SRC/dgeqp3rk.f(3)

NAME
SRC/dgeqp3rk.f

SYNOPSIS

  Functions/Subroutines
subroutine dgeqp3rk (m, n, nrhs, kmax, abstol, reltol, a, lda, k, maxc2nrmk, relmaxc2nrmk, jpiv, tau, work, lwork, iwork, info)
DGEQP3RK computes a truncated Householder QR factorization with column pivoting of a real m-by-n matrix A by using Level 3 BLAS and overwrites a real m-by-nrhs matrix B with Q**T * B.

Function/Subroutine Documentation

  subroutine dgeqp3rk (integer m, integer n, integer nrhs, integer kmax, double precision abstol, double precision reltol, double precision, dimension( lda, * ) a, integer lda, integer k, double precision maxc2nrmk, double precision relmaxc2nrmk, integer, dimension( * ) jpiv, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

DGEQP3RK computes a truncated Householder QR factorization with column pivoting of a real m-by-n matrix A by using Level 3 BLAS and overwrites a real m-by-nrhs matrix B with Q**T * B.

Purpose:

    !>
    !> DGEQP3RK performs two tasks simultaneously:
    !>
    !> Task 1: The routine computes a truncated (rank K) or full rank
    !> Householder QR factorization with column pivoting of a real
    !> 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 orthogonal 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 real M-by-NRHS
    !> matrix B with  Q(K)**T * 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)**T * B, the matrix B with the orthogonal
    !>        transformation Q(K)**T 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 orthogonal 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**T,
    !>
    !> where 1 <= j <= K and
    !>   I    is an M-by-M identity matrix,
    !>   tau  is a real scalar,
    !>   v    is a real 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.
    !>

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.
    !>

    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( max(M,N)*EPS, sqrt(EPS) ) <= RELTOL
    !>

    A

    !>          A is DOUBLE PRECISION 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
    !>                 orthogonal 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)**T * 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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.
    !>          LWORK >= 1, if MIN(M,N) = 0, and
    !>          LWORK >= (3*N+NRHS-1), otherwise.
    !>          For optimal performance LWORK >= (2*N + NB*( N+NRHS+1 )),
    !>          where NB is the optimal block size for DGEQP3RK 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.
    !>

    IWORK

    !>          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 in the blocked step auxiliary subroutine DLAQP3RK ).
    !>

    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 ).
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !> DGEQP3RK is based on the same BLAS3 Householder QR factorization
    !> algorithm with column pivoting as in DGEQP3 routine which uses
    !> DLARFG 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           )
    !>
    !>

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 583 of file dgeqp3rk.f.

Author

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