SUBROUTINE DGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
$ WORK, LWORK, INFO )
*
* -- LAPACK driver routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DGELSY 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' [ inv(T11)*Q1'*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.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of
* columns of matrices B and X. NRHS >= 0.
*
* A (input/output) DOUBLE PRECISION 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.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
* On entry, the M-by-NRHS right hand side matrix B.
* On exit, the N-by-NRHS solution matrix X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,M,N).
*
* JPVT (input/output) 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.
*
* RCOND (input) DOUBLE PRECISION
* 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.
*
* RANK (output) 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.
*
* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) 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 DGEQP3, DTZRZF, STZRQF, DORMQR,
* and DORMRZ.
*
* 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.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: If INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* Based on contributions by
* A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
* E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
* G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
*
* =====================================================================
*
* .. Parameters ..
INTEGER IMAX, IMIN
PARAMETER ( IMAX = 1, IMIN = 2 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, LWKMIN,
$ LWKOPT, MN, NB, NB1, NB2, NB3, NB4
DOUBLE PRECISION ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX,
$ SMAXPR, SMIN, SMINPR, SMLNUM, WSIZE
* ..
* .. External Functions ..
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL ILAENV, DLAMCH, DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEQP3, DLABAD, DLAIC1, DLASCL, DLASET,
$ DORMQR, DORMRZ, DTRSM, DTZRZF, 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, 'DGEQRF', ' ', M, N, -1, -1 )
NB2 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
NB3 = ILAENV( 1, 'DORMQR', ' ', M, N, NRHS, -1 )
NB4 = ILAENV( 1, 'DORMRQ', ' ', 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( 'DGELSY', -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 = DLAMCH( 'S' ) / DLAMCH( 'P' )
BIGNUM = ONE / SMLNUM
CALL DLABAD( SMLNUM, BIGNUM )
*
* Scale A, B if max entries 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 )
RANK = 0
GO TO 70
END IF
*
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
*
* Compute QR factorization with column pivoting of A:
* A * P = Q * R
*
CALL DGEQP3( 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 DLASET( '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 DLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
$ A( I, I ), SMINPR, S1, C1 )
CALL DLAIC1( 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 DTZRZF( 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' * B(1:M,1:NRHS)
*
CALL DORMQR( '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 DTRSM( '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' * B(1:N,1:NRHS)
*
IF( RANK.LT.N ) THEN
CALL DORMRZ( '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 DCOPY( N, WORK( 1 ), 1, B( 1, J ), 1 )
60 CONTINUE
*
* workspace: N.
*
* Undo scaling
*
IF( IASCL.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
CALL DLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
$ INFO )
ELSE IF( IASCL.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
CALL DLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
$ 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
*
70 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DGELSY
*
END