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OpenBLAS/lapack-netlib/SRC/dbdsdc.f

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*> \brief \b DBDSDC
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DBDSDC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dbdsdc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dbdsdc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dbdsdc.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DBDSDC( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ,
* WORK, IWORK, INFO )
*
* .. 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( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> 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.
*>
*> 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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': B is upper bidiagonal.
*> = 'L': B is lower bidiagonal.
*> \endverbatim
*>
*> \param[in] COMPQ
*> \verbatim
*> COMPQ is 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.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is 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.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is 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.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is 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.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= 1.
*> If singular vectors are desired, then LDU >= max( 1, N ).
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension (LDVT,N)
*> If COMPQ = 'I', then:
*> On exit, if INFO = 0, VT**T contains the right singular
*> vectors of the bidiagonal matrix.
*> For other values of COMPQ, VT is not referenced.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT. LDVT >= 1.
*> If singular vectors are desired, then LDVT >= max( 1, N ).
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is 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.
*> \endverbatim
*>
*> \param[out] IQ
*> \verbatim
*> IQ is 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.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is 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).
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: The algorithm failed to compute a singular value.
*> The update process of divide and conquer failed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DBDSDC( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ,
$ WORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. 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( * )
* ..
*
* =====================================================================
* 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
IF( ICOMPQ .EQ. 2 ) 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
* Ignore WSTART, instead using WORK( 1 ), since the two vectors
* for CS and -SN above are added only if ICOMPQ == 2,
* and adding them exceeds documented WORK size of 4*n.
CALL DLASDQ( 'U', 0, N, 0, 0, 0, D, E, VT, LDVT, U, LDU, U,
$ LDU, WORK( 1 ), 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 = (0.9D+0)*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 )
END IF
IF( INFO.NE.0 ) THEN
RETURN
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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