*> \brief \b SLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLAQPS + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLAQPS( 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( * ) * REAL A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ), * $ VN1( * ), VN2( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLAQPS 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 REAL 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 REAL array, dimension (KB) *> The scalar factors of the elementary reflectors. *> \endverbatim *> *> \param[in,out] VN1 *> \verbatim *> VN1 is REAL array, dimension (N) *> The vector with the partial column norms. *> \endverbatim *> *> \param[in,out] VN2 *> \verbatim *> VN2 is REAL array, dimension (N) *> The vector with the exact column norms. *> \endverbatim *> *> \param[in,out] AUXV *> \verbatim *> AUXV is REAL array, dimension (NB) *> Auxiliary vector. *> \endverbatim *> *> \param[in,out] F *> \verbatim *> F is REAL 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. * *> \ingroup realOTHERauxiliary * *> \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 *> [PDF] *> \endhtmlonly * * ===================================================================== SUBROUTINE SLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, $ VN2, AUXV, F, LDF ) * * -- LAPACK auxiliary 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 KB, LDA, LDF, M, N, NB, OFFSET * .. * .. Array Arguments .. INTEGER JPVT( * ) REAL A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ), $ VN1( * ), VN2( * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) * .. * .. Local Scalars .. INTEGER ITEMP, J, K, LASTRK, LSTICC, PVT, RK REAL AKK, TEMP, TEMP2, TOL3Z * .. * .. External Subroutines .. EXTERNAL SGEMM, SGEMV, SLARFG, SSWAP * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, NINT, REAL, SQRT * .. * .. External Functions .. INTEGER ISAMAX REAL SLAMCH, SNRM2 EXTERNAL ISAMAX, SLAMCH, SNRM2 * .. * .. Executable Statements .. * LASTRK = MIN( M, N+OFFSET ) LSTICC = 0 K = 0 TOL3Z = SQRT(SLAMCH('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 ) + ISAMAX( N-K+1, VN1( K ), 1 ) IF( PVT.NE.K ) THEN CALL SSWAP( M, A( 1, PVT ), 1, A( 1, K ), 1 ) CALL SSWAP( 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 SGEMV( '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 SLARFG( M-RK+1, A( RK, K ), A( RK+1, K ), 1, TAU( K ) ) ELSE CALL SLARFG( 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 SGEMV( '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 SGEMV( 'Transpose', M-RK+1, K-1, -TAU( K ), A( RK, 1 ), $ LDA, A( RK, K ), 1, ZERO, AUXV( 1 ), 1 ) * CALL SGEMV( '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 SGEMV( '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 ) = REAL( 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 SGEMM( '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 ) = SNRM2( 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 SLAQPS * END