SGELSY computes the minimum-norm solution to a real linear least


 squares problem:


     minimize || A * X - B ||


 using a complete orthogonal factorization of A.  A is an M-by-N


 matrix which may be rank-deficient.


 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.


 The routine first computes a QR factorization with column pivoting:


     A * P = Q * [ R11 R12 ]


                 [  0  R22 ]


 with R11 defined as the largest leading submatrix whose estimated


 condition number is less than 1/RCOND.  The order of R11, RANK,


 is the effective rank of A.


 Then, R22 is considered to be negligible, and R12 is annihilated


 by orthogonal transformations from the right, arriving at the


 complete orthogonal factorization:


    A * P = Q * [ T11 0 ] * Z


                [  0  0 ]


 The minimum-norm solution is then


    X = P * Z**T [ inv(T11)*Q1**T*B ]


                 [        0         ]


 where Q1 consists of the first RANK columns of Q.


 This routine is basically identical to the original xGELSX except


 three differences:


   o The call to the subroutine xGEQPF has been substituted by the


     the call to the subroutine xGEQP3. This subroutine is a Blas-3


     version of the QR factorization with column pivoting.


   o Matrix B (the right hand side) is updated with Blas-3.


   o The permutation of matrix B (the right hand side) is faster and


     more simple.

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 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 has been overwritten by details of its


          complete orthogonal factorization.

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 M-by-NRHS right hand side matrix B.


          On exit, the N-by-NRHS solution matrix X.


          If M = 0 or N = 0, B is not referenced.

LDB

          LDB is INTEGER


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

JPVT

          JPVT is INTEGER array, dimension (N)


          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted


          to the front of AP, otherwise column i is a free column.


          On exit, if JPVT(i) = k, then the i-th column of AP


          was the k-th column of A.

RCOND

          RCOND is REAL


          RCOND is used to determine the effective rank of A, which


          is defined as the order of the largest leading triangular


          submatrix R11 in the QR factorization with pivoting of A,


          whose estimated condition number < 1/RCOND.

RANK

          RANK is INTEGER


          The effective rank of A, i.e., the order of the submatrix


          R11.  This is the same as the order of the submatrix T11


          in the complete orthogonal factorization of A.


          If NRHS = 0, RANK = 0 on output.

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.


          The unblocked strategy requires that:


             LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),


          where MN = min( M, N ).


          The block algorithm requires that:


             LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),


          where NB is an upper bound on the blocksize returned


          by ILAENV for the routines SGEQP3, STZRZF, STZRQF, SORMQR,


          and SORMRZ.


          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.

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain