*> \brief \b SGEQP3 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SGEQP3 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SGEQP3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, LWORK, M, N * .. * .. Array Arguments .. * INTEGER JPVT( * ) * REAL A( LDA, * ), TAU( * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SGEQP3 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 REAL 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 REAL array, dimension (min(M,N)) *> The scalar factors of the elementary reflectors. *> \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. 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. * *> \ingroup realGEcomputational * *> \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 SGEQP3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO ) * * -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. INTEGER INFO, LDA, LWORK, M, N * .. * .. Array Arguments .. INTEGER JPVT( * ) REAL 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 SGEQRF, SLAQP2, SLAQPS, SORMQR, SSWAP, XERBLA * .. * .. External Functions .. INTEGER ILAENV REAL SNRM2 EXTERNAL ILAENV, SNRM2 * .. * .. Intrinsic Functions .. INTRINSIC INT, MAX, MIN * 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, 'SGEQRF', ' ', 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( 'SGEQP3', -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 SSWAP( 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 SGEQR2( M, NA, A, LDA, TAU, WORK, INFO ) CALL SGEQRF( M, NA, A, LDA, TAU, WORK, LWORK, INFO ) IWS = MAX( IWS, INT( WORK( 1 ) ) ) IF( NA.LT.N ) THEN *CC CALL SORM2R( 'Left', 'Transpose', M, N-NA, NA, A, LDA, *CC \$ TAU, A( 1, NA+1 ), LDA, WORK, INFO ) CALL SORMQR( '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, 'SGEQRF', ' ', 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, 'SGEQRF', ' ', 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, 'SGEQRF', ' ', 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 ) = SNRM2( 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 SLAQPS( 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 SLAQP2( 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 SGEQP3 * END