!>
!>  ZUNHR_COL takes an M-by-N complex 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 ZGEQRT).
!> 

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 COMPLEX*16 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
!>             ZGEQRT). 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.
!> 

LDA

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

T

!>          T is COMPLEX*16 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 ZGEQRT).
!>          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 COMPLEX*16 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
!> 

!>
!> The computed M-by-M unitary factor Q_out is defined implicitly as
!> a product of unitary 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
!> unitary 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.
!> 

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NAG Ltd.

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