#include "ESMF_LapackBlas.inc" !> \brief \b DLAQPS ! ! =========== DOCUMENTATION =========== ! ! Online html documentation available at ! http://www.netlib.org/lapack/explore-html/ ! !> \htmlonly !> Download DLAQPS + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqps.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqps.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqps.f"> !> [TXT]</a> !> \endhtmlonly ! ! Definition: ! =========== ! ! SUBROUTINE DLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, ! VN2, AUXV, F, LDF ) ! ! .. Scalar Arguments .. ! INTEGER KB, LDA, LDF, M, N, NB, OFFSET ! .. ! .. Array Arguments .. ! INTEGER JPVT( * ) ! DOUBLE PRECISION A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ), ! $ VN1( * ), VN2( * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> DLAQPS computes a step of QR factorization with column pivoting !> of a real M-by-N matrix A by using Blas-3. It tries to factorize !> NB columns from A starting from the row OFFSET+1, and updates all !> of the matrix with Blas-3 xGEMM. !> !> In some cases, due to catastrophic cancellations, it cannot !> factorize NB columns. Hence, the actual number of factorized !> columns is returned in KB. !> !> Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized. !> \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] OFFSET !> \verbatim !> OFFSET is INTEGER !> The number of rows of A that have been factorized in !> previous steps. !> \endverbatim !> !> \param[in] NB !> \verbatim !> NB is INTEGER !> The number of columns to factorize. !> \endverbatim !> !> \param[out] KB !> \verbatim !> KB is INTEGER !> The number of columns actually factorized. !> \endverbatim !> !> \param[in,out] A !> \verbatim !> A is DOUBLE PRECISION array, dimension (LDA,N) !> On entry, the M-by-N matrix A. !> On exit, block A(OFFSET+1:M,1:KB) is the triangular !> factor obtained and block A(1:OFFSET,1:N) has been !> accordingly pivoted, but no factorized. !> The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has !> been updated. !> \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) !> JPVT(I) = K <==> Column K of the full matrix A has been !> permuted into position I in AP. !> \endverbatim !> !> \param[out] TAU !> \verbatim !> TAU is DOUBLE PRECISION array, dimension (KB) !> The scalar factors of the elementary reflectors. !> \endverbatim !> !> \param[in,out] VN1 !> \verbatim !> VN1 is DOUBLE PRECISION array, dimension (N) !> The vector with the partial column norms. !> \endverbatim !> !> \param[in,out] VN2 !> \verbatim !> VN2 is DOUBLE PRECISION array, dimension (N) !> The vector with the exact column norms. !> \endverbatim !> !> \param[in,out] AUXV !> \verbatim !> AUXV is DOUBLE PRECISION array, dimension (NB) !> Auxiliar vector. !> \endverbatim !> !> \param[in,out] F !> \verbatim !> F is DOUBLE PRECISION array, dimension (LDF,NB) !> Matrix F**T = L*Y**T*A. !> \endverbatim !> !> \param[in] LDF !> \verbatim !> LDF is INTEGER !> The leading dimension of the array F. LDF >= max(1,N). !> \endverbatim ! ! Authors: ! ======== ! !> \author Univ. of Tennessee !> \author Univ. of California Berkeley !> \author Univ. of Colorado Denver !> \author NAG Ltd. ! !> \date November 2011 ! !> \ingroup doubleOTHERauxiliary ! !> \par Contributors: ! ================== !> !> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain !> X. Sun, Computer Science Dept., Duke University, USA !> \n !> Partial column norm updating strategy modified on April 2011 !> Z. Drmac and Z. Bujanovic, Dept. of Mathematics, !> University of Zagreb, Croatia. ! !> \par References: ! ================ !> !> LAPACK Working Note 176 ! !> \htmlonly !> <a href="http://www.netlib.org/lapack/lawnspdf/lawn176.pdf">[PDF]</a> !> \endhtmlonly ! ! ===================================================================== SUBROUTINE DLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, & & VN2, AUXV, F, LDF ) ! ! -- LAPACK auxiliary 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 KB, LDA, LDF, M, N, NB, OFFSET ! .. ! .. Array Arguments .. INTEGER JPVT( * ) DOUBLE PRECISION A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ), & & VN1( * ), VN2( * ) ! .. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER ITEMP, J, K, LASTRK, LSTICC, PVT, RK DOUBLE PRECISION AKK, TEMP, TEMP2, TOL3Z ! .. ! .. External Subroutines .. EXTERNAL DGEMM, DGEMV, DLARFG, DSWAP ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX, MIN, NINT, SQRT ! .. ! .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DLAMCH, DNRM2 EXTERNAL IDAMAX, DLAMCH, DNRM2 ! .. ! .. Executable Statements .. ! LASTRK = MIN( M, N+OFFSET ) LSTICC = 0 K = 0 TOL3Z = SQRT(DLAMCH('Epsilon')) ! ! Beginning of while loop. ! 10 CONTINUE IF( ( K.LT.NB ) .AND. ( LSTICC.EQ.0 ) ) THEN K = K + 1 RK = OFFSET + K ! ! Determine ith pivot column and swap if necessary ! PVT = ( K-1 ) + IDAMAX( N-K+1, VN1( K ), 1 ) IF( PVT.NE.K ) THEN CALL DSWAP( M, A( 1, PVT ), 1, A( 1, K ), 1 ) CALL DSWAP( K-1, F( PVT, 1 ), LDF, F( K, 1 ), LDF ) ITEMP = JPVT( PVT ) JPVT( PVT ) = JPVT( K ) JPVT( K ) = ITEMP VN1( PVT ) = VN1( K ) VN2( PVT ) = VN2( K ) END IF ! ! Apply previous Householder reflectors to column K: ! A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)**T. ! IF( K.GT.1 ) THEN CALL DGEMV( 'No transpose', M-RK+1, K-1, -ONE, A( RK, 1 ), & & LDA, F( K, 1 ), LDF, ONE, A( RK, K ), 1 ) END IF ! ! Generate elementary reflector H(k). ! IF( RK.LT.M ) THEN CALL DLARFG( M-RK+1, A( RK, K ), A( RK+1, K ), 1, TAU( K ) ) ELSE CALL DLARFG( 1, A( RK, K ), A( RK, K ), 1, TAU( K ) ) END IF ! AKK = A( RK, K ) A( RK, K ) = ONE ! ! Compute Kth column of F: ! ! Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)**T*A(RK:M,K). ! IF( K.LT.N ) THEN CALL DGEMV( 'Transpose', M-RK+1, N-K, TAU( K ), & & A( RK, K+1 ), LDA, A( RK, K ), 1, ZERO, & & F( K+1, K ), 1 ) END IF ! ! Padding F(1:K,K) with zeros. ! DO 20 J = 1, K F( J, K ) = ZERO 20 CONTINUE ! ! Incremental updating of F: ! F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)**T ! *A(RK:M,K). ! IF( K.GT.1 ) THEN CALL DGEMV( 'Transpose', M-RK+1, K-1, -TAU( K ), A( RK, 1 ), & & LDA, A( RK, K ), 1, ZERO, AUXV( 1 ), 1 ) ! CALL DGEMV( 'No transpose', N, K-1, ONE, F( 1, 1 ), LDF, & & AUXV( 1 ), 1, ONE, F( 1, K ), 1 ) END IF ! ! Update the current row of A: ! A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)**T. ! IF( K.LT.N ) THEN CALL DGEMV( 'No transpose', N-K, K, -ONE, F( K+1, 1 ), LDF, & & A( RK, 1 ), LDA, ONE, A( RK, K+1 ), LDA ) END IF ! ! Update partial column norms. ! IF( RK.LT.LASTRK ) THEN DO 30 J = K + 1, N IF( VN1( J ).NE.ZERO ) THEN ! ! NOTE: The following 4 lines follow from the analysis in ! Lapack Working Note 176. ! TEMP = ABS( A( RK, J ) ) / VN1( J ) TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) ) TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2 IF( TEMP2 .LE. TOL3Z ) THEN VN2( J ) = DBLE( LSTICC ) LSTICC = J ELSE VN1( J ) = VN1( J )*SQRT( TEMP ) END IF END IF 30 CONTINUE END IF ! A( RK, K ) = AKK ! ! End of while loop. ! GO TO 10 END IF KB = K RK = OFFSET + KB ! ! Apply the block reflector to the rest of the matrix: ! A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) - ! A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)**T. ! IF( KB.LT.MIN( N, M-OFFSET ) ) THEN CALL DGEMM( 'No transpose', 'Transpose', M-RK, N-KB, KB, -ONE, & & A( RK+1, 1 ), LDA, F( KB+1, 1 ), LDF, ONE, & & A( RK+1, KB+1 ), LDA ) END IF ! ! Recomputation of difficult columns. ! 40 CONTINUE IF( LSTICC.GT.0 ) THEN ITEMP = NINT( VN2( LSTICC ) ) VN1( LSTICC ) = DNRM2( M-RK, A( RK+1, LSTICC ), 1 ) ! ! NOTE: The computation of VN1( LSTICC ) relies on the fact that ! SNRM2 does not fail on vectors with norm below the value of ! SQRT(DLAMCH('S')) ! VN2( LSTICC ) = VN1( LSTICC ) LSTICC = ITEMP GO TO 40 END IF ! RETURN ! ! End of DLAQPS ! END