Purpose
*  =======
*
*  SHSEIN uses inverse iteration to find specified right and/or left
*  eigenvectors of a real upper Hessenberg matrix H.
*
*  The right eigenvector x and the left eigenvector y of the matrix H
*  corresponding to an eigenvalue w are defined by:
*
*               H * x = w * x,     y**h * H = w * y**h
*
*  where y**h denotes the conjugate transpose of the vector y.
*
*  Arguments
*  =========
*
*  SIDE    (input) CHARACTER*1
*          = 'R': compute right eigenvectors only;
*          = 'L': compute left eigenvectors only;
*          = 'B': compute both right and left eigenvectors.
*
*  EIGSRC  (input) CHARACTER*1
*          Specifies the source of eigenvalues supplied in (WR,WI):
*          = 'Q': the eigenvalues were found using SHSEQR; thus, if
*                 H has zero subdiagonal elements, and so is
*                 block-triangular, then the j-th eigenvalue can be
*                 assumed to be an eigenvalue of the block containing
*                 the j-th row/column.  This property allows SHSEIN to
*                 perform inverse iteration on just one diagonal block.
*          = 'N': no assumptions are made on the correspondence
*                 between eigenvalues and diagonal blocks.  In this
*                 case, SHSEIN must always perform inverse iteration
*                 using the whole matrix H.
*
*  INITV   (input) CHARACTER*1
*          = 'N': no initial vectors are supplied;
*          = 'U': user-supplied initial vectors are stored in the arrays
*                 VL and/or VR.
*
*  SELECT  (input/output) LOGICAL array, dimension (N)
*          Specifies the eigenvectors to be computed. To select the
*          real eigenvector corresponding to a real eigenvalue WR(j),
*          SELECT(j) must be set to .TRUE.. To select the complex
*          eigenvector corresponding to a complex eigenvalue
*          (WR(j),WI(j)), with complex conjugate (WR(j+1),WI(j+1)),
*          either SELECT(j) or SELECT(j+1) or both must be set to
*          .TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is
*          .FALSE..
*
*  N       (input) INTEGER
*          The order of the matrix H.  N >= 0.
*
*  H       (input) REAL array, dimension (LDH,N)
*          The upper Hessenberg matrix H.
*
*  LDH     (input) INTEGER
*          The leading dimension of the array H.  LDH >= max(1,N).
*
*  WR      (input/output) REAL array, dimension (N)
*  WI      (input) REAL array, dimension (N)
*          On entry, the real and imaginary parts of the eigenvalues of
*          H; a complex conjugate pair of eigenvalues must be stored in
*          consecutive elements of WR and WI.
*          On exit, WR may have been altered since close eigenvalues
*          are perturbed slightly in searching for independent
*          eigenvectors.
*
*  VL      (input/output) REAL array, dimension (LDVL,MM)
*          On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
*          contain starting vectors for the inverse iteration for the
*          left eigenvectors; the starting vector for each eigenvector
*          must be in the same column(s) in which the eigenvector will
*          be stored.
*          On exit, if SIDE = 'L' or 'B', the left eigenvectors
*          specified by SELECT will be stored consecutively in the
*          columns of VL, in the same order as their eigenvalues. A
*          complex eigenvector corresponding to a complex eigenvalue is
*          stored in two consecutive columns, the first holding the real
*          part and the second the imaginary part.
*          If SIDE = 'R', VL is not referenced.
*
*  LDVL    (input) INTEGER
*          The leading dimension of the array VL.
*          LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
*
*  VR      (input/output) REAL array, dimension (LDVR,MM)
*          On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
*          contain starting vectors for the inverse iteration for the
*          right eigenvectors; the starting vector for each eigenvector
*          must be in the same column(s) in which the eigenvector will
*          be stored.
*          On exit, if SIDE = 'R' or 'B', the right eigenvectors
*          specified by SELECT will be stored consecutively in the
*          columns of VR, in the same order as their eigenvalues. A
*          complex eigenvector corresponding to a complex eigenvalue is
*          stored in two consecutive columns, the first holding the real
*          part and the second the imaginary part.
*          If SIDE = 'L', VR is not referenced.
*
*  LDVR    (input) INTEGER
*          The leading dimension of the array VR.
*          LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
*
*  MM      (input) INTEGER
*          The number of columns in the arrays VL and/or VR. MM >= M.
*
*  M       (output) INTEGER
*          The number of columns in the arrays VL and/or VR required to
*          store the eigenvectors; each selected real eigenvector
*          occupies one column and each selected complex eigenvector
*          occupies two columns.
*
*  WORK    (workspace) REAL array, dimension ((N+2)*N)
*
*  IFAILL  (output) INTEGER array, dimension (MM)
*          If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
*          eigenvector in the i-th column of VL (corresponding to the
*          eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
*          eigenvector converged satisfactorily. If the i-th and (i+1)th
*          columns of VL hold a complex eigenvector, then IFAILL(i) and
*          IFAILL(i+1) are set to the same value.
*          If SIDE = 'R', IFAILL is not referenced.
*
*  IFAILR  (output) INTEGER array, dimension (MM)
*          If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
*          eigenvector in the i-th column of VR (corresponding to the
*          eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
*          eigenvector converged satisfactorily. If the i-th and (i+1)th
*          columns of VR hold a complex eigenvector, then IFAILR(i) and
*          IFAILR(i+1) are set to the same value.
*          If SIDE = 'L', IFAILR is not referenced.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  if INFO = i, i is the number of eigenvectors which
*                failed to converge; see IFAILL and IFAILR for further
*                details.
*
*  Further Details
*  ===============
*
*  Each eigenvector is normalized so that the element of largest
*  magnitude has magnitude 1; here the magnitude of a complex number
*  (x,y) is taken to be |x|+|y|.
*
*  =====================================================================
*
*     .. Parameters ..
REAL               ZERO, ONE
PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
*     ..
*     .. Local Scalars ..
LOGICAL            BOTHV, FROMQR, LEFTV, NOINIT, PAIR, RIGHTV
INTEGER            I, IINFO, K, KL, KLN, KR, KSI, KSR, LDWORK
REAL               BIGNUM, EPS3, HNORM, SMLNUM, ULP, UNFL, WKI,
\$                   WKR
*     ..
*     .. External Functions ..
LOGICAL            LSAME
REAL               SLAMCH, SLANHS
EXTERNAL           LSAME, SLAMCH, SLANHS
*     ..
*     .. External Subroutines ..
EXTERNAL           SLAEIN, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters.
*
BOTHV = LSAME( SIDE, 'B' )
RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV
LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV
*
FROMQR = LSAME( EIGSRC, 'Q' )
*
NOINIT = LSAME( INITV, 'N' )
*
*     Set M to the number of columns required to store the selected
*     eigenvectors, and standardize the array SELECT.
*
M = 0
PAIR = .FALSE.
DO 10 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
SELECT( K ) = .FALSE.
ELSE
IF( WI( K ).EQ.ZERO ) THEN
IF( SELECT( K ) )
\$            M = M + 1
ELSE
PAIR = .TRUE.
IF( SELECT( K ) .OR. SELECT( K+1 ) ) THEN
SELECT( K ) = .TRUE.
M = M + 2
END IF
END IF
END IF
10 CONTINUE
*
INFO = 0
IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
INFO = -1
ELSE IF( .NOT.FROMQR .AND. .NOT.LSAME( EIGSRC, 'N' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOINIT .AND. .NOT.LSAME( INITV, 'U' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN
INFO = -11
ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN
INFO = -13
ELSE IF( MM.LT.M ) THEN
INFO = -14
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SHSEIN', -INFO )
RETURN
END IF
*
*     Quick return if possible.
*
IF( N.EQ.0 )
\$   RETURN
*
*     Set machine-dependent constants.
*
UNFL = SLAMCH( 'Safe minimum' )
ULP = SLAMCH( 'Precision' )
SMLNUM = UNFL*( N / ULP )
BIGNUM = ( ONE-ULP ) / SMLNUM
*
LDWORK = N + 1
*
KL = 1
KLN = 0
IF( FROMQR ) THEN
KR = 0
ELSE
KR = N
END IF
KSR = 1
*
DO 120 K = 1, N
IF( SELECT( K ) ) THEN
*
*           Compute eigenvector(s) corresponding to W(K).
*
IF( FROMQR ) THEN
*
*              If affiliation of eigenvalues is known, check whether
*              the matrix splits.
*
*              Determine KL and KR such that 1 <= KL <= K <= KR <= N
*              and H(KL,KL-1) and H(KR+1,KR) are zero (or KL = 1 or
*              KR = N).
*
*              Then inverse iteration can be performed with the
*              submatrix H(KL:N,KL:N) for a left eigenvector, and with
*              the submatrix H(1:KR,1:KR) for a right eigenvector.
*
DO 20 I = K, KL + 1, -1
IF( H( I, I-1 ).EQ.ZERO )
\$               GO TO 30
20          CONTINUE
30          CONTINUE
KL = I
IF( K.GT.KR ) THEN
DO 40 I = K, N - 1
IF( H( I+1, I ).EQ.ZERO )
\$                  GO TO 50
40             CONTINUE
50             CONTINUE
KR = I
END IF
END IF
*
IF( KL.NE.KLN ) THEN
KLN = KL
*
*              Compute infinity-norm of submatrix H(KL:KR,KL:KR) if it
*              has not ben computed before.
*
HNORM = SLANHS( 'I', KR-KL+1, H( KL, KL ), LDH, WORK )
IF( HNORM.GT.ZERO ) THEN
EPS3 = HNORM*ULP
ELSE
EPS3 = SMLNUM
END IF
END IF
*
*           Perturb eigenvalue if it is close to any previous
*           selected eigenvalues affiliated to the submatrix
*           H(KL:KR,KL:KR). Close roots are modified by EPS3.
*
WKR = WR( K )
WKI = WI( K )
60       CONTINUE
DO 70 I = K - 1, KL, -1
IF( SELECT( I ) .AND. ABS( WR( I )-WKR )+
\$             ABS( WI( I )-WKI ).LT.EPS3 ) THEN
WKR = WKR + EPS3
GO TO 60
END IF
70       CONTINUE
WR( K ) = WKR
*
PAIR = WKI.NE.ZERO
IF( PAIR ) THEN
KSI = KSR + 1
ELSE
KSI = KSR
END IF
IF( LEFTV ) THEN
*
*              Compute left eigenvector.
*
CALL SLAEIN( .FALSE., NOINIT, N-KL+1, H( KL, KL ), LDH,
\$                      WKR, WKI, VL( KL, KSR ), VL( KL, KSI ),
\$                      WORK, LDWORK, WORK( N*N+N+1 ), EPS3, SMLNUM,
\$                      BIGNUM, IINFO )
IF( IINFO.GT.0 ) THEN
IF( PAIR ) THEN
INFO = INFO + 2
ELSE
INFO = INFO + 1
END IF
IFAILL( KSR ) = K
IFAILL( KSI ) = K
ELSE
IFAILL( KSR ) = 0
IFAILL( KSI ) = 0
END IF
DO 80 I = 1, KL - 1
VL( I, KSR ) = ZERO
80          CONTINUE
IF( PAIR ) THEN
DO 90 I = 1, KL - 1
VL( I, KSI ) = ZERO
90             CONTINUE
END IF
END IF
IF( RIGHTV ) THEN
*
*              Compute right eigenvector.
*
CALL SLAEIN( .TRUE., NOINIT, KR, H, LDH, WKR, WKI,
\$                      VR( 1, KSR ), VR( 1, KSI ), WORK, LDWORK,
\$                      WORK( N*N+N+1 ), EPS3, SMLNUM, BIGNUM,
\$                      IINFO )
IF( IINFO.GT.0 ) THEN
IF( PAIR ) THEN
INFO = INFO + 2
ELSE
INFO = INFO + 1
END IF
IFAILR( KSR ) = K
IFAILR( KSI ) = K
ELSE
IFAILR( KSR ) = 0
IFAILR( KSI ) = 0
END IF
DO 100 I = KR + 1, N
VR( I, KSR ) = ZERO
100          CONTINUE
IF( PAIR ) THEN
DO 110 I = KR + 1, N
VR( I, KSI ) = ZERO
110             CONTINUE
END IF
END IF
*
IF( PAIR ) THEN
KSR = KSR + 2
ELSE
KSR = KSR + 1
END IF
END IF
120 CONTINUE
*
RETURN
*
*     End of SHSEIN
*
END

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