ZGGLSE(3)      LAPACK routine of NEC Numeric Library Collection      ZGGLSE(3)



NAME
       ZGGLSE

SYNOPSIS
       SUBROUTINE ZGGLSE (M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, INFO)



PURPOSE
            ZGGLSE 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.




ARGUMENTS
           M         (input)
                     M is INTEGER
                     The number of rows of the matrix A.  M >= 0.

           N         (input)
                     N is INTEGER
                     The number of columns of the matrices A and B. N >= 0.

           P         (input)
                     P is INTEGER
                     The number of rows of the matrix B. 0 <= P <= N <= M+P.

           A         (input/output)
                     A is COMPLEX*16 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       (input)
                     LDA is INTEGER
                     The leading dimension of the array A. LDA >= max(1,M).

           B         (input/output)
                     B is COMPLEX*16 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       (input)
                     LDB is INTEGER
                     The leading dimension of the array B. LDB >= max(1,P).

           C         (input/output)
                     C is COMPLEX*16 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         (input/output)
                     D is COMPLEX*16 array, dimension (P)
                     On entry, D contains the right hand side vector for the
                     constrained equation.
                     On exit, D is destroyed.

           X         (output)
                     X is COMPLEX*16 array, dimension (N)
                     On exit, X is the solution of the LSE problem.

           WORK      (output)
                     WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK     (input)
                     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
                     ZGEQRF, CGERQF, ZUNMQR and CUNMRQ.

                     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      (output)
                     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.



LAPACK routine                  31 October 2017                      ZGGLSE(3)