*> \brief \b DGEQP3
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
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*
* Definition:
* ===========
*
* SUBROUTINE DGEQP3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* INTEGER JPVT( * )
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEQP3 computes a QR factorization with column pivoting of a
*> matrix A: A*P = Q*R using Level 3 BLAS.
*> \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,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the upper triangle of the array contains the
*> min(M,N)-by-N upper trapezoidal matrix R; the elements below
*> the diagonal, together with the array TAU, represent the
*> orthogonal matrix Q as a product of min(M,N) elementary
*> reflectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*> JPVT is INTEGER array, dimension (N)
*> On entry, if JPVT(J).ne.0, the J-th column of A is permuted
*> to the front of A*P (a leading column); if JPVT(J)=0,
*> the J-th column of A is a free column.
*> On exit, if JPVT(J)=K, then the J-th column of A*P was the
*> the K-th column of A.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors.
*> \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 >= 3*N+1.
*> For optimal performance LWORK >= 2*N+( N+1 )*NB, where NB
*> is the optimal blocksize.
*>
*> 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 2015
*
*> \ingroup doubleGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real/complex vector
*> with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
*> A(i+1:m,i), and tau in TAU(i).
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
*> X. Sun, Computer Science Dept., Duke University, USA
*>
* =====================================================================
SUBROUTINE DGEQP3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2015
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER INB, INBMIN, IXOVER
PARAMETER ( INB = 1, INBMIN = 2, IXOVER = 3 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER FJB, IWS, J, JB, LWKOPT, MINMN, MINWS, NA, NB,
$ NBMIN, NFXD, NX, SM, SMINMN, SN, TOPBMN
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DLAQP2, DLAQPS, DORMQR, DSWAP, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
DOUBLE PRECISION DNRM2
EXTERNAL ILAENV, DNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test 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( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
*
IF( INFO.EQ.0 ) THEN
MINMN = MIN( M, N )
IF( MINMN.EQ.0 ) THEN
IWS = 1
LWKOPT = 1
ELSE
IWS = 3*N + 1
NB = ILAENV( INB, 'DGEQRF', ' ', M, N, -1, -1 )
LWKOPT = 2*N + ( N + 1 )*NB
END IF
WORK( 1 ) = LWKOPT
*
IF( ( LWORK.LT.IWS ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQP3', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Move initial columns up front.
*
NFXD = 1
DO 10 J = 1, N
IF( JPVT( J ).NE.0 ) THEN
IF( J.NE.NFXD ) THEN
CALL DSWAP( M, A( 1, J ), 1, A( 1, NFXD ), 1 )
JPVT( J ) = JPVT( NFXD )
JPVT( NFXD ) = J
ELSE
JPVT( J ) = J
END IF
NFXD = NFXD + 1
ELSE
JPVT( J ) = J
END IF
10 CONTINUE
NFXD = NFXD - 1
*
* Factorize fixed columns
* =======================
*
* Compute the QR factorization of fixed columns and update
* remaining columns.
*
IF( NFXD.GT.0 ) THEN
NA = MIN( M, NFXD )
*CC CALL DGEQR2( M, NA, A, LDA, TAU, WORK, INFO )
CALL DGEQRF( M, NA, A, LDA, TAU, WORK, LWORK, INFO )
IWS = MAX( IWS, INT( WORK( 1 ) ) )
IF( NA.LT.N ) THEN
*CC CALL DORM2R( 'Left', 'Transpose', M, N-NA, NA, A, LDA,
*CC $ TAU, A( 1, NA+1 ), LDA, WORK, INFO )
CALL DORMQR( 'Left', 'Transpose', M, N-NA, NA, A, LDA, TAU,
$ A( 1, NA+1 ), LDA, WORK, LWORK, INFO )
IWS = MAX( IWS, INT( WORK( 1 ) ) )
END IF
END IF
*
* Factorize free columns
* ======================
*
IF( NFXD.LT.MINMN ) THEN
*
SM = M - NFXD
SN = N - NFXD
SMINMN = MINMN - NFXD
*
* Determine the block size.
*
NB = ILAENV( INB, 'DGEQRF', ' ', SM, SN, -1, -1 )
NBMIN = 2
NX = 0
*
IF( ( NB.GT.1 ) .AND. ( NB.LT.SMINMN ) ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( IXOVER, 'DGEQRF', ' ', SM, SN, -1,
$ -1 ) )
*
*
IF( NX.LT.SMINMN ) THEN
*
* Determine if workspace is large enough for blocked code.
*
MINWS = 2*SN + ( SN+1 )*NB
IWS = MAX( IWS, MINWS )
IF( LWORK.LT.MINWS ) THEN
*
* Not enough workspace to use optimal NB: Reduce NB and
* determine the minimum value of NB.
*
NB = ( LWORK-2*SN ) / ( SN+1 )
NBMIN = MAX( 2, ILAENV( INBMIN, 'DGEQRF', ' ', SM, SN,
$ -1, -1 ) )
*
*
END IF
END IF
END IF
*
* Initialize partial column norms. The first N elements of work
* store the exact column norms.
*
DO 20 J = NFXD + 1, N
WORK( J ) = DNRM2( SM, A( NFXD+1, J ), 1 )
WORK( N+J ) = WORK( J )
20 CONTINUE
*
IF( ( NB.GE.NBMIN ) .AND. ( NB.LT.SMINMN ) .AND.
$ ( NX.LT.SMINMN ) ) THEN
*
* Use blocked code initially.
*
J = NFXD + 1
*
* Compute factorization: while loop.
*
*
TOPBMN = MINMN - NX
30 CONTINUE
IF( J.LE.TOPBMN ) THEN
JB = MIN( NB, TOPBMN-J+1 )
*
* Factorize JB columns among columns J:N.
*
CALL DLAQPS( M, N-J+1, J-1, JB, FJB, A( 1, J ), LDA,
$ JPVT( J ), TAU( J ), WORK( J ), WORK( N+J ),
$ WORK( 2*N+1 ), WORK( 2*N+JB+1 ), N-J+1 )
*
J = J + FJB
GO TO 30
END IF
ELSE
J = NFXD + 1
END IF
*
* Use unblocked code to factor the last or only block.
*
*
IF( J.LE.MINMN )
$ CALL DLAQP2( M, N-J+1, J-1, A( 1, J ), LDA, JPVT( J ),
$ TAU( J ), WORK( J ), WORK( N+J ),
$ WORK( 2*N+1 ) )
*
END IF
*
WORK( 1 ) = IWS
RETURN
*
* End of DGEQP3
*
END