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SUBROUTINE DLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
$ VN2, AUXV, F, LDF ) * * -- LAPACK auxiliary routine (version 3.3.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * -- April 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( * ) * .. * * 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**T = L*Y**T*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. * -- April 2011 -- * 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)**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 |