DGGGLM(1) LAPACK driver routine (version 3.2) DGGGLM(1)NAME
DGGGLM - solves a general Gauss-Markov linear model (GLM) problem
SYNOPSIS
SUBROUTINE DGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK, INFO
)
INTEGER INFO, LDA, LDB, LWORK, M, N, P
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), D( * ), WORK( *
), X( * ), Y( * )
PURPOSE
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.
ARGUMENTS
N (input) INTEGER
The number of rows of the matrices A and B. N >= 0.
M (input) INTEGER
The number of columns of the matrix A. 0 <= M <= N.
P (input) INTEGER
The number of columns of the matrix B. P >= N-M.
A (input/output) 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 (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) 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 (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
D (input/output) DOUBLE PRECISION array, dimension (N)
On entry, D is the left hand side of the GLM equation. On
exit, D is destroyed.
X (output) DOUBLE PRECISION array, dimension (M)
Y (output) DOUBLE PRECISION array, dimension (P) On exit,
X and Y are the solutions of the GLM problem.
WORK (workspace/output) DOUBLE PRECISION array, dimension
(MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) 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 (output) 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 com‐
puted. = 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 driver routine (version 3November 2008 DGGGLM(1)
