sorhr_col.f(3)

NAME
sorhr_col.f

SYNOPSIS

  Functions/Subroutines
subroutine sorhr_col (m, n, nb, a, lda, t, ldt, d, info)
SORHR_COL

Function/Subroutine Documentation

  subroutine sorhr_col (integer m, integer n, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( ldt, * ) t, integer ldt, real, dimension( * ) d, integer info)

SORHR_COL

Purpose:

      SORHR_COL takes an M-by-N real matrix Q_in with orthonormal columns
      as input, stored in A, and performs Householder Reconstruction (HR),
      i.e. reconstructs Householder vectors V(i) implicitly representing
      another M-by-N matrix Q_out, with the property that Q_in = Q_out*S,
      where S is an N-by-N diagonal matrix with diagonal entries
      equal to +1 or -1. The Householder vectors (columns V(i) of V) are
      stored in A on output, and the diagonal entries of S are stored in D.
      Block reflectors are also returned in T
      (same output format as SGEQRT).

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. M >= N >= 0.

    NB

              NB is INTEGER
              The column block size to be used in the reconstruction
              of Householder column vector blocks in the array A and
              corresponding block reflectors in the array T. NB >= 1.
              (Note that if NB > N, then N is used instead of NB
              as the column block size.)

    A

              A is REAL array, dimension (LDA,N)
              On entry:
                 The array A contains an M-by-N orthonormal matrix Q_in,
                 i.e the columns of A are orthogonal unit vectors.
              On exit:
                 The elements below the diagonal of A represent the unit
                 lower-trapezoidal matrix V of Householder column vectors
                 V(i). The unit diagonal entries of V are not stored
                 (same format as the output below the diagonal in A from
                 SGEQRT). The matrix T and the matrix V stored on output
                 in A implicitly define Q_out.
                 The elements above the diagonal contain the factor U
                 of the 'modified' LU-decomposition:
                    Q_in - ( S ) = V * U
                           ( 0 )
                 where 0 is a (M-N)-by-(M-N) zero matrix.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    T

              T is REAL array,
              dimension (LDT, N)
              Let NOCB = Number_of_output_col_blocks
                       = CEIL(N/NB)
              On exit, T(1:NB, 1:N) contains NOCB upper-triangular
              block reflectors used to define Q_out stored in compact
              form as a sequence of upper-triangular NB-by-NB column
              blocks (same format as the output T in SGEQRT).
              The matrix T and the matrix V stored on output in A
              implicitly define Q_out. NOTE: The lower triangles
              below the upper-triangular blocks will be filled with
              zeros. See Further Details.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.
              LDT >= max(1,min(NB,N)).

    D

              D is REAL array, dimension min(M,N).
              The elements can be only plus or minus one.
              D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where
              1 <= i <= min(M,N), and Q_in_i is Q_in after performing
              i-1 steps of “modified” Gaussian elimination.
              See Further Details.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Further Details:

     The computed M-by-M orthogonal factor Q_out is defined implicitly as
     a product of orthogonal matrices Q_out(i). Each Q_out(i) is stored in
     the compact WY-representation format in the corresponding blocks of
     matrices V (stored in A) and T.
     The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N
     matrix A contains the column vectors V(i) in NB-size column
     blocks VB(j). For example, VB(1) contains the columns
     V(1), V(2), ... V(NB). NOTE: The unit entries on
     the diagonal of Y are not stored in A.
     The number of column blocks is
         NOCB = Number_of_output_col_blocks = CEIL(N/NB)
     where each block is of order NB except for the last block, which
     is of order LAST_NB = N - (NOCB-1)*NB.
     For example, if M=6,  N=5 and NB=2, the matrix V is
         V = (    VB(1),   VB(2), VB(3) ) =
           = (   1                      )
             ( v21    1                 )
             ( v31  v32    1            )
             ( v41  v42  v43   1        )
             ( v51  v52  v53  v54    1  )
             ( v61  v62  v63  v54   v65 )
     For each of the column blocks VB(i), an upper-triangular block
     reflector TB(i) is computed. These blocks are stored as
     a sequence of upper-triangular column blocks in the NB-by-N
     matrix T. The size of each TB(i) block is NB-by-NB, except
     for the last block, whose size is LAST_NB-by-LAST_NB.
     For example, if M=6,  N=5 and NB=2, the matrix T is
         T  = (    TB(1),    TB(2), TB(3) ) =
            = ( t11  t12  t13  t14   t15  )
              (      t22       t24        )
     The M-by-M factor Q_out is given as a product of NOCB
     orthogonal M-by-M matrices Q_out(i).
         Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB),
     where each matrix Q_out(i) is given by the WY-representation
     using corresponding blocks from the matrices V and T:
         Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T,
     where I is the identity matrix. Here is the formula with matrix
     dimensions:
      Q(i){M-by-M} = I{M-by-M} -
        VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M},
     where INB = NB, except for the last block NOCB
     for which INB=LAST_NB.
     =====
     NOTE:
     =====
     If Q_in is the result of doing a QR factorization
     B = Q_in * R_in, then:
     B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out.
     So if one wants to interpret Q_out as the result
     of the QR factorization of B, then the corresponding R_out
     should be equal to R_out = S * R_in, i.e. some rows of R_in
     should be multiplied by -1.
     For the details of the algorithm, see [1].
     [1] 'Reconstructing Householder vectors from tall-skinny QR',
         G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
         E. Solomonik, J. Parallel Distrib. Comput.,
         vol. 85, pp. 3-31, 2015.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

     November   2019, Igor Kozachenko,
                Computer Science Division,
                University of California, Berkeley

Definition at line 258 of file sorhr_col.f.

Author

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