LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ sggglm()

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

SGGGLM

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

Purpose:
 SGGGLM 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 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 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 REAL 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 REAL 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 REAL array, dimension (N)
          On entry, D is the left hand side of the GLM equation.
          On exit, D is destroyed.
[out]X
          X is REAL array, dimension (M)
[out]Y
          Y is REAL array, dimension (P)

          On exit, X and Y are the solutions of the GLM 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,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
          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 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.