sgglse.f3

!>
!> SGGLSE solves the linear equality-constrained least squares (LSE)
!> problem:
!>
!>         minimize || c - A*x ||_2   subject to   B*x = d
!>
!> where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
!> M-vector, and d is a given P-vector. It is assumed that
!> P <= N <= M+P, and
!>
!>          rank(B) = P and  rank( (A) ) = N.
!>                               ( (B) )
!>
!> These conditions ensure that the LSE problem has a unique solution,
!> which is obtained using a generalized RQ factorization of the
!> matrices (B, A) given by
!>
!>    B = (0 R)*Q,   A = Z*T*Q.
!>
!> Callers of this subroutine should note that the singularity/rank-deficiency checks
!> implemented in this subroutine are rudimentary. The STRTRS 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 RQ
!> factorization of the pair (B, A) 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.
!> 
!> 

M

!>          M is INTEGER
!>          The number of rows of the matrix A.  M >= 0.
!> 

N

!>          N is INTEGER
!>          The number of columns of the matrices A and B. N >= 0.
!> 

P

!>          P is INTEGER
!>          The number of rows of the matrix B. 0 <= P <= N <= M+P.
!> 

A

!>          A is REAL array, dimension (LDA,N)
!>          On entry, the M-by-N matrix A.
!>          On exit, the elements on and above the diagonal of the array
!>          contain the min(M,N)-by-N upper trapezoidal matrix T.
!> 

LDA

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

B

!>          B is REAL array, dimension (LDB,N)
!>          On entry, the P-by-N matrix B.
!>          On exit, the upper triangle of the subarray B(1:P,N-P+1:N)
!>          contains the P-by-P upper triangular matrix R.
!> 

LDB

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

C

!>          C is REAL array, dimension (M)
!>          On entry, C contains the right hand side vector for the
!>          least squares part of the LSE problem.
!>          On exit, the residual sum of squares for the solution
!>          is given by the sum of squares of elements N-P+1 to M of
!>          vector C.
!> 

D

!>          D is REAL array, dimension (P)
!>          On entry, D contains the right hand side vector for the
!>          constrained equation.
!>          On exit, D is destroyed.
!> 

X

!>          X is REAL array, dimension (N)
!>          On exit, X is the solution of the LSE problem.
!> 

WORK

!>          WORK is REAL 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,M+N+P).
!>          For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,
!>          where NB is an upper bound for the optimal blocksizes for
!>          SGEQRF, SGERQF, SORMQR 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 B in the
!>                generalized RQ factorization of the pair (B, A) is exactly
!>                singular, so that rank(B) < P; the least squares
!>                solution could not be computed.
!>          = 2:  the (N-P) by (N-P) part of the upper trapezoidal factor
!>                T associated with A in the generalized RQ factorization
!>                of the pair (B, A) is exactly singular, so that
!>                rank( (A) ) < N; the least squares solution could not
!>                    ( (B) )
!>                be computed.
!> 

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