SUBROUTINE ZLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U, $ LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, $ GIVCOL, LDGCOL, PERM, GIVNUM, C, S, RWORK, $ IWORK, INFO ) * * -- LAPACK routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS, $ SMLSIZ * .. * .. Array Arguments .. INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), $ K( * ), PERM( LDGCOL, * ) DOUBLE PRECISION C( * ), DIFL( LDU, * ), DIFR( LDU, * ), $ GIVNUM( LDU, * ), POLES( LDU, * ), RWORK( * ), $ S( * ), U( LDU, * ), VT( LDU, * ), Z( LDU, * ) COMPLEX*16 B( LDB, * ), BX( LDBX, * ) * .. * * Purpose * ======= * * ZLALSA 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, ZLALSA applies the inverse of the left singular vector * matrix of an upper bidiagonal matrix to the right hand side; and if * ICOMPQ = 1, ZLALSA applies the right singular vector matrix to the * right hand side. The singular vector matrices were generated in * compact form by ZLALSA. * * Arguments * ========= * * ICOMPQ (input) INTEGER * Specifies whether the left or the right singular vector * matrix is involved. * = 0: Left singular vector matrix * = 1: Right singular vector matrix * * SMLSIZ (input) INTEGER * The maximum size of the subproblems at the bottom of the * computation tree. * * N (input) INTEGER * The row and column dimensions of the upper bidiagonal matrix. * * NRHS (input) INTEGER * The number of columns of B and BX. NRHS must be at least 1. * * B (input/output) COMPLEX*16 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. * * LDB (input) INTEGER * The leading dimension of B in the calling subprogram. * LDB must be at least max(1,MAX( M, N ) ). * * BX (output) COMPLEX*16 array, dimension ( LDBX, NRHS ) * On exit, the result of applying the left or right singular * vector matrix to B. * * LDBX (input) INTEGER * The leading dimension of BX. * * U (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ). * On entry, U contains the left singular vector matrices of all * subproblems at the bottom level. * * LDU (input) INTEGER, LDU = > N. * The leading dimension of arrays U, VT, DIFL, DIFR, * POLES, GIVNUM, and Z. * * VT (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ). * On entry, VT' contains the right singular vector matrices of * all subproblems at the bottom level. * * K (input) INTEGER array, dimension ( N ). * * DIFL (input) DOUBLE PRECISION array, dimension ( LDU, NLVL ). * where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1. * * DIFR (input) DOUBLE PRECISION 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. * * Z (input) DOUBLE PRECISION 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. * * POLES (input) DOUBLE PRECISION 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. * * GIVPTR (input) INTEGER array, dimension ( N ). * On entry, GIVPTR( I ) records the number of Givens * rotations performed on the I-th problem on the computation * tree. * * GIVCOL (input) 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. * * LDGCOL (input) INTEGER, LDGCOL = > N. * The leading dimension of arrays GIVCOL and PERM. * * PERM (input) INTEGER array, dimension ( LDGCOL, NLVL ). * On entry, PERM(*, I) records permutations done on the I-th * level of the computation tree. * * GIVNUM (input) DOUBLE PRECISION 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. * * C (input) DOUBLE PRECISION 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. * * S (input) DOUBLE PRECISION 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. * * RWORK (workspace) DOUBLE PRECISION array, dimension at least * max ( N, (SMLSZ+1)*NRHS*3 ). * * IWORK (workspace) INTEGER array. * The dimension must be at least 3 * N * * INFO (output) INTEGER * = 0: successful exit. * < 0: if INFO = -i, the i-th argument had an illegal value. * * Further Details * =============== * * Based on contributions by * Ming Gu and Ren-Cang Li, Computer Science Division, University of * California at Berkeley, USA * Osni Marques, LBNL/NERSC, USA * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) * .. * .. Local Scalars .. INTEGER I, I1, IC, IM1, INODE, J, JCOL, JIMAG, JREAL, $ JROW, LF, LL, LVL, LVL2, ND, NDB1, NDIML, $ NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, SQRE * .. * .. External Subroutines .. EXTERNAL DGEMM, DLASDT, XERBLA, ZCOPY, ZLALS0 * .. * .. Intrinsic Functions .. INTRINSIC DBLE, DCMPLX, DIMAG * .. * .. 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( 'ZLALSA', -INFO ) RETURN END IF * * Book-keeping and setting up the computation tree. * INODE = 1 NDIML = INODE + N NDIMR = NDIML + N * CALL DLASDT( 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 170. * IF( ICOMPQ.EQ.1 ) THEN GO TO 170 END IF * * The nodes on the bottom level of the tree were solved * by DLASDQ. 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 130 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 * * Since B and BX are complex, the following call to DGEMM * is performed in two steps (real and imaginary parts). * * CALL DGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU, * $ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX ) * J = NL*NRHS*2 DO 20 JCOL = 1, NRHS DO 10 JROW = NLF, NLF + NL - 1 J = J + 1 RWORK( J ) = DBLE( B( JROW, JCOL ) ) 10 CONTINUE 20 CONTINUE CALL DGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU, $ RWORK( 1+NL*NRHS*2 ), NL, ZERO, RWORK( 1 ), NL ) J = NL*NRHS*2 DO 40 JCOL = 1, NRHS DO 30 JROW = NLF, NLF + NL - 1 J = J + 1 RWORK( J ) = DIMAG( B( JROW, JCOL ) ) 30 CONTINUE 40 CONTINUE CALL DGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU, $ RWORK( 1+NL*NRHS*2 ), NL, ZERO, RWORK( 1+NL*NRHS ), $ NL ) JREAL = 0 JIMAG = NL*NRHS DO 60 JCOL = 1, NRHS DO 50 JROW = NLF, NLF + NL - 1 JREAL = JREAL + 1 JIMAG = JIMAG + 1 BX( JROW, JCOL ) = DCMPLX( RWORK( JREAL ), $ RWORK( JIMAG ) ) 50 CONTINUE 60 CONTINUE * * Since B and BX are complex, the following call to DGEMM * is performed in two steps (real and imaginary parts). * * CALL DGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU, * $ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX ) * J = NR*NRHS*2 DO 80 JCOL = 1, NRHS DO 70 JROW = NRF, NRF + NR - 1 J = J + 1 RWORK( J ) = DBLE( B( JROW, JCOL ) ) 70 CONTINUE 80 CONTINUE CALL DGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU, $ RWORK( 1+NR*NRHS*2 ), NR, ZERO, RWORK( 1 ), NR ) J = NR*NRHS*2 DO 100 JCOL = 1, NRHS DO 90 JROW = NRF, NRF + NR - 1 J = J + 1 RWORK( J ) = DIMAG( B( JROW, JCOL ) ) 90 CONTINUE 100 CONTINUE CALL DGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU, $ RWORK( 1+NR*NRHS*2 ), NR, ZERO, RWORK( 1+NR*NRHS ), $ NR ) JREAL = 0 JIMAG = NR*NRHS DO 120 JCOL = 1, NRHS DO 110 JROW = NRF, NRF + NR - 1 JREAL = JREAL + 1 JIMAG = JIMAG + 1 BX( JROW, JCOL ) = DCMPLX( RWORK( JREAL ), $ RWORK( JIMAG ) ) 110 CONTINUE 120 CONTINUE * 130 CONTINUE * * Next copy the rows of B that correspond to unchanged rows * in the bidiagonal matrix to BX. * DO 140 I = 1, ND IC = IWORK( INODE+I-1 ) CALL ZCOPY( NRHS, B( IC, 1 ), LDB, BX( IC, 1 ), LDBX ) 140 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 160 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 150 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 ZLALS0( 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 ), RWORK, $ INFO ) 150 CONTINUE 160 CONTINUE GO TO 330 * * ICOMPQ = 1: applying back the right singular vector factors. * 170 CONTINUE * * First now go through the right singular vector matrices of all * the tree nodes top-down. * J = 0 DO 190 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 180 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 ZLALS0( 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 ), RWORK, $ INFO ) 180 CONTINUE 190 CONTINUE * * The nodes on the bottom level of the tree were solved * by DLASDQ. The corresponding right singular vector * matrices are in explicit form. Apply them back. * NDB1 = ( ND+1 ) / 2 DO 320 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 * * Since B and BX are complex, the following call to DGEMM is * performed in two steps (real and imaginary parts). * * CALL DGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ), LDU, * $ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX ) * J = NLP1*NRHS*2 DO 210 JCOL = 1, NRHS DO 200 JROW = NLF, NLF + NLP1 - 1 J = J + 1 RWORK( J ) = DBLE( B( JROW, JCOL ) ) 200 CONTINUE 210 CONTINUE CALL DGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ), LDU, $ RWORK( 1+NLP1*NRHS*2 ), NLP1, ZERO, RWORK( 1 ), $ NLP1 ) J = NLP1*NRHS*2 DO 230 JCOL = 1, NRHS DO 220 JROW = NLF, NLF + NLP1 - 1 J = J + 1 RWORK( J ) = DIMAG( B( JROW, JCOL ) ) 220 CONTINUE 230 CONTINUE CALL DGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ), LDU, $ RWORK( 1+NLP1*NRHS*2 ), NLP1, ZERO, $ RWORK( 1+NLP1*NRHS ), NLP1 ) JREAL = 0 JIMAG = NLP1*NRHS DO 250 JCOL = 1, NRHS DO 240 JROW = NLF, NLF + NLP1 - 1 JREAL = JREAL + 1 JIMAG = JIMAG + 1 BX( JROW, JCOL ) = DCMPLX( RWORK( JREAL ), $ RWORK( JIMAG ) ) 240 CONTINUE 250 CONTINUE * * Since B and BX are complex, the following call to DGEMM is * performed in two steps (real and imaginary parts). * * CALL DGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ), LDU, * $ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX ) * J = NRP1*NRHS*2 DO 270 JCOL = 1, NRHS DO 260 JROW = NRF, NRF + NRP1 - 1 J = J + 1 RWORK( J ) = DBLE( B( JROW, JCOL ) ) 260 CONTINUE 270 CONTINUE CALL DGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ), LDU, $ RWORK( 1+NRP1*NRHS*2 ), NRP1, ZERO, RWORK( 1 ), $ NRP1 ) J = NRP1*NRHS*2 DO 290 JCOL = 1, NRHS DO 280 JROW = NRF, NRF + NRP1 - 1 J = J + 1 RWORK( J ) = DIMAG( B( JROW, JCOL ) ) 280 CONTINUE 290 CONTINUE CALL DGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ), LDU, $ RWORK( 1+NRP1*NRHS*2 ), NRP1, ZERO, $ RWORK( 1+NRP1*NRHS ), NRP1 ) JREAL = 0 JIMAG = NRP1*NRHS DO 310 JCOL = 1, NRHS DO 300 JROW = NRF, NRF + NRP1 - 1 JREAL = JREAL + 1 JIMAG = JIMAG + 1 BX( JROW, JCOL ) = DCMPLX( RWORK( JREAL ), $ RWORK( JIMAG ) ) 300 CONTINUE 310 CONTINUE * 320 CONTINUE * 330 CONTINUE * RETURN * * End of ZLALSA * END