```      SUBROUTINE DBDSDC( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ,
\$                   WORK, IWORK, INFO )
*
*  -- LAPACK routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
CHARACTER          COMPQ, UPLO
INTEGER            INFO, LDU, LDVT, N
*     ..
*     .. Array Arguments ..
INTEGER            IQ( * ), IWORK( * )
DOUBLE PRECISION   D( * ), E( * ), Q( * ), U( LDU, * ),
\$                   VT( LDVT, * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DBDSDC computes the singular value decomposition (SVD) of a real
*  N-by-N (upper or lower) bidiagonal matrix B:  B = U * S * VT,
*  using a divide and conquer method, where S is a diagonal matrix
*  with non-negative diagonal elements (the singular values of B), and
*  U and VT are orthogonal matrices of left and right singular vectors,
*  respectively. DBDSDC can be used to compute all singular values,
*  and optionally, singular vectors or singular vectors in compact form.
*
*  This code makes very mild assumptions about floating point
*  arithmetic. It will work on machines with a guard digit in
*  add/subtract, or on those binary machines without guard digits
*  which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
*  It could conceivably fail on hexadecimal or decimal machines
*  without guard digits, but we know of none.  See DLASD3 for details.
*
*  The code currently calls DLASDQ if singular values only are desired.
*  However, it can be slightly modified to compute singular values
*  using the divide and conquer method.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  B is upper bidiagonal.
*          = 'L':  B is lower bidiagonal.
*
*  COMPQ   (input) CHARACTER*1
*          Specifies whether singular vectors are to be computed
*          as follows:
*          = 'N':  Compute singular values only;
*          = 'P':  Compute singular values and compute singular
*                  vectors in compact form;
*          = 'I':  Compute singular values and singular vectors.
*
*  N       (input) INTEGER
*          The order of the matrix B.  N >= 0.
*
*  D       (input/output) DOUBLE PRECISION array, dimension (N)
*          On entry, the n diagonal elements of the bidiagonal matrix B.
*          On exit, if INFO=0, the singular values of B.
*
*  E       (input/output) DOUBLE PRECISION array, dimension (N-1)
*          On entry, the elements of E contain the offdiagonal
*          elements of the bidiagonal matrix whose SVD is desired.
*          On exit, E has been destroyed.
*
*  U       (output) DOUBLE PRECISION array, dimension (LDU,N)
*          If  COMPQ = 'I', then:
*             On exit, if INFO = 0, U contains the left singular vectors
*             of the bidiagonal matrix.
*          For other values of COMPQ, U is not referenced.
*
*  LDU     (input) INTEGER
*          The leading dimension of the array U.  LDU >= 1.
*          If singular vectors are desired, then LDU >= max( 1, N ).
*
*  VT      (output) DOUBLE PRECISION array, dimension (LDVT,N)
*          If  COMPQ = 'I', then:
*             On exit, if INFO = 0, VT' contains the right singular
*             vectors of the bidiagonal matrix.
*          For other values of COMPQ, VT is not referenced.
*
*  LDVT    (input) INTEGER
*          The leading dimension of the array VT.  LDVT >= 1.
*          If singular vectors are desired, then LDVT >= max( 1, N ).
*
*  Q       (output) DOUBLE PRECISION array, dimension (LDQ)
*          If  COMPQ = 'P', then:
*             On exit, if INFO = 0, Q and IQ contain the left
*             and right singular vectors in a compact form,
*             requiring O(N log N) space instead of 2*N**2.
*             In particular, Q contains all the DOUBLE PRECISION data in
*             LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))
*             words of memory, where SMLSIZ is returned by ILAENV and
*             is equal to the maximum size of the subproblems at the
*             bottom of the computation tree (usually about 25).
*          For other values of COMPQ, Q is not referenced.
*
*  IQ      (output) INTEGER array, dimension (LDIQ)
*          If  COMPQ = 'P', then:
*             On exit, if INFO = 0, Q and IQ contain the left
*             and right singular vectors in a compact form,
*             requiring O(N log N) space instead of 2*N**2.
*             In particular, IQ contains all INTEGER data in
*             LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
*             words of memory, where SMLSIZ is returned by ILAENV and
*             is equal to the maximum size of the subproblems at the
*             bottom of the computation tree (usually about 25).
*          For other values of COMPQ, IQ is not referenced.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*          If COMPQ = 'N' then LWORK >= (4 * N).
*          If COMPQ = 'P' then LWORK >= (6 * N).
*          If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).
*
*  IWORK   (workspace) INTEGER array, dimension (8*N)
*
*  INFO    (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*          > 0:  The algorithm failed to compute an singular value.
*                The update process of divide and conquer failed.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*  Changed dimension statement in comment describing E from (N) to
*  (N-1).  Sven, 17 Feb 05.
*  =====================================================================
*
*     .. Parameters ..
DOUBLE PRECISION   ZERO, ONE, TWO
PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
*     ..
*     .. Local Scalars ..
INTEGER            DIFL, DIFR, GIVCOL, GIVNUM, GIVPTR, I, IC,
\$                   ICOMPQ, IERR, II, IS, IU, IUPLO, IVT, J, K, KK,
\$                   MLVL, NM1, NSIZE, PERM, POLES, QSTART, SMLSIZ,
\$                   SMLSZP, SQRE, START, WSTART, Z
DOUBLE PRECISION   CS, EPS, ORGNRM, P, R, SN
*     ..
*     .. External Functions ..
LOGICAL            LSAME
INTEGER            ILAENV
DOUBLE PRECISION   DLAMCH, DLANST
EXTERNAL           LSAME, ILAENV, DLAMCH, DLANST
*     ..
*     .. External Subroutines ..
EXTERNAL           DCOPY, DLARTG, DLASCL, DLASD0, DLASDA, DLASDQ,
\$                   DLASET, DLASR, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          ABS, DBLE, INT, LOG, SIGN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
INFO = 0
*
IUPLO = 0
IF( LSAME( UPLO, 'U' ) )
\$   IUPLO = 1
IF( LSAME( UPLO, 'L' ) )
\$   IUPLO = 2
IF( LSAME( COMPQ, 'N' ) ) THEN
ICOMPQ = 0
ELSE IF( LSAME( COMPQ, 'P' ) ) THEN
ICOMPQ = 1
ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
ICOMPQ = 2
ELSE
ICOMPQ = -1
END IF
IF( IUPLO.EQ.0 ) THEN
INFO = -1
ELSE IF( ICOMPQ.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ( LDU.LT.1 ) .OR. ( ( ICOMPQ.EQ.2 ) .AND. ( LDU.LT.
\$         N ) ) ) THEN
INFO = -7
ELSE IF( ( LDVT.LT.1 ) .OR. ( ( ICOMPQ.EQ.2 ) .AND. ( LDVT.LT.
\$         N ) ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DBDSDC', -INFO )
RETURN
END IF
*
*     Quick return if possible
*
IF( N.EQ.0 )
\$   RETURN
SMLSIZ = ILAENV( 9, 'DBDSDC', ' ', 0, 0, 0, 0 )
IF( N.EQ.1 ) THEN
IF( ICOMPQ.EQ.1 ) THEN
Q( 1 ) = SIGN( ONE, D( 1 ) )
Q( 1+SMLSIZ*N ) = ONE
ELSE IF( ICOMPQ.EQ.2 ) THEN
U( 1, 1 ) = SIGN( ONE, D( 1 ) )
VT( 1, 1 ) = ONE
END IF
D( 1 ) = ABS( D( 1 ) )
RETURN
END IF
NM1 = N - 1
*
*     If matrix lower bidiagonal, rotate to be upper bidiagonal
*     by applying Givens rotations on the left
*
WSTART = 1
QSTART = 3
IF( ICOMPQ.EQ.1 ) THEN
CALL DCOPY( N, D, 1, Q( 1 ), 1 )
CALL DCOPY( N-1, E, 1, Q( N+1 ), 1 )
END IF
IF( IUPLO.EQ.2 ) THEN
QSTART = 5
WSTART = 2*N - 1
DO 10 I = 1, N - 1
CALL DLARTG( D( I ), E( I ), CS, SN, R )
D( I ) = R
E( I ) = SN*D( I+1 )
D( I+1 ) = CS*D( I+1 )
IF( ICOMPQ.EQ.1 ) THEN
Q( I+2*N ) = CS
Q( I+3*N ) = SN
ELSE IF( ICOMPQ.EQ.2 ) THEN
WORK( I ) = CS
WORK( NM1+I ) = -SN
END IF
10    CONTINUE
END IF
*
*     If ICOMPQ = 0, use DLASDQ to compute the singular values.
*
IF( ICOMPQ.EQ.0 ) THEN
CALL DLASDQ( 'U', 0, N, 0, 0, 0, D, E, VT, LDVT, U, LDU, U,
\$                LDU, WORK( WSTART ), INFO )
GO TO 40
END IF
*
*     If N is smaller than the minimum divide size SMLSIZ, then solve
*     the problem with another solver.
*
IF( N.LE.SMLSIZ ) THEN
IF( ICOMPQ.EQ.2 ) THEN
CALL DLASET( 'A', N, N, ZERO, ONE, U, LDU )
CALL DLASET( 'A', N, N, ZERO, ONE, VT, LDVT )
CALL DLASDQ( 'U', 0, N, N, N, 0, D, E, VT, LDVT, U, LDU, U,
\$                   LDU, WORK( WSTART ), INFO )
ELSE IF( ICOMPQ.EQ.1 ) THEN
IU = 1
IVT = IU + N
CALL DLASET( 'A', N, N, ZERO, ONE, Q( IU+( QSTART-1 )*N ),
\$                   N )
CALL DLASET( 'A', N, N, ZERO, ONE, Q( IVT+( QSTART-1 )*N ),
\$                   N )
CALL DLASDQ( 'U', 0, N, N, N, 0, D, E,
\$                   Q( IVT+( QSTART-1 )*N ), N,
\$                   Q( IU+( QSTART-1 )*N ), N,
\$                   Q( IU+( QSTART-1 )*N ), N, WORK( WSTART ),
\$                   INFO )
END IF
GO TO 40
END IF
*
IF( ICOMPQ.EQ.2 ) THEN
CALL DLASET( 'A', N, N, ZERO, ONE, U, LDU )
CALL DLASET( 'A', N, N, ZERO, ONE, VT, LDVT )
END IF
*
*     Scale.
*
ORGNRM = DLANST( 'M', N, D, E )
IF( ORGNRM.EQ.ZERO )
\$   RETURN
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, IERR )
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, NM1, 1, E, NM1, IERR )
*
EPS = DLAMCH( 'Epsilon' )
*
MLVL = INT( LOG( DBLE( N ) / DBLE( SMLSIZ+1 ) ) / LOG( TWO ) ) + 1
SMLSZP = SMLSIZ + 1
*
IF( ICOMPQ.EQ.1 ) THEN
IU = 1
IVT = 1 + SMLSIZ
DIFL = IVT + SMLSZP
DIFR = DIFL + MLVL
Z = DIFR + MLVL*2
IC = Z + MLVL
IS = IC + 1
POLES = IS + 1
GIVNUM = POLES + 2*MLVL
*
K = 1
GIVPTR = 2
PERM = 3
GIVCOL = PERM + MLVL
END IF
*
DO 20 I = 1, N
IF( ABS( D( I ) ).LT.EPS ) THEN
D( I ) = SIGN( EPS, D( I ) )
END IF
20 CONTINUE
*
START = 1
SQRE = 0
*
DO 30 I = 1, NM1
IF( ( ABS( E( I ) ).LT.EPS ) .OR. ( I.EQ.NM1 ) ) THEN
*
*        Subproblem found. First determine its size and then
*        apply divide and conquer on it.
*
IF( I.LT.NM1 ) THEN
*
*        A subproblem with E(I) small for I < NM1.
*
NSIZE = I - START + 1
ELSE IF( ABS( E( I ) ).GE.EPS ) THEN
*
*        A subproblem with E(NM1) not too small but I = NM1.
*
NSIZE = N - START + 1
ELSE
*
*        A subproblem with E(NM1) small. This implies an
*        1-by-1 subproblem at D(N). Solve this 1-by-1 problem
*        first.
*
NSIZE = I - START + 1
IF( ICOMPQ.EQ.2 ) THEN
U( N, N ) = SIGN( ONE, D( N ) )
VT( N, N ) = ONE
ELSE IF( ICOMPQ.EQ.1 ) THEN
Q( N+( QSTART-1 )*N ) = SIGN( ONE, D( N ) )
Q( N+( SMLSIZ+QSTART-1 )*N ) = ONE
END IF
D( N ) = ABS( D( N ) )
END IF
IF( ICOMPQ.EQ.2 ) THEN
CALL DLASD0( NSIZE, SQRE, D( START ), E( START ),
\$                      U( START, START ), LDU, VT( START, START ),
\$                      LDVT, SMLSIZ, IWORK, WORK( WSTART ), INFO )
ELSE
CALL DLASDA( ICOMPQ, SMLSIZ, NSIZE, SQRE, D( START ),
\$                      E( START ), Q( START+( IU+QSTART-2 )*N ), N,
\$                      Q( START+( IVT+QSTART-2 )*N ),
\$                      IQ( START+K*N ), Q( START+( DIFL+QSTART-2 )*
\$                      N ), Q( START+( DIFR+QSTART-2 )*N ),
\$                      Q( START+( Z+QSTART-2 )*N ),
\$                      Q( START+( POLES+QSTART-2 )*N ),
\$                      IQ( START+GIVPTR*N ), IQ( START+GIVCOL*N ),
\$                      N, IQ( START+PERM*N ),
\$                      Q( START+( GIVNUM+QSTART-2 )*N ),
\$                      Q( START+( IC+QSTART-2 )*N ),
\$                      Q( START+( IS+QSTART-2 )*N ),
\$                      WORK( WSTART ), IWORK, INFO )
IF( INFO.NE.0 ) THEN
RETURN
END IF
END IF
START = I + 1
END IF
30 CONTINUE
*
*     Unscale
*
CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, IERR )
40 CONTINUE
*
*     Use Selection Sort to minimize swaps of singular vectors
*
DO 60 II = 2, N
I = II - 1
KK = I
P = D( I )
DO 50 J = II, N
IF( D( J ).GT.P ) THEN
KK = J
P = D( J )
END IF
50    CONTINUE
IF( KK.NE.I ) THEN
D( KK ) = D( I )
D( I ) = P
IF( ICOMPQ.EQ.1 ) THEN
IQ( I ) = KK
ELSE IF( ICOMPQ.EQ.2 ) THEN
CALL DSWAP( N, U( 1, I ), 1, U( 1, KK ), 1 )
CALL DSWAP( N, VT( I, 1 ), LDVT, VT( KK, 1 ), LDVT )
END IF
ELSE IF( ICOMPQ.EQ.1 ) THEN
IQ( I ) = I
END IF
60 CONTINUE
*
*     If ICOMPQ = 1, use IQ(N,1) as the indicator for UPLO
*
IF( ICOMPQ.EQ.1 ) THEN
IF( IUPLO.EQ.1 ) THEN
IQ( N ) = 1
ELSE
IQ( N ) = 0
END IF
END IF
*
*     If B is lower bidiagonal, update U by those Givens rotations
*     which rotated B to be upper bidiagonal
*
IF( ( IUPLO.EQ.2 ) .AND. ( ICOMPQ.EQ.2 ) )
\$   CALL DLASR( 'L', 'V', 'B', N, N, WORK( 1 ), WORK( N ), U, LDU )
*
RETURN
*
*     End of DBDSDC
*
END

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