### Estimating parameters of a Wiener system using Levenberg-Marquardt algorithm

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

Purpose

```  To compute a set of parameters for approximating a Wiener system
in a least-squares sense, using a neural network approach and a
Levenberg-Marquardt algorithm. Conjugate gradients (CG) or
Cholesky algorithms are used to solve linear systems of equations.
The Wiener system is represented as

x(t+1) = A*x(t) + B*u(t)
z(t)   = C*x(t) + D*u(t),

y(t)   = f(z(t),wb(1:L)),

where t = 1, 2, ..., NSMP, and f is a nonlinear function,
evaluated by the SLICOT Library routine NF01AY. The parameter
vector X is partitioned as X = ( wb(1), ..., wb(L), theta ),
where wb(i), i = 1 : L, correspond to the nonlinear part, and
theta corresponds to the linear part. See SLICOT Library routine
NF01AD for further details.

The sum of squares of the error functions, defined by

e(t) = y(t) - Y(t),  t = 1, 2, ..., NSMP,

is minimized, where Y(t) is the measured output vector. The
functions and their Jacobian matrices are evaluated by SLICOT
Library routine NF01BB (the FCN routine in the call of MD03AD).

```
Specification
```      SUBROUTINE IB03AD( INIT, ALG, STOR, NOBR, M, L, NSMP, N, NN,
\$                   ITMAX1, ITMAX2, NPRINT, U, LDU, Y, LDY, X, LX,
\$                   TOL1, TOL2, IWORK, DWORK, LDWORK, IWARN, INFO )
C     .. Scalar Arguments ..
CHARACTER         ALG, INIT, STOR
INTEGER           INFO, ITMAX1, ITMAX2, IWARN, L, LDU, LDWORK,
\$                  LDY, LX, M, N, NN, NOBR, NPRINT, NSMP
DOUBLE PRECISION  TOL1, TOL2
C     .. Array Arguments ..
DOUBLE PRECISION  DWORK(*), U(LDU, *), X(*), Y(LDY, *)
INTEGER           IWORK(*)

```
Arguments

Mode Parameters

```  INIT    CHARACTER*1
Specifies which parts have to be initialized, as follows:
= 'L' : initialize the linear part only, X already
contains an initial approximation of the
nonlinearity;
= 'S' : initialize the static nonlinearity only, X
already contains an initial approximation of the
linear part;
= 'B' : initialize both linear and nonlinear parts;
= 'N' : do not initialize anything, X already contains
an initial approximation.
If INIT = 'S' or 'B', the error functions for the
nonlinear part, and their Jacobian matrices, are evaluated
by SLICOT Library routine NF01BA (used as a second FCN
routine in the MD03AD call for the initialization step,
see METHOD).

ALG     CHARACTER*1
Specifies the algorithm used for solving the linear
systems involving a Jacobian matrix J, as follows:
= 'D' :  a direct algorithm, which computes the Cholesky
factor of the matrix J'*J + par*I is used, where
par is the Levenberg factor;
= 'I' :  an iterative Conjugate Gradients algorithm, which
only needs the matrix J, is used.
In both cases, matrix J is stored in a compressed form.

STOR    CHARACTER*1
If ALG = 'D', specifies the storage scheme for the
symmetric matrix J'*J, as follows:
= 'F' :  full storage is used;
= 'P' :  packed storage is used.
The option STOR = 'F' usually ensures a faster execution.
This parameter is not relevant if ALG = 'I'.

```
Input/Output Parameters
```  NOBR    (input) INTEGER
If INIT = 'L' or 'B', NOBR is the number of block rows, s,
in the input and output block Hankel matrices to be
processed for estimating the linear part.  NOBR > 0.
(In the MOESP theory,  NOBR  should be larger than  n,
the estimated dimension of state vector.)
This parameter is ignored if INIT is 'S' or 'N'.

M       (input) INTEGER
The number of system inputs.  M >= 0.

L       (input) INTEGER
The number of system outputs.  L >= 0, and L > 0, if
INIT = 'L' or 'B'.

NSMP    (input) INTEGER
The number of input and output samples, t.  NSMP >= 0, and
NSMP >= 2*(M+L+1)*NOBR - 1, if INIT = 'L' or 'B'.

N       (input/output) INTEGER
The order of the linear part.
If INIT = 'L' or 'B', and N < 0 on entry, the order is
assumed unknown and it will be found by the routine.
Otherwise, the input value will be used. If INIT = 'S'
or 'N', N must be non-negative. The values N >= NOBR,
or N = 0, are not acceptable if INIT = 'L' or 'B'.

NN      (input) INTEGER
The number of neurons which shall be used to approximate
the nonlinear part.  NN >= 0.

ITMAX1  (input) INTEGER
The maximum number of iterations for the initialization of
the static nonlinearity.
This parameter is ignored if INIT is 'N' or 'L'.
Otherwise, ITMAX1 >= 0.

ITMAX2  (input) INTEGER
The maximum number of iterations.  ITMAX2 >= 0.

NPRINT  (input) INTEGER
This parameter enables controlled printing of iterates if
it is positive. In this case, FCN is called with IFLAG = 0
at the beginning of the first iteration and every NPRINT
iterations thereafter and immediately prior to return,
and the current error norm is printed. Other intermediate
results could be printed by modifying the corresponding
FCN routine (NF01BA and/or NF01BB). If NPRINT <= 0, no
special calls of FCN with IFLAG = 0 are made.

U       (input) DOUBLE PRECISION array, dimension (LDU, M)
The leading NSMP-by-M part of this array must contain the
set of input samples,
U = ( U(1,1),...,U(1,M); ...; U(NSMP,1),...,U(NSMP,M) ).

LDU     INTEGER
The leading dimension of array U.  LDU >= MAX(1,NSMP).

Y       (input) DOUBLE PRECISION array, dimension (LDY, L)
The leading NSMP-by-L part of this array must contain the
set of output samples,
Y = ( Y(1,1),...,Y(1,L); ...; Y(NSMP,1),...,Y(NSMP,L) ).

LDY     INTEGER
The leading dimension of array Y.  LDY >= MAX(1,NSMP).

X       (input/output) DOUBLE PRECISION array dimension (LX)
On entry, if INIT = 'L', the leading (NN*(L+2) + 1)*L part
of this array must contain the initial parameters for
the nonlinear part of the system.
On entry, if INIT = 'S', the elements lin1 : lin2 of this
array must contain the initial parameters for the linear
part of the system, corresponding to the output normal
form, computed by SLICOT Library routine TB01VD, where
lin1 = (NN*(L+2) + 1)*L + 1;
lin2 = (NN*(L+2) + 1)*L + N*(L+M+1) + L*M.
On entry, if INIT = 'N', the elements 1 : lin2 of this
array must contain the initial parameters for the
nonlinear part followed by the initial parameters for the
linear part of the system, as specified above.
This array need not be set on entry if INIT = 'B'.
On exit, the elements 1 : lin2 of this array contain the
optimal parameters for the nonlinear part followed by the
optimal parameters for the linear part of the system, as
specified above.

LX      (input/output) INTEGER
On entry, this parameter must contain the intended length
of X. If N >= 0, then LX >= NX := lin2 (see parameter X).
If N is unknown (N < 0 on entry), a large enough estimate
of N should be used in the formula of lin2.
On exit, if N < 0 on entry, but LX is not large enough,
then this parameter contains the actual length of X,
corresponding to the computed N. Otherwise, its value
is unchanged.

```
Tolerances
```  TOL1    DOUBLE PRECISION
If INIT = 'S' or 'B' and TOL1 >= 0, TOL1 is the tolerance
which measures the relative error desired in the sum of
squares, for the initialization step of nonlinear part.
Termination occurs when the actual relative reduction in
the sum of squares is at most TOL1. In addition, if
ALG = 'I', TOL1 also measures the relative residual of
the solutions computed by the CG algorithm (for the
initialization step). Termination of a CG process occurs
when the relative residual is at most TOL1.
If the user sets  TOL1 < 0,  then  SQRT(EPS)  is used
instead TOL1, where EPS is the machine precision
(see LAPACK Library routine DLAMCH).
This parameter is ignored if INIT is 'N' or 'L'.

TOL2    DOUBLE PRECISION
If TOL2 >= 0, TOL2 is the tolerance which measures the
relative error desired in the sum of squares, for the
whole optimization process. Termination occurs when the
actual relative reduction in the sum of squares is at
most TOL2.
If ALG = 'I', TOL2 also measures the relative residual of
the solutions computed by the CG algorithm (for the whole
optimization). Termination of a CG process occurs when the
relative residual is at most TOL2.
If the user sets  TOL2 < 0,  then  SQRT(EPS)  is used
instead TOL2. This default value could require many
iterations, especially if TOL1 is larger. If INIT = 'S'
or 'B', it is advisable that TOL2 be larger than TOL1,
and spend more time with cheaper iterations.

```
Workspace
```  IWORK   INTEGER array, dimension (MAX( 3, LIW1, LIW2 )), where
LIW1 = LIW2 = 0,  if INIT = 'S' or 'N'; otherwise,
LIW1 = M+L;
LIW2 = MAX(M*NOBR+N,M*(N+L)).
On output, if INFO = 0, IWORK(1) and IWORK(2) return the
(total) number of function and Jacobian evaluations,
respectively (including the initialization step, if it was
performed), and if INIT = 'L' or INIT = 'B', IWORK(3)
specifies how many locations of DWORK contain reciprocal
condition number estimates (see below); otherwise,
IWORK(3) = 0.

DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On entry, if desired, and if INIT = 'S' or 'B', the
entries DWORK(1:4) are set to initialize the random
numbers generator for the nonlinear part parameters (see
the description of the argument XINIT of SLICOT Library
routine MD03AD); this enables to obtain reproducible
results. The same seed is used for all outputs.
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, DWORK(2) returns the residual error norm (the
sum of squares), DWORK(3) returns the number of iterations
performed, DWORK(4) returns the number of conjugate
gradients iterations performed, and DWORK(5) returns the
final Levenberg factor, for optimizing the parameters of
both the linear part and the static nonlinearity part.
If INIT = 'S' or INIT = 'B' and INFO = 0, then the
elements DWORK(6) to DWORK(10) contain the corresponding
five values for the initialization step (see METHOD).
(If L > 1, DWORK(10) contains the maximum of the Levenberg
factors for all outputs.) If INIT = 'L' or INIT = 'B', and
INFO = 0, DWORK(11) to DWORK(10+IWORK(3)) contain
reciprocal condition number estimates set by SLICOT
Library routines IB01AD, IB01BD, and IB01CD.
On exit, if  INFO = -23,  DWORK(1)  returns the minimum
value of LDWORK.

LDWORK  INTEGER
The length of the array DWORK.
In the formulas below, N should be taken not larger than
NOBR - 1, if N < 0 on entry.
LDWORK = MAX( LW1, LW2, LW3, LW4 ), where
LW1 = 0, if INIT = 'S' or 'N'; otherwise,
LW1 = MAX( 2*(M+L)*NOBR*(2*(M+L)*(NOBR+1)+3) + L*NOBR,
4*(M+L)*NOBR*(M+L)*NOBR + (N+L)*(N+M) +
MAX( LDW1, LDW2 ),
(N+L)*(N+M) + N + N*N + 2 + N*(N+M+L) +
MAX( 5*N, 2, MIN( LDW3, LDW4 ), LDW5, LDW6 ),
where,
LDW1 >= MAX( 2*(L*NOBR-L)*N+2*N, (L*NOBR-L)*N+N*N+7*N,
L*NOBR*N +
MAX( (L*NOBR-L)*N+2*N + (2*M+L)*NOBR+L,
2*(L*NOBR-L)*N+N*N+8*N,
N+4*(M*NOBR+N)+1, M*NOBR+3*N+L ) )
LDW2 >= 0,                                  if M = 0;
LDW2 >= L*NOBR*N + M*NOBR*(N+L)*(M*(N+L)+1) +
MAX( (N+L)**2, 4*M*(N+L)+1 ),       if M > 0;
LDW3 = NSMP*L*(N+1) + 2*N + MAX( 2*N*N, 4*N ),
LDW4 = N*(N+1) + 2*N +
MAX( N*L*(N+1) + 2*N*N + L*N, 4*N );
LDW5 = NSMP*L + (N+L)*(N+M) + 3*N+M+L;
LDW6 = NSMP*L + (N+L)*(N+M) + N +
MAX(1, N*N*L + N*L + N, N*N +
MAX(N*N + N*MAX(N,L) + 6*N + MIN(N,L),
N*M));
LW2 = LW3 = 0, if INIT = 'L' or 'N'; otherwise,
LW2 = NSMP*L +
MAX( 5, NSMP + 2*BSN + NSMP*BSN +
MAX( 2*NN + BSN, LDW7 ) );
LDW7 = BSN*BSN,       if ALG = 'D' and STOR = 'F';
LDW7 = BSN*(BSN+1)/2, if ALG = 'D' and STOR = 'P';
LDW7 = 3*BSN + NSMP,  if ALG = 'I';
LW3 = MAX( LDW8, NSMP*L + (N+L)*(2*N+M) + 2*N );
LDW8 = NSMP*L + (N+L)*(N+M) + 3*N+M+L,  if M > 0;
LDW8 = NSMP*L + (N+L)*N + 2*N+L,        if M = 0;
LW4 = MAX( 5, NSMP*L + 2*NX + NSMP*L*( BSN + LTHS ) +
MAX( L1 + NX, NSMP*L + L1, L2 ) ),
L0 = MAX( N*(N+L), N+M+L ),    if M > 0;
L0 = MAX( N*(N+L), L ),        if M = 0;
L1 = NSMP*L + MAX( 2*NN, (N+L)*(N+M) + 2*N + L0);
L2 = NX*NX,          if ALG = 'D' and STOR = 'F';
L2 = NX*(NX+1)/2,    if ALG = 'D' and STOR = 'P';
L2 = 3*NX + NSMP*L,  if ALG = 'I',
with BSN  = NN*( L + 2 ) + 1,
LTHS = N*( L + M + 1 ) + L*M.
For optimum performance LDWORK should be larger.

```
Warning Indicator
```  IWARN   INTEGER
= 0:  no warning;
< 0:  the user set IFLAG = IWARN in (one of) the
subroutine(s) FCN, i.e., NF01BA, if INIT = 'S'
or 'B', and/or NF01BB; this value cannot be returned
without changing the FCN routine(s);
otherwise, IWARN has the value k*100 + j*10 + i,
where k is defined below, i refers to the whole
optimization process, and j refers to the
initialization step (j = 0, if INIT = 'L' or 'N'),
and the possible values for i and j have the
following meaning (where TOL* denotes TOL1 or TOL2,
and similarly for ITMAX*):
= 1:  the number of iterations has reached ITMAX* without
satisfying the convergence condition;
= 2:  if alg = 'I' and in an iteration of the Levenberg-
Marquardt algorithm, the CG algorithm finished
after 3*NX iterations (or 3*(lin1-1) iterations, for
the initialization phase), without achieving the
precision required in the call;
= 3:  the cosine of the angle between the vector of error
function values and any column of the Jacobian is at
most FACTOR*EPS in absolute value (FACTOR = 100);
= 4:  TOL* is too small: no further reduction in the sum
of squares is possible.
The digit k is normally 0, but if INIT = 'L' or 'B', it
can have a value in the range 1 to 6 (see IB01AD, IB01BD
and IB01CD). In all these cases, the entries DWORK(1:5),
DWORK(6:10) (if INIT = 'S' or 'B'), and
DWORK(11:10+IWORK(3)) (if INIT = 'L' or 'B'), are set as
described above.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
otherwise, INFO has the value k*100 + j*10 + i,
where k is defined below, i refers to the whole
optimization process, and j refers to the
initialization step (j = 0, if INIT = 'L' or 'N'),
and the possible values for i and j have the
following meaning:
= 1:  the routine FCN returned with INFO <> 0 for
IFLAG = 1;
= 2:  the routine FCN returned with INFO <> 0 for
IFLAG = 2;
= 3:  ALG = 'D' and SLICOT Library routines MB02XD or
NF01BU (or NF01BV, if INIT = 'S' or 'B') or
ALG = 'I' and SLICOT Library routines MB02WD or
NF01BW (or NF01BX, if INIT = 'S' or 'B') returned
with INFO <> 0.
In addition, if INIT = 'L' or 'B', i could also be
= 4:  if a Lyapunov equation could not be solved;
= 5:  if the identified linear system is unstable;
= 6:  if the QR algorithm failed on the state matrix
of the identified linear system.
The digit k is normally 0, but if INIT = 'L' or 'B', it
can have a value in the range 1 to 10 (see IB01AD/IB01BD).

```
Method
```  If INIT = 'L' or 'B', the linear part of the system is
approximated using the combined MOESP and N4SID algorithm. If
necessary, this algorithm can also choose the order, but it is
advantageous if the order is already known.

If INIT = 'S' or 'B', the output of the approximated linear part
is computed and used to calculate an approximation of the static
nonlinearity using the Levenberg-Marquardt algorithm [1].
This step is referred to as the (nonlinear) initialization step.

As last step, the Levenberg-Marquardt algorithm is used again to
optimize the parameters of the linear part and the static
nonlinearity as a whole. Therefore, it is necessary to parametrise
the matrices of the linear part. The output normal form [2]
parameterisation is used.

The Jacobian is computed analytically, for the nonlinear part, and
numerically, for the linear part.

```
References
```  [1] Kelley, C.T.
Iterative Methods for Optimization.
Society for Industrial and Applied Mathematics (SIAM),

[2] Peeters, R.L.M., Hanzon, B., and Olivi, M.
Balanced realizations of discrete-time stable all-pass
systems and the tangential Schur algorithm.
Proceedings of the European Control Conference,
31 August - 3 September 1999, Karlsruhe, Germany.
Session CP-6, Discrete-time Systems, 1999.

```
```  None
```
Example

Program Text

```*     IB03AD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER           NIN, NOUT
PARAMETER         ( NIN = 5, NOUT = 6 )
INTEGER           LDU, LDY, LIWORK, LMAX, MMAX, NMAX, NNMAX,
\$                  NOBRMX, NSMPMX
PARAMETER         ( LMAX = 2, MMAX = 3, NOBRMX = 10, NNMAX = 12,
\$                    NMAX = 4, NSMPMX = 1024,
\$                    LDU  = NSMPMX, LDY = NSMPMX,
\$                    LIWORK = MAX( MMAX + LMAX, MMAX*NOBRMX + NMAX,
\$                                  MMAX*( NMAX + LMAX ) ) )
INTEGER           BSNM, L0, L1M, L2M, LDW1, LDW2, LDW3, LDW4,
\$                  LDW5, LDW6, LDW7, LDW8, LDWORK, LTHS, LW1, LW2,
\$                  LW3, LW4, LXM
PARAMETER         ( BSNM = NNMAX*( LMAX + 2 ) + 1,
\$                    LTHS = NMAX*( LMAX + MMAX + 1 ) + LMAX*MMAX,
\$                    L0   = MAX( NMAX*( NMAX + LMAX ),
\$                                NMAX + MMAX + LMAX ),
\$                    L1M  = NSMPMX*LMAX +
\$                           MAX( 2*NNMAX,
\$                                ( NMAX + LMAX )*( NMAX + MMAX ) +
\$                                2*NMAX + L0 ),
\$                    LXM  = BSNM*LMAX + LTHS,
\$                    L2M  = MAX( LXM*LXM, 3*LXM + NSMPMX*LMAX ),
\$                    LDW1 = MAX( 2*( LMAX*NOBRMX - LMAX )*NMAX +
\$                                2*NMAX,
\$                                ( LMAX*NOBRMX - LMAX )*NMAX +
\$                                NMAX*NMAX + 7*NMAX,
\$                                LMAX*NOBRMX*NMAX +
\$                                MAX( ( LMAX*NOBRMX - LMAX )*NMAX +
\$                                     2*NMAX + LMAX +
\$                                     ( 2*MMAX + LMAX )*NOBRMX,
\$                                     2*( LMAX*NOBRMX - LMAX )*NMAX
\$                                   + NMAX*NMAX + 8*NMAX,
\$                                     NMAX + 4*( MMAX*NOBRMX +
\$                                                NMAX ) + 1,
\$                                     MMAX*NOBRMX + 3*NMAX + LMAX )
\$                              ),
\$                    LDW2 = LMAX*NOBRMX*NMAX +
\$                           MMAX*NOBRMX*( NMAX + LMAX )*
\$                           ( MMAX*( NMAX + LMAX ) + 1 ) +
\$                           MAX( ( NMAX + LMAX )**2,
\$                           4*MMAX*( NMAX + LMAX ) + 1 ),
\$                    LDW3 = NSMPMX*LMAX*( NMAX + 1 ) + 2*NMAX +
\$                           MAX( 2*NMAX*NMAX, 4*NMAX ),
\$                    LDW4 = NMAX*( NMAX + 1 ) + 2*NMAX +
\$                           MAX( NMAX*LMAX*( NMAX + 1 ) +
\$                           2*NMAX*NMAX + LMAX*NMAX, 4*NMAX ),
\$                    LDW5 = NSMPMX*LMAX + ( NMAX + LMAX )*
\$                           ( NMAX + MMAX ) + 3*NMAX + MMAX + LMAX,
\$                    LDW6 = NSMPMX*LMAX + ( NMAX + LMAX )*
\$                           ( NMAX + MMAX ) + NMAX +
\$                           MAX( 1, NMAX*NMAX*LMAX + NMAX*LMAX +
\$                                NMAX, NMAX*NMAX +
\$                                MAX( NMAX*NMAX +
\$                                     NMAX*MAX( NMAX, LMAX ) +
\$                                     6*NMAX + MIN( NMAX, LMAX ),
\$                                     NMAX*MMAX ) ),
\$                    LDW7 = MAX( BSNM*BSNM, 3*BSNM + NSMPMX ),
\$                    LDW8 = NSMPMX*LMAX + ( NMAX + LMAX )*
\$                           ( NMAX + MMAX ) + 3*NMAX + MMAX + LMAX,
\$                    LW1  = MAX( 2*( MMAX + LMAX )*NOBRMX*
\$                                ( 2*( MMAX + LMAX )*( NOBRMX + 1 )
\$                                  + 3 ) + LMAX*NOBRMX,
\$                                4*( MMAX + LMAX )*NOBRMX*
\$                                ( MMAX + LMAX )*NOBRMX +
\$                                ( NMAX + LMAX )*( NMAX + MMAX ) +
\$                                MAX( LDW1, LDW2 ),
\$                                ( NMAX + LMAX )*( NMAX + MMAX ) +
\$                                NMAX + NMAX*NMAX + 2 +
\$                                NMAX*( NMAX + MMAX + LMAX ) +
\$                                MAX( 5*NMAX, 2, MIN( LDW3, LDW4 ),
\$                                     LDW5, LDW6 ) ),
\$                    LW2  = NSMPMX*LMAX +
\$                           MAX( 5, NSMPMX + 2*BSNM + NSMPMX*BSNM +
\$                                   MAX( 2*NNMAX + BSNM, LDW7 ) ),
\$                    LW3  = MAX( LDW8, NSMPMX*LMAX +
\$                                ( NMAX + LMAX )*( 2*NMAX + MMAX )+
\$                                2*NMAX ),
\$                    LW4  = MAX( 5, NSMPMX*LMAX + 2*LXM +
\$                                NSMPMX*LMAX*( BSNM + LTHS ) +
\$                                MAX( L1M + LXM, NSMPMX*LMAX + L1M,
\$                                     L2M ) ),
\$                    LDWORK = MAX( LW1, LW2, LW3, LW4 ) )
*     .. Local Scalars ..
LOGICAL           INIT1, INITB, INITL, INITN, INITS
CHARACTER*1       ALG, INIT, STOR
INTEGER           BSN, I, INFO, INI, ITER, ITERCG, ITMAX1, ITMAX2,
\$                  IWARN, J, L, L1, L2, LPAR, LX, M, N, NN, NOBR,
\$                  NPRINT, NS, NSMP
DOUBLE PRECISION  TOL1, TOL2
*     .. Array Arguments ..
INTEGER           IWORK(LIWORK)
DOUBLE PRECISION  DWORK(LDWORK), U(LDU,MMAX), X(LXM), Y(LDY,LMAX)
*     .. External Functions ..
LOGICAL           LSAME
EXTERNAL          LSAME
*     .. External Subroutines ..
*     .. Intrinsic Functions ..
INTRINSIC         MAX, MIN
*     .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) NOBR, M, L, NSMP, N, NN, ITMAX1, ITMAX2,
\$                      NPRINT, TOL1, TOL2, INIT, ALG, STOR
INITL = LSAME( INIT, 'L' )
INITS = LSAME( INIT, 'S' )
INITB = LSAME( INIT, 'B' )
INITN = LSAME( INIT, 'N' )
INIT1 = INITL .OR. INITB
IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
IF( L.LE.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) L
ELSE
NS = N
IF( INIT1 ) THEN
IF( NOBR.LE.0 .OR. NOBR.GT.NOBRMX ) THEN
WRITE ( NOUT, FMT = 99991 ) NOBR
STOP
ELSEIF( NSMP.LT.2*( M + L + 1 )*NOBR - 1 ) THEN
WRITE ( NOUT, FMT = 99990 ) NSMP
STOP
ELSEIF( N.EQ.0 .OR. N.GE.NOBR ) THEN
WRITE ( NOUT, FMT = 99989 ) N
STOP
END IF
IF ( N.LT.0 )
\$            N = NOBR - 1
ELSE
IF( NSMP.LT.0 ) THEN
WRITE ( NOUT, FMT = 99990 ) NSMP
STOP
ELSEIF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) N
STOP
END IF
END IF
IF( NN.LT.0 .OR. NN.GT.NNMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) NN
ELSE
BSN = NN*( L + 2 ) + 1
L1  = BSN*L
L2  = N*( L + M + 1 ) + L*M
LX  = L1 + L2
INI = 1
IF ( INITL ) THEN
LPAR = L1
ELSEIF ( INITS ) THEN
INI  = L1 + 1
LPAR = L2
ELSEIF ( INITN ) THEN
LPAR = LX
END IF
IF( INIT1 )
\$            N = NS
*              Read the input-output data, initial parameters, and seed.
READ ( NIN, FMT = * ) ( ( U(I,J), J = 1,M ), I = 1,NSMP )
READ ( NIN, FMT = * ) ( ( Y(I,J), J = 1,L ), I = 1,NSMP )
IF ( .NOT.INITB )
\$            READ ( NIN, FMT = * ) ( X(I), I = INI,INI+LPAR-1 )
IF ( INITS .OR. INITB )
\$            READ ( NIN, FMT = * ) ( DWORK(I), I = 1,4 )
*              Solve a Wiener system identification problem.
CALL IB03AD( INIT, ALG, STOR, NOBR, M, L, NSMP, N, NN,
\$                      ITMAX1, ITMAX2, NPRINT, U, LDU, Y, LDY,
\$                      X, LX, TOL1, TOL2, IWORK, DWORK, LDWORK,
\$                      IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( IWARN.NE.0 ) WRITE ( NOUT, FMT = 99987 ) IWARN
ITER   = DWORK(3)
ITERCG = DWORK(4)
WRITE ( NOUT, FMT = 99997 ) DWORK(2)
WRITE ( NOUT, FMT = 99996 ) ITER, ITERCG,
\$                                        IWORK(1), IWORK(2)
*                 Recompute LX is necessary.
IF ( INIT1 .AND. NS.LT.0 )
\$               LX = L1 + N*( L + M + 1 ) + L*M
WRITE ( NOUT, FMT = 99994 )
WRITE ( NOUT, FMT = 99995 ) ( X(I), I = 1, LX )
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' IB03AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from IB03AD = ',I4)
99997 FORMAT (/' Final 2-norm of the residuals = ',D15.7)
99996 FORMAT (/' Number of iterations                     = ', I7,
\$        /' Number of conjugate gradients iterations = ', I7,
\$        /' Number of function evaluations           = ', I7,
\$        /' Number of Jacobian evaluations           = ', I7)
99995 FORMAT (10(1X,F8.4))
99994 FORMAT (/' Final approximate solution is ' )
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' L is out of range.',/' L = ',I5)
99991 FORMAT (/' NOBR is out of range.',/' NOBR = ',I5)
99990 FORMAT (/' NSMP is out of range.',/' NSMP = ',I5)
99989 FORMAT (/' N is out of range.',/' N = ',I5)
99988 FORMAT (/' NN is out of range.',/' NN = ',I5)
99987 FORMAT (' IWARN on exit from IB03AD = ',I4)
END
```
Program Data
``` IB03AD EXAMPLE PROGRAM DATA
10     1     1  1024    4   12   500  1000     0  .00001  .00001   B   D   F
2.2183165e-01
3.9027807e-02
-5.0295887e-02
8.5386224e-03
7.2431159e-02
-1.7082198e-03
-1.7176287e-01
-2.6198104e-01
-1.7194108e-01
1.8566868e-02
1.5625362e-01
1.7463811e-01
1.1564450e-01
2.8779248e-02
-8.4265993e-02
-2.0978501e-01
-2.6591828e-01
-1.7268680e-01
2.1525013e-02
1.4363602e-01
7.3101431e-02
-1.0259212e-01
-1.6380473e-01
-1.0021167e-02
2.0263451e-01
2.1983417e-01
-2.1636523e-02
-3.0986057e-01
-3.8521982e-01
-2.1785179e-01
-1.4761096e-02
3.7005180e-02
-2.8119028e-02
-4.2167901e-02
5.2117694e-02
1.2023747e-01
1.8863385e-02
-1.9506434e-01
-3.0192175e-01
-1.7000747e-01
8.0740471e-02
2.0188076e-01
8.5108288e-02
-1.3270970e-01
-2.3646822e-01
-1.6505385e-01
-4.7448014e-02
-2.7886815e-02
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8.3519614e-02
1.5926650e-01
8.6142760e-02
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-3.2883874e-01
-2.5908050e-01
-1.1648940e-01
-3.0653766e-03
1.0326675e-02
-5.3445909e-02
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1.1133075e-01
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2.4258524e-01
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2.3481991e-02
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-1.6716420e-01
1998.   1999.   2000.   2001.
```
Program Results
``` IB03AD EXAMPLE PROGRAM RESULTS

Final 2-norm of the residuals =   0.2970365D+00

Number of iterations                     =      87
Number of conjugate gradients iterations =       0
Number of function evaluations           =    1322
Number of Jacobian evaluations           =     105

Final approximate solution is
-0.9728   0.6465  -1.2888  -0.4296  -0.8530   0.3181   0.9778   0.4570  -0.1420   0.8984
-0.6031   0.0697  -1.0822   0.4465   0.6036   0.3792   0.2532  -0.0285   0.4129   0.4833
0.1746   0.5626   0.2150  -0.3343   0.4013  -0.3679   0.5653   0.8092  -0.2363  -0.6361
-0.6818   0.6110  -0.5506   0.9914   0.0352   0.1968  -0.2502   7.0067 -10.7378   2.6900
-59.8756  -0.9898  -0.8296   2.3429   1.3456  -0.2531  -1.1265   0.0326   0.5617   0.1045
```