SGETSLS solves overdetermined or underdetermined real linear systems


 involving an M-by-N matrix A, using a tall skinny QR or short wide LQ


 factorization of A.  It is assumed that A has full rank.


 The following options are provided:


 1. If TRANS = 'N' and m >= n:  find the least squares solution of


    an overdetermined system, i.e., solve the least squares problem


                 minimize || B - A*X ||.


 2. If TRANS = 'N' and m < n:  find the minimum norm solution of


    an underdetermined system A * X = B.


 3. If TRANS = 'T' and m >= n:  find the minimum norm solution of


    an undetermined system A**T * X = B.


 4. If TRANS = 'T' and m < n:  find the least squares solution of


    an overdetermined system, i.e., solve the least squares problem


                 minimize || B - A**T * X ||.


 Several right hand side vectors b and solution vectors x can be


 handled in a single call; they are stored as the columns of the


 M-by-NRHS right hand side matrix B and the N-by-NRHS solution


 matrix X.

TRANS

          TRANS is CHARACTER*1


          = 'N': the linear system involves A;


          = 'T': the linear system involves A**T.

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.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of


          columns of the matrices B and X. NRHS >=0.

A

          A is REAL array, dimension (LDA,N)


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


          On exit,


          A is overwritten by details of its QR or LQ


          factorization as returned by SGEQR or SGELQ.

LDA

          LDA is INTEGER


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

B

          B is REAL array, dimension (LDB,NRHS)


          On entry, the matrix B of right hand side vectors, stored


          columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS


          if TRANS = 'T'.


          On exit, if INFO = 0, B is overwritten by the solution


          vectors, stored columnwise:


          if TRANS = 'N' and m >= n, rows 1 to n of B contain the least


          squares solution vectors.


          if TRANS = 'N' and m < n, rows 1 to N of B contain the


          minimum norm solution vectors;


          if TRANS = 'T' and m >= n, rows 1 to M of B contain the


          minimum norm solution vectors;


          if TRANS = 'T' and m < n, rows 1 to M of B contain the


          least squares solution vectors.

LDB

          LDB is INTEGER


          The leading dimension of the array B. LDB >= MAX(1,M,N).

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.


          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


          > 0:  if INFO =  i, the i-th diagonal element of the


                triangular factor of A is zero, so that A does not have


                full rank; the least squares solution could not be


                computed.

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