man dggglm (Fonctions bibliothèques) - solve a general Gauss-Markov linear model (GLM) problem

NAME

DGGGLM - solve 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 A and B.

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, A is destroyed.
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, B is destroyed.
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 (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.