!>
!> ZLAQP3RK computes a step of truncated QR factorization with column
!> pivoting of a complex 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)**H * 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 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 ZGEQP3RK. 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 ZGEQP3RK.
!>          MAXC2NRM >= 0.
!> 

A

!>          A is COMPLEX*16 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 unitary 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)**H.
!> 

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

!>          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 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 COMPLEX*16 array, dimension (min(M-IOFFSET,N))
!>          The scalar factors of the elementary reflectors.
!> 

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

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

!>          AUXV is COMPLEX*16 array, dimension (NB)
!>          Auxiliary vector.
!> 

!>          F is COMPLEX*16 array, dimension (LDF,NB)
!>          Matrix F**H = L*(Y**H)*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  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 392 of file zlaqp3rk.f.