DGGGLM 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.

N

          N is INTEGER


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

M

          M is INTEGER


          The number of columns of the matrix A.  0 <= M <= N.

P

          P is INTEGER


          The number of columns of the matrix B.  P >= N-M.

A

          A is DOUBLE PRECISION 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

          LDA is INTEGER


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

B

          B is DOUBLE PRECISION 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

          LDB is INTEGER


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

D

          D is DOUBLE PRECISION array, dimension (N)


          On entry, D is the left hand side of the GLM equation.


          On exit, D is destroyed.

X

          X is DOUBLE PRECISION array, dimension (M)

Y

          Y is DOUBLE PRECISION array, dimension (P)


          On exit, X and Y are the solutions of the GLM problem.

WORK

          WORK is DOUBLE PRECISION 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,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


          DGEQRF, SGERQF, DORMQR 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 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.

Univ. of Tennessee

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NAG Ltd.

December 2016