*> \brief \b SLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K,
* DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,
* PERM, GIVNUM, C, S, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
* $ K( * ), PERM( LDGCOL, * )
* REAL C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ),
* $ E( * ), GIVNUM( LDU, * ), POLES( LDU, * ),
* $ S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ),
* $ Z( LDU, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Using a divide and conquer approach, SLASDA 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, SLASD0, computes the singular values and
*> the singular vectors in explicit form.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is 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.
*> \endverbatim
*>
*> \param[in] SMLSIZ
*> \verbatim
*> SMLSIZ is INTEGER
*> The maximum size of the subproblems at the bottom of the
*> computation tree.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The row dimension of the upper bidiagonal matrix. This is
*> also the dimension of the main diagonal array D.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*> SQRE is 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.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is REAL array, dimension ( N )
*> On entry D contains the main diagonal of the bidiagonal
*> matrix. On exit D, if INFO = 0, contains its singular values.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is REAL array, dimension ( M-1 )
*> Contains the subdiagonal entries of the bidiagonal matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is REAL 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.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER, LDU = > N.
*> The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
*> GIVNUM, and Z.
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*> VT is REAL array,
*> dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced
*> if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right
*> singular vector matrices of all subproblems at the bottom
*> level.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*> K is 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.
*> \endverbatim
*>
*> \param[out] DIFL
*> \verbatim
*> DIFL is REAL array, dimension ( LDU, NLVL ),
*> where NLVL = floor(log_2 (N/SMLSIZ))).
*> \endverbatim
*>
*> \param[out] DIFR
*> \verbatim
*> DIFR is REAL 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 SLASD8 for details.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is REAL 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.
*> \endverbatim
*>
*> \param[out] POLES
*> \verbatim
*> POLES is REAL 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.
*> \endverbatim
*>
*> \param[out] GIVPTR
*> \verbatim
*> GIVPTR is 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.
*> \endverbatim
*>
*> \param[out] GIVCOL
*> \verbatim
*> GIVCOL is 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.
*> \endverbatim
*>
*> \param[in] LDGCOL
*> \verbatim
*> LDGCOL is INTEGER, LDGCOL = > N.
*> The leading dimension of arrays GIVCOL and PERM.
*> \endverbatim
*>
*> \param[out] PERM
*> \verbatim
*> PERM is 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.
*> \endverbatim
*>
*> \param[out] GIVNUM
*> \verbatim
*> GIVNUM is REAL 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.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*> C is REAL 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.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is REAL 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.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension
*> (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (7*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: if INFO = 1, a singular value did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup OTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE SLASDA( 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 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE
* ..
* .. Array Arguments ..
INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
$ K( * ), PERM( LDGCOL, * )
REAL C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ),
$ E( * ), GIVNUM( LDU, * ), POLES( LDU, * ),
$ S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ),
$ Z( LDU, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+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
REAL ALPHA, BETA
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SLASD6, SLASDQ, SLASDT, SLASET, 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( 'SLASDA', -INFO )
RETURN
END IF
*
M = N + SQRE
*
* If the input matrix is too small, call SLASDQ to find the SVD.
*
IF( N.LE.SMLSIZ ) THEN
IF( ICOMPQ.EQ.0 ) THEN
CALL SLASDQ( 'U', SQRE, N, 0, 0, 0, D, E, VT, LDU, U, LDU,
$ U, LDU, WORK, INFO )
ELSE
CALL SLASDQ( '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 SLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),
$ IWORK( NDIMR ), SMLSIZ )
*
* for the nodes on bottom level of the tree, solve
* their subproblems by SLASDQ.
*
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 SLASET( 'A', NLP1, NLP1, ZERO, ONE, WORK( NWORK1 ),
$ SMLSZP )
CALL SLASDQ( '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 SCOPY( NLP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )
CALL SCOPY( NLP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )
ELSE
CALL SLASET( 'A', NL, NL, ZERO, ONE, U( NLF, 1 ), LDU )
CALL SLASET( 'A', NLP1, NLP1, ZERO, ONE, VT( NLF, 1 ), LDU )
CALL SLASDQ( '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 SCOPY( NLP1, VT( NLF, 1 ), 1, WORK( VFI ), 1 )
CALL SCOPY( 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 SLASET( 'A', NRP1, NRP1, ZERO, ONE, WORK( NWORK1 ),
$ SMLSZP )
CALL SLASDQ( '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 SCOPY( NRP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )
CALL SCOPY( NRP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )
ELSE
CALL SLASET( 'A', NR, NR, ZERO, ONE, U( NRF, 1 ), LDU )
CALL SLASET( 'A', NRP1, NRP1, ZERO, ONE, VT( NRF, 1 ), LDU )
CALL SLASDQ( '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 SCOPY( NRP1, VT( NRF, 1 ), 1, WORK( VFI ), 1 )
CALL SCOPY( 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 SLASD6( 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 SLASD6( 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 SLASDA
*
END