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



NAME
       ZGGGLM

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



PURPOSE
            ZGGGLM solves a general Gauss-Markov linear model (GLM) problem:

                    minimize || y ||_2   subject to   d = A*x + B*y
                        x

            where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
            given N-vector. It is assumed that M <= N <= M+P, and

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

            Under these assumptions, the constrained equation is always
            consistent, and there is a unique solution x and a minimal 2-norm
            solution y, which is obtained using a generalized QR factorization
            of the matrices (A, B) given by

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

            In particular, if matrix B is square nonsingular, then the problem
            GLM is equivalent to the following weighted linear least squares
            problem

                         minimize || inv(B)*(d-A*x) ||_2
                             x

            where inv(B) denotes the inverse of B.




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

           M         (input)
                     M is INTEGER
                     The number of columns of the matrix A.  0 <= M <= N.

           P         (input)
                     P is INTEGER
                     The number of columns of the matrix B.  P >= N-M.

           A         (input/output)
                     A is COMPLEX*16 array, dimension (LDA,M)
                     On entry, the N-by-M matrix A.
                     On exit, the upper triangular part of the array A contains
                     the M-by-M upper triangular matrix R.

           LDA       (input)
                     LDA is INTEGER
                     The leading dimension of the array A. LDA >= max(1,N).

           B         (input/output)
                     B is COMPLEX*16 array, dimension (LDB,P)
                     On entry, the N-by-P matrix B.
                     On exit, if N <= P, the upper triangle of the subarray
                     B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
                     if N > P, the elements on and above the (N-P)th subdiagonal
                     contain the N-by-P upper trapezoidal matrix T.

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

           D         (input/output)
                     D is COMPLEX*16 array, dimension (N)
                     On entry, D is the left hand side of the GLM equation.
                     On exit, D is destroyed.

           X         (output)
                     X is COMPLEX*16 array, dimension (M)

           Y         (output)
                     Y is COMPLEX*16 array, dimension (P)

                     On exit, X and Y are the solutions of the GLM 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,N+M+P).
                     For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
                     where NB is an upper bound for the optimal blocksizes for
                     ZGEQRF, ZGERQF, ZUNMQR and ZUNMRQ.

                     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 A in the
                           generalized QR factorization of the pair (A, B) is
                           singular, so that rank(A) < M; the least squares
                           solution could not be computed.
                     = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
                           factor T associated with B in the generalized QR
                           factorization of the pair (A, B) is singular, so that
                           rank( A B ) < N; the least squares solution could not
                           be computed.



LAPACK routine                  31 October 2017                      ZGGGLM(3)