zggglm()

subroutine zggglm	(	integer	n,
		integer	m,
		integer	p,
		complex*16, dimension( lda, * )	a,
		integer	lda,
		complex*16, dimension( ldb, * )	b,
		integer	ldb,
		complex*16, dimension( * )	d,
		complex*16, dimension( * )	x,
		complex*16, dimension( * )	y,
		complex*16, dimension( * )	work,
		integer	lwork,
		integer	info )

ZGGGLM

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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.
!>
!> Callers of this subroutine should note that the singularity/rank-deficiency checks
!> implemented in this subroutine are rudimentary. The ZTRTRS subroutine called by this
!> subroutine only signals a failure due to singularity if the problem is exactly singular.
!>
!> It is conceivable for one (or more) of the factors involved in the generalized QR
!> factorization of the pair (A, B) to be subnormally close to singularity without this
!> subroutine signalling an error. The solutions computed for such almost-rank-deficient
!> problems may be less accurate due to a loss of numerical precision.
!> 
!> 

Parameters

[in]	N	!> N is INTEGER !> The number of rows of the matrices A and B. N >= 0. !>
[in]	M	!> M is INTEGER !> The number of columns of the matrix A. 0 <= M <= N. !>
[in]	P	!> P is INTEGER !> The number of columns of the matrix B. P >= N-M. !>
[in,out]	A	!> 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. !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !>
[in,out]	B	!> 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. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !>
[in,out]	D	!> D is COMPLEX*16 array, dimension (N) !> On entry, D is the left hand side of the GLM equation. !> On exit, D is destroyed. !>
[out]	X	!> X is COMPLEX*16 array, dimension (M) !>
[out]	Y	!> Y is COMPLEX*16 array, dimension (P) !> !> On exit, X and Y are the solutions of the GLM problem. !>
[out]	WORK	!> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !>
[in]	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 !> 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. !>
[out]	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 exactly !> 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 exactly singular, so that !> rank( A B ) < N; the least squares solution could not !> be computed. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.