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## sgglse()

subroutine sgglse | ( | integer | m,
|  | integer | n,
|  | integer | p,
|  | real, dimension( lda, * ) | a,
|  | integer | lda,
|  | real, dimension( ldb, * ) | b,
|  | integer | ldb,
|  | real, dimension( * ) | c,
|  | real, dimension( * ) | d,
|  | real, dimension( * ) | x,
|  | real, dimension( * ) | work,
|  | integer | lwork,
|  | integer | info )

SGGLSE solves overdetermined or underdetermined systems for OTHER matrices

Download SGGLSE + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> 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.
      !>
      !>

Parameters

  [in]

  M

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

  [in]

  N

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

  [in]

  P

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

  [in,out]

  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.
      !>

  [in]

  LDA

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

  [in,out]

  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.
      !>

  [in]

  LDB

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

  [in,out]

  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.
      !>

  [in,out]

  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.
      !>

  [out]

  X

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

  [out]

  WORK

      !>          WORK is REAL 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,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.
      !>

  [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 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.
      !>

Author
  Univ. of Tennessee
  Univ. of California Berkeley
  Univ. of Colorado Denver
  NAG Ltd.