SGELSS computes the minimum norm solution to a real linear least


 squares problem:


 Minimize 2-norm(| b - A*x |).


 using the singular value decomposition (SVD) 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 effective rank of A is determined by treating as zero those


 singular values which are less than RCOND times the largest singular


 value.

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, the first min(m,n) rows of A are overwritten with


          its right singular vectors, stored rowwise.

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, B is overwritten by the N-by-NRHS solution


          matrix X.  If m >= n and RANK = n, the residual


          sum-of-squares for the solution in the i-th column is given


          by the sum of squares of elements n+1:m in that column.

LDB

          LDB is INTEGER


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

S

          S is REAL array, dimension (min(M,N))


          The singular values of A in decreasing order.


          The condition number of A in the 2-norm = S(1)/S(min(m,n)).

RCOND

          RCOND is REAL


          RCOND is used to determine the effective rank of A.


          Singular values S(i) <= RCOND*S(1) are treated as zero.


          If RCOND < 0, machine precision is used instead.

RANK

          RANK is INTEGER


          The effective rank of A, i.e., the number of singular values


          which are greater than RCOND*S(1).

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 >= 1, and also:


          LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )


          For good performance, LWORK should generally be larger.


          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:  the algorithm for computing the SVD failed to converge;


                if INFO = i, i off-diagonal elements of an intermediate


                bidiagonal form did not converge to zero.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016