*> \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
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaqps.f">
*> [TXT]</a>
*> \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.
*
*> \date December 2016
*
*> \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
*> <a href="http://www.netlib.org/lapack/lawnspdf/lawn176.pdf">[PDF]</a>
*> \endhtmlonly
*
* =====================================================================
SUBROUTINE SLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
$ VN2, AUXV, F, LDF )
*
* -- LAPACK auxiliary routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. 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