SLAQP3RK 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.

          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 REAL, 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 REAL, 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 SGEQP3RK. 1 <= KP1 <= N_orig.

MAXC2NRM

          MAXC2NRM is REAL


          The maximum column 2-norm of the whole original


          matrix A_orig computed in the main routine SGEQP3RK.


          MAXC2NRM >= 0.

A

          A is REAL 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 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 is REAL


          The maximum column 2-norm of the residual matrix,


          when the factorization stopped at rank KB. MAXC2NRMK >= 0.

          RELMAXC2NRMK is REAL


          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 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 is REAL array, dimension (min(M-IOFFSET,N))


          The scalar factors of the elementary reflectors.

          VN1 is REAL array, dimension (N)


          The vector with the partial column norms.

          VN2 is REAL array, dimension (N)


          The vector with the exact column norms.

          AUXV is REAL array, dimension (NB)


          Auxiliary vector.

          F is REAL array, dimension (LDF,NB)


          Matrix F**T = L*(Y**T)*A.

          LDF is INTEGER


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

          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 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

References:

[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 slaqp3rk.f.