SUBROUTINE SGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND,
$ RANK, WORK, LWORK, IWORK, INFO )
*
* -- LAPACK driver routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* June 30, 1999
* 8-15-00: Improve consistency of WS calculations (eca)
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
REAL RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
REAL A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* SGELSD 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 tranformations 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.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of A. M >= 0.
*
* N (input) INTEGER
* The number of columns of A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrices B and X. NRHS >= 0.
*
* A (input/output) REAL array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, the first min(m,n) rows of A are overwritten with
* its right singular vectors, stored rowwise.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) REAL 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.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,max(M,N)).
*
* S (output) REAL 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)).
*
* RCOND (input) REAL
* 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.
*
* RANK (output) INTEGER
* The effective rank of A, i.e., the number of singular values
* which are greater than RCOND*S(1).
*
* WORK (workspace/output) REAL array, dimension (LWORK)
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= 1.
* The exact minimum amount of workspace needed depends on M,
* N and NRHS.
* If M >= N, LWORK >= 11*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS.
* If M < N, LWORK >= 11*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS.
* 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 = INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1
* For good performance, LWORK should generally be larger.
*
* If LWORK = -1, a workspace query is assumed. The optimal
* size for the WORK array is calculated and stored in WORK(1),
* and no other work except argument checking is performed.
*
* IWORK (workspace) INTEGER array, dimension (LIWORK)
* LIWORK >= 3 * MINMN * NLVL + 11 * MINMN,
* where MINMN = MIN( M,N ).
*
* INFO (output) 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.
*
* 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 ..
INTEGER LQUERV
PARAMETER ( LQUERV = -1 )
REAL ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0 )
* ..
* .. Local Scalars ..
INTEGER IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
$ LDWORK, MAXMN, MAXWRK, MINMN, MINWRK, MM,
$ MNTHR, NLVL, NWORK, SMLSIZ
REAL ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
* ..
* .. External Subroutines ..
EXTERNAL SGEBRD, SGELQF, SGEQRF, SLABAD, SLACPY, SLALSD,
$ SLASCL, SLASET, SORMBR, SORMLQ, SORMQR, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
REAL SLAMCH, SLANGE
EXTERNAL SLAMCH, SLANGE, ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC REAL, INT, LOG, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments.
*
INFO = 0
MINMN = MIN( M, N )
MAXMN = MAX( M, N )
MNTHR = ILAENV( 6, 'SGELSD', ' ', M, N, NRHS, -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, 'SGELSD', ' ', 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
MINMN = MAX( 1, MINMN )
NLVL = INT( LOG( REAL( MINMN ) / REAL( SMLSIZ+1 ) ) /
$ LOG( TWO ) ) + 1
*
IF( INFO.EQ.0 ) THEN
MAXWRK = 0
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, 'SGEQRF', ' ', M, N,
$ -1, -1 ) )
MAXWRK = MAX( MAXWRK, N+NRHS*
$ ILAENV( 1, 'SORMQR', '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, 'SGEBRD', ' ', MM, N, -1, -1 ) )
MAXWRK = MAX( MAXWRK, 3*N+NRHS*
$ ILAENV( 1, 'SORMBR', 'QLT', MM, NRHS, N, -1 ) )
MAXWRK = MAX( MAXWRK, 3*N+( N-1 )*
$ ILAENV( 1, 'SORMBR', 'PLN', N, NRHS, N, -1 ) )
MAXWRK = MAX( MAXWRK, 3*N+8*N+2*N*SMLSIZ+8*N*NLVL+N*NRHS )
MINWRK = MAX( 3*N+MM, 3*N+NRHS,
$ 3*N+8*N+2*N*SMLSIZ+8*N*NLVL+N*NRHS )
END IF
IF( N.GT.M ) THEN
IF( N.GE.MNTHR ) THEN
*
* Path 2a - underdetermined, with many more columns
* than rows.
*
MAXWRK = M + M*ILAENV( 1, 'SGELQF', ' ', M, N, -1, -1 )
MAXWRK = MAX( MAXWRK, M*M+4*M+2*M*
$ ILAENV( 1, 'SGEBRD', ' ', M, M, -1, -1 ) )
MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS*
$ ILAENV( 1, 'SORMBR', 'QLT', M, NRHS, M, -1 ) )
MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )*
$ ILAENV( 1, 'SORMBR', '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, 'SORMLQ', 'LT', N, NRHS, M, -1 ) )
MAXWRK = MAX( MAXWRK, M*M+4*M+8*M+2*M*SMLSIZ+8*M*NLVL+M*
$ NRHS )
ELSE
*
* Path 2 - remaining underdetermined cases.
*
MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'SGEBRD', ' ', M, N,
$ -1, -1 )
MAXWRK = MAX( MAXWRK, 3*M+NRHS*
$ ILAENV( 1, 'SORMBR', 'QLT', M, NRHS, N, -1 ) )
MAXWRK = MAX( MAXWRK, 3*M+M*
$ ILAENV( 1, 'SORMBR', 'PLN', N, NRHS, M, -1 ) )
MAXWRK = MAX( MAXWRK, 3*M+8*M+2*M*SMLSIZ+8*M*NLVL+M*
$ NRHS )
END IF
MINWRK = MAX( 3*M+NRHS, 3*M+M,
$ 3*M+8*M+2*M*SMLSIZ+8*M*NLVL+M*NRHS )
END IF
MINWRK = MIN( MINWRK, MAXWRK )
WORK( 1 ) = MAXWRK
IF( LWORK.LT.MINWRK .AND. LWORK.NE.LQUERV )
$ INFO = -12
END IF
*
* Quick returns
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGELSD', -INFO )
RETURN
END IF
IF( LWORK.EQ.LQUERV ) RETURN
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RANK = 0
RETURN
END IF
*
* Get machine parameters.
*
EPS = SLAMCH( 'P' )
SFMIN = SLAMCH( 'S' )
SMLNUM = SFMIN / EPS
BIGNUM = ONE / SMLNUM
CALL SLABAD( SMLNUM, BIGNUM )
*
* Scale A if max entry outside range [SMLNUM,BIGNUM].
*
ANRM = SLANGE( 'M', M, N, A, LDA, WORK )
IASCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM.
*
CALL SLASCL( '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 SLASCL( '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 SLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
CALL SLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )
RANK = 0
GO TO 10
END IF
*
* Scale B if max entry outside range [SMLNUM,BIGNUM].
*
BNRM = SLANGE( 'M', M, NRHS, B, LDB, WORK )
IBSCL = 0
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM.
*
CALL SLASCL( '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 SLASCL( '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 SLASET( '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 SGEQRF( 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 SORMQR( '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 SLASET( '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 SGEBRD( 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 SORMBR( '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 SLALSD( '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 SORMBR( '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 ) ) 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 ) )LDWORK = LDA
ITAU = 1
NWORK = M + 1
*
* Compute A=L*Q.
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL SGELQF( 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 SLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
CALL SLASET( '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 SGEBRD( 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 SORMBR( '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 SLALSD( '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 SORMBR( '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 SLASET( '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 SORMLQ( '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 SGEBRD( 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 SORMBR( '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 SLALSD( '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 SORMBR( '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 SLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
CALL SLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
$ INFO )
ELSE IF( IASCL.EQ.2 ) THEN
CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
CALL SLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
$ INFO )
END IF
IF( IBSCL.EQ.1 ) THEN
CALL SLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
ELSE IF( IBSCL.EQ.2 ) THEN
CALL SLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
END IF
*
10 CONTINUE
WORK( 1 ) = MAXWRK
RETURN
*
* End of SGELSD
*
END