You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
631 lines
19 KiB
631 lines
19 KiB
*> \brief \b DLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download DLASD2 + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd2.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd2.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd2.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE DLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT,
|
|
* LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX,
|
|
* IDXC, IDXQ, COLTYP, INFO )
|
|
*
|
|
* .. Scalar Arguments ..
|
|
* INTEGER INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE
|
|
* DOUBLE PRECISION ALPHA, BETA
|
|
* ..
|
|
* .. Array Arguments ..
|
|
* INTEGER COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ),
|
|
* $ IDXQ( * )
|
|
* DOUBLE PRECISION D( * ), DSIGMA( * ), U( LDU, * ),
|
|
* $ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
|
|
* $ Z( * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> DLASD2 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.
|
|
*>
|
|
*> DLASD2 is called from DLASD1.
|
|
*> \endverbatim
|
|
*
|
|
* Arguments:
|
|
* ==========
|
|
*
|
|
*> \param[in] NL
|
|
*> \verbatim
|
|
*> NL is INTEGER
|
|
*> The row dimension of the upper block. NL >= 1.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] NR
|
|
*> \verbatim
|
|
*> NR is INTEGER
|
|
*> The row dimension of the lower block. NR >= 1.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] SQRE
|
|
*> \verbatim
|
|
*> SQRE is 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.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] K
|
|
*> \verbatim
|
|
*> K is INTEGER
|
|
*> Contains the dimension of the non-deflated matrix,
|
|
*> This is the order of the related secular equation. 1 <= K <=N.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] D
|
|
*> \verbatim
|
|
*> D is DOUBLE PRECISION 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.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] Z
|
|
*> \verbatim
|
|
*> Z is DOUBLE PRECISION array, dimension(N)
|
|
*> On exit Z contains the updating row vector in the secular
|
|
*> equation.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] ALPHA
|
|
*> \verbatim
|
|
*> ALPHA is DOUBLE PRECISION
|
|
*> Contains the diagonal element associated with the added row.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] BETA
|
|
*> \verbatim
|
|
*> BETA is DOUBLE PRECISION
|
|
*> Contains the off-diagonal element associated with the added
|
|
*> row.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] U
|
|
*> \verbatim
|
|
*> U is DOUBLE PRECISION 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.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDU
|
|
*> \verbatim
|
|
*> LDU is INTEGER
|
|
*> The leading dimension of the array U. LDU >= N.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] VT
|
|
*> \verbatim
|
|
*> VT is DOUBLE PRECISION array, dimension(LDVT,M)
|
|
*> On entry VT**T 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**T 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.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDVT
|
|
*> \verbatim
|
|
*> LDVT is INTEGER
|
|
*> The leading dimension of the array VT. LDVT >= M.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] DSIGMA
|
|
*> \verbatim
|
|
*> DSIGMA is DOUBLE PRECISION array, dimension (N)
|
|
*> Contains a copy of the diagonal elements (K-1 singular values
|
|
*> and one zero) in the secular equation.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] U2
|
|
*> \verbatim
|
|
*> U2 is DOUBLE PRECISION array, dimension(LDU2,N)
|
|
*> Contains a copy of the first K-1 left singular vectors which
|
|
*> will be used by DLASD3 in a matrix multiply (DGEMM) 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.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDU2
|
|
*> \verbatim
|
|
*> LDU2 is INTEGER
|
|
*> The leading dimension of the array U2. LDU2 >= N.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] VT2
|
|
*> \verbatim
|
|
*> VT2 is DOUBLE PRECISION array, dimension(LDVT2,N)
|
|
*> VT2**T contains a copy of the first K right singular vectors
|
|
*> which will be used by DLASD3 in a matrix multiply (DGEMM) 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.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDVT2
|
|
*> \verbatim
|
|
*> LDVT2 is INTEGER
|
|
*> The leading dimension of the array VT2. LDVT2 >= M.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] IDXP
|
|
*> \verbatim
|
|
*> IDXP is 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.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] IDX
|
|
*> \verbatim
|
|
*> IDX is INTEGER array, dimension(N)
|
|
*> This will contain the permutation used to sort the contents of
|
|
*> D into ascending order.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] IDXC
|
|
*> \verbatim
|
|
*> IDXC is 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.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] IDXQ
|
|
*> \verbatim
|
|
*> IDXQ is 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.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] COLTYP
|
|
*> \verbatim
|
|
*> COLTYP is 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.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] INFO
|
|
*> \verbatim
|
|
*> INFO is INTEGER
|
|
*> = 0: successful exit.
|
|
*> < 0: if INFO = -i, the i-th argument had an illegal value.
|
|
*> \endverbatim
|
|
*
|
|
* Authors:
|
|
* ========
|
|
*
|
|
*> \author Univ. of Tennessee
|
|
*> \author Univ. of California Berkeley
|
|
*> \author Univ. of Colorado Denver
|
|
*> \author NAG Ltd.
|
|
*
|
|
*> \ingroup OTHERauxiliary
|
|
*
|
|
*> \par Contributors:
|
|
* ==================
|
|
*>
|
|
*> Ming Gu and Huan Ren, Computer Science Division, University of
|
|
*> California at Berkeley, USA
|
|
*>
|
|
* =====================================================================
|
|
SUBROUTINE DLASD2( 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 --
|
|
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
|
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
|
*
|
|
* .. Scalar Arguments ..
|
|
INTEGER INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE
|
|
DOUBLE PRECISION ALPHA, BETA
|
|
* ..
|
|
* .. Array Arguments ..
|
|
INTEGER COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ),
|
|
$ IDXQ( * )
|
|
DOUBLE PRECISION D( * ), DSIGMA( * ), U( LDU, * ),
|
|
$ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
|
|
$ Z( * )
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
DOUBLE PRECISION ZERO, ONE, TWO, EIGHT
|
|
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
|
|
$ EIGHT = 8.0D+0 )
|
|
* ..
|
|
* .. Local Arrays ..
|
|
INTEGER CTOT( 4 ), PSM( 4 )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
INTEGER CT, I, IDXI, IDXJ, IDXJP, J, JP, JPREV, K2, M,
|
|
$ N, NLP1, NLP2
|
|
DOUBLE PRECISION C, EPS, HLFTOL, S, TAU, TOL, Z1
|
|
* ..
|
|
* .. External Functions ..
|
|
DOUBLE PRECISION DLAMCH, DLAPY2
|
|
EXTERNAL DLAMCH, DLAPY2
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL DCOPY, DLACPY, DLAMRG, DLASET, DROT, 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( 'DLASD2', -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 DLAMRG( 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 = DLAMCH( '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 = DLAPY2( 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 DROT( N, U( 1, IDXJP ), 1, U( 1, IDXJ ), 1, C, S )
|
|
CALL DROT( 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 DCOPY( N, U( 1, IDXJ ), 1, U2( 1, J ), 1 )
|
|
CALL DCOPY( 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 ) = DLAPY2( 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 DCOPY( 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 DLASET( '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 DCOPY( M, VT( NLP1, 1 ), LDVT, VT2( 1, 1 ), LDVT2 )
|
|
END IF
|
|
IF( M.GT.N ) THEN
|
|
CALL DCOPY( 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 DCOPY( N-K, DSIGMA( K+1 ), 1, D( K+1 ), 1 )
|
|
CALL DLACPY( 'A', N, N-K, U2( 1, K+1 ), LDU2, U( 1, K+1 ),
|
|
$ LDU )
|
|
CALL DLACPY( 'A', N-K, M, VT2( K+1, 1 ), LDVT2, VT( K+1, 1 ),
|
|
$ LDVT )
|
|
END IF
|
|
*
|
|
* Copy CTOT into COLTYP for referencing in DLASD3.
|
|
*
|
|
DO 190 J = 1, 4
|
|
COLTYP( J ) = CTOT( J )
|
|
190 CONTINUE
|
|
*
|
|
RETURN
|
|
*
|
|
* End of DLASD2
|
|
*
|
|
END
|
|
|