!>
!> DLATSQR computes a blocked Tall-Skinny QR factorization of
!> a real M-by-N matrix A for M >= N:
!>
!>    A = Q * ( R ),
!>            ( 0 )
!>
!> where:
!>
!>    Q is a M-by-M orthogonal matrix, stored on exit in an implicit
!>    form in the elements below the diagonal of the array A and in
!>    the elements of the array T;
!>
!>    R is an upper-triangular N-by-N matrix, stored on exit in
!>    the elements on and above the diagonal of the array A.
!>
!>    0 is a (M-N)-by-N zero matrix, and is not stored.
!>
!> 

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.
!> 

MB

!>          MB is INTEGER
!>          The row block size to be used in the blocked QR.
!>          MB > 0.
!> 

NB

!>          NB is INTEGER
!>          The column block size to be used in the blocked QR.
!>          N >= NB >= 1.
!> 

A

!>          A is DOUBLE PRECISION array, dimension (LDA,N)
!>          On entry, the M-by-N matrix A.
!>          On exit, the elements on and above the diagonal
!>          of the array contain the N-by-N upper triangular matrix R;
!>          the elements below the diagonal represent Q by the columns
!>          of blocked V (see Further Details).
!> 

LDA

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

T

!>          T is DOUBLE PRECISION array,
!>          dimension (LDT, N * Number_of_row_blocks)
!>          where Number_of_row_blocks = CEIL((M-N)/(MB-N))
!>          The blocked upper triangular block reflectors stored in compact form
!>          as a sequence of upper triangular blocks.
!>          See Further Details below.
!> 

LDT

!>          LDT is INTEGER
!>          The leading dimension of the array T.  LDT >= NB.
!> 

WORK

!>         (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
!> 

LWORK

!>          LWORK is INTEGER
!>          The dimension of the array WORK.  LWORK >= NB*N.
!>          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.
!> 

INFO

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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

!> Tall-Skinny QR (TSQR) performs QR by a sequence of orthogonal transformations,
!> representing Q as a product of other orthogonal matrices
!>   Q = Q(1) * Q(2) * . . . * Q(k)
!> where each Q(i) zeros out subdiagonal entries of a block of MB rows of A:
!>   Q(1) zeros out the subdiagonal entries of rows 1:MB of A
!>   Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
!>   Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
!>   . . .
!>
!> Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors
!> stored under the diagonal of rows 1:MB of A, and by upper triangular
!> block reflectors, stored in array T(1:LDT,1:N).
!> For more information see Further Details in GEQRT.
!>
!> Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors
!> stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular
!> block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N).
!> The last Q(k) may use fewer rows.
!> For more information see Further Details in TPQRT.
!>
!> For more details of the overall algorithm, see the description of
!> Sequential TSQR in Section 2.2 of [1].
!>
!> [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
!>     J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
!>     SIAM J. Sci. Comput, vol. 34, no. 1, 2012
!>