## NF01BP

### Levenberg-Marquardt parameter for Wiener system identification

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

Purpose

      To determine a value for the Levenberg-Marquardt parameter PAR
      such that if x solves the system

            J*x = b ,     sqrt(PAR)*D*x = 0 ,

      in the least squares sense, where J is an m-by-n matrix, D is an
      n-by-n nonsingular diagonal matrix, and b is an m-vector, and if
      DELTA is a positive number, DXNORM is the Euclidean norm of D*x,
      then either PAR is zero and

            ( DXNORM - DELTA ) .LE. 0.1*DELTA ,

      or PAR is positive and

            ABS( DXNORM - DELTA ) .LE. 0.1*DELTA .

      The matrix J is the current Jacobian matrix 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 NF01BP needs the upper triangle of R
      in compressed form, the permutation matrix P, and the first
      n components of Q'*b (' denotes the transpose). On output,
      NF01BP also provides a compressed representation of an upper
      triangular matrix S, such that

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

      Matrix S is used in the solution process. 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 NF01BP( COND, N, IPAR, LIPAR, R, LDR, IPVT, DIAG, QTB,
         $                   DELTA, PAR, RANKS, X, RX, TOL, DWORK, LDWORK,
         $                   INFO )
    C     .. Scalar Arguments ..
          CHARACTER         COND
          INTEGER           INFO, LDR, LDWORK, LIPAR, N
          DOUBLE PRECISION  DELTA, PAR, TOL
    C     .. Array Arguments ..
          INTEGER           IPAR(*), IPVT(*), RANKS(*)
          DOUBLE PRECISION  DIAG(*), DWORK(*), QTB(*), R(LDR,*), RX(*), X(*)

Arguments

Mode Parameters

      COND    CHARACTER*1
              Specifies whether the condition of the diagonal blocks R_k
              and S_k of the matrices R and S should be estimated,
              as follows:
              = 'E' :  use incremental condition estimation for each
                       diagonal block of R_k and S_k to find its
                       numerical rank;
              = 'N' :  do not use condition estimation, but check the
                       diagonal entries of R_k and S_k for zero values;
              = 'U' :  use the ranks already stored in RANKS (for R).

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 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.  DIAG(I) <> 0, I = 1,...,N.

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

      DELTA   (input) DOUBLE PRECISION
              An upper bound on the Euclidean norm of D*x.  DELTA > 0.

      PAR     (input/output) DOUBLE PRECISION
              On entry, PAR must contain an initial estimate of the
              Levenberg-Marquardt parameter.  PAR >= 0.
              On exit, it contains the final estimate of this parameter.

      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 R_k, k = 1:l(+1).
              On exit, if N > 0, this array contains the numerical ranks
              of the submatrices S_k, k = 1:l(+1).

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

      RX      (output) DOUBLE PRECISION array, dimension (N)
              This array contains the matrix-vector product -R*P'*x.

Tolerances

      TOL     DOUBLE PRECISION
              If COND = 'E', the tolerance to be used for finding the
              ranks of the submatrices R_k and 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.
              If BN > 1 and BSN > 0, the elements N+1 : 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

      The algorithm computes the Gauss-Newton direction. An approximate
      basic least squares solution is found if the Jacobian is rank
      deficient. The computations exploit the special structure and
      storage scheme of the matrix R. If one or more of the submatrices
      R_k or 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 or S_k, k = 1:l+1 (with adapted right
      hand sides).

      If the Gauss-Newton direction is not acceptable, then an iterative
      algorithm obtains improved lower and upper bounds for the
      Levenberg-Marquardt parameter PAR. Only a few iterations are
      generally needed for convergence of the algorithm. If, however,
      the limit of ITMAX = 10 iterations is reached, then the output PAR
      will contain the best value obtained so far. If the Gauss-Newton
      step is acceptable, it is stored in x, and PAR is set to zero,
      hence S = R.

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 LMPAR from the MINPACK package [1], and with optional condition
      estimation. The option COND = 'U' is useful when dealing with
      several right-hand side vectors, but RANKS array should be reset.
      If COND = 'E', but the matrix S is guaranteed to be nonsingular
      and well conditioned relative to TOL, i.e., rank(R) = N, and
      min(DIAG) > 0, then its condition is not estimated.

Example

Program Text

      None

Program Data

      None

Program Results

      None

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