SGELS solves overdetermined or underdetermined real linear systems


 involving an M-by-N matrix A, or its transpose, using a QR or 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,


            if M >= N, A is overwritten by details of its QR


                       factorization as returned by SGEQRF;


            if M <  N, A is overwritten by details of its LQ


                       factorization as returned by SGELQF.

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; the residual sum of squares for the


          solution in each column is given by the sum of squares of


          elements N+1 to M in that column;


          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; the residual sum of squares


          for the solution in each column is given by the sum of


          squares of elements M+1 to N in that column.

LDB

          LDB is INTEGER


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

WORK

          WORK is REAL array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.


          LWORK >= max( 1, MN + max( MN, NRHS ) ).


          For optimal performance,


          LWORK >= max( 1, MN + max( MN, NRHS )*NB ).


          where MN = min(M,N) and NB is the optimum block size.


          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


          > 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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011