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.

```
```  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

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Program Data
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Program Results
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