*> \brief \b DBDSVDX
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DBDSVDX + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DBDSVDX( UPLO, JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
* $ NS, S, Z, LDZ, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, RANGE, UPLO
* INTEGER IL, INFO, IU, LDZ, N, NS
* DOUBLE PRECISION VL, VU
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), S( * ), WORK( * ),
* Z( LDZ, * )
* ..
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DBDSVDX computes the singular value decomposition (SVD) of a real
*> N-by-N (upper or lower) bidiagonal matrix B, B = U * S * VT,
*> 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.
*>
*> Given an upper bidiagonal B with diagonal D = [ d_1 d_2 ... d_N ]
*> and superdiagonal E = [ e_1 e_2 ... e_N-1 ], DBDSVDX computes the
*> singular value decompositon of B through the eigenvalues and
*> eigenvectors of the N*2-by-N*2 tridiagonal matrix
*>
*> | 0 d_1 |
*> | d_1 0 e_1 |
*> TGK = | e_1 0 d_2 |
*> | d_2 . . |
*> | . . . |
*>
*> If (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then
*> (+/-s,q), ||q|| = 1, are eigenpairs of TGK, with q = P * ( u' +/-v' ) /
*> sqrt(2) = ( v_1 u_1 v_2 u_2 ... v_n u_n ) / sqrt(2), and
*> P = [ e_{n+1} e_{1} e_{n+2} e_{2} ... ].
*>
*> Given a TGK matrix, one can either a) compute -s,-v and change signs
*> so that the singular values (and corresponding vectors) are already in
*> descending order (as in DGESVD/DGESDD) or b) compute s,v and reorder
*> the values (and corresponding vectors). DBDSVDX implements a) by
*> calling DSTEVX (bisection plus inverse iteration, to be replaced
*> with a version of the Multiple Relative Robust Representation
*> algorithm. (See P. Willems and B. Lang, A framework for the MR^3
*> algorithm: theory and implementation, SIAM J. Sci. Comput.,
*> 35:740-766, 2013.)
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': B is upper bidiagonal;
*> = 'L': B is lower bidiagonal.
*> \endverbatim
*>
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute singular values only;
*> = 'V': Compute singular values and singular vectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': all singular values will be found.
*> = 'V': all singular values in the half-open interval [VL,VU)
*> will be found.
*> = 'I': the IL-th through IU-th singular values will be found.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the bidiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the bidiagonal matrix B.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (max(1,N-1))
*> The (n-1) superdiagonal elements of the bidiagonal matrix
*> B in elements 1 to N-1.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*> VL >=0.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*> If RANGE='V', the lower and upper bounds of the interval to
*> be searched for singular values. VU > VL.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*> If RANGE='I', the indices (in ascending order) of the
*> smallest and largest singular values to be returned.
*> 1 <= IL <= IU <= min(M,N), if min(M,N) > 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[out] NS
*> \verbatim
*> NS is INTEGER
*> The total number of singular values found. 0 <= NS <= N.
*> If RANGE = 'A', NS = N, and if RANGE = 'I', NS = IU-IL+1.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> The first NS elements contain the selected singular values in
*> ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (2*N,K) )
*> If JOBZ = 'V', then if INFO = 0 the first NS columns of Z
*> contain the singular vectors of the matrix B corresponding to
*> the selected singular values, with U in rows 1 to N and V
*> in rows N+1 to N*2, i.e.
*> Z = [ U ]
*> [ V ]
*> If JOBZ = 'N', then Z is not referenced.
*> Note: The user must ensure that at least K = NS+1 columns are
*> supplied in the array Z; if RANGE = 'V', the exact value of
*> NS is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(2,N*2).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (14*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (12*N)
*> If JOBZ = 'V', then if INFO = 0, the first NS elements of
*> IWORK are zero. If INFO > 0, then IWORK contains the indices
*> of the eigenvectors that failed to converge in DSTEVX.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, then i eigenvectors failed to converge
*> in DSTEVX. The indices of the eigenvectors
*> (as returned by DSTEVX) are stored in the
*> array IWORK.
*> if INFO = N*2 + 1, an internal error occurred.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHEReigen
*
* =====================================================================
SUBROUTINE DBDSVDX( UPLO, JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
$ NS, S, Z, LDZ, WORK, IWORK, INFO)
*
* -- LAPACK driver routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2016
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, LDZ, N, NS
DOUBLE PRECISION VL, VU
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION D( * ), E( * ), S( * ), WORK( * ),
$ Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TEN, HNDRD, MEIGTH
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TEN = 10.0D0,
$ HNDRD = 100.0D0, MEIGTH = -0.1250D0 )
DOUBLE PRECISION FUDGE
PARAMETER ( FUDGE = 2.0D0 )
* ..
* .. Local Scalars ..
CHARACTER RNGVX
LOGICAL ALLSV, INDSV, LOWER, SPLIT, SVEQ0, VALSV, WANTZ
INTEGER I, ICOLZ, IDBEG, IDEND, IDTGK, IDPTR, IEPTR,
$ IETGK, IIFAIL, IIWORK, ILTGK, IROWU, IROWV,
$ IROWZ, ISBEG, ISPLT, ITEMP, IUTGK, J, K,
$ NTGK, NRU, NRV, NSL
DOUBLE PRECISION ABSTOL, EPS, EMIN, MU, NRMU, NRMV, ORTOL, SMAX,
$ SMIN, SQRT2, THRESH, TOL, ULP,
$ VLTGK, VUTGK, ZJTJI
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DDOT, DLAMCH, DNRM2
EXTERNAL IDAMAX, LSAME, DAXPY, DDOT, DLAMCH, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLASET, DSCAL, DSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
ALLSV = LSAME( RANGE, 'A' )
VALSV = LSAME( RANGE, 'V' )
INDSV = LSAME( RANGE, 'I' )
WANTZ = LSAME( JOBZ, 'V' )
LOWER = LSAME( UPLO, 'L' )
*
INFO = 0
IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LOWER ) THEN
INFO = -1
ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( ALLSV .OR. VALSV .OR. INDSV ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( N.GT.0 ) THEN
IF( VALSV ) THEN
IF( VL.LT.ZERO ) THEN
INFO = -7
ELSE IF( VU.LE.VL ) THEN
INFO = -8
END IF
ELSE IF( INDSV ) THEN
IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -10
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N*2 ) ) INFO = -14
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DBDSVDX', -INFO )
RETURN
END IF
*
* Quick return if possible (N.LE.1)
*
NS = 0
IF( N.EQ.0 ) RETURN
*
IF( N.EQ.1 ) THEN
IF( ALLSV .OR. INDSV ) THEN
NS = 1
S( 1 ) = ABS( D( 1 ) )
ELSE
IF( VL.LT.ABS( D( 1 ) ) .AND. VU.GE.ABS( D( 1 ) ) ) THEN
NS = 1
S( 1 ) = ABS( D( 1 ) )
END IF
END IF
IF( WANTZ ) THEN
Z( 1, 1 ) = SIGN( ONE, D( 1 ) )
Z( 2, 1 ) = ONE
END IF
RETURN
END IF
*
ABSTOL = 2*DLAMCH( 'Safe Minimum' )
ULP = DLAMCH( 'Precision' )
EPS = DLAMCH( 'Epsilon' )
SQRT2 = SQRT( 2.0D0 )
ORTOL = SQRT( ULP )
*
* Criterion for splitting is taken from DBDSQR when singular
* values are computed to relative accuracy TOL. (See J. Demmel and
* W. Kahan, Accurate singular values of bidiagonal matrices, SIAM
* J. Sci. and Stat. Comput., 11:873–912, 1990.)
*
TOL = MAX( TEN, MIN( HNDRD, EPS**MEIGTH ) )*EPS
*
* Compute approximate maximum, minimum singular values.
*
I = IDAMAX( N, D, 1 )
SMAX = ABS( D( I ) )
I = IDAMAX( N-1, E, 1 )
SMAX = MAX( SMAX, ABS( E( I ) ) )
*
* Compute threshold for neglecting D's and E's.
*
SMIN = ABS( D( 1 ) )
IF( SMIN.NE.ZERO ) THEN
MU = SMIN
DO I = 2, N
MU = ABS( D( I ) )*( MU / ( MU+ABS( E( I-1 ) ) ) )
SMIN = MIN( SMIN, MU )
IF( SMIN.EQ.ZERO ) EXIT
END DO
END IF
SMIN = SMIN / SQRT( DBLE( N ) )
THRESH = TOL*SMIN
*
* Check for zeros in D and E (splits), i.e. submatrices.
*
DO I = 1, N-1
IF( ABS( D( I ) ).LE.THRESH ) D( I ) = ZERO
IF( ABS( E( I ) ).LE.THRESH ) E( I ) = ZERO
END DO
IF( ABS( D( N ) ).LE.THRESH ) D( N ) = ZERO
*
* Pointers for arrays used by DSTEVX.
*
IDTGK = 1
IETGK = IDTGK + N*2
ITEMP = IETGK + N*2
IIFAIL = 1
IIWORK = IIFAIL + N*2
*
* Set RNGVX, which corresponds to RANGE for DSTEVX in TGK mode.
* VL,VU or IL,IU are redefined to conform to implementation a)
* described in the leading comments.
*
ILTGK = 0
IUTGK = 0
VLTGK = ZERO
VUTGK = ZERO
*
IF( ALLSV ) THEN
*
* All singular values will be found. We aim at -s (see
* leading comments) with RNGVX = 'I'. IL and IU are set
* later (as ILTGK and IUTGK) according to the dimension
* of the active submatrix.
*
RNGVX = 'I'
IF( WANTZ ) CALL DLASET( 'F', N*2, N+1, ZERO, ZERO, Z, LDZ )
ELSE IF( VALSV ) THEN
*
* Find singular values in a half-open interval. We aim
* at -s (see leading comments) and we swap VL and VU
* (as VUTGK and VLTGK), changing their signs.
*
RNGVX = 'V'
VLTGK = -VU
VUTGK = -VL
WORK( IDTGK:IDTGK+2*N-1 ) = ZERO
CALL DCOPY( N, D, 1, WORK( IETGK ), 2 )
CALL DCOPY( N-1, E, 1, WORK( IETGK+1 ), 2 )
CALL DSTEVX( 'N', 'V', N*2, WORK( IDTGK ), WORK( IETGK ),
$ VLTGK, VUTGK, ILTGK, ILTGK, ABSTOL, NS, S,
$ Z, LDZ, WORK( ITEMP ), IWORK( IIWORK ),
$ IWORK( IIFAIL ), INFO )
IF( NS.EQ.0 ) THEN
RETURN
ELSE
IF( WANTZ ) CALL DLASET( 'F', N*2, NS, ZERO, ZERO, Z, LDZ )
END IF
ELSE IF( INDSV ) THEN
*
* Find the IL-th through the IU-th singular values. We aim
* at -s (see leading comments) and indices are mapped into
* values, therefore mimicking DSTEBZ, where
*
* GL = GL - FUDGE*TNORM*ULP*N - FUDGE*TWO*PIVMIN
* GU = GU + FUDGE*TNORM*ULP*N + FUDGE*PIVMIN
*
ILTGK = IL
IUTGK = IU
RNGVX = 'V'
WORK( IDTGK:IDTGK+2*N-1 ) = ZERO
CALL DCOPY( N, D, 1, WORK( IETGK ), 2 )
CALL DCOPY( N-1, E, 1, WORK( IETGK+1 ), 2 )
CALL DSTEVX( 'N', 'I', N*2, WORK( IDTGK ), WORK( IETGK ),
$ VLTGK, VLTGK, ILTGK, ILTGK, ABSTOL, NS, S,
$ Z, LDZ, WORK( ITEMP ), IWORK( IIWORK ),
$ IWORK( IIFAIL ), INFO )
VLTGK = S( 1 ) - FUDGE*SMAX*ULP*N
WORK( IDTGK:IDTGK+2*N-1 ) = ZERO
CALL DCOPY( N, D, 1, WORK( IETGK ), 2 )
CALL DCOPY( N-1, E, 1, WORK( IETGK+1 ), 2 )
CALL DSTEVX( 'N', 'I', N*2, WORK( IDTGK ), WORK( IETGK ),
$ VUTGK, VUTGK, IUTGK, IUTGK, ABSTOL, NS, S,
$ Z, LDZ, WORK( ITEMP ), IWORK( IIWORK ),
$ IWORK( IIFAIL ), INFO )
VUTGK = S( 1 ) + FUDGE*SMAX*ULP*N
VUTGK = MIN( VUTGK, ZERO )
*
* If VLTGK=VUTGK, DSTEVX returns an error message,
* so if needed we change VUTGK slightly.
*
IF( VLTGK.EQ.VUTGK ) VLTGK = VLTGK - TOL
*
IF( WANTZ ) CALL DLASET( 'F', N*2, IU-IL+1, ZERO, ZERO, Z, LDZ)
END IF
*
* Initialize variables and pointers for S, Z, and WORK.
*
* NRU, NRV: number of rows in U and V for the active submatrix
* IDBEG, ISBEG: offsets for the entries of D and S
* IROWZ, ICOLZ: offsets for the rows and columns of Z
* IROWU, IROWV: offsets for the rows of U and V
*
NS = 0
NRU = 0
NRV = 0
IDBEG = 1
ISBEG = 1
IROWZ = 1
ICOLZ = 1
IROWU = 2
IROWV = 1
SPLIT = .FALSE.
SVEQ0 = .FALSE.
*
* Form the tridiagonal TGK matrix.
*
S( 1:N ) = ZERO
WORK( IETGK+2*N-1 ) = ZERO
WORK( IDTGK:IDTGK+2*N-1 ) = ZERO
CALL DCOPY( N, D, 1, WORK( IETGK ), 2 )
CALL DCOPY( N-1, E, 1, WORK( IETGK+1 ), 2 )
*
*
* Check for splits in two levels, outer level
* in E and inner level in D.
*
DO IEPTR = 2, N*2, 2
IF( WORK( IETGK+IEPTR-1 ).EQ.ZERO ) THEN
*
* Split in E (this piece of B is square) or bottom
* of the (input bidiagonal) matrix.
*
ISPLT = IDBEG
IDEND = IEPTR - 1
DO IDPTR = IDBEG, IDEND, 2
IF( WORK( IETGK+IDPTR-1 ).EQ.ZERO ) THEN
*
* Split in D (rectangular submatrix). Set the number
* of rows in U and V (NRU and NRV) accordingly.
*
IF( IDPTR.EQ.IDBEG ) THEN
*
* D=0 at the top.
*
SVEQ0 = .TRUE.
IF( IDBEG.EQ.IDEND) THEN
NRU = 1
NRV = 1
END IF
ELSE IF( IDPTR.EQ.IDEND ) THEN
*
* D=0 at the bottom.
*
SVEQ0 = .TRUE.
NRU = (IDEND-ISPLT)/2 + 1
NRV = NRU
IF( ISPLT.NE.IDBEG ) THEN
NRU = NRU + 1
END IF
ELSE
IF( ISPLT.EQ.IDBEG ) THEN
*
* Split: top rectangular submatrix.
*
NRU = (IDPTR-IDBEG)/2
NRV = NRU + 1
ELSE
*
* Split: middle square submatrix.
*
NRU = (IDPTR-ISPLT)/2 + 1
NRV = NRU
END IF
END IF
ELSE IF( IDPTR.EQ.IDEND ) THEN
*
* Last entry of D in the active submatrix.
*
IF( ISPLT.EQ.IDBEG ) THEN
*
* No split (trivial case).
*
NRU = (IDEND-IDBEG)/2 + 1
NRV = NRU
ELSE
*
* Split: bottom rectangular submatrix.
*
NRV = (IDEND-ISPLT)/2 + 1
NRU = NRV + 1
END IF
END IF
*
NTGK = NRU + NRV
*
IF( NTGK.GT.0 ) THEN
*
* Compute eigenvalues/vectors of the active
* submatrix according to RANGE:
* if RANGE='A' (ALLSV) then RNGVX = 'I'
* if RANGE='V' (VALSV) then RNGVX = 'V'
* if RANGE='I' (INDSV) then RNGVX = 'V'
*
ILTGK = 1
IUTGK = NTGK / 2
IF( ALLSV .OR. VUTGK.EQ.ZERO ) THEN
IF( SVEQ0 .OR.
$ SMIN.LT.EPS .OR.
$ MOD(NTGK,2).GT.0 ) THEN
* Special case: eigenvalue equal to zero or very
* small, additional eigenvector is needed.
IUTGK = IUTGK + 1
END IF
END IF
*
* Workspace needed by DSTEVX:
* WORK( ITEMP: ): 2*5*NTGK
* IWORK( 1: ): 2*6*NTGK
*
CALL DSTEVX( JOBZ, RNGVX, NTGK, WORK( IDTGK+ISPLT-1 ),
$ WORK( IETGK+ISPLT-1 ), VLTGK, VUTGK,
$ ILTGK, IUTGK, ABSTOL, NSL, S( ISBEG ),
$ Z( IROWZ,ICOLZ ), LDZ, WORK( ITEMP ),
$ IWORK( IIWORK ), IWORK( IIFAIL ),
$ INFO )
IF( INFO.NE.0 ) THEN
* Exit with the error code from DSTEVX.
RETURN
END IF
EMIN = ABS( MAXVAL( S( ISBEG:ISBEG+NSL-1 ) ) )
*
IF( NSL.GT.0 .AND. WANTZ ) THEN
*
* Normalize u=Z([2,4,...],:) and v=Z([1,3,...],:),
* changing the sign of v as discussed in the leading
* comments. The norms of u and v may be (slightly)
* different from 1/sqrt(2) if the corresponding
* eigenvalues are very small or too close. We check
* those norms and, if needed, reorthogonalize the
* vectors.
*
IF( NSL.GT.1 .AND.
$ VUTGK.EQ.ZERO .AND.
$ MOD(NTGK,2).EQ.0 .AND.
$ EMIN.EQ.0 .AND. .NOT.SPLIT ) THEN
*
* D=0 at the top or bottom of the active submatrix:
* one eigenvalue is equal to zero; concatenate the
* eigenvectors corresponding to the two smallest
* eigenvalues.
*
Z( IROWZ:IROWZ+NTGK-1,ICOLZ+NSL-2 ) =
$ Z( IROWZ:IROWZ+NTGK-1,ICOLZ+NSL-2 ) +
$ Z( IROWZ:IROWZ+NTGK-1,ICOLZ+NSL-1 )
Z( IROWZ:IROWZ+NTGK-1,ICOLZ+NSL-1 ) =
$ ZERO
* IF( IUTGK*2.GT.NTGK ) THEN
* Eigenvalue equal to zero or very small.
* NSL = NSL - 1
* END IF
END IF
*
DO I = 0, MIN( NSL-1, NRU-1 )
NRMU = DNRM2( NRU, Z( IROWU, ICOLZ+I ), 2 )
IF( NRMU.EQ.ZERO ) THEN
INFO = N*2 + 1
RETURN
END IF
CALL DSCAL( NRU, ONE/NRMU,
$ Z( IROWU,ICOLZ+I ), 2 )
IF( NRMU.NE.ONE .AND.
$ ABS( NRMU-ORTOL )*SQRT2.GT.ONE )
$ THEN
DO J = 0, I-1
ZJTJI = -DDOT( NRU, Z( IROWU, ICOLZ+J ),
$ 2, Z( IROWU, ICOLZ+I ), 2 )
CALL DAXPY( NRU, ZJTJI,
$ Z( IROWU, ICOLZ+J ), 2,
$ Z( IROWU, ICOLZ+I ), 2 )
END DO
NRMU = DNRM2( NRU, Z( IROWU, ICOLZ+I ), 2 )
CALL DSCAL( NRU, ONE/NRMU,
$ Z( IROWU,ICOLZ+I ), 2 )
END IF
END DO
DO I = 0, MIN( NSL-1, NRV-1 )
NRMV = DNRM2( NRV, Z( IROWV, ICOLZ+I ), 2 )
IF( NRMV.EQ.ZERO ) THEN
INFO = N*2 + 1
RETURN
END IF
CALL DSCAL( NRV, -ONE/NRMV,
$ Z( IROWV,ICOLZ+I ), 2 )
IF( NRMV.NE.ONE .AND.
$ ABS( NRMV-ORTOL )*SQRT2.GT.ONE )
$ THEN
DO J = 0, I-1
ZJTJI = -DDOT( NRV, Z( IROWV, ICOLZ+J ),
$ 2, Z( IROWV, ICOLZ+I ), 2 )
CALL DAXPY( NRU, ZJTJI,
$ Z( IROWV, ICOLZ+J ), 2,
$ Z( IROWV, ICOLZ+I ), 2 )
END DO
NRMV = DNRM2( NRV, Z( IROWV, ICOLZ+I ), 2 )
CALL DSCAL( NRV, ONE/NRMV,
$ Z( IROWV,ICOLZ+I ), 2 )
END IF
END DO
IF( VUTGK.EQ.ZERO .AND.
$ IDPTR.LT.IDEND .AND.
$ MOD(NTGK,2).GT.0 ) THEN
*
* D=0 in the middle of the active submatrix (one
* eigenvalue is equal to zero): save the corresponding
* eigenvector for later use (when bottom of the
* active submatrix is reached).
*
SPLIT = .TRUE.
Z( IROWZ:IROWZ+NTGK-1,N+1 ) =
$ Z( IROWZ:IROWZ+NTGK-1,NS+NSL )
Z( IROWZ:IROWZ+NTGK-1,NS+NSL ) =
$ ZERO
END IF
END IF !** WANTZ **!
*
NSL = MIN( NSL, NRU )
SVEQ0 = .FALSE.
*
* Absolute values of the eigenvalues of TGK.
*
DO I = 0, NSL-1
S( ISBEG+I ) = ABS( S( ISBEG+I ) )
END DO
*
* Update pointers for TGK, S and Z.
*
ISBEG = ISBEG + NSL
IROWZ = IROWZ + NTGK
ICOLZ = ICOLZ + NSL
IROWU = IROWZ
IROWV = IROWZ + 1
ISPLT = IDPTR + 1
NS = NS + NSL
NRU = 0
NRV = 0
END IF !** NTGK.GT.0 **!
IF( IROWZ.LT.N*2 .AND. WANTZ ) THEN
Z( 1:IROWZ-1, ICOLZ ) = ZERO
END IF
END DO !** IDPTR loop **!
IF( SPLIT .AND. WANTZ ) THEN
*
* Bring back eigenvector corresponding
* to eigenvalue equal to zero.
*
Z( IDBEG:IDEND-NTGK+1,ISBEG-1 ) =
$ Z( IDBEG:IDEND-NTGK+1,ISBEG-1 ) +
$ Z( IDBEG:IDEND-NTGK+1,N+1 )
Z( IDBEG:IDEND-NTGK+1,N+1 ) = 0
END IF
IROWV = IROWV - 1
IROWU = IROWU + 1
IDBEG = IEPTR + 1
SVEQ0 = .FALSE.
SPLIT = .FALSE.
END IF !** Check for split in E **!
END DO !** IEPTR loop **!
*
* Sort the singular values into decreasing order (insertion sort on
* singular values, but only one transposition per singular vector)
*
DO I = 1, NS-1
K = 1
SMIN = S( 1 )
DO J = 2, NS + 1 - I
IF( S( J ).LE.SMIN ) THEN
K = J
SMIN = S( J )
END IF
END DO
IF( K.NE.NS+1-I ) THEN
S( K ) = S( NS+1-I )
S( NS+1-I ) = SMIN
IF( WANTZ ) CALL DSWAP( N*2, Z( 1,K ), 1, Z( 1,NS+1-I ), 1 )
END IF
END DO
*
* If RANGE=I, check for singular values/vectors to be discarded.
*
IF( INDSV ) THEN
K = IU - IL + 1
IF( K.LT.NS ) THEN
S( K+1:NS ) = ZERO
IF( WANTZ ) Z( 1:N*2,K+1:NS ) = ZERO
NS = K
END IF
END IF
*
* Reorder Z: U = Z( 1:N,1:NS ), V = Z( N+1:N*2,1:NS ).
* If B is a lower diagonal, swap U and V.
*
IF( WANTZ ) THEN
DO I = 1, NS
CALL DCOPY( N*2, Z( 1,I ), 1, WORK, 1 )
IF( LOWER ) THEN
CALL DCOPY( N, WORK( 2 ), 2, Z( N+1,I ), 1 )
CALL DCOPY( N, WORK( 1 ), 2, Z( 1 ,I ), 1 )
ELSE
CALL DCOPY( N, WORK( 2 ), 2, Z( 1 ,I ), 1 )
CALL DCOPY( N, WORK( 1 ), 2, Z( N+1,I ), 1 )
END IF
END DO
END IF
*
RETURN
*
* End of DBDSVDX
*
END