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 digonal of the array A and in


    the elemenst 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 > N.

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

          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