SUBROUTINE DLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
$ VN2, AUXV, F, LDF )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER KB, LDA, LDF, M, N, NB, OFFSET
* ..
* .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ),
$ VN1( * ), VN2( * )
* ..
*
* Purpose
* =======
*
* 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.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0
*
* OFFSET (input) INTEGER
* The number of rows of A that have been factorized in
* previous steps.
*
* NB (input) INTEGER
* The number of columns to factorize.
*
* KB (output) INTEGER
* The number of columns actually factorized.
*
* A (input/output) 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.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* JPVT (input/output) INTEGER array, dimension (N)
* JPVT(I) = K <==> Column K of the full matrix A has been
* permuted into position I in AP.
*
* TAU (output) DOUBLE PRECISION array, dimension (KB)
* The scalar factors of the elementary reflectors.
*
* VN1 (input/output) DOUBLE PRECISION array, dimension (N)
* The vector with the partial column norms.
*
* VN2 (input/output) DOUBLE PRECISION array, dimension (N)
* The vector with the exact column norms.
*
* AUXV (input/output) DOUBLE PRECISION array, dimension (NB)
* Auxiliar vector.
*
* F (input/output) DOUBLE PRECISION array, dimension (LDF,NB)
* Matrix F' = L*Y'*A.
*
* LDF (input) INTEGER
* The leading dimension of the array F. LDF >= max(1,N).
*
* Further Details
* ===============
*
* Based on contributions by
* G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
* X. Sun, Computer Science Dept., Duke University, USA
*
* Partial column norm updating strategy modified by
* Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
* University of Zagreb, Croatia.
* June 2006.
* For more details see LAPACK Working Note 176.
* =====================================================================
*
* .. 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)'.
*
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)'*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)'
* *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)'.
*
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)'.
*
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