unhr_col(3)

NAME
unhr_col - {un,or}hr_col: Householder reconstruction

SYNOPSIS

  Functions

subroutine cunhr_col (m, n, nb, a, lda, t, ldt, d, info)
CUNHR_COL
subroutine dorhr_col (m, n, nb, a, lda, t, ldt, d, info)
DORHR_COL
subroutine sorhr_col (m, n, nb, a, lda, t, ldt, d, info)
SORHR_COL
subroutine zunhr_col (m, n, nb, a, lda, t, ldt, d, info)
ZUNHR_COL

Detailed Description

Function Documentation

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

CUNHR_COL

Purpose:

    !>
    !>  CUNHR_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 CGEQRT).
    !>

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 COMPLEX 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
    !>             CGEQRT). 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 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 CGEQRT).
    !>          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 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 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.
    !>

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
    !>
    !>

  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

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
    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 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.
    !>

    LDA

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

    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.
    !>

    LDT

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

    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.
    !>

    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
    !>
    !>

  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  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] ,
    !>     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
    !>
    !>

  subroutine zunhr_col (integer m, integer n, integer nb, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( * ) d, integer info)

ZUNHR_COL

Purpose:

    !>
    !>  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).
    !>

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 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
    !>

Further Details:

    !>
    !> 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.
    !>

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
    !>
    !>

Author

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