dlaqp3rk.f - Man Page

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.

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.