Levenberg-Marquardt parameter for Wiener system identification

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


  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.

     $                   DELTA, PAR, RANKS, X, RX, TOL, DWORK, LDWORK,
     $                   INFO )
C     .. Scalar Arguments ..
      CHARACTER         COND
C     .. Array Arguments ..
      INTEGER           IPAR(*), IPVT(*), RANKS(*)
      DOUBLE PRECISION  DIAG(*), DWORK(*), QTB(*), R(LDR,*), RX(*), X(*)


Mode Parameters

          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.

          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

  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.

          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.

          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'.

  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.

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

  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.

  [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.


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