dlaqp2rk.f

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*> \brief \b DLAQP2RK computes truncated QR factorization with column pivoting of a real matrix block using Level 2 BLAS and overwrites a real m-by-nrhs matrix B with Q**T * B.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*      SUBROUTINE DLAQP2RK( M, N, NRHS, IOFFSET, KMAX, ABSTOL, RELTOL,
*     $                     KP1, MAXC2NRM, A, LDA, K, MAXC2NRMK,
*     $                     RELMAXC2NRMK, JPIV, TAU, VN1, VN2, WORK,
*     $                     INFO )
*      IMPLICIT NONE
*
*     .. Scalar Arguments ..
*      INTEGER            INFO, IOFFSET, KP1, K, KMAX, LDA, M, N, NRHS
*      DOUBLE PRECISION   ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK,
*     $                   RELTOL
*     ..
*     .. Array Arguments ..
*      INTEGER            JPIV( * )
*      DOUBLE PRECISION   A( LDA, * ), TAU( * ), VN1( * ), VN2( * ),
*     $                   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAQP2RK computes a truncated (rank K) or full rank Householder QR
*> factorization with column pivoting of a real matrix
*> block A(IOFFSET+1:M,1:N) as
*>
*>   A * P(K) = Q(K) * R(K).
*>
*> The routine uses Level 2 BLAS. The block A(1:IOFFSET,1:N)
*> is accordingly pivoted, but not factorized.
*>
*> The routine also overwrites the right-hand-sides matrix block B
*> stored in A(IOFFSET+1:M,N+1:N+NRHS) with Q(K)**T * B.
*> \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] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of
*>          columns of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] IOFFSET
*> \verbatim
*>          IOFFSET is INTEGER
*>          The number of rows of the matrix A that must be pivoted
*>          but not factorized. IOFFSET >= 0.
*>
*>          IOFFSET also represents the number of columns of the whole
*>          original matrix A_orig that have been factorized
*>          in the previous steps.
*> \endverbatim
*>
*> \param[in] KMAX
*> \verbatim
*>          KMAX is INTEGER
*>
*>          The first factorization stopping criterion. KMAX >= 0.
*>
*>          The maximum number of columns of the matrix A to factorize,
*>          i.e. the maximum factorization rank.
*>
*>          a) If KMAX >= min(M-IOFFSET,N), then this stopping
*>                criterion is not used, factorize columns
*>                depending on ABSTOL and RELTOL.
*>
*>          b) If KMAX = 0, then this stopping criterion is
*>             satisfied on input and the routine exits immediately.
*>             This means that the factorization is not performed,
*>             the matrices A and B and the arrays TAU, IPIV
*>             are not modified.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is DOUBLE PRECISION, cannot be NaN.
*>
*>          The second factorization stopping criterion.
*>
*>          The absolute tolerance (stopping threshold) for
*>          maximum column 2-norm of the residual matrix.
*>          The algorithm converges (stops the factorization) when
*>          the maximum column 2-norm of the residual matrix
*>          is less than or equal to ABSTOL.
*>
*>          a) If ABSTOL < 0.0, then this stopping criterion is not
*>                used, the routine factorizes columns depending
*>                on KMAX and RELTOL.
*>                This includes the case ABSTOL = -Inf.
*>
*>          b) If 0.0 <= ABSTOL then the input value
*>                of ABSTOL is used.
*> \endverbatim
*>
*> \param[in] RELTOL
*> \verbatim
*>          RELTOL is DOUBLE PRECISION, cannot be NaN.
*>
*>          The third factorization stopping criterion.
*>
*>          The tolerance (stopping threshold) for the ratio of the
*>          maximum column 2-norm of the residual matrix to the maximum
*>          column 2-norm of the original matrix A_orig. The algorithm
*>          converges (stops the factorization), when this ratio is
*>          less than or equal to RELTOL.
*>
*>          a) If RELTOL < 0.0, then this stopping criterion is not
*>                used, the routine factorizes columns depending
*>                on KMAX and ABSTOL.
*>                This includes the case RELTOL = -Inf.
*>
*>          d) If 0.0 <= RELTOL then the input value of RELTOL
*>                is used.
*> \endverbatim
*>
*> \param[in] KP1
*> \verbatim
*>          KP1 is INTEGER
*>          The index of the column with the maximum 2-norm in
*>          the whole original matrix A_orig determined in the
*>          main routine DGEQP3RK. 1 <= KP1 <= N_orig_mat.
*> \endverbatim
*>
*> \param[in] MAXC2NRM
*> \verbatim
*>          MAXC2NRM is DOUBLE PRECISION
*>          The maximum column 2-norm of the whole original
*>          matrix A_orig computed in the main routine DGEQP3RK.
*>          MAXC2NRM >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N+NRHS)
*>          On entry:
*>              the M-by-N matrix A and M-by-NRHS matrix B, as in
*>
*>                                  N     NRHS
*>              array_A   =   M  [ mat_A, mat_B ]
*>
*>          On exit:
*>          1. The elements in block A(IOFFSET+1:M,1:K) below
*>             the diagonal together with the array TAU represent
*>             the orthogonal matrix Q(K) as a product of elementary
*>             reflectors.
*>          2. The upper triangular block of the matrix A stored
*>             in A(IOFFSET+1:M,1:K) is the triangular factor obtained.
*>          3. The block of the matrix A stored in A(1:IOFFSET,1:N)
*>             has been accordingly pivoted, but not factorized.
*>          4. The rest of the array A, block A(IOFFSET+1:M,K+1:N+NRHS).
*>             The left part A(IOFFSET+1:M,K+1:N) of this block
*>             contains the residual of the matrix A, and,
*>             if NRHS > 0, the right part of the block
*>             A(IOFFSET+1:M,N+1:N+NRHS) contains the block of
*>             the right-hand-side matrix B. Both these blocks have been
*>             updated by multiplication from the left by Q(K)**T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*>          K is INTEGER
*>          Factorization rank of the matrix A, i.e. the rank of
*>          the factor R, which is the same as the number of non-zero
*>          rows of the factor R. 0 <= K <= min(M-IOFFSET,KMAX,N).
*>
*>          K also represents the number of non-zero Householder
*>          vectors.
*> \endverbatim
*>
*> \param[out] MAXC2NRMK
*> \verbatim
*>          MAXC2NRMK is DOUBLE PRECISION
*>          The maximum column 2-norm of the residual matrix,
*>          when the factorization stopped at rank K. MAXC2NRMK >= 0.
*> \endverbatim
*>
*> \param[out] RELMAXC2NRMK
*> \verbatim
*>          RELMAXC2NRMK is DOUBLE PRECISION
*>          The ratio MAXC2NRMK / MAXC2NRM of the maximum column
*>          2-norm of the residual matrix (when the factorization
*>          stopped at rank K) to the maximum column 2-norm of the
*>          whole original matrix A. RELMAXC2NRMK >= 0.
*> \endverbatim
*>
*> \param[out] JPIV
*> \verbatim
*>          JPIV is INTEGER array, dimension (N)
*>          Column pivot indices, for 1 <= j <= N, column j
*>          of the matrix A was interchanged with column JPIV(j).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (min(M-IOFFSET,N))
*>          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[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N-1)
*>          Used in DLARF1F subroutine to apply an elementary
*>          reflector from the left.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          1) INFO = 0: successful exit.
*>          2) If INFO = j_1, where 1 <= j_1 <= N, then NaN was
*>             detected and the routine stops the computation.
*>             The j_1-th column of the matrix A or the j_1-th
*>             element of array TAU contains the first occurrence
*>             of NaN in the factorization step K+1 ( when K columns
*>             have been factorized ).
*>
*>             On exit:
*>             K                  is set to the number of
*>                                   factorized columns without
*>                                   exception.
*>             MAXC2NRMK          is set to NaN.
*>             RELMAXC2NRMK       is set to NaN.
*>             TAU(K+1:min(M,N))  is not set and contains undefined
*>                                   elements. If j_1=K+1, TAU(K+1)
*>                                   may contain NaN.
*>          3) If INFO = j_2, where N+1 <= j_2 <= 2*N, then no NaN
*>             was detected, but +Inf (or -Inf) was detected and
*>             the routine continues the computation until completion.
*>             The (j_2-N)-th column of the matrix A contains the first
*>             occurrence of +Inf (or -Inf) in the factorization
*>             step K+1 ( when K columns have been factorized ).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqp2rk
*
*> \par References:
*  ================
*> [1] A Level 3 BLAS QR factorization algorithm with column pivoting developed in 1996.
*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain.
*> X. Sun, Computer Science Dept., Duke University, USA.
*> C. H. Bischof, Math. and Comp. Sci. Div., Argonne National Lab, USA.
*> A BLAS-3 version of the QR factorization with column pivoting.
*> LAPACK Working Note 114
*> <a href="https://www.netlib.org/lapack/lawnspdf/lawn114.pdf">https://www.netlib.org/lapack/lawnspdf/lawn114.pdf</a>
*> and in
*> SIAM J. Sci. Comput., 19(5):1486-1494, Sept. 1998.
*> <a href="https://doi.org/10.1137/S1064827595296732">https://doi.org/10.1137/S1064827595296732</a>
*>
*> [2] A partial column norm updating strategy developed in 2006.
*> Z. Drmac and Z. Bujanovic, Dept. of Math., University of Zagreb, Croatia.
*> On the failure of rank revealing QR factorization software – a case study.
*> LAPACK Working Note 176.
*> <a href="http://www.netlib.org/lapack/lawnspdf/lawn176.pdf">http://www.netlib.org/lapack/lawnspdf/lawn176.pdf</a>
*> and in
*> ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages.
*> <a href="https://doi.org/10.1145/1377612.1377616">https://doi.org/10.1145/1377612.1377616</a>
*
*> \par Contributors:
*  ==================
*>
*> \verbatim
*>
*>  November  2023, Igor Kozachenko, James Demmel,
*>                  EECS Department,
*>                  University of California, Berkeley, USA.
*>
*> \endverbatim
*
*  =====================================================================
      SUBROUTINE dlaqp2rk( M, N, NRHS, IOFFSET, KMAX, ABSTOL, RELTOL,
     $                     KP1, MAXC2NRM, A, LDA, K, MAXC2NRMK,
     $                     RELMAXC2NRMK, JPIV, TAU, VN1, VN2, WORK,
     $                     INFO )
      IMPLICIT NONE
*
*  -- 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            INFO, IOFFSET, KP1, K, KMAX, LDA, M, N, NRHS
      DOUBLE PRECISION   ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK,
     $                   RELTOL
*     ..
*     .. Array Arguments ..
      INTEGER            JPIV( * )
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), VN1( * ), VN2( * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0d+0, one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, ITEMP, J, JMAXC2NRM, KK, KP, MINMNFACT,
     $                   MINMNUPDT
      DOUBLE PRECISION   HUGEVAL, TEMP, TEMP2, TOL3Z
*     ..
*     .. External Subroutines ..
      EXTERNAL           dlarf1f, dlarfg, dswap
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min, sqrt
*     ..
*     .. External Functions ..
      LOGICAL            DISNAN
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH, DNRM2
      EXTERNAL           disnan, dlamch, idamax, dnrm2
*     ..
*     .. Executable Statements ..
*
*     Initialize INFO
*
      info = 0
*
*     MINMNFACT in the smallest dimension of the submatrix
*     A(IOFFSET+1:M,1:N) to be factorized.
*
*     MINMNUPDT is the smallest dimension
*     of the subarray A(IOFFSET+1:M,1:N+NRHS) to be udated, which
*     contains the submatrices A(IOFFSET+1:M,1:N) and
*     B(IOFFSET+1:M,1:NRHS) as column blocks.
*
      minmnfact = min( m-ioffset, n )
      minmnupdt = min( m-ioffset, n+nrhs )
      kmax = min( kmax, minmnfact )
      tol3z = sqrt( dlamch( 'Epsilon' ) )
      hugeval = dlamch( 'Overflow' )
*
*     Compute the factorization, KK is the lomn loop index.
*
      DO kk = 1, kmax
*
         i = ioffset + kk
*
         IF( i.EQ.1 ) THEN
*
*           ============================================================
*
*           We are at the first column of the original whole matrix A,
*           therefore we use the computed KP1 and MAXC2NRM from the
*           main routine.
*
 
            kp = kp1
*
*           ============================================================
*
         ELSE
*
*           ============================================================
*
*           Determine the pivot column in KK-th step, i.e. the index
*           of the column with the maximum 2-norm in the
*           submatrix A(I:M,K:N).
*
            kp = ( kk-1 ) + idamax( n-kk+1, vn1( kk ), 1 )
*
*           Determine the maximum column 2-norm and the relative maximum
*           column 2-norm of the submatrix A(I:M,KK:N) in step KK.
*           RELMAXC2NRMK  will be computed later, after somecondition
*           checks on MAXC2NRMK.
*
            maxc2nrmk = vn1( kp )
*
*           ============================================================
*
*           Check if the submatrix A(I:M,KK:N) contains NaN, and set
*           INFO parameter to the column number, where the first NaN
*           is found and return from the routine.
*           We need to check the condition only if the
*           column index (same as row index) of the original whole
*           matrix is larger than 1, since the condition for whole
*           original matrix is checked in the main routine.
*
            IF( disnan( maxc2nrmk ) ) THEN
*
*              Set K, the number of factorized columns.
*              that are not zero.
*
                k = kk - 1
                info = k + kp
*
*               Set RELMAXC2NRMK to NaN.
*
                relmaxc2nrmk = maxc2nrmk
*
*               Array TAU(K+1:MINMNFACT) is not set and contains
*               undefined elements.
*
               RETURN
            END IF
*
*           ============================================================
*
*           Quick return, if the submatrix A(I:M,KK:N) is
*           a zero matrix.
*           We need to check the condition only if the
*           column index (same as row index) of the original whole
*           matrix is larger than 1, since the condition for whole
*           original matrix is checked in the main routine.
*
            IF( maxc2nrmk.EQ.zero ) THEN
*
*              Set K, the number of factorized columns.
*              that are not zero.
*
               k = kk - 1
               relmaxc2nrmk = zero
*
*              Set TAUs corresponding to the columns that were not
*              factorized to ZERO, i.e. set TAU(KK:MINMNFACT) to ZERO.
*
               DO j = kk, minmnfact
                  tau( j ) = zero
               END DO
*
*              Return from the routine.
*
               RETURN
*
            END IF
*
*           ============================================================
*
*           Check if the submatrix A(I:M,KK:N) contains Inf,
*           set INFO parameter to the column number, where
*           the first Inf is found plus N, and continue
*           the computation.
*           We need to check the condition only if the
*           column index (same as row index) of the original whole
*           matrix is larger than 1, since the condition for whole
*           original matrix is checked in the main routine.
*
            IF( info.EQ.0 .AND. maxc2nrmk.GT.hugeval ) THEN
               info = n + kk - 1 + kp
            END IF
*
*           ============================================================
*
*           Test for the second and third stopping criteria.
*           NOTE: There is no need to test for ABSTOL >= ZERO, since
*           MAXC2NRMK is non-negative. Similarly, there is no need
*           to test for RELTOL >= ZERO, since RELMAXC2NRMK is
*           non-negative.
*           We need to check the condition only if the
*           column index (same as row index) of the original whole
*           matrix is larger than 1, since the condition for whole
*           original matrix is checked in the main routine.
 
            relmaxc2nrmk =  maxc2nrmk / maxc2nrm
*
            IF( maxc2nrmk.LE.abstol .OR. relmaxc2nrmk.LE.reltol ) THEN
*
*              Set K, the number of factorized columns.
*
               k = kk - 1
*
*              Set TAUs corresponding to the columns that were not
*              factorized to ZERO, i.e. set TAU(KK:MINMNFACT) to ZERO.
*
               DO j = kk, minmnfact
                  tau( j ) = zero
               END DO
*
*              Return from the routine.
*
               RETURN
*
            END IF
*
*           ============================================================
*
*           End ELSE of IF(I.EQ.1)
*
         END IF
*
*        ===============================================================
*
*        If the pivot column is not the first column of the
*        subblock A(1:M,KK:N):
*        1) swap the KK-th column and the KP-th pivot column
*           in A(1:M,1:N);
*        2) copy the KK-th element into the KP-th element of the partial
*           and exact 2-norm vectors VN1 and VN2. ( Swap is not needed
*           for VN1 and VN2 since we use the element with the index
*           larger than KK in the next loop step.)
*        3) Save the pivot interchange with the indices relative to the
*           the original matrix A, not the block A(1:M,1:N).
*
         IF( kp.NE.kk ) THEN
            CALL dswap( m, a( 1, kp ), 1, a( 1, kk ), 1 )
            vn1( kp ) = vn1( kk )
            vn2( kp ) = vn2( kk )
            itemp = jpiv( kp )
            jpiv( kp ) = jpiv( kk )
            jpiv( kk ) = itemp
         END IF
*
*        Generate elementary reflector H(KK) using the column A(I:M,KK),
*        if the column has more than one element, otherwise
*        the elementary reflector would be an identity matrix,
*        and TAU(KK) = ZERO.
*
         IF( i.LT.m ) THEN
            CALL dlarfg( m-i+1, a( i, kk ), a( i+1, kk ), 1,
     $                   tau( kk ) )
         ELSE
            tau( kk ) = zero
         END IF
*
*        Check if TAU(KK) contains NaN, set INFO parameter
*        to the column number where NaN is found and return from
*        the routine.
*        NOTE: There is no need to check TAU(KK) for Inf,
*        since DLARFG cannot produce TAU(KK) or Householder vector
*        below the diagonal containing Inf. Only BETA on the diagonal,
*        returned by DLARFG can contain Inf, which requires
*        TAU(KK) to contain NaN. Therefore, this case of generating Inf
*        by DLARFG is covered by checking TAU(KK) for NaN.
*
         IF( disnan( tau(kk) ) ) THEN
            k = kk - 1
            info = kk
*
*           Set MAXC2NRMK and  RELMAXC2NRMK to NaN.
*
            maxc2nrmk = tau( kk )
            relmaxc2nrmk = tau( kk )
*
*           Array TAU(KK:MINMNFACT) is not set and contains
*           undefined elements, except the first element TAU(KK) = NaN.
*
            RETURN
         END IF
*
*        Apply H(KK)**T to A(I:M,KK+1:N+NRHS) from the left.
*        ( If M >= N, then at KK = N there is no residual matrix,
*         i.e. no columns of A to update, only columns of B.
*         If M < N, then at KK = M-IOFFSET, I = M and we have a
*         one-row residual matrix in A and the elementary
*         reflector is a unit matrix, TAU(KK) = ZERO, i.e. no update
*         is needed for the residual matrix in A and the
*         right-hand-side-matrix in B.
*         Therefore, we update only if
*         KK < MINMNUPDT = min(M-IOFFSET, N+NRHS)
*         condition is satisfied, not only KK < N+NRHS )
*
         IF( kk.LT.minmnupdt ) THEN
            CALL dlarf1f( 'Left', m-i+1, n+nrhs-kk, a( i, kk ), 1,
     $                  tau( kk ), a( i, kk+1 ), lda, work( 1 ) )
         END IF
*
         IF( kk.LT.minmnfact ) THEN
*
*           Update the partial column 2-norms for the residual matrix,
*           only if the residual matrix A(I+1:M,KK+1:N) exists, i.e.
*           when KK < min(M-IOFFSET, N).
*
            DO j = kk + 1, n
               IF( vn1( j ).NE.zero ) THEN
*
*                 NOTE: The following lines follow from the analysis in
*                 Lapack Working Note 176.
*
                  temp = one - ( abs( a( i, j ) ) / vn1( j ) )**2
                  temp = max( temp, zero )
                  temp2 = temp*( vn1( j ) / vn2( j ) )**2
                  IF( temp2 .LE. tol3z ) THEN
*
*                    Compute the column 2-norm for the partial
*                    column A(I+1:M,J) by explicitly computing it,
*                    and store it in both partial 2-norm vector VN1
*                    and exact column 2-norm vector VN2.
*
                     vn1( j ) = dnrm2( m-i, a( i+1, j ), 1 )
                     vn2( j ) = vn1( j )
*
                  ELSE
*
*                    Update the column 2-norm for the partial
*                    column A(I+1:M,J) by removing one
*                    element A(I,J) and store it in partial
*                    2-norm vector VN1.
*
                     vn1( j ) = vn1( j )*sqrt( temp )
*
                  END IF
               END IF
            END DO
*
         END IF
*
*     End factorization loop
*
      END DO
*
*     If we reached this point, all colunms have been factorized,
*     i.e. no condition was triggered to exit the routine.
*     Set the number of factorized columns.
*
      k = kmax
*
*     We reached the end of the loop, i.e. all KMAX columns were
*     factorized, we need to set MAXC2NRMK and RELMAXC2NRMK before
*     we return.
*
      IF( k.LT.minmnfact ) THEN
*
         jmaxc2nrm = k + idamax( n-k, vn1( k+1 ), 1 )
         maxc2nrmk = vn1( jmaxc2nrm )
*
         IF( k.EQ.0 ) THEN
            relmaxc2nrmk = one
         ELSE
            relmaxc2nrmk = maxc2nrmk / maxc2nrm
         END IF
*
      ELSE
         maxc2nrmk = zero
         relmaxc2nrmk = zero
      END IF
*
*     We reached the end of the loop, i.e. all KMAX columns were
*     factorized, set TAUs corresponding to the columns that were
*     not factorized to ZERO, i.e. TAU(K+1:MINMNFACT) set to ZERO.
*
      DO j = k + 1, minmnfact
         tau( j ) = zero
      END DO
*
      RETURN
*
*     End of DLAQP2RK
*
      END