SGGLSE solves the linear equality-constrained least squares (LSE)


 problem:


         minimize || c - A*x ||_2   subject to   B*x = d


 where A is an M-by-N matrix, B is a P-by-N matrix, c is a given


 M-vector, and d is a given P-vector. It is assumed that


 P <= N <= M+P, and


          rank(B) = P and  rank( (A) ) = N.


                               ( (B) )


 These conditions ensure that the LSE problem has a unique solution,


 which is obtained using a generalized RQ factorization of the


 matrices (B, A) given by


    B = (0 R)*Q,   A = Z*T*Q.

M

          M is INTEGER


          The number of rows of the matrix A.  M >= 0.

N

          N is INTEGER


          The number of columns of the matrices A and B. N >= 0.

P

          P is INTEGER


          The number of rows of the matrix B. 0 <= P <= N <= M+P.

A

          A is REAL array, dimension (LDA,N)


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


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


          contain the min(M,N)-by-N upper trapezoidal matrix T.

LDA

          LDA is INTEGER


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

B

          B is REAL array, dimension (LDB,N)


          On entry, the P-by-N matrix B.


          On exit, the upper triangle of the subarray B(1:P,N-P+1:N)


          contains the P-by-P upper triangular matrix R.

LDB

          LDB is INTEGER


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

C

          C is REAL array, dimension (M)


          On entry, C contains the right hand side vector for the


          least squares part of the LSE problem.


          On exit, the residual sum of squares for the solution


          is given by the sum of squares of elements N-P+1 to M of


          vector C.

D

          D is REAL array, dimension (P)


          On entry, D contains the right hand side vector for the


          constrained equation.


          On exit, D is destroyed.

X

          X is REAL array, dimension (N)


          On exit, X is the solution of the LSE problem.

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,M+N+P).


          For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,


          where NB is an upper bound for the optimal blocksizes for


          SGEQRF, SGERQF, SORMQR and SORMRQ.


          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.


          = 1:  the upper triangular factor R associated with B in the


                generalized RQ factorization of the pair (B, A) is


                singular, so that rank(B) < P; the least squares


                solution could not be computed.


          = 2:  the (N-P) by (N-P) part of the upper trapezoidal factor


                T associated with A in the generalized RQ factorization


                of the pair (B, A) is singular, so that


                rank( (A) ) < N; the least squares solution could not


                    ( (B) )


                be computed.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011