*> \brief \b DLALSD uses the singular value decomposition of A to solve the least squares problem. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DLALSD + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, * RANK, WORK, IWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER INFO, LDB, N, NRHS, RANK, SMLSIZ * DOUBLE PRECISION RCOND * .. * .. Array Arguments .. * INTEGER IWORK( * ) * DOUBLE PRECISION B( LDB, * ), D( * ), E( * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DLALSD uses the singular value decomposition of A to solve the least *> squares problem of finding X to minimize the Euclidean norm of each *> column of A*X-B, where A is N-by-N upper bidiagonal, and X and B *> are N-by-NRHS. The solution X overwrites B. *> *> The singular values of A smaller than RCOND times the largest *> singular value are treated as zero in solving the least squares *> problem; in this case a minimum norm solution is returned. *> The actual singular values are returned in D in ascending order. *> *> This code makes very mild assumptions about floating point *> arithmetic. It will work on machines with a guard digit in *> add/subtract, or on those binary machines without guard digits *> which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2. *> It could conceivably fail on hexadecimal or decimal machines *> without guard digits, but we know of none. *> \endverbatim * * Arguments: * ========== * *> \param[in] UPLO *> \verbatim *> UPLO is CHARACTER*1 *> = 'U': D and E define an upper bidiagonal matrix. *> = 'L': D and E define a lower bidiagonal 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 dimension of the bidiagonal matrix. N >= 0. *> \endverbatim *> *> \param[in] NRHS *> \verbatim *> NRHS is INTEGER *> The number of columns of B. NRHS must be at least 1. *> \endverbatim *> *> \param[in,out] D *> \verbatim *> D is DOUBLE PRECISION array, dimension (N) *> On entry D contains the main diagonal of the bidiagonal *> matrix. On exit, if INFO = 0, D contains its singular values. *> \endverbatim *> *> \param[in,out] E *> \verbatim *> E is DOUBLE PRECISION array, dimension (N-1) *> Contains the super-diagonal entries of the bidiagonal matrix. *> On exit, E has been destroyed. *> \endverbatim *> *> \param[in,out] B *> \verbatim *> B is DOUBLE PRECISION array, dimension (LDB,NRHS) *> On input, B contains the right hand sides of the least *> squares problem. On output, B contains the solution X. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of B in the calling subprogram. *> LDB must be at least max(1,N). *> \endverbatim *> *> \param[in] RCOND *> \verbatim *> RCOND is DOUBLE PRECISION *> The singular values of A less than or equal to RCOND times *> the largest singular value are treated as zero in solving *> the least squares problem. If RCOND is negative, *> machine precision is used instead. *> For example, if diag(S)*X=B were the least squares problem, *> where diag(S) is a diagonal matrix of singular values, the *> solution would be X(i) = B(i) / S(i) if S(i) is greater than *> RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to *> RCOND*max(S). *> \endverbatim *> *> \param[out] RANK *> \verbatim *> RANK is INTEGER *> The number of singular values of A greater than RCOND times *> the largest singular value. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension at least *> (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2), *> where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1). *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension at least *> (3*N*NLVL + 11*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 while *> working on the submatrix lying in rows and columns *> INFO/(N+1) through MOD(INFO,N+1). *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup doubleOTHERcomputational * *> \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 DLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, \$ RANK, 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 .. CHARACTER UPLO INTEGER INFO, LDB, N, NRHS, RANK, SMLSIZ DOUBLE PRECISION RCOND * .. * .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION B( LDB, * ), D( * ), E( * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 ) * .. * .. Local Scalars .. INTEGER BX, BXST, C, DIFL, DIFR, GIVCOL, GIVNUM, \$ GIVPTR, I, ICMPQ1, ICMPQ2, IWK, J, K, NLVL, \$ NM1, NSIZE, NSUB, NWORK, PERM, POLES, S, SIZEI, \$ SMLSZP, SQRE, ST, ST1, U, VT, Z DOUBLE PRECISION CS, EPS, ORGNRM, R, RCND, SN, TOL * .. * .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DLAMCH, DLANST EXTERNAL IDAMAX, DLAMCH, DLANST * .. * .. External Subroutines .. EXTERNAL DCOPY, DGEMM, DLACPY, DLALSA, DLARTG, DLASCL, \$ DLASDA, DLASDQ, DLASET, DLASRT, DROT, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, INT, LOG, SIGN * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 * IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.1 ) THEN INFO = -4 ELSE IF( ( LDB.LT.1 ) .OR. ( LDB.LT.N ) ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLALSD', -INFO ) RETURN END IF * EPS = DLAMCH( 'Epsilon' ) * * Set up the tolerance. * IF( ( RCOND.LE.ZERO ) .OR. ( RCOND.GE.ONE ) ) THEN RCND = EPS ELSE RCND = RCOND END IF * RANK = 0 * * Quick return if possible. * IF( N.EQ.0 ) THEN RETURN ELSE IF( N.EQ.1 ) THEN IF( D( 1 ).EQ.ZERO ) THEN CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, B, LDB ) ELSE RANK = 1 CALL DLASCL( 'G', 0, 0, D( 1 ), ONE, 1, NRHS, B, LDB, INFO ) D( 1 ) = ABS( D( 1 ) ) END IF RETURN END IF * * Rotate the matrix if it is lower bidiagonal. * IF( UPLO.EQ.'L' ) THEN 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( NRHS.EQ.1 ) THEN CALL DROT( 1, B( I, 1 ), 1, B( I+1, 1 ), 1, CS, SN ) ELSE WORK( I*2-1 ) = CS WORK( I*2 ) = SN END IF 10 CONTINUE IF( NRHS.GT.1 ) THEN DO 30 I = 1, NRHS DO 20 J = 1, N - 1 CS = WORK( J*2-1 ) SN = WORK( J*2 ) CALL DROT( 1, B( J, I ), 1, B( J+1, I ), 1, CS, SN ) 20 CONTINUE 30 CONTINUE END IF END IF * * Scale. * NM1 = N - 1 ORGNRM = DLANST( 'M', N, D, E ) IF( ORGNRM.EQ.ZERO ) THEN CALL DLASET( 'A', N, NRHS, ZERO, ZERO, B, LDB ) RETURN END IF * CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO ) CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, NM1, 1, E, NM1, INFO ) * * If N is smaller than the minimum divide size SMLSIZ, then solve * the problem with another solver. * IF( N.LE.SMLSIZ ) THEN NWORK = 1 + N*N CALL DLASET( 'A', N, N, ZERO, ONE, WORK, N ) CALL DLASDQ( 'U', 0, N, N, 0, NRHS, D, E, WORK, N, WORK, N, B, \$ LDB, WORK( NWORK ), INFO ) IF( INFO.NE.0 ) THEN RETURN END IF TOL = RCND*ABS( D( IDAMAX( N, D, 1 ) ) ) DO 40 I = 1, N IF( D( I ).LE.TOL ) THEN CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB ) ELSE CALL DLASCL( 'G', 0, 0, D( I ), ONE, 1, NRHS, B( I, 1 ), \$ LDB, INFO ) RANK = RANK + 1 END IF 40 CONTINUE CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB, ZERO, \$ WORK( NWORK ), N ) CALL DLACPY( 'A', N, NRHS, WORK( NWORK ), N, B, LDB ) * * Unscale. * CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO ) CALL DLASRT( 'D', N, D, INFO ) CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, NRHS, B, LDB, INFO ) * RETURN END IF * * Book-keeping and setting up some constants. * NLVL = INT( LOG( DBLE( N ) / DBLE( SMLSIZ+1 ) ) / LOG( TWO ) ) + 1 * SMLSZP = SMLSIZ + 1 * U = 1 VT = 1 + SMLSIZ*N DIFL = VT + SMLSZP*N DIFR = DIFL + NLVL*N Z = DIFR + NLVL*N*2 C = Z + NLVL*N S = C + N POLES = S + N GIVNUM = POLES + 2*NLVL*N BX = GIVNUM + 2*NLVL*N NWORK = BX + N*NRHS * SIZEI = 1 + N K = SIZEI + N GIVPTR = K + N PERM = GIVPTR + N GIVCOL = PERM + NLVL*N IWK = GIVCOL + NLVL*N*2 * ST = 1 SQRE = 0 ICMPQ1 = 1 ICMPQ2 = 0 NSUB = 0 * DO 50 I = 1, N IF( ABS( D( I ) ).LT.EPS ) THEN D( I ) = SIGN( EPS, D( I ) ) END IF 50 CONTINUE * DO 60 I = 1, NM1 IF( ( ABS( E( I ) ).LT.EPS ) .OR. ( I.EQ.NM1 ) ) THEN NSUB = NSUB + 1 IWORK( NSUB ) = ST * * 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 - ST + 1 IWORK( SIZEI+NSUB-1 ) = NSIZE ELSE IF( ABS( E( I ) ).GE.EPS ) THEN * * A subproblem with E(NM1) not too small but I = NM1. * NSIZE = N - ST + 1 IWORK( SIZEI+NSUB-1 ) = NSIZE ELSE * * A subproblem with E(NM1) small. This implies an * 1-by-1 subproblem at D(N), which is not solved * explicitly. * NSIZE = I - ST + 1 IWORK( SIZEI+NSUB-1 ) = NSIZE NSUB = NSUB + 1 IWORK( NSUB ) = N IWORK( SIZEI+NSUB-1 ) = 1 CALL DCOPY( NRHS, B( N, 1 ), LDB, WORK( BX+NM1 ), N ) END IF ST1 = ST - 1 IF( NSIZE.EQ.1 ) THEN * * This is a 1-by-1 subproblem and is not solved * explicitly. * CALL DCOPY( NRHS, B( ST, 1 ), LDB, WORK( BX+ST1 ), N ) ELSE IF( NSIZE.LE.SMLSIZ ) THEN * * This is a small subproblem and is solved by DLASDQ. * CALL DLASET( 'A', NSIZE, NSIZE, ZERO, ONE, \$ WORK( VT+ST1 ), N ) CALL DLASDQ( 'U', 0, NSIZE, NSIZE, 0, NRHS, D( ST ), \$ E( ST ), WORK( VT+ST1 ), N, WORK( NWORK ), \$ N, B( ST, 1 ), LDB, WORK( NWORK ), INFO ) IF( INFO.NE.0 ) THEN RETURN END IF CALL DLACPY( 'A', NSIZE, NRHS, B( ST, 1 ), LDB, \$ WORK( BX+ST1 ), N ) ELSE * * A large problem. Solve it using divide and conquer. * CALL DLASDA( ICMPQ1, SMLSIZ, NSIZE, SQRE, D( ST ), \$ E( ST ), WORK( U+ST1 ), N, WORK( VT+ST1 ), \$ IWORK( K+ST1 ), WORK( DIFL+ST1 ), \$ WORK( DIFR+ST1 ), WORK( Z+ST1 ), \$ WORK( POLES+ST1 ), IWORK( GIVPTR+ST1 ), \$ IWORK( GIVCOL+ST1 ), N, IWORK( PERM+ST1 ), \$ WORK( GIVNUM+ST1 ), WORK( C+ST1 ), \$ WORK( S+ST1 ), WORK( NWORK ), IWORK( IWK ), \$ INFO ) IF( INFO.NE.0 ) THEN RETURN END IF BXST = BX + ST1 CALL DLALSA( ICMPQ2, SMLSIZ, NSIZE, NRHS, B( ST, 1 ), \$ LDB, WORK( BXST ), N, WORK( U+ST1 ), N, \$ WORK( VT+ST1 ), IWORK( K+ST1 ), \$ WORK( DIFL+ST1 ), WORK( DIFR+ST1 ), \$ WORK( Z+ST1 ), WORK( POLES+ST1 ), \$ IWORK( GIVPTR+ST1 ), IWORK( GIVCOL+ST1 ), N, \$ IWORK( PERM+ST1 ), WORK( GIVNUM+ST1 ), \$ WORK( C+ST1 ), WORK( S+ST1 ), WORK( NWORK ), \$ IWORK( IWK ), INFO ) IF( INFO.NE.0 ) THEN RETURN END IF END IF ST = I + 1 END IF 60 CONTINUE * * Apply the singular values and treat the tiny ones as zero. * TOL = RCND*ABS( D( IDAMAX( N, D, 1 ) ) ) * DO 70 I = 1, N * * Some of the elements in D can be negative because 1-by-1 * subproblems were not solved explicitly. * IF( ABS( D( I ) ).LE.TOL ) THEN CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, WORK( BX+I-1 ), N ) ELSE RANK = RANK + 1 CALL DLASCL( 'G', 0, 0, D( I ), ONE, 1, NRHS, \$ WORK( BX+I-1 ), N, INFO ) END IF D( I ) = ABS( D( I ) ) 70 CONTINUE * * Now apply back the right singular vectors. * ICMPQ2 = 1 DO 80 I = 1, NSUB ST = IWORK( I ) ST1 = ST - 1 NSIZE = IWORK( SIZEI+I-1 ) BXST = BX + ST1 IF( NSIZE.EQ.1 ) THEN CALL DCOPY( NRHS, WORK( BXST ), N, B( ST, 1 ), LDB ) ELSE IF( NSIZE.LE.SMLSIZ ) THEN CALL DGEMM( 'T', 'N', NSIZE, NRHS, NSIZE, ONE, \$ WORK( VT+ST1 ), N, WORK( BXST ), N, ZERO, \$ B( ST, 1 ), LDB ) ELSE CALL DLALSA( ICMPQ2, SMLSIZ, NSIZE, NRHS, WORK( BXST ), N, \$ B( ST, 1 ), LDB, WORK( U+ST1 ), N, \$ WORK( VT+ST1 ), IWORK( K+ST1 ), \$ WORK( DIFL+ST1 ), WORK( DIFR+ST1 ), \$ WORK( Z+ST1 ), WORK( POLES+ST1 ), \$ IWORK( GIVPTR+ST1 ), IWORK( GIVCOL+ST1 ), N, \$ IWORK( PERM+ST1 ), WORK( GIVNUM+ST1 ), \$ WORK( C+ST1 ), WORK( S+ST1 ), WORK( NWORK ), \$ IWORK( IWK ), INFO ) IF( INFO.NE.0 ) THEN RETURN END IF END IF 80 CONTINUE * * Unscale and sort the singular values. * CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO ) CALL DLASRT( 'D', N, D, INFO ) CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, NRHS, B, LDB, INFO ) * RETURN * * End of DLALSD * END