SRC/dlaqp3rk.f(3)

NAME
SRC/dlaqp3rk.f

SYNOPSIS

  Functions/Subroutines
subroutine dlaqp3rk (m, n, nrhs, ioffset, nb, abstol, reltol, kp1, maxc2nrm, a, lda, done, kb, maxc2nrmk, relmaxc2nrmk, jpiv, tau, vn1, vn2, auxv, f, ldf, iwork, info)
DLAQP3RK computes a step of truncated QR factorization with column pivoting of a real m-by-n matrix A using Level 3 BLAS and overwrites a real m-by-nrhs matrix B with Q**T * B.

Function/Subroutine Documentation

  subroutine dlaqp3rk (integer m, integer n, integer nrhs, integer ioffset, integer nb, double precision abstol, double precision reltol, integer kp1, double precision maxc2nrm, double precision, dimension( lda, * ) a, integer lda, logical done, integer kb, double precision maxc2nrmk, double precision relmaxc2nrmk, integer, dimension( * ) jpiv, double precision, dimension( * ) tau, double precision, dimension( * ) vn1, double precision, dimension( * ) vn2, double precision, dimension( * ) auxv, double precision, dimension( ldf, * ) f, integer ldf, integer, dimension( * ) iwork, integer info)

DLAQP3RK computes a step of truncated QR factorization with column pivoting of a real m-by-n matrix A using Level 3 BLAS and overwrites a real m-by-nrhs matrix B with Q**T * B.

Purpose:

     DLAQP3RK computes a step of truncated QR factorization with column
     pivoting of a real M-by-N matrix A block A(IOFFSET+1:M,1:N)
     by using Level 3 BLAS as
       A * P(KB) = Q(KB) * R(KB).
     The routine tries to factorize NB columns from A starting from
     the row IOFFSET+1 and updates the residual matrix with BLAS 3
     xGEMM. The number of actually factorized columns is returned
     is smaller than NB.
     Block A(1:IOFFSET,1:N) is accordingly pivoted, but not factorized.
     The routine also overwrites the right-hand-sides B matrix stored
     in A(IOFFSET+1:M,1:N+1:N+NRHS) with Q(KB)**T * B.
     Cases when the number of factorized columns KB < NB:
     (1) In some cases, due to catastrophic cancellations, it cannot
     factorize all NB columns and need to update the residual matrix.
     Hence, the actual number of factorized columns in the block returned
     in KB is smaller than NB. The logical DONE is returned as FALSE.
     The factorization of the whole original matrix A_orig must proceed
     with the next block.
     (2) Whenever the stopping criterion ABSTOL or RELTOL is satisfied,
     the factorization of the whole original matrix A_orig is stopped,
     the logical DONE is returned as TRUE. The number of factorized
     columns which is smaller than NB is returned in KB.
     (3) In case both stopping criteria ABSTOL or RELTOL are not used,
     and when the residual matrix is a zero matrix in some factorization
     step KB, the factorization of the whole original matrix A_orig is
     stopped, the logical DONE is returned as TRUE. The number of
     factorized columns which is smaller than NB is returned in KB.
     (4) Whenever NaN is detected in the matrix A or in the array TAU,
     the factorization of the whole original matrix A_orig is stopped,
     the logical DONE is returned as TRUE. The number of factorized
     columns which is smaller than NB is returned in KB. The INFO
     parameter is set to the column index of the first NaN occurrence.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix A. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix A. N >= 0

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of
              columns of the matrix B. NRHS >= 0.

    IOFFSET

              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.

    NB

              NB is INTEGER
              Factorization block size, i.e the number of columns
              to factorize in the matrix A. 0 <= NB
              If NB = 0, then 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.

    ABSTOL

              ABSTOL is DOUBLE PRECISION, cannot be NaN.
              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 NB and RELTOL.
                    This includes the case ABSTOL = -Inf.
              b) If 0.0 <= ABSTOL then the input value
                    of ABSTOL is used.

    RELTOL

              RELTOL is DOUBLE PRECISION, cannot be NaN.
              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 NB and ABSTOL.
                    This includes the case RELTOL = -Inf.
              d) If 0.0 <= RELTOL then the input value of RELTOL
                    is used.

    KP1

              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.

    MAXC2NRM

              MAXC2NRM is DOUBLE PRECISION
              The maximum column 2-norm of the whole original
              matrix A_orig computed in the main routine DGEQP3RK.
              MAXC2NRM >= 0.

    A

              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:KB) below
                 the diagonal together with the array TAU represent
                 the orthogonal matrix Q(KB) as a product of elementary
                 reflectors.
              2. The upper triangular block of the matrix A stored
                 in A(IOFFSET+1:M,1:KB) 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,KB+1:N+NRHS).
                 The left part A(IOFFSET+1:M,KB+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(KB)**T.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,M).

    .RE

    verbatim DONE is LOGICAL TRUE: a) if the factorization completed before processing all min(M-IOFFSET,NB,N) columns due to ABSTOL or RELTOL criterion, b) if the factorization completed before processing all min(M-IOFFSET,NB,N) columns due to the residual matrix being a ZERO matrix. c) when NaN was detected in the matrix A or in the array TAU. FALSE: otherwise.

    Parameters
        KB

                  KB 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 <= KB <= min(M-IOFFSET,NB,N).
                  KB also represents the number of non-zero Householder
                  vectors.

        MAXC2NRMK

                  MAXC2NRMK is DOUBLE PRECISION
                  The maximum column 2-norm of the residual matrix,
                  when the factorization stopped at rank KB. MAXC2NRMK >= 0.

        RELMAXC2NRMK

                  RELMAXC2NRMK is DOUBLE PRECISION
                  The ratio MAXC2NRMK / MAXC2NRM of the maximum column
                  2-norm of the residual matrix (when the factorization
                  stopped at rank KB) to the maximum column 2-norm of the
                  original matrix A_orig. RELMAXC2NRMK >= 0.

        JPIV

                  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).

        TAU

                  TAU is DOUBLE PRECISION array, dimension (min(M-IOFFSET,N))
                  The scalar factors of the elementary reflectors.

        VN1

                  VN1 is DOUBLE PRECISION array, dimension (N)
                  The vector with the partial column norms.

        VN2

                  VN2 is DOUBLE PRECISION array, dimension (N)
                  The vector with the exact column norms.

        AUXV

                  AUXV is DOUBLE PRECISION array, dimension (NB)
                  Auxiliary vector.

        F

                  F is DOUBLE PRECISION array, dimension (LDF,NB)
                  Matrix F**T = L*(Y**T)*A.

        LDF

                  LDF is INTEGER
                  The leading dimension of the array F. LDF >= max(1,N+NRHS).

        IWORK

                  IWORK is INTEGER array, dimension (N-1).
                  Is a work array. ( IWORK is used to store indices
                  of 'bad' columns for norm downdating in the residual
                  matrix ).

        INFO

                  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 KB+1 ( when KB columns
                     have been factorized ).
                     On exit:
                     KB                  is set to the number of
                                            factorized columns without
                                            exception.
                     MAXC2NRMK           is set to NaN.
                     RELMAXC2NRMK        is set to NaN.
                     TAU(KB+1:min(M,N))     is not set and contains undefined
                                            elements. If j_1=KB+1, TAU(KB+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 actorization
                     step KB+1 ( when KB columns have been factorized ).

    Author
        Univ. of Tennessee

        Univ. of California Berkeley

        Univ. of Colorado Denver

        NAG Ltd.

    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 and in SIAM J. Sci. Comput., 19(5):1486-1494, Sept. 1998.

    [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. and in ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages.

    Contributors:

          November  2023, Igor Kozachenko, James Demmel,
                          EECS Department,
                          University of California, Berkeley, USA.

    Definition at line 398 of file dlaqp3rk.f.

Author

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