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## dorhr_col()

subroutine dorhr_col | ( | integer | m,
|  | integer | n,
|  | integer | nb,
|  | double precision, dimension( lda, * ) | a,
|  | integer | lda,
|  | double precision, dimension( ldt, * ) | t,
|  | integer | ldt,
|  | double precision, dimension( * ) | d,
|  | integer | info )

DORHR_COL

Download DORHR_COL + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !>  DORHR_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 DGEQRT).
      !>

Parameters

  [in]

  M

      !>          M is INTEGER
      !>          The number of rows of the matrix A. M >= 0.
      !>

  [in]

  N

      !>          N is INTEGER
      !>          The number of columns of the matrix A. M >= N >= 0.
      !>

  [in]

  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.)
      !>

  [in,out]

  A

      !>          A is DOUBLE PRECISION 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
      !>             DGEQRT). 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  LU-decomposition:
      !>                Q_in - ( S ) = V * U
      !>                       ( 0 )
      !>             where 0 is a (M-N)-by-(M-N) zero matrix.
      !>

  [in]

  LDA

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

  [out]

  T

      !>          T is DOUBLE PRECISION 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 DGEQRT).
      !>          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.
      !>

  [in]

  LDT

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

  [out]

  D

      !>          D is DOUBLE PRECISION 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.
      !>

  [out]

  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] ,
      !>     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
      !>
      !>