## NF01BQ

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

[Specification] [Arguments] [Method] [References] [Comments] [Example]

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.

Specification

          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(*)

Arguments

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).

Input/Output Parameters

      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.

Tolerances

      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'.

Workspace

      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'.

Error Indicator

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

Method

      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).

References

      [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.

Numerical Aspects

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

Further Comments

      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.

Example

Program Text

      None

Program Data

      None

Program Results

      None

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