```      SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K,
\$                   DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,
\$                   PERM, GIVNUM, C, S, WORK, IWORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
INTEGER            ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE
*     ..
*     .. Array Arguments ..
INTEGER            GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
\$                   K( * ), PERM( LDGCOL, * )
DOUBLE PRECISION   C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ),
\$                   E( * ), GIVNUM( LDU, * ), POLES( LDU, * ),
\$                   S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ),
\$                   Z( LDU, * )
*     ..
*
*  Purpose
*  =======
*
*  Using a divide and conquer approach, DLASDA computes the singular
*  value decomposition (SVD) of a real upper bidiagonal N-by-M matrix
*  B with diagonal D and offdiagonal E, where M = N + SQRE. The
*  algorithm computes the singular values in the SVD B = U * S * VT.
*  The orthogonal matrices U and VT are optionally computed in
*  compact form.
*
*  A related subroutine, DLASD0, computes the singular values and
*  the singular vectors in explicit form.
*
*  Arguments
*  =========
*
*  ICOMPQ (input) INTEGER
*         Specifies whether singular vectors are to be computed
*         in compact form, as follows
*         = 0: Compute singular values only.
*         = 1: Compute singular vectors of upper bidiagonal
*              matrix in compact form.
*
*  SMLSIZ (input) INTEGER
*         The maximum size of the subproblems at the bottom of the
*         computation tree.
*
*  N      (input) INTEGER
*         The row dimension of the upper bidiagonal matrix. This is
*         also the dimension of the main diagonal array D.
*
*  SQRE   (input) INTEGER
*         Specifies the column dimension of the bidiagonal matrix.
*         = 0: The bidiagonal matrix has column dimension M = N;
*         = 1: The bidiagonal matrix has column dimension M = N + 1.
*
*  D      (input/output) DOUBLE PRECISION array, dimension ( N )
*         On entry D contains the main diagonal of the bidiagonal
*         matrix. On exit D, if INFO = 0, contains its singular values.
*
*  E      (input) DOUBLE PRECISION array, dimension ( M-1 )
*         Contains the subdiagonal entries of the bidiagonal matrix.
*         On exit, E has been destroyed.
*
*  U      (output) DOUBLE PRECISION array,
*         dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced
*         if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left
*         singular vector matrices of all subproblems at the bottom
*         level.
*
*  LDU    (input) INTEGER, LDU = > N.
*         The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
*         GIVNUM, and Z.
*
*  VT     (output) DOUBLE PRECISION array,
*         dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced
*         if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains the right
*         singular vector matrices of all subproblems at the bottom
*         level.
*
*  K      (output) INTEGER array,
*         dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0.
*         If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th
*         secular equation on the computation tree.
*
*  DIFL   (output) DOUBLE PRECISION array, dimension ( LDU, NLVL ),
*         where NLVL = floor(log_2 (N/SMLSIZ))).
*
*  DIFR   (output) DOUBLE PRECISION array,
*                  dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and
*                  dimension ( N ) if ICOMPQ = 0.
*         If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1)
*         record distances between singular values on the I-th
*         level and singular values on the (I -1)-th level, and
*         DIFR(1:N, 2 * I ) contains the normalizing factors for
*         the right singular vector matrix. See DLASD8 for details.
*
*  Z      (output) DOUBLE PRECISION array,
*                  dimension ( LDU, NLVL ) if ICOMPQ = 1 and
*                  dimension ( N ) if ICOMPQ = 0.
*         The first K elements of Z(1, I) contain the components of
*         the deflation-adjusted updating row vector for subproblems
*         on the I-th level.
*
*  POLES  (output) DOUBLE PRECISION array,
*         dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced
*         if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and
*         POLES(1, 2*I) contain  the new and old singular values
*         involved in the secular equations on the I-th level.
*
*  GIVPTR (output) INTEGER array,
*         dimension ( N ) if ICOMPQ = 1, and not referenced if
*         ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records
*         the number of Givens rotations performed on the I-th
*         problem on the computation tree.
*
*  GIVCOL (output) INTEGER array,
*         dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not
*         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
*         GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations
*         of Givens rotations performed on the I-th level on the
*         computation tree.
*
*  LDGCOL (input) INTEGER, LDGCOL = > N.
*         The leading dimension of arrays GIVCOL and PERM.
*
*  PERM   (output) INTEGER array,
*         dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced
*         if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records
*         permutations done on the I-th level of the computation tree.
*
*  GIVNUM (output) DOUBLE PRECISION array,
*         dimension ( LDU,  2 * NLVL ) if ICOMPQ = 1, and not
*         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
*         GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S-
*         values of Givens rotations performed on the I-th level on
*         the computation tree.
*
*  C      (output) DOUBLE PRECISION array,
*         dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0.
*         If ICOMPQ = 1 and the I-th subproblem is not square, on exit,
*         C( I ) contains the C-value of a Givens rotation related to
*         the right null space of the I-th subproblem.
*
*  S      (output) DOUBLE PRECISION array, dimension ( N ) if
*         ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1
*         and the I-th subproblem is not square, on exit, S( I )
*         contains the S-value of a Givens rotation related to
*         the right null space of the I-th subproblem.
*
*  WORK   (workspace) DOUBLE PRECISION array, dimension
*         (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).
*
*  IWORK  (workspace) INTEGER array.
*         Dimension must be at least (7 * N).
*
*  INFO   (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*          > 0:  if INFO = 1, an singular value did not converge
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*
*     .. Parameters ..
DOUBLE PRECISION   ZERO, ONE
PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
INTEGER            I, I1, IC, IDXQ, IDXQI, IM1, INODE, ITEMP, IWK,
\$                   J, LF, LL, LVL, LVL2, M, NCC, ND, NDB1, NDIML,
\$                   NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, NRU,
\$                   NWORK1, NWORK2, SMLSZP, SQREI, VF, VFI, VL, VLI
DOUBLE PRECISION   ALPHA, BETA
*     ..
*     .. External Subroutines ..
EXTERNAL           DCOPY, DLASD6, DLASDQ, DLASDT, DLASET, XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
INFO = 0
*
IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
INFO = -1
ELSE IF( SMLSIZ.LT.3 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
INFO = -4
ELSE IF( LDU.LT.( N+SQRE ) ) THEN
INFO = -8
ELSE IF( LDGCOL.LT.N ) THEN
INFO = -17
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASDA', -INFO )
RETURN
END IF
*
M = N + SQRE
*
*     If the input matrix is too small, call DLASDQ to find the SVD.
*
IF( N.LE.SMLSIZ ) THEN
IF( ICOMPQ.EQ.0 ) THEN
CALL DLASDQ( 'U', SQRE, N, 0, 0, 0, D, E, VT, LDU, U, LDU,
\$                   U, LDU, WORK, INFO )
ELSE
CALL DLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDU, U, LDU,
\$                   U, LDU, WORK, INFO )
END IF
RETURN
END IF
*
*     Book-keeping and  set up the computation tree.
*
INODE = 1
NDIML = INODE + N
NDIMR = NDIML + N
IDXQ = NDIMR + N
IWK = IDXQ + N
*
NCC = 0
NRU = 0
*
SMLSZP = SMLSIZ + 1
VF = 1
VL = VF + M
NWORK1 = VL + M
NWORK2 = NWORK1 + SMLSZP*SMLSZP
*
CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),
\$             IWORK( NDIMR ), SMLSIZ )
*
*     for the nodes on bottom level of the tree, solve
*     their subproblems by DLASDQ.
*
NDB1 = ( ND+1 ) / 2
DO 30 I = NDB1, ND
*
*        IC : center row of each node
*        NL : number of rows of left  subproblem
*        NR : number of rows of right subproblem
*        NLF: starting row of the left   subproblem
*        NRF: starting row of the right  subproblem
*
I1 = I - 1
IC = IWORK( INODE+I1 )
NL = IWORK( NDIML+I1 )
NLP1 = NL + 1
NR = IWORK( NDIMR+I1 )
NLF = IC - NL
NRF = IC + 1
IDXQI = IDXQ + NLF - 2
VFI = VF + NLF - 1
VLI = VL + NLF - 1
SQREI = 1
IF( ICOMPQ.EQ.0 ) THEN
CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, WORK( NWORK1 ),
\$                   SMLSZP )
CALL DLASDQ( 'U', SQREI, NL, NLP1, NRU, NCC, D( NLF ),
\$                   E( NLF ), WORK( NWORK1 ), SMLSZP,
\$                   WORK( NWORK2 ), NL, WORK( NWORK2 ), NL,
\$                   WORK( NWORK2 ), INFO )
ITEMP = NWORK1 + NL*SMLSZP
CALL DCOPY( NLP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )
CALL DCOPY( NLP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )
ELSE
CALL DLASET( 'A', NL, NL, ZERO, ONE, U( NLF, 1 ), LDU )
CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, VT( NLF, 1 ), LDU )
CALL DLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ),
\$                   E( NLF ), VT( NLF, 1 ), LDU, U( NLF, 1 ), LDU,
\$                   U( NLF, 1 ), LDU, WORK( NWORK1 ), INFO )
CALL DCOPY( NLP1, VT( NLF, 1 ), 1, WORK( VFI ), 1 )
CALL DCOPY( NLP1, VT( NLF, NLP1 ), 1, WORK( VLI ), 1 )
END IF
IF( INFO.NE.0 ) THEN
RETURN
END IF
DO 10 J = 1, NL
IWORK( IDXQI+J ) = J
10    CONTINUE
IF( ( I.EQ.ND ) .AND. ( SQRE.EQ.0 ) ) THEN
SQREI = 0
ELSE
SQREI = 1
END IF
IDXQI = IDXQI + NLP1
VFI = VFI + NLP1
VLI = VLI + NLP1
NRP1 = NR + SQREI
IF( ICOMPQ.EQ.0 ) THEN
CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, WORK( NWORK1 ),
\$                   SMLSZP )
CALL DLASDQ( 'U', SQREI, NR, NRP1, NRU, NCC, D( NRF ),
\$                   E( NRF ), WORK( NWORK1 ), SMLSZP,
\$                   WORK( NWORK2 ), NR, WORK( NWORK2 ), NR,
\$                   WORK( NWORK2 ), INFO )
ITEMP = NWORK1 + ( NRP1-1 )*SMLSZP
CALL DCOPY( NRP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )
CALL DCOPY( NRP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )
ELSE
CALL DLASET( 'A', NR, NR, ZERO, ONE, U( NRF, 1 ), LDU )
CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, VT( NRF, 1 ), LDU )
CALL DLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ),
\$                   E( NRF ), VT( NRF, 1 ), LDU, U( NRF, 1 ), LDU,
\$                   U( NRF, 1 ), LDU, WORK( NWORK1 ), INFO )
CALL DCOPY( NRP1, VT( NRF, 1 ), 1, WORK( VFI ), 1 )
CALL DCOPY( NRP1, VT( NRF, NRP1 ), 1, WORK( VLI ), 1 )
END IF
IF( INFO.NE.0 ) THEN
RETURN
END IF
DO 20 J = 1, NR
IWORK( IDXQI+J ) = J
20    CONTINUE
30 CONTINUE
*
*     Now conquer each subproblem bottom-up.
*
J = 2**NLVL
DO 50 LVL = NLVL, 1, -1
LVL2 = LVL*2 - 1
*
*        Find the first node LF and last node LL on
*        the current level LVL.
*
IF( LVL.EQ.1 ) THEN
LF = 1
LL = 1
ELSE
LF = 2**( LVL-1 )
LL = 2*LF - 1
END IF
DO 40 I = LF, LL
IM1 = I - 1
IC = IWORK( INODE+IM1 )
NL = IWORK( NDIML+IM1 )
NR = IWORK( NDIMR+IM1 )
NLF = IC - NL
NRF = IC + 1
IF( I.EQ.LL ) THEN
SQREI = SQRE
ELSE
SQREI = 1
END IF
VFI = VF + NLF - 1
VLI = VL + NLF - 1
IDXQI = IDXQ + NLF - 1
ALPHA = D( IC )
BETA = E( IC )
IF( ICOMPQ.EQ.0 ) THEN
CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ),
\$                      WORK( VFI ), WORK( VLI ), ALPHA, BETA,
\$                      IWORK( IDXQI ), PERM, GIVPTR( 1 ), GIVCOL,
\$                      LDGCOL, GIVNUM, LDU, POLES, DIFL, DIFR, Z,
\$                      K( 1 ), C( 1 ), S( 1 ), WORK( NWORK1 ),
\$                      IWORK( IWK ), INFO )
ELSE
J = J - 1
CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ),
\$                      WORK( VFI ), WORK( VLI ), ALPHA, BETA,
\$                      IWORK( IDXQI ), PERM( NLF, LVL ),
\$                      GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL,
\$                      GIVNUM( NLF, LVL2 ), LDU,
\$                      POLES( NLF, LVL2 ), DIFL( NLF, LVL ),
\$                      DIFR( NLF, LVL2 ), Z( NLF, LVL ), K( J ),
\$                      C( J ), S( J ), WORK( NWORK1 ),
\$                      IWORK( IWK ), INFO )
END IF
IF( INFO.NE.0 ) THEN
RETURN
END IF
40    CONTINUE
50 CONTINUE
*
RETURN
*
*     End of DLASDA
*
END

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