ZGEQP3RK performs two tasks simultaneously:
 Task 1: The routine computes a truncated (rank K) or full rank
 Householder QR factorization with column pivoting of a complex
 M-by-N matrix A using Level 3 BLAS. K is the number of columns
 that were factorized, i.e. factorization rank of the
 factor R, K <= min(M,N).
  A * P(K) = Q(K) * R(K)  =
        = Q(K) * ( R11(K) R12(K) ) = Q(K) * (   R(K)_approx    )
                 ( 0      R22(K) )          ( 0  R(K)_residual ),
 where:
  P(K)            is an N-by-N permutation matrix;
  Q(K)            is an M-by-M unitary matrix;
  R(K)_approx   = ( R11(K), R12(K) ) is a rank K approximation of the
                    full rank factor R with K-by-K upper-triangular
                    R11(K) and K-by-N rectangular R12(K). The diagonal
                    entries of R11(K) appear in non-increasing order
                    of absolute value, and absolute values of all of
                    them exceed the maximum column 2-norm of R22(K)
                    up to roundoff error.
  R(K)_residual = R22(K) is the residual of a rank K approximation
                    of the full rank factor R. It is a
                    an (M-K)-by-(N-K) rectangular matrix;
  0               is a an (M-K)-by-K zero matrix.
 Task 2: At the same time, the routine overwrites a complex M-by-NRHS
 matrix B with  Q(K)**H * B  using Level 3 BLAS.
 =====================================================================
 The matrices A and B are stored on input in the array A as
 the left and right blocks A(1:M,1:N) and A(1:M, N+1:N+NRHS)
 respectively.
                                  N     NRHS
             array_A   =   M  [ mat_A, mat_B ]
 The truncation criteria (i.e. when to stop the factorization)
 can be any of the following:
   1) The input parameter KMAX, the maximum number of columns
      KMAX to factorize, i.e. the factorization rank is limited
      to KMAX. If KMAX >= min(M,N), the criterion is not used.
   2) The input parameter ABSTOL, the absolute tolerance for
      the maximum column 2-norm of the residual matrix R22(K). This
      means that the factorization stops if this norm is less or
      equal to ABSTOL. If ABSTOL < 0.0, the criterion is not used.
   3) The input parameter RELTOL, the tolerance for the maximum
      column 2-norm matrix of the residual matrix R22(K) divided
      by the maximum column 2-norm of the original matrix A, which
      is equal to abs(R(1,1)). This means that the factorization stops
      when the ratio of the maximum column 2-norm of R22(K) to
      the maximum column 2-norm of A is less than or equal to RELTOL.
      If RELTOL < 0.0, the criterion is not used.
   4) In case both stopping criteria ABSTOL or RELTOL are not used,
      and when the residual matrix R22(K) is a zero matrix in some
      factorization step K. ( This stopping criterion is implicit. )
  The algorithm stops when any of these conditions is first
  satisfied, otherwise the whole matrix A is factorized.
  To factorize the whole matrix A, use the values
  KMAX >= min(M,N), ABSTOL < 0.0 and RELTOL < 0.0.
  The routine returns:
     a) Q(K), R(K)_approx = ( R11(K), R12(K) ),
        R(K)_residual = R22(K), P(K), i.e. the resulting matrices
        of the factorization; P(K) is represented by JPIV,
        ( if K = min(M,N), R(K)_approx is the full factor R,
        and there is no residual matrix R(K)_residual);
     b) K, the number of columns that were factorized,
        i.e. factorization rank;
     c) MAXC2NRMK, the maximum column 2-norm of the residual
        matrix R(K)_residual = R22(K),
        ( if K = min(M,N), MAXC2NRMK = 0.0 );
     d) RELMAXC2NRMK equals MAXC2NRMK divided by MAXC2NRM, the maximum
        column 2-norm of the original matrix A, which is equal
        to abs(R(1,1)), ( if K = min(M,N), RELMAXC2NRMK = 0.0 );
     e) Q(K)**H * B, the matrix B with the unitary
        transformation Q(K)**H applied on the left.
 The N-by-N permutation matrix P(K) is stored in a compact form in
 the integer array JPIV. For 1 <= j <= N, column j
 of the matrix A was interchanged with column JPIV(j).
 The M-by-M unitary matrix Q is represented as a product
 of elementary Householder reflectors
     Q(K) = H(1) *  H(2) * . . . * H(K),
 where K is the number of columns that were factorized.
 Each H(j) has the form
     H(j) = I - tau * v * v**H,
 where 1 <= j <= K and
   I    is an M-by-M identity matrix,
   tau  is a complex scalar,
   v    is a complex vector with v(1:j-1) = 0 and v(j) = 1.
 v(j+1:M) is stored on exit in A(j+1:M,j) and tau in TAU(j).
 See the Further Details section for more information.

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.

KMAX

          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,N), then this stopping criterion
                is not used, the routine factorizes 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 are not modified, and
                the matrix A is itself the residual.

ABSTOL

          ABSTOL is DOUBLE PRECISION
          The second factorization stopping criterion, cannot be NaN.
          The absolute tolerance (stopping threshold) for
          maximum column 2-norm of the residual matrix R22(K).
          The algorithm converges (stops the factorization) when
          the maximum column 2-norm of the residual matrix R22(K)
          is less than or equal to ABSTOL. Let SAFMIN = DLAMCH('S').
          a) If ABSTOL is NaN, then no computation is performed
                and an error message ( INFO = -5 ) is issued
                by XERBLA.
          b) 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.
          c) If 0.0 <= ABSTOL < 2*SAFMIN, then ABSTOL = 2*SAFMIN
                is used. This includes the case ABSTOL = -0.0.
          d) If 2*SAFMIN <= ABSTOL then the input value
                of ABSTOL is used.
          Let MAXC2NRM be the maximum column 2-norm of the
          whole original matrix A.
          If ABSTOL chosen above is >= MAXC2NRM, then this
          stopping criterion is satisfied on input and routine exits
          immediately after MAXC2NRM is computed. The routine
          returns MAXC2NRM in MAXC2NORMK,
          and 1.0 in RELMAXC2NORMK.
          This includes the case ABSTOL = +Inf. This means that the
          factorization is not performed, the matrices A and B are not
          modified, and the matrix A is itself the residual.

RELTOL

          RELTOL is DOUBLE PRECISION
          The third factorization stopping criterion, cannot be NaN.
          The tolerance (stopping threshold) for the ratio
          abs(R(K+1,K+1))/abs(R(1,1)) of the maximum column 2-norm of
          the residual matrix R22(K) to the maximum column 2-norm of
          the original matrix A. The algorithm converges (stops the
          factorization), when abs(R(K+1,K+1))/abs(R(1,1)) A is less
          than or equal to RELTOL. Let EPS = DLAMCH('E').
          a) If RELTOL is NaN, then no computation is performed
                and an error message ( INFO = -6 ) is issued
                by XERBLA.
          b) 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.
          c) If 0.0 <= RELTOL < EPS, then RELTOL = EPS is used.
                This includes the case RELTOL = -0.0.
          d) If EPS <= RELTOL then the input value of RELTOL
                is used.
          Let MAXC2NRM be the maximum column 2-norm of the
          whole original matrix A.
          If RELTOL chosen above is >= 1.0, then this stopping
          criterion is satisfied on input and routine exits
          immediately after MAXC2NRM is computed.
          The routine returns MAXC2NRM in MAXC2NORMK,
          and 1.0 in RELMAXC2NORMK.
          This includes the case RELTOL = +Inf. This means that the
          factorization is not performed, the matrices A and B are not
          modified, and the matrix A is itself the residual.
          NOTE: We recommend that RELTOL satisfy
                min( 10*max(M,N)*EPS, sqrt(EPS) ) <= RELTOL

A

          A is COMPLEX*16 array, dimension (LDA,N+NRHS)
          On entry:
          a) The subarray A(1:M,1:N) contains the M-by-N matrix A.
          b) The subarray A(1:M,N+1:N+NRHS) contains the M-by-NRHS
             matrix B.
                                  N     NRHS
              array_A   =   M  [ mat_A, mat_B ]
          On exit:
          a) The subarray A(1:M,1:N) contains parts of the factors
             of the matrix A:
            1) If K = 0, A(1:M,1:N) contains the original matrix A.
            2) If K > 0, A(1:M,1:N) contains parts of the
            factors:
              1. The elements below the diagonal of the subarray
                 A(1:M,1:K) together with TAU(1:K) represent the
                 unitary matrix Q(K) as a product of K Householder
                 elementary reflectors.
              2. The elements on and above the diagonal of
                 the subarray A(1:K,1:N) contain K-by-N
                 upper-trapezoidal matrix
                 R(K)_approx = ( R11(K), R12(K) ).
                 NOTE: If K=min(M,N), i.e. full rank factorization,
                       then R_approx(K) is the full factor R which
                       is upper-trapezoidal. If, in addition, M>=N,
                       then R is upper-triangular.
              3. The subarray A(K+1:M,K+1:N) contains (M-K)-by-(N-K)
                 rectangular matrix R(K)_residual = R22(K).
          b) If NRHS > 0, the subarray A(1:M,N+1:N+NRHS) contains
             the M-by-NRHS product Q(K)**H * B.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).
          This is the leading dimension for both matrices, A and B.

K

          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,KMAX,N).
          K also represents the number of non-zero Householder
          vectors.
          NOTE: If K = 0, a) the arrays A and B are not modified;
                          b) the array TAU(1:min(M,N)) is set to ZERO,
                             if the matrix A does not contain NaN,
                             otherwise the elements TAU(1:min(M,N))
                             are undefined;
                          c) the elements of the array JPIV are set
                             as follows: for j = 1:N, JPIV(j) = j.

MAXC2NRMK

          MAXC2NRMK is DOUBLE PRECISION
          The maximum column 2-norm of the residual matrix R22(K),
          when the factorization stopped at rank K. MAXC2NRMK >= 0.
          a) If K = 0, i.e. the factorization was not performed,
             the matrix A was not modified and is itself a residual
             matrix, then MAXC2NRMK equals the maximum column 2-norm
             of the original matrix A.
          b) If 0 < K < min(M,N), then MAXC2NRMK is returned.
          c) If K = min(M,N), i.e. the whole matrix A was
             factorized and there is no residual matrix,
             then MAXC2NRMK = 0.0.
          NOTE: MAXC2NRMK in the factorization step K would equal
                R(K+1,K+1) in the next factorization step K+1.

RELMAXC2NRMK

          RELMAXC2NRMK is DOUBLE PRECISION
          The ratio MAXC2NRMK / MAXC2NRM of the maximum column
          2-norm of the residual matrix R22(K) (when the factorization
          stopped at rank K) to the maximum column 2-norm of the
          whole original matrix A. RELMAXC2NRMK >= 0.
          a) If K = 0, i.e. the factorization was not performed,
             the matrix A was not modified and is itself a residual
             matrix, then RELMAXC2NRMK = 1.0.
          b) If 0 < K < min(M,N), then
                RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM is returned.
          c) If K = min(M,N), i.e. the whole matrix A was
             factorized and there is no residual matrix,
             then RELMAXC2NRMK = 0.0.
         NOTE: RELMAXC2NRMK in the factorization step K would equal
               abs(R(K+1,K+1))/abs(R(1,1)) in the next factorization
               step K+1.

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).
          The elements of the array JPIV(1:N) are always set
          by the routine, for example, even  when no columns
          were factorized, i.e. when K = 0, the elements are
          set as JPIV(j) = j for j = 1:N.

TAU

          TAU is COMPLEX*16 array, dimension (min(M,N))
          The scalar factors of the elementary reflectors.
          If 0 < K <= min(M,N), only the elements TAU(1:K) of
          the array TAU are modified by the factorization.
          After the factorization computed, if no NaN was found
          during the factorization, the remaining elements
          TAU(K+1:min(M,N)) are set to zero, otherwise the
          elements TAU(K+1:min(M,N)) are not set and therefore
          undefined.
          ( If K = 0, all elements of TAU are set to zero, if
          the matrix A does not contain NaN. )

WORK

          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
For optimal performance LWORK >= NB*( N+NRHS+1 ),
          where NB is the optimal block size for ZGEQP3RK returned
          by ILAENV. Minimal block size MINNB=2.
          NOTE: The decision, whether to use unblocked BLAS 2
          or blocked BLAS 3 code is based not only on the dimension
          LWORK of the availbale workspace WORK, but also also on the
          matrix A dimension N via crossover point NX returned
          by ILAENV. (For N less than NX, unblocked code should be
          used.)
          If LWORK = -1, then a workspace query is assumed;
          the routine only calculates the optimal size of the WORK
          array, returns this value as the first entry of the WORK
          array, and no error message related to LWORK is issued
          by XERBLA.

RWORK

          RWORK is DOUBLE PRECISION array, dimension (2*N)

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 in the blocked step auxiliary subroutine ZLAQP3RK ).

INFO

          INFO is INTEGER
          1) INFO = 0: successful exit.
          2) INFO < 0: if INFO = -i, the i-th argument had an
                       illegal value.
          3) 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.
          4) 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 ).

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 ZGEQP3RK is based on the same BLAS3 Householder QR factorization
 algorithm with column pivoting as in ZGEQP3 routine which uses
 ZLARFG routine to generate Householder reflectors
 for QR factorization.
 We can also write:
   A = A_approx(K) + A_residual(K)
 The low rank approximation matrix A(K)_approx from
 the truncated QR factorization of rank K of the matrix A is:
   A(K)_approx = Q(K) * ( R(K)_approx ) * P(K)**T
                        (     0     0 )
               = Q(K) * ( R11(K) R12(K) ) * P(K)**T
                        (      0      0 )
 The residual A_residual(K) of the matrix A is:
   A_residual(K) = Q(K) * ( 0              0 ) * P(K)**T =
                          ( 0  R(K)_residual )
                 = Q(K) * ( 0        0 ) * P(K)**T
                          ( 0   R22(K) )
 The truncated (rank K) factorization guarantees that
 the maximum column 2-norm of A_residual(K) is less than
 or equal to MAXC2NRMK up to roundoff error.
 NOTE: An approximation of the null vectors
       of A can be easily computed from R11(K)
       and R12(K):
       Null( A(K) )_approx = P * ( inv(R11(K)) * R12(K) )
                                 (         -I           )

[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.

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