```      SUBROUTINE SLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT,
\$                   LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX,
\$                   IDXC, IDXQ, COLTYP, INFO )
*
*  -- LAPACK auxiliary routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
INTEGER            INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE
REAL               ALPHA, BETA
*     ..
*     .. Array Arguments ..
INTEGER            COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ),
\$                   IDXQ( * )
REAL               D( * ), DSIGMA( * ), U( LDU, * ),
\$                   U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
\$                   Z( * )
*     ..
*
*  Purpose
*  =======
*
*  SLASD2 merges the two sets of singular values together into a single
*  sorted set.  Then it tries to deflate the size of the problem.
*  There are two ways in which deflation can occur:  when two or more
*  singular values are close together or if there is a tiny entry in the
*  Z vector.  For each such occurrence the order of the related secular
*  equation problem is reduced by one.
*
*  SLASD2 is called from SLASD1.
*
*  Arguments
*  =========
*
*  NL     (input) INTEGER
*         The row dimension of the upper block.  NL >= 1.
*
*  NR     (input) INTEGER
*         The row dimension of the lower block.  NR >= 1.
*
*  SQRE   (input) INTEGER
*         = 0: the lower block is an NR-by-NR square matrix.
*         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*
*         The bidiagonal matrix has N = NL + NR + 1 rows and
*         M = N + SQRE >= N columns.
*
*  K      (output) INTEGER
*         Contains the dimension of the non-deflated matrix,
*         This is the order of the related secular equation. 1 <= K <=N.
*
*  D      (input/output) REAL array, dimension (N)
*         On entry D contains the singular values of the two submatrices
*         to be combined.  On exit D contains the trailing (N-K) updated
*         singular values (those which were deflated) sorted into
*         increasing order.
*
*  Z      (output) REAL array, dimension (N)
*         On exit Z contains the updating row vector in the secular
*         equation.
*
*  ALPHA  (input) REAL
*         Contains the diagonal element associated with the added row.
*
*  BETA   (input) REAL
*         Contains the off-diagonal element associated with the added
*         row.
*
*  U      (input/output) REAL array, dimension (LDU,N)
*         On entry U contains the left singular vectors of two
*         submatrices in the two square blocks with corners at (1,1),
*         (NL, NL), and (NL+2, NL+2), (N,N).
*         On exit U contains the trailing (N-K) updated left singular
*         vectors (those which were deflated) in its last N-K columns.
*
*  LDU    (input) INTEGER
*         The leading dimension of the array U.  LDU >= N.
*
*  VT     (input/output) REAL array, dimension (LDVT,M)
*         On entry VT' contains the right singular vectors of two
*         submatrices in the two square blocks with corners at (1,1),
*         (NL+1, NL+1), and (NL+2, NL+2), (M,M).
*         On exit VT' contains the trailing (N-K) updated right singular
*         vectors (those which were deflated) in its last N-K columns.
*         In case SQRE =1, the last row of VT spans the right null
*         space.
*
*  LDVT   (input) INTEGER
*         The leading dimension of the array VT.  LDVT >= M.
*
*  DSIGMA (output) REAL array, dimension (N)
*         Contains a copy of the diagonal elements (K-1 singular values
*         and one zero) in the secular equation.
*
*  U2     (output) REAL array, dimension (LDU2,N)
*         Contains a copy of the first K-1 left singular vectors which
*         will be used by SLASD3 in a matrix multiply (SGEMM) to solve
*         for the new left singular vectors. U2 is arranged into four
*         blocks. The first block contains a column with 1 at NL+1 and
*         zero everywhere else; the second block contains non-zero
*         entries only at and above NL; the third contains non-zero
*         entries only below NL+1; and the fourth is dense.
*
*  LDU2   (input) INTEGER
*         The leading dimension of the array U2.  LDU2 >= N.
*
*  VT2    (output) REAL array, dimension (LDVT2,N)
*         VT2' contains a copy of the first K right singular vectors
*         which will be used by SLASD3 in a matrix multiply (SGEMM) to
*         solve for the new right singular vectors. VT2 is arranged into
*         three blocks. The first block contains a row that corresponds
*         to the special 0 diagonal element in SIGMA; the second block
*         contains non-zeros only at and before NL +1; the third block
*         contains non-zeros only at and after  NL +2.
*
*  LDVT2  (input) INTEGER
*         The leading dimension of the array VT2.  LDVT2 >= M.
*
*  IDXP   (workspace) INTEGER array, dimension (N)
*         This will contain the permutation used to place deflated
*         values of D at the end of the array. On output IDXP(2:K)
*         points to the nondeflated D-values and IDXP(K+1:N)
*         points to the deflated singular values.
*
*  IDX    (workspace) INTEGER array, dimension (N)
*         This will contain the permutation used to sort the contents of
*         D into ascending order.
*
*  IDXC   (output) INTEGER array, dimension (N)
*         This will contain the permutation used to arrange the columns
*         of the deflated U matrix into three groups:  the first group
*         contains non-zero entries only at and above NL, the second
*         contains non-zero entries only below NL+2, and the third is
*         dense.
*
*  IDXQ   (input/output) INTEGER array, dimension (N)
*         This contains the permutation which separately sorts the two
*         sub-problems in D into ascending order.  Note that entries in
*         the first hlaf of this permutation must first be moved one
*         position backward; and entries in the second half
*         must first have NL+1 added to their values.
*
*  COLTYP (workspace/output) INTEGER array, dimension (N)
*         As workspace, this will contain a label which will indicate
*         which of the following types a column in the U2 matrix or a
*         row in the VT2 matrix is:
*         1 : non-zero in the upper half only
*         2 : non-zero in the lower half only
*         3 : dense
*         4 : deflated
*
*         On exit, it is an array of dimension 4, with COLTYP(I) being
*         the dimension of the I-th type columns.
*
*  INFO   (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*
*     .. Parameters ..
REAL               ZERO, ONE, TWO, EIGHT
PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0,
\$                   EIGHT = 8.0E+0 )
*     ..
*     .. Local Arrays ..
INTEGER            CTOT( 4 ), PSM( 4 )
*     ..
*     .. Local Scalars ..
INTEGER            CT, I, IDXI, IDXJ, IDXJP, J, JP, JPREV, K2, M,
\$                   N, NLP1, NLP2
REAL               C, EPS, HLFTOL, S, TAU, TOL, Z1
*     ..
*     .. External Functions ..
REAL               SLAMCH, SLAPY2
EXTERNAL           SLAMCH, SLAPY2
*     ..
*     .. External Subroutines ..
EXTERNAL           SCOPY, SLACPY, SLAMRG, SLASET, SROT, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
INFO = 0
*
IF( NL.LT.1 ) THEN
INFO = -1
ELSE IF( NR.LT.1 ) THEN
INFO = -2
ELSE IF( ( SQRE.NE.1 ) .AND. ( SQRE.NE.0 ) ) THEN
INFO = -3
END IF
*
N = NL + NR + 1
M = N + SQRE
*
IF( LDU.LT.N ) THEN
INFO = -10
ELSE IF( LDVT.LT.M ) THEN
INFO = -12
ELSE IF( LDU2.LT.N ) THEN
INFO = -15
ELSE IF( LDVT2.LT.M ) THEN
INFO = -17
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SLASD2', -INFO )
RETURN
END IF
*
NLP1 = NL + 1
NLP2 = NL + 2
*
*     Generate the first part of the vector Z; and move the singular
*     values in the first part of D one position backward.
*
Z1 = ALPHA*VT( NLP1, NLP1 )
Z( 1 ) = Z1
DO 10 I = NL, 1, -1
Z( I+1 ) = ALPHA*VT( I, NLP1 )
D( I+1 ) = D( I )
IDXQ( I+1 ) = IDXQ( I ) + 1
10 CONTINUE
*
*     Generate the second part of the vector Z.
*
DO 20 I = NLP2, M
Z( I ) = BETA*VT( I, NLP2 )
20 CONTINUE
*
*     Initialize some reference arrays.
*
DO 30 I = 2, NLP1
COLTYP( I ) = 1
30 CONTINUE
DO 40 I = NLP2, N
COLTYP( I ) = 2
40 CONTINUE
*
*     Sort the singular values into increasing order
*
DO 50 I = NLP2, N
IDXQ( I ) = IDXQ( I ) + NLP1
50 CONTINUE
*
*     DSIGMA, IDXC, IDXC, and the first column of U2
*     are used as storage space.
*
DO 60 I = 2, N
DSIGMA( I ) = D( IDXQ( I ) )
U2( I, 1 ) = Z( IDXQ( I ) )
IDXC( I ) = COLTYP( IDXQ( I ) )
60 CONTINUE
*
CALL SLAMRG( NL, NR, DSIGMA( 2 ), 1, 1, IDX( 2 ) )
*
DO 70 I = 2, N
IDXI = 1 + IDX( I )
D( I ) = DSIGMA( IDXI )
Z( I ) = U2( IDXI, 1 )
COLTYP( I ) = IDXC( IDXI )
70 CONTINUE
*
*     Calculate the allowable deflation tolerance
*
EPS = SLAMCH( 'Epsilon' )
TOL = MAX( ABS( ALPHA ), ABS( BETA ) )
TOL = EIGHT*EPS*MAX( ABS( D( N ) ), TOL )
*
*     There are 2 kinds of deflation -- first a value in the z-vector
*     is small, second two (or more) singular values are very close
*     together (their difference is small).
*
*     If the value in the z-vector is small, we simply permute the
*     array so that the corresponding singular value is moved to the
*     end.
*
*     If two values in the D-vector are close, we perform a two-sided
*     rotation designed to make one of the corresponding z-vector
*     entries zero, and then permute the array so that the deflated
*     singular value is moved to the end.
*
*     If there are multiple singular values then the problem deflates.
*     Here the number of equal singular values are found.  As each equal
*     singular value is found, an elementary reflector is computed to
*     rotate the corresponding singular subspace so that the
*     corresponding components of Z are zero in this new basis.
*
K = 1
K2 = N + 1
DO 80 J = 2, N
IF( ABS( Z( J ) ).LE.TOL ) THEN
*
*           Deflate due to small z component.
*
K2 = K2 - 1
IDXP( K2 ) = J
COLTYP( J ) = 4
IF( J.EQ.N )
\$         GO TO 120
ELSE
JPREV = J
GO TO 90
END IF
80 CONTINUE
90 CONTINUE
J = JPREV
100 CONTINUE
J = J + 1
IF( J.GT.N )
\$   GO TO 110
IF( ABS( Z( J ) ).LE.TOL ) THEN
*
*        Deflate due to small z component.
*
K2 = K2 - 1
IDXP( K2 ) = J
COLTYP( J ) = 4
ELSE
*
*        Check if singular values are close enough to allow deflation.
*
IF( ABS( D( J )-D( JPREV ) ).LE.TOL ) THEN
*
*           Deflation is possible.
*
S = Z( JPREV )
C = Z( J )
*
*           Find sqrt(a**2+b**2) without overflow or
*           destructive underflow.
*
TAU = SLAPY2( C, S )
C = C / TAU
S = -S / TAU
Z( J ) = TAU
Z( JPREV ) = ZERO
*
*           Apply back the Givens rotation to the left and right
*           singular vector matrices.
*
IDXJP = IDXQ( IDX( JPREV )+1 )
IDXJ = IDXQ( IDX( J )+1 )
IF( IDXJP.LE.NLP1 ) THEN
IDXJP = IDXJP - 1
END IF
IF( IDXJ.LE.NLP1 ) THEN
IDXJ = IDXJ - 1
END IF
CALL SROT( N, U( 1, IDXJP ), 1, U( 1, IDXJ ), 1, C, S )
CALL SROT( M, VT( IDXJP, 1 ), LDVT, VT( IDXJ, 1 ), LDVT, C,
\$                 S )
IF( COLTYP( J ).NE.COLTYP( JPREV ) ) THEN
COLTYP( J ) = 3
END IF
COLTYP( JPREV ) = 4
K2 = K2 - 1
IDXP( K2 ) = JPREV
JPREV = J
ELSE
K = K + 1
U2( K, 1 ) = Z( JPREV )
DSIGMA( K ) = D( JPREV )
IDXP( K ) = JPREV
JPREV = J
END IF
END IF
GO TO 100
110 CONTINUE
*
*     Record the last singular value.
*
K = K + 1
U2( K, 1 ) = Z( JPREV )
DSIGMA( K ) = D( JPREV )
IDXP( K ) = JPREV
*
120 CONTINUE
*
*     Count up the total number of the various types of columns, then
*     form a permutation which positions the four column types into
*     four groups of uniform structure (although one or more of these
*     groups may be empty).
*
DO 130 J = 1, 4
CTOT( J ) = 0
130 CONTINUE
DO 140 J = 2, N
CT = COLTYP( J )
CTOT( CT ) = CTOT( CT ) + 1
140 CONTINUE
*
*     PSM(*) = Position in SubMatrix (of types 1 through 4)
*
PSM( 1 ) = 2
PSM( 2 ) = 2 + CTOT( 1 )
PSM( 3 ) = PSM( 2 ) + CTOT( 2 )
PSM( 4 ) = PSM( 3 ) + CTOT( 3 )
*
*     Fill out the IDXC array so that the permutation which it induces
*     will place all type-1 columns first, all type-2 columns next,
*     then all type-3's, and finally all type-4's, starting from the
*     second column. This applies similarly to the rows of VT.
*
DO 150 J = 2, N
JP = IDXP( J )
CT = COLTYP( JP )
IDXC( PSM( CT ) ) = J
PSM( CT ) = PSM( CT ) + 1
150 CONTINUE
*
*     Sort the singular values and corresponding singular vectors into
*     DSIGMA, U2, and VT2 respectively.  The singular values/vectors
*     which were not deflated go into the first K slots of DSIGMA, U2,
*     and VT2 respectively, while those which were deflated go into the
*     last N - K slots, except that the first column/row will be treated
*     separately.
*
DO 160 J = 2, N
JP = IDXP( J )
DSIGMA( J ) = D( JP )
IDXJ = IDXQ( IDX( IDXP( IDXC( J ) ) )+1 )
IF( IDXJ.LE.NLP1 ) THEN
IDXJ = IDXJ - 1
END IF
CALL SCOPY( N, U( 1, IDXJ ), 1, U2( 1, J ), 1 )
CALL SCOPY( M, VT( IDXJ, 1 ), LDVT, VT2( J, 1 ), LDVT2 )
160 CONTINUE
*
*     Determine DSIGMA(1), DSIGMA(2) and Z(1)
*
DSIGMA( 1 ) = ZERO
HLFTOL = TOL / TWO
IF( ABS( DSIGMA( 2 ) ).LE.HLFTOL )
\$   DSIGMA( 2 ) = HLFTOL
IF( M.GT.N ) THEN
Z( 1 ) = SLAPY2( Z1, Z( M ) )
IF( Z( 1 ).LE.TOL ) THEN
C = ONE
S = ZERO
Z( 1 ) = TOL
ELSE
C = Z1 / Z( 1 )
S = Z( M ) / Z( 1 )
END IF
ELSE
IF( ABS( Z1 ).LE.TOL ) THEN
Z( 1 ) = TOL
ELSE
Z( 1 ) = Z1
END IF
END IF
*
*     Move the rest of the updating row to Z.
*
CALL SCOPY( K-1, U2( 2, 1 ), 1, Z( 2 ), 1 )
*
*     Determine the first column of U2, the first row of VT2 and the
*     last row of VT.
*
CALL SLASET( 'A', N, 1, ZERO, ZERO, U2, LDU2 )
U2( NLP1, 1 ) = ONE
IF( M.GT.N ) THEN
DO 170 I = 1, NLP1
VT( M, I ) = -S*VT( NLP1, I )
VT2( 1, I ) = C*VT( NLP1, I )
170    CONTINUE
DO 180 I = NLP2, M
VT2( 1, I ) = S*VT( M, I )
VT( M, I ) = C*VT( M, I )
180    CONTINUE
ELSE
CALL SCOPY( M, VT( NLP1, 1 ), LDVT, VT2( 1, 1 ), LDVT2 )
END IF
IF( M.GT.N ) THEN
CALL SCOPY( M, VT( M, 1 ), LDVT, VT2( M, 1 ), LDVT2 )
END IF
*
*     The deflated singular values and their corresponding vectors go
*     into the back of D, U, and V respectively.
*
IF( N.GT.K ) THEN
CALL SCOPY( N-K, DSIGMA( K+1 ), 1, D( K+1 ), 1 )
CALL SLACPY( 'A', N, N-K, U2( 1, K+1 ), LDU2, U( 1, K+1 ),
\$                LDU )
CALL SLACPY( 'A', N-K, M, VT2( K+1, 1 ), LDVT2, VT( K+1, 1 ),
\$                LDVT )
END IF
*
*     Copy CTOT into COLTYP for referencing in SLASD3.
*
DO 190 J = 1, 4
COLTYP( J ) = CTOT( J )
190 CONTINUE
*
RETURN
*
*     End of SLASD2
*
END

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