*> \brief SGELSY solves overdetermined or underdetermined systems for GE matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SGELSY + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, * WORK, LWORK, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK * REAL RCOND * .. * .. Array Arguments .. * INTEGER JPVT( * ) * REAL A( LDA, * ), B( LDB, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SGELSY computes the minimum-norm solution to a real linear least *> squares problem: *> minimize || A * X - B || *> using a complete orthogonal factorization 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 routine first computes a QR factorization with column pivoting: *> A * P = Q * [ R11 R12 ] *> [ 0 R22 ] *> with R11 defined as the largest leading submatrix whose estimated *> condition number is less than 1/RCOND. The order of R11, RANK, *> is the effective rank of A. *> *> Then, R22 is considered to be negligible, and R12 is annihilated *> by orthogonal transformations from the right, arriving at the *> complete orthogonal factorization: *> A * P = Q * [ T11 0 ] * Z *> [ 0 0 ] *> The minimum-norm solution is then *> X = P * Z**T [ inv(T11)*Q1**T*B ] *> [ 0 ] *> where Q1 consists of the first RANK columns of Q. *> *> This routine is basically identical to the original xGELSX except *> three differences: *> o The call to the subroutine xGEQPF has been substituted by the *> the call to the subroutine xGEQP3. This subroutine is a Blas-3 *> version of the QR factorization with column pivoting. *> o Matrix B (the right hand side) is updated with Blas-3. *> o The permutation of matrix B (the right hand side) is faster and *> more simple. *> \endverbatim * * Arguments: * ========== * *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows of the matrix A. M >= 0. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] NRHS *> \verbatim *> NRHS is INTEGER *> The number of right hand sides, i.e., the number of *> columns of matrices B and X. NRHS >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is REAL array, dimension (LDA,N) *> On entry, the M-by-N matrix A. *> On exit, A has been overwritten by details of its *> complete orthogonal factorization. *> \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 REAL array, dimension (LDB,NRHS) *> On entry, the M-by-NRHS right hand side matrix B. *> On exit, the N-by-NRHS solution matrix X. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= max(1,M,N). *> \endverbatim *> *> \param[in,out] JPVT *> \verbatim *> JPVT is INTEGER array, dimension (N) *> On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted *> to the front of AP, otherwise column i is a free column. *> On exit, if JPVT(i) = k, then the i-th column of AP *> was the k-th column of A. *> \endverbatim *> *> \param[in] RCOND *> \verbatim *> RCOND is REAL *> RCOND is used to determine the effective rank of A, which *> is defined as the order of the largest leading triangular *> submatrix R11 in the QR factorization with pivoting of A, *> whose estimated condition number < 1/RCOND. *> \endverbatim *> *> \param[out] RANK *> \verbatim *> RANK is INTEGER *> The effective rank of A, i.e., the order of the submatrix *> R11. This is the same as the order of the submatrix T11 *> in the complete orthogonal factorization of A. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL 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. *> The unblocked strategy requires that: *> LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ), *> where MN = min( M, N ). *> The block algorithm requires that: *> LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ), *> where NB is an upper bound on the blocksize returned *> by ILAENV for the routines SGEQP3, STZRZF, STZRQF, SORMQR, *> and SORMRZ. *> *> 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] 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 November 2011 * *> \ingroup realGEsolve * *> \par Contributors: * ================== *> *> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA \n *> E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n *> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n *> * ===================================================================== SUBROUTINE SGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, $ WORK, LWORK, INFO ) * * -- LAPACK driver routine (version 3.4.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2011 * * .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK REAL RCOND * .. * .. Array Arguments .. INTEGER JPVT( * ) REAL A( LDA, * ), B( LDB, * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. INTEGER IMAX, IMIN PARAMETER ( IMAX = 1, IMIN = 2 ) REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) * .. * .. Local Scalars .. LOGICAL LQUERY INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, LWKMIN, $ LWKOPT, MN, NB, NB1, NB2, NB3, NB4 REAL ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX, $ SMAXPR, SMIN, SMINPR, SMLNUM, WSIZE * .. * .. External Functions .. INTEGER ILAENV REAL SLAMCH, SLANGE EXTERNAL ILAENV, SLAMCH, SLANGE * .. * .. External Subroutines .. EXTERNAL SCOPY, SGEQP3, SLABAD, SLAIC1, SLASCL, SLASET, $ SORMQR, SORMRZ, STRSM, STZRZF, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN * .. * .. Executable Statements .. * MN = MIN( M, N ) ISMIN = MN + 1 ISMAX = 2*MN + 1 * * Test the input arguments. * INFO = 0 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, M, N ) ) THEN INFO = -7 END IF * * Figure out optimal block size * IF( INFO.EQ.0 ) THEN IF( MN.EQ.0 .OR. NRHS.EQ.0 ) THEN LWKMIN = 1 LWKOPT = 1 ELSE NB1 = ILAENV( 1, 'SGEQRF', ' ', M, N, -1, -1 ) NB2 = ILAENV( 1, 'SGERQF', ' ', M, N, -1, -1 ) NB3 = ILAENV( 1, 'SORMQR', ' ', M, N, NRHS, -1 ) NB4 = ILAENV( 1, 'SORMRQ', ' ', M, N, NRHS, -1 ) NB = MAX( NB1, NB2, NB3, NB4 ) LWKMIN = MN + MAX( 2*MN, N + 1, MN + NRHS ) LWKOPT = MAX( LWKMIN, $ MN + 2*N + NB*( N + 1 ), 2*MN + NB*NRHS ) END IF WORK( 1 ) = LWKOPT * IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN INFO = -12 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGELSY', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( MN.EQ.0 .OR. NRHS.EQ.0 ) THEN RANK = 0 RETURN END IF * * Get machine parameters * SMLNUM = SLAMCH( 'S' ) / SLAMCH( 'P' ) BIGNUM = ONE / SMLNUM CALL SLABAD( SMLNUM, BIGNUM ) * * Scale A, B if max entries 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 ) RANK = 0 GO TO 70 END IF * 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 * * Compute QR factorization with column pivoting of A: * A * P = Q * R * CALL SGEQP3( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ), $ LWORK-MN, INFO ) WSIZE = MN + WORK( MN+1 ) * * workspace: MN+2*N+NB*(N+1). * Details of Householder rotations stored in WORK(1:MN). * * Determine RANK using incremental condition estimation * WORK( ISMIN ) = ONE WORK( ISMAX ) = ONE SMAX = ABS( A( 1, 1 ) ) SMIN = SMAX IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN RANK = 0 CALL SLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) GO TO 70 ELSE RANK = 1 END IF * 10 CONTINUE IF( RANK.LT.MN ) THEN I = RANK + 1 CALL SLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ), $ A( I, I ), SMINPR, S1, C1 ) CALL SLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ), $ A( I, I ), SMAXPR, S2, C2 ) * IF( SMAXPR*RCOND.LE.SMINPR ) THEN DO 20 I = 1, RANK WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 ) WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 ) 20 CONTINUE WORK( ISMIN+RANK ) = C1 WORK( ISMAX+RANK ) = C2 SMIN = SMINPR SMAX = SMAXPR RANK = RANK + 1 GO TO 10 END IF END IF * * workspace: 3*MN. * * Logically partition R = [ R11 R12 ] * [ 0 R22 ] * where R11 = R(1:RANK,1:RANK) * * [R11,R12] = [ T11, 0 ] * Y * IF( RANK.LT.N ) $ CALL STZRZF( RANK, N, A, LDA, WORK( MN+1 ), WORK( 2*MN+1 ), $ LWORK-2*MN, INFO ) * * workspace: 2*MN. * Details of Householder rotations stored in WORK(MN+1:2*MN) * * B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS) * CALL SORMQR( 'Left', 'Transpose', M, NRHS, MN, A, LDA, WORK( 1 ), $ B, LDB, WORK( 2*MN+1 ), LWORK-2*MN, INFO ) WSIZE = MAX( WSIZE, 2*MN+WORK( 2*MN+1 ) ) * * workspace: 2*MN+NB*NRHS. * * B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS) * CALL STRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK, $ NRHS, ONE, A, LDA, B, LDB ) * DO 40 J = 1, NRHS DO 30 I = RANK + 1, N B( I, J ) = ZERO 30 CONTINUE 40 CONTINUE * * B(1:N,1:NRHS) := Y**T * B(1:N,1:NRHS) * IF( RANK.LT.N ) THEN CALL SORMRZ( 'Left', 'Transpose', N, NRHS, RANK, N-RANK, A, $ LDA, WORK( MN+1 ), B, LDB, WORK( 2*MN+1 ), $ LWORK-2*MN, INFO ) END IF * * workspace: 2*MN+NRHS. * * B(1:N,1:NRHS) := P * B(1:N,1:NRHS) * DO 60 J = 1, NRHS DO 50 I = 1, N WORK( JPVT( I ) ) = B( I, J ) 50 CONTINUE CALL SCOPY( N, WORK( 1 ), 1, B( 1, J ), 1 ) 60 CONTINUE * * workspace: N. * * Undo scaling * IF( IASCL.EQ.1 ) THEN CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO ) CALL SLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA, $ INFO ) ELSE IF( IASCL.EQ.2 ) THEN CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO ) CALL SLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA, $ 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 * 70 CONTINUE WORK( 1 ) = LWKOPT * RETURN * * End of SGELSY * END