*> \brief DGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DGELSD + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, * WORK, LWORK, IWORK, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK * DOUBLE PRECISION RCOND * .. * .. Array Arguments .. * INTEGER IWORK( * ) * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DGELSD computes the minimum-norm solution to a real linear least *> squares problem: *> minimize 2-norm(| b - A*x |) *> using the singular value decomposition (SVD) of A. A is an M-by-N *> matrix which may be rank-deficient. *> *> Several right hand side vectors b and solution vectors x can be *> handled in a single call; they are stored as the columns of the *> M-by-NRHS right hand side matrix B and the N-by-NRHS solution *> matrix X. *> *> The problem is solved in three steps: *> (1) Reduce the coefficient matrix A to bidiagonal form with *> Householder transformations, reducing the original problem *> into a "bidiagonal least squares problem" (BLS) *> (2) Solve the BLS using a divide and conquer approach. *> (3) Apply back all the Householder transformations to solve *> the original least squares problem. *> *> The effective rank of A is determined by treating as zero those *> singular values which are less than RCOND times the largest singular *> value. *> *> The divide and conquer algorithm 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 X-MP, Cray Y-MP, 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] M *> \verbatim *> M is INTEGER *> The number of rows of A. M >= 0. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of A. N >= 0. *> \endverbatim *> *> \param[in] NRHS *> \verbatim *> NRHS is INTEGER *> The number of right hand sides, i.e., the number of columns *> of the matrices B and X. NRHS >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is DOUBLE PRECISION array, dimension (LDA,N) *> On entry, the M-by-N matrix A. *> On exit, A has been destroyed. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,M). *> \endverbatim *> *> \param[in,out] B *> \verbatim *> B is DOUBLE PRECISION array, dimension (LDB,NRHS) *> On entry, the M-by-NRHS right hand side matrix B. *> On exit, B is overwritten by the N-by-NRHS solution *> matrix X. If m >= n and RANK = n, the residual *> sum-of-squares for the solution in the i-th column is given *> by the sum of squares of elements n+1:m in that column. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= max(1,max(M,N)). *> \endverbatim *> *> \param[out] S *> \verbatim *> S is DOUBLE PRECISION array, dimension (min(M,N)) *> The singular values of A in decreasing order. *> The condition number of A in the 2-norm = S(1)/S(min(m,n)). *> \endverbatim *> *> \param[in] RCOND *> \verbatim *> RCOND is DOUBLE PRECISION *> RCOND is used to determine the effective rank of A. *> Singular values S(i) <= RCOND*S(1) are treated as zero. *> If RCOND < 0, machine precision is used instead. *> \endverbatim *> *> \param[out] RANK *> \verbatim *> RANK is INTEGER *> The effective rank of A, i.e., the number of singular values *> which are greater than RCOND*S(1). *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK. LWORK must be at least 1. *> The exact minimum amount of workspace needed depends on M, *> N and NRHS. As long as LWORK is at least *> 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2, *> if M is greater than or equal to N or *> 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, *> if M is less than N, the code will execute correctly. *> 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), and *> NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) *> For good performance, LWORK should generally be larger. *> *> If LWORK = -1, then a workspace query is assumed; the routine *> only calculates the optimal size of the WORK array, returns *> this value as the first entry of the WORK array, and no error *> message related to LWORK is issued by XERBLA. *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) *> LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN), *> where MINMN = MIN( M,N ). *> On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK. *> \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 for computing the SVD failed to converge; *> if INFO = i, i off-diagonal elements of an intermediate *> bidiagonal form did not converge to zero. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date June 2017 * *> \ingroup doubleGEsolve * *> \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 DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, \$ WORK, LWORK, IWORK, INFO ) * * -- LAPACK driver routine (version 3.7.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * June 2017 * * .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK DOUBLE PRECISION RCOND * .. * .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 ) * .. * .. Local Scalars .. LOGICAL LQUERY INTEGER IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ, \$ LDWORK, LIWORK, MAXMN, MAXWRK, MINMN, MINWRK, \$ MM, MNTHR, NLVL, NWORK, SMLSIZ, WLALSD DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM * .. * .. External Subroutines .. EXTERNAL DGEBRD, DGELQF, DGEQRF, DLABAD, DLACPY, DLALSD, \$ DLASCL, DLASET, DORMBR, DORMLQ, DORMQR, XERBLA * .. * .. External Functions .. INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL ILAENV, DLAMCH, DLANGE * .. * .. Intrinsic Functions .. INTRINSIC DBLE, INT, LOG, MAX, MIN * .. * .. Executable Statements .. * * Test the input arguments. * INFO = 0 MINMN = MIN( M, N ) MAXMN = MAX( M, N ) MNTHR = ILAENV( 6, 'DGELSD', ' ', M, N, NRHS, -1 ) LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN INFO = -7 END IF * SMLSIZ = ILAENV( 9, 'DGELSD', ' ', 0, 0, 0, 0 ) * * Compute workspace. * (Note: Comments in the code beginning "Workspace:" describe the * minimal amount of workspace needed at that point in the code, * as well as the preferred amount for good performance. * NB refers to the optimal block size for the immediately * following subroutine, as returned by ILAENV.) * MINWRK = 1 LIWORK = 1 MINMN = MAX( 1, MINMN ) NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ+1 ) ) / \$ LOG( TWO ) ) + 1, 0 ) * IF( INFO.EQ.0 ) THEN MAXWRK = 0 LIWORK = 3*MINMN*NLVL + 11*MINMN MM = M IF( M.GE.N .AND. M.GE.MNTHR ) THEN * * Path 1a - overdetermined, with many more rows than columns. * MM = N MAXWRK = MAX( MAXWRK, N+N*ILAENV( 1, 'DGEQRF', ' ', M, N, \$ -1, -1 ) ) MAXWRK = MAX( MAXWRK, N+NRHS* \$ ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, -1 ) ) END IF IF( M.GE.N ) THEN * * Path 1 - overdetermined or exactly determined. * MAXWRK = MAX( MAXWRK, 3*N+( MM+N )* \$ ILAENV( 1, 'DGEBRD', ' ', MM, N, -1, -1 ) ) MAXWRK = MAX( MAXWRK, 3*N+NRHS* \$ ILAENV( 1, 'DORMBR', 'QLT', MM, NRHS, N, -1 ) ) MAXWRK = MAX( MAXWRK, 3*N+( N-1 )* \$ ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, N, -1 ) ) WLALSD = 9*N+2*N*SMLSIZ+8*N*NLVL+N*NRHS+(SMLSIZ+1)**2 MAXWRK = MAX( MAXWRK, 3*N+WLALSD ) MINWRK = MAX( 3*N+MM, 3*N+NRHS, 3*N+WLALSD ) END IF IF( N.GT.M ) THEN WLALSD = 9*M+2*M*SMLSIZ+8*M*NLVL+M*NRHS+(SMLSIZ+1)**2 IF( N.GE.MNTHR ) THEN * * Path 2a - underdetermined, with many more columns * than rows. * MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) MAXWRK = MAX( MAXWRK, M*M+4*M+2*M* \$ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS* \$ ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )* \$ ILAENV( 1, 'DORMBR', 'PLN', M, NRHS, M, -1 ) ) IF( NRHS.GT.1 ) THEN MAXWRK = MAX( MAXWRK, M*M+M+M*NRHS ) ELSE MAXWRK = MAX( MAXWRK, M*M+2*M ) END IF MAXWRK = MAX( MAXWRK, M+NRHS* \$ ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+WLALSD ) ! XXX: Ensure the Path 2a case below is triggered. The workspace ! calculation should use queries for all routines eventually. MAXWRK = MAX( MAXWRK, \$ 4*M+M*M+MAX( M, 2*M-4, NRHS, N-3*M ) ) ELSE * * Path 2 - remaining underdetermined cases. * MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'DGEBRD', ' ', M, N, \$ -1, -1 ) MAXWRK = MAX( MAXWRK, 3*M+NRHS* \$ ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, N, -1 ) ) MAXWRK = MAX( MAXWRK, 3*M+M* \$ ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, 3*M+WLALSD ) END IF MINWRK = MAX( 3*M+NRHS, 3*M+M, 3*M+WLALSD ) END IF MINWRK = MIN( MINWRK, MAXWRK ) WORK( 1 ) = MAXWRK IWORK( 1 ) = LIWORK IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -12 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGELSD', -INFO ) RETURN ELSE IF( LQUERY ) THEN GO TO 10 END IF * * Quick return if possible. * IF( M.EQ.0 .OR. N.EQ.0 ) THEN RANK = 0 RETURN END IF * * Get machine parameters. * EPS = DLAMCH( 'P' ) SFMIN = DLAMCH( 'S' ) SMLNUM = SFMIN / EPS BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) * * Scale A if max entry outside range [SMLNUM,BIGNUM]. * ANRM = DLANGE( 'M', M, N, A, LDA, WORK ) IASCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN * * Scale matrix norm up to SMLNUM. * CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO ) IASCL = 1 ELSE IF( ANRM.GT.BIGNUM ) THEN * * Scale matrix norm down to BIGNUM. * CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO ) IASCL = 2 ELSE IF( ANRM.EQ.ZERO ) THEN * * Matrix all zero. Return zero solution. * CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 ) RANK = 0 GO TO 10 END IF * * Scale B if max entry outside range [SMLNUM,BIGNUM]. * BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK ) IBSCL = 0 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN * * Scale matrix norm up to SMLNUM. * CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO ) IBSCL = 1 ELSE IF( BNRM.GT.BIGNUM ) THEN * * Scale matrix norm down to BIGNUM. * CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO ) IBSCL = 2 END IF * * If M < N make sure certain entries of B are zero. * IF( M.LT.N ) \$ CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB ) * * Overdetermined case. * IF( M.GE.N ) THEN * * Path 1 - overdetermined or exactly determined. * MM = M IF( M.GE.MNTHR ) THEN * * Path 1a - overdetermined, with many more rows than columns. * MM = N ITAU = 1 NWORK = ITAU + N * * Compute A=Q*R. * (Workspace: need 2*N, prefer N+N*NB) * CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), \$ LWORK-NWORK+1, INFO ) * * Multiply B by transpose(Q). * (Workspace: need N+NRHS, prefer N+NRHS*NB) * CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ), B, \$ LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) * * Zero out below R. * IF( N.GT.1 ) THEN CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA ) END IF END IF * IE = 1 ITAUQ = IE + N ITAUP = ITAUQ + N NWORK = ITAUP + N * * Bidiagonalize R in A. * (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB) * CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), \$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, \$ INFO ) * * Multiply B by transpose of left bidiagonalizing vectors of R. * (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB) * CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ), \$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) * * Solve the bidiagonal least squares problem. * CALL DLALSD( 'U', SMLSIZ, N, NRHS, S, WORK( IE ), B, LDB, \$ RCOND, RANK, WORK( NWORK ), IWORK, INFO ) IF( INFO.NE.0 ) THEN GO TO 10 END IF * * Multiply B by right bidiagonalizing vectors of R. * CALL DORMBR( 'P', 'L', 'N', N, NRHS, N, A, LDA, WORK( ITAUP ), \$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) * ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+ \$ MAX( M, 2*M-4, NRHS, N-3*M, WLALSD ) ) THEN * * Path 2a - underdetermined, with many more columns than rows * and sufficient workspace for an efficient algorithm. * LDWORK = M IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ), \$ M*LDA+M+M*NRHS, 4*M+M*LDA+WLALSD ) )LDWORK = LDA ITAU = 1 NWORK = M + 1 * * Compute A=L*Q. * (Workspace: need 2*M, prefer M+M*NB) * CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), \$ LWORK-NWORK+1, INFO ) IL = NWORK * * Copy L to WORK(IL), zeroing out above its diagonal. * CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ), \$ LDWORK ) IE = IL + LDWORK*M ITAUQ = IE + M ITAUP = ITAUQ + M NWORK = ITAUP + M * * Bidiagonalize L in WORK(IL). * (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB) * CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ), \$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ), \$ LWORK-NWORK+1, INFO ) * * Multiply B by transpose of left bidiagonalizing vectors of L. * (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB) * CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK, \$ WORK( ITAUQ ), B, LDB, WORK( NWORK ), \$ LWORK-NWORK+1, INFO ) * * Solve the bidiagonal least squares problem. * CALL DLALSD( 'U', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB, \$ RCOND, RANK, WORK( NWORK ), IWORK, INFO ) IF( INFO.NE.0 ) THEN GO TO 10 END IF * * Multiply B by right bidiagonalizing vectors of L. * CALL DORMBR( 'P', 'L', 'N', M, NRHS, M, WORK( IL ), LDWORK, \$ WORK( ITAUP ), B, LDB, WORK( NWORK ), \$ LWORK-NWORK+1, INFO ) * * Zero out below first M rows of B. * CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB ) NWORK = ITAU + M * * Multiply transpose(Q) by B. * (Workspace: need M+NRHS, prefer M+NRHS*NB) * CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B, \$ LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) * ELSE * * Path 2 - remaining underdetermined cases. * IE = 1 ITAUQ = IE + M ITAUP = ITAUQ + M NWORK = ITAUP + M * * Bidiagonalize A. * (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB) * CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), \$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, \$ INFO ) * * Multiply B by transpose of left bidiagonalizing vectors. * (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB) * CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ), \$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) * * Solve the bidiagonal least squares problem. * CALL DLALSD( 'L', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB, \$ RCOND, RANK, WORK( NWORK ), IWORK, INFO ) IF( INFO.NE.0 ) THEN GO TO 10 END IF * * Multiply B by right bidiagonalizing vectors of A. * CALL DORMBR( 'P', 'L', 'N', N, NRHS, M, A, LDA, WORK( ITAUP ), \$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) * END IF * * Undo scaling. * IF( IASCL.EQ.1 ) THEN CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO ) CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN, \$ INFO ) ELSE IF( IASCL.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO ) CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN, \$ INFO ) END IF IF( IBSCL.EQ.1 ) THEN CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO ) ELSE IF( IBSCL.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO ) END IF * 10 CONTINUE WORK( 1 ) = MAXWRK IWORK( 1 ) = LIWORK RETURN * * End of DGELSD * END