SGELQ computes an LQ factorization of a real M-by-N matrix A:


    A = ( L 0 ) *  Q


 where:


    Q is a N-by-N orthogonal matrix;


    L is a lower-triangular M-by-M matrix;


    0 is a M-by-(N-M) zero matrix, if M < N.

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.  N >= 0.

A

          A is REAL array, dimension (LDA,N)


          On entry, the M-by-N matrix A.


          On exit, the elements on and below the diagonal of the array


          contain the M-by-min(M,N) lower trapezoidal matrix L


          (L is lower triangular if M <= N);


          the elements above the diagonal are used to store part of the 


          data structure to represent Q.

LDA

          LDA is INTEGER


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

T

          T is REAL array, dimension (MAX(5,TSIZE))


          On exit, if INFO = 0, T(1) returns optimal (or either minimal 


          or optimal, if query is assumed) TSIZE. See TSIZE for details.


          Remaining T contains part of the data structure used to represent Q.


          If one wants to apply or construct Q, then one needs to keep T 


          (in addition to A) and pass it to further subroutines.

TSIZE

          TSIZE is INTEGER


          If TSIZE >= 5, the dimension of the array T.


          If TSIZE = -1 or -2, then a workspace query is assumed. The routine


          only calculates the sizes of the T and WORK arrays, returns these


          values as the first entries of the T and WORK arrays, and no error


          message related to T or WORK is issued by XERBLA.


          If TSIZE = -1, the routine calculates optimal size of T for the 


          optimum performance and returns this value in T(1).


          If TSIZE = -2, the routine calculates minimal size of T and 


          returns this value in T(1).

WORK

          (workspace) REAL array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) contains optimal (or either minimal


          or optimal, if query was assumed) LWORK.


          See LWORK for details.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.


          If LWORK = -1 or -2, then a workspace query is assumed. The routine


          only calculates the sizes of the T and WORK arrays, returns these


          values as the first entries of the T and WORK arrays, and no error


          message related to T or WORK is issued by XERBLA.


          If LWORK = -1, the routine calculates optimal size of WORK for the


          optimal performance and returns this value in WORK(1).


          If LWORK = -2, the routine calculates minimal size of WORK and 


          returns this value in WORK(1).

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value

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 The goal of the interface is to give maximum freedom to the developers for


 creating any LQ factorization algorithm they wish. The triangular 


 (trapezoidal) L has to be stored in the lower part of A. The lower part of A


 and the array T can be used to store any relevant information for applying or


 constructing the Q factor. The WORK array can safely be discarded after exit.


 Caution: One should not expect the sizes of T and WORK to be the same from one 


 LAPACK implementation to the other, or even from one execution to the other.


 A workspace query (for T and WORK) is needed at each execution. However, 


 for a given execution, the size of T and WORK are fixed and will not change 


 from one query to the next.

 These details are particular for this LAPACK implementation. Users should not 


 take them for granted. These details may change in the future, and are not likely


 true for another LAPACK implementation. These details are relevant if one wants


 to try to understand the code. They are not part of the interface.


 In this version,


          T(2): row block size (MB)


          T(3): column block size (NB)


          T(6:TSIZE): data structure needed for Q, computed by


                           SLASWLQ or SGELQT


  Depending on the matrix dimensions M and N, and row and column


  block sizes MB and NB returned by ILAENV, SGELQ will use either


  SLASWLQ (if the matrix is short-and-wide) or SGELQT to compute


  the LQ factorization.