*> \brief \b SLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLALSA + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U, * LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, * GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK, * IWORK, INFO ) * * .. Scalar Arguments .. * INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS, * $ SMLSIZ * .. * .. Array Arguments .. * INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), * $ K( * ), PERM( LDGCOL, * ) * REAL B( LDB, * ), BX( LDBX, * ), C( * ), * $ DIFL( LDU, * ), DIFR( LDU, * ), * $ GIVNUM( LDU, * ), POLES( LDU, * ), S( * ), * $ U( LDU, * ), VT( LDU, * ), WORK( * ), * $ Z( LDU, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLALSA is an itermediate step in solving the least squares problem *> by computing the SVD of the coefficient matrix in compact form (The *> singular vectors are computed as products of simple orthorgonal *> matrices.). *> *> If ICOMPQ = 0, SLALSA applies the inverse of the left singular vector *> matrix of an upper bidiagonal matrix to the right hand side; and if *> ICOMPQ = 1, SLALSA applies the right singular vector matrix to the *> right hand side. The singular vector matrices were generated in *> compact form by SLALSA. *> \endverbatim * * Arguments: * ========== * *> \param[in] ICOMPQ *> \verbatim *> ICOMPQ is INTEGER *> Specifies whether the left or the right singular vector *> matrix is involved. *> = 0: Left singular vector matrix *> = 1: Right singular vector matrix *> \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 and column dimensions of the upper bidiagonal matrix. *> \endverbatim *> *> \param[in] NRHS *> \verbatim *> NRHS is INTEGER *> The number of columns of B and BX. NRHS must be at least 1. *> \endverbatim *> *> \param[in,out] B *> \verbatim *> B is REAL array, dimension ( LDB, NRHS ) *> On input, B contains the right hand sides of the least *> squares problem in rows 1 through M. *> On output, B contains the solution X in rows 1 through N. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of B in the calling subprogram. *> LDB must be at least max(1,MAX( M, N ) ). *> \endverbatim *> *> \param[out] BX *> \verbatim *> BX is REAL array, dimension ( LDBX, NRHS ) *> On exit, the result of applying the left or right singular *> vector matrix to B. *> \endverbatim *> *> \param[in] LDBX *> \verbatim *> LDBX is INTEGER *> The leading dimension of BX. *> \endverbatim *> *> \param[in] U *> \verbatim *> U is REAL array, dimension ( LDU, SMLSIZ ). *> On entry, 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[in] VT *> \verbatim *> VT is REAL array, dimension ( LDU, SMLSIZ+1 ). *> On entry, VT**T contains the right singular vector matrices of *> all subproblems at the bottom level. *> \endverbatim *> *> \param[in] K *> \verbatim *> K is INTEGER array, dimension ( N ). *> \endverbatim *> *> \param[in] DIFL *> \verbatim *> DIFL is REAL array, dimension ( LDU, NLVL ). *> where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1. *> \endverbatim *> *> \param[in] DIFR *> \verbatim *> DIFR is REAL array, dimension ( LDU, 2 * NLVL ). *> On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record *> distances between singular values on the I-th level and *> singular values on the (I -1)-th level, and DIFR(*, 2 * I) *> record the normalizing factors of the right singular vectors *> matrices of subproblems on I-th level. *> \endverbatim *> *> \param[in] Z *> \verbatim *> Z is REAL array, dimension ( LDU, NLVL ). *> On entry, Z(1, I) contains the components of the deflation- *> adjusted updating row vector for subproblems on the I-th *> level. *> \endverbatim *> *> \param[in] POLES *> \verbatim *> POLES is REAL array, dimension ( LDU, 2 * NLVL ). *> On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old *> singular values involved in the secular equations on the I-th *> level. *> \endverbatim *> *> \param[in] GIVPTR *> \verbatim *> GIVPTR is INTEGER array, dimension ( N ). *> On entry, GIVPTR( I ) records the number of Givens *> rotations performed on the I-th problem on the computation *> tree. *> \endverbatim *> *> \param[in] GIVCOL *> \verbatim *> GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ). *> On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records 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[in] PERM *> \verbatim *> PERM is INTEGER array, dimension ( LDGCOL, NLVL ). *> On entry, PERM(*, I) records permutations done on the I-th *> level of the computation tree. *> \endverbatim *> *> \param[in] GIVNUM *> \verbatim *> GIVNUM is REAL array, dimension ( LDU, 2 * NLVL ). *> On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S- *> values of Givens rotations performed on the I-th level on the *> computation tree. *> \endverbatim *> *> \param[in] C *> \verbatim *> C is REAL array, dimension ( N ). *> On entry, if the I-th subproblem is not square, *> C( I ) contains the C-value of a Givens rotation related to *> the right null space of the I-th subproblem. *> \endverbatim *> *> \param[in] S *> \verbatim *> S is REAL array, dimension ( N ). *> On entry, if the I-th subproblem is not square, *> 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. *> The dimension must be at least N. *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array. *> The dimension must be at least 3 * N *> \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. * *> \date December 2016 * *> \ingroup realOTHERcomputational * *> \par Contributors: * ================== *> *> Ming Gu and Ren-Cang Li, Computer Science Division, University of *> California at Berkeley, USA \n *> Osni Marques, LBNL/NERSC, USA \n * * ===================================================================== SUBROUTINE SLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U, $ LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, $ GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK, $ IWORK, INFO ) * * -- LAPACK computational 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, LDB, LDBX, LDGCOL, LDU, N, NRHS, $ SMLSIZ * .. * .. Array Arguments .. INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), $ K( * ), PERM( LDGCOL, * ) REAL B( LDB, * ), BX( LDBX, * ), C( * ), $ DIFL( LDU, * ), DIFR( LDU, * ), $ GIVNUM( LDU, * ), POLES( LDU, * ), S( * ), $ U( LDU, * ), VT( LDU, * ), WORK( * ), $ Z( LDU, * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) * .. * .. Local Scalars .. INTEGER I, I1, IC, IM1, INODE, J, LF, LL, LVL, LVL2, $ ND, NDB1, NDIML, NDIMR, NL, NLF, NLP1, NLVL, $ NR, NRF, NRP1, SQRE * .. * .. External Subroutines .. EXTERNAL SCOPY, SGEMM, SLALS0, SLASDT, 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.SMLSIZ ) THEN INFO = -3 ELSE IF( NRHS.LT.1 ) THEN INFO = -4 ELSE IF( LDB.LT.N ) THEN INFO = -6 ELSE IF( LDBX.LT.N ) THEN INFO = -8 ELSE IF( LDU.LT.N ) THEN INFO = -10 ELSE IF( LDGCOL.LT.N ) THEN INFO = -19 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SLALSA', -INFO ) RETURN END IF * * Book-keeping and setting up the computation tree. * INODE = 1 NDIML = INODE + N NDIMR = NDIML + N * CALL SLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ), $ IWORK( NDIMR ), SMLSIZ ) * * The following code applies back the left singular vector factors. * For applying back the right singular vector factors, go to 50. * IF( ICOMPQ.EQ.1 ) THEN GO TO 50 END IF * * The nodes on the bottom level of the tree were solved * by SLASDQ. The corresponding left and right singular vector * matrices are in explicit form. First apply back the left * singular vector matrices. * NDB1 = ( ND+1 ) / 2 DO 10 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 ) NR = IWORK( NDIMR+I1 ) NLF = IC - NL NRF = IC + 1 CALL SGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU, $ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX ) CALL SGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU, $ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX ) 10 CONTINUE * * Next copy the rows of B that correspond to unchanged rows * in the bidiagonal matrix to BX. * DO 20 I = 1, ND IC = IWORK( INODE+I-1 ) CALL SCOPY( NRHS, B( IC, 1 ), LDB, BX( IC, 1 ), LDBX ) 20 CONTINUE * * Finally go through the left singular vector matrices of all * the other subproblems bottom-up on the tree. * J = 2**NLVL SQRE = 0 * DO 40 LVL = NLVL, 1, -1 LVL2 = 2*LVL - 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 30 I = LF, LL IM1 = I - 1 IC = IWORK( INODE+IM1 ) NL = IWORK( NDIML+IM1 ) NR = IWORK( NDIMR+IM1 ) NLF = IC - NL NRF = IC + 1 J = J - 1 CALL SLALS0( ICOMPQ, NL, NR, SQRE, NRHS, BX( NLF, 1 ), LDBX, $ B( NLF, 1 ), LDB, 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, $ INFO ) 30 CONTINUE 40 CONTINUE GO TO 90 * * ICOMPQ = 1: applying back the right singular vector factors. * 50 CONTINUE * * First now go through the right singular vector matrices of all * the tree nodes top-down. * J = 0 DO 70 LVL = 1, NLVL LVL2 = 2*LVL - 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 60 I = LL, LF, -1 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 SQRE = 0 ELSE SQRE = 1 END IF J = J + 1 CALL SLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B( NLF, 1 ), LDB, $ BX( NLF, 1 ), LDBX, 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, $ INFO ) 60 CONTINUE 70 CONTINUE * * The nodes on the bottom level of the tree were solved * by SLASDQ. The corresponding right singular vector * matrices are in explicit form. Apply them back. * NDB1 = ( ND+1 ) / 2 DO 80 I = NDB1, ND I1 = I - 1 IC = IWORK( INODE+I1 ) NL = IWORK( NDIML+I1 ) NR = IWORK( NDIMR+I1 ) NLP1 = NL + 1 IF( I.EQ.ND ) THEN NRP1 = NR ELSE NRP1 = NR + 1 END IF NLF = IC - NL NRF = IC + 1 CALL SGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ), LDU, $ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX ) CALL SGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ), LDU, $ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX ) 80 CONTINUE * 90 CONTINUE * RETURN * * End of SLALSA * END