NF01BQ

Solving the linear system J x = b, D x = 0, D diagonal, for Wiener system identification

Purpose

  To determine a vector x which solves the system of linear
  equations

        J*x = b ,     D*x = 0 ,

  in the least squares sense, where J is an m-by-n matrix,
  D is an n-by-n diagonal matrix, and b is an m-vector. The matrix J
  is the current Jacobian of a nonlinear least squares problem,
  provided in a compressed form by SLICOT Library routine NF01BD.
  It is assumed that a block QR factorization, with column pivoting,
  of J is available, that is, J*P = Q*R, where P is a permutation
  matrix, Q has orthogonal columns, and R is an upper triangular
  matrix with diagonal elements of nonincreasing magnitude for each
  block, as returned by SLICOT Library routine NF01BS. The routine
  NF01BQ needs the upper triangle of R in compressed form, the
  permutation matrix P, and the first n components of Q'*b
  (' denotes the transpose). The system J*x = b, D*x = 0, is then
  equivalent to

        R*z = Q'*b ,  P'*D*P*z = 0 ,                             (1)

  where x = P*z. If this system does not have full rank, then an
  approximate least squares solution is obtained (see METHOD).
  On output, NF01BQ also provides an upper triangular matrix S
  such that

        P'*(J'*J + D*D)*P = S'*S .

  The system (1) is equivalent to S*z = c , where c contains the
  first n components of the vector obtained by applying to
  [ (Q'*b)'  0 ]' the transformations which triangularized
  [ R'  P'*D*P ]', getting S.

  The matrix R has the following structure

      /   R_1    0    ..   0   |   L_1   \
      |    0    R_2   ..   0   |   L_2   |
      |    :     :    ..   :   |    :    | ,
      |    0     0    ..  R_l  |   L_l   |
      \    0     0    ..   0   |  R_l+1  /

  where the submatrices R_k, k = 1:l, have the same order BSN,
  and R_k, k = 1:l+1, are square and upper triangular. This matrix
  is stored in the compressed form

           /   R_1  |   L_1   \
           |   R_2  |   L_2   |
    Rc =   |    :   |    :    | ,
           |   R_l  |   L_l   |
           \    X   |  R_l+1  /

  where the submatrix X is irrelevant. The matrix S has the same
  structure as R, and its diagonal blocks are denoted by S_k,
  k = 1:l+1.

  If l <= 1, then the full upper triangle of the matrix R is stored.

      SUBROUTINE NF01BQ( COND, N, IPAR, LIPAR, R, LDR, IPVT, DIAG, QTB,
     $                   RANKS, X, TOL, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         COND
      INTEGER           INFO, LDR, LDWORK, LIPAR, N
      DOUBLE PRECISION  TOL
C     .. Array Arguments ..
      INTEGER           IPAR(*), IPVT(*), RANKS(*)
      DOUBLE PRECISION  DIAG(*), DWORK(*), QTB(*), R(LDR,*), X(*)

Mode Parameters

  COND    CHARACTER*1
          Specifies whether the condition of the matrices S_k should
          be estimated, as follows:
          = 'E' :  use incremental condition estimation and store
                   the numerical rank of S_k in the array entry
                   RANKS(k), for k = 1:l+1;
          = 'N' :  do not use condition estimation, but check the
                   diagonal entries of S_k for zero values;
          = 'U' :  use the ranks already stored in RANKS(1:l+1).

  N       (input) INTEGER
          The order of the matrix R.  N = BN*BSN + ST >= 0.
          (See parameter description below.)

  IPAR    (input) INTEGER array, dimension (LIPAR)
          The integer parameters describing the structure of the
          matrix R, as follows:
          IPAR(1) must contain ST, the number of columns of the
                  submatrices L_k and the order of R_l+1.  ST >= 0.
          IPAR(2) must contain BN, the number of blocks, l, in the
                  block diagonal part of R.  BN >= 0.
          IPAR(3) must contain BSM, the number of rows of the blocks
                  R_k, k = 1:l.  BSM >= 0.
          IPAR(4) must contain BSN, the number of columns of the
                  blocks R_k, k = 1:l.  BSN >= 0.
          BSM is not used by this routine, but assumed equal to BSN.

  LIPAR   (input) INTEGER
          The length of the array IPAR.  LIPAR >= 4.

  R       (input/output) DOUBLE PRECISION array, dimension (LDR, NC)
          where NC = N if BN <= 1, and NC = BSN+ST, if BN > 1.
          On entry, the leading N-by-NC part of this array must
          contain the (compressed) representation (Rc) of the upper
          triangular matrix R. If BN > 1, the submatrix X in Rc is
          not referenced. The zero strict lower triangles of R_k,
          k = 1:l+1, need not be set. If BN <= 1 or BSN = 0, then
          the full upper triangle of R must be stored.
          On exit, the full upper triangles of R_k, k = 1:l+1, and
          L_k, k = 1:l, are unaltered, and the strict lower
          triangles of R_k, k = 1:l+1, contain the corresponding
          strict upper triangles (transposed) of the upper
          triangular matrix S.
          If BN <= 1 or BSN = 0, then the transpose of the strict
          upper triangle of S is stored in the strict lower triangle
          of R.

  LDR     INTEGER
          The leading dimension of the array R.  LDR >= MAX(1,N).

  IPVT    (input) INTEGER array, dimension (N)
          This array must define the permutation matrix P such that
          J*P = Q*R. Column j of P is column IPVT(j) of the identity
          matrix.

  DIAG    (input) DOUBLE PRECISION array, dimension (N)
          This array must contain the diagonal elements of the
          matrix D.

  QTB     (input) DOUBLE PRECISION array, dimension (N)
          This array must contain the first n elements of the
          vector Q'*b.

  RANKS   (input or output) INTEGER array, dimension (r), where
          r = BN + 1,  if ST > 0, BSN > 0, and BN > 1;
          r = BN,      if ST = 0 and BSN > 0;
          r = 1,       if ST > 0 and ( BSN = 0 or BN <= 1 );
          r = 0,       if ST = 0 and BSN = 0.
          On entry, if COND = 'U' and N > 0, this array must contain
          the numerical ranks of the submatrices S_k, k = 1:l(+1).
          On exit, if COND = 'E' or 'N' and N > 0, this array
          contains the numerical ranks of the submatrices S_k,
          k = 1:l(+1), estimated according to the value of COND.

  X       (output) DOUBLE PRECISION array, dimension (N)
          This array contains the least squares solution of the
          system J*x = b, D*x = 0.

  TOL     DOUBLE PRECISION
          If COND = 'E', the tolerance to be used for finding the
          ranks of the submatrices S_k. If the user sets TOL > 0,
          then the given value of TOL is used as a lower bound for
          the reciprocal condition number;  a (sub)matrix whose
          estimated condition number is less than 1/TOL is
          considered to be of full rank.  If the user sets TOL <= 0,
          then an implicitly computed, default tolerance, defined by
          TOLDEF = N*EPS,  is used instead, where EPS is the machine
          precision (see LAPACK Library routine DLAMCH).
          This parameter is not relevant if COND = 'U' or 'N'.

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, the first N elements of this array contain the
          diagonal elements of the upper triangular matrix S, and
          the next N elements contain the solution z.
          If BN > 1 and BSN > 0, the elements 2*N+1 : 2*N+ST*(N-ST)
          contain the submatrix (S(1:N-ST,N-ST+1:N))' of the
          matrix S.

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= 2*N,              if BN <= 1 or  BSN = 0 and
                                                     COND <> 'E';
          LDWORK >= 4*N,              if BN <= 1 or  BSN = 0 and
                                                     COND =  'E';
          LDWORK >= ST*(N-ST) + 2*N,  if BN >  1 and BSN > 0 and
                                                     COND <> 'E';
          LDWORK >= ST*(N-ST) + 2*N + 2*MAX(BSN,ST),
                                      if BN >  1 and BSN > 0 and
                                                     COND = 'E'.

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value.

  Standard plane rotations are used to annihilate the elements of
  the diagonal matrix D, updating the upper triangular matrix R
  and the first n elements of the vector Q'*b. A basic least squares
  solution is computed. The computations exploit the special
  structure and storage scheme of the matrix R. If one or more of
  the submatrices S_k, k = 1:l+1, is singular, then the computed
  result is not the basic least squares solution for the whole
  problem, but a concatenation of (least squares) solutions of the
  individual subproblems involving R_k, k = 1:l+1 (with adapted
  right hand sides).

  [1] More, J.J., Garbow, B.S, and Hillstrom, K.E.
      User's Guide for MINPACK-1.
      Applied Math. Division, Argonne National Laboratory, Argonne,
      Illinois, Report ANL-80-74, 1980.

  The algorithm requires 0(N*(BSN+ST)) operations and is backward
  stable, if R is nonsingular.

  This routine is a structure-exploiting, LAPACK-based modification
  of QRSOLV from the MINPACK package [1], and with optional
  condition estimation.
  The option COND = 'U' is useful when dealing with several
  right-hand side vectors.

Program Text

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