IB03BD

Wiener system identification using a MINPACK-like 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
MINPACK-like Levenberg-Marquardt algorithm. The Wiener system
consists of a linear part and a static nonlinearity, and it 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 theta corresponds to the linear part, and wb(i), i = 1 : L,
correspond to the nonlinear part. See SLICOT Library routine

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 NF01BF (the FCN routine in the call of MD03BD).

```
Specification
```      SUBROUTINE IB03BD( INIT, 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         INIT
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 NF01BE (used as a second FCN
routine in the MD03BD call for the initialization step,
see METHOD).

```
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 (NF01BE and/or NF01BF). 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, as well as the relative error desired in the
approximate solution, for the initialization step of
nonlinear part. Termination occurs when either both the
actual and predicted relative reductions in the sum of
squares, or the relative error between two consecutive
iterates are 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, as well as
the relative error desired in the approximate solution,
for the whole optimization process. Termination occurs
when either both the actual and predicted relative
reductions in the sum of squares, or the relative error
between two consecutive iterates are 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( LIW1, LIW2, LIW3 )), where
LIW1 = LIW2 = 0,  if INIT = 'S' or 'N'; otherwise,
LIW1 = M+L;
LIW2 = MAX(M*NOBR+N,M*(N+L));
LIW3 = 3+MAX(NN*(L+2)+2,NX+L), if INIT = 'S' or 'B';
LIW3 = 3+NX+L,                 if INIT = 'L' or 'N'.
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. If INFO = 0, the entries 4 to 3+NX of IWORK
define a permutation matrix P such that J*P = Q*R, where
J is the final calculated Jacobian, Q is an orthogonal
matrix (not stored), and R is upper triangular with
diagonal elements of nonincreasing magnitude (possibly
for each block column of J). Column j of P is column
IWORK(3+j) of the identity matrix. Moreover, the entries
4+NX:3+NX+L of this array contain the ranks of the final
submatrices S_k (see description of LMPARM in MD03BD).

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 MD03BD); 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, and DWORK(4) 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(5) to
DWORK(8) contain the corresponding four values for the
initialization step (see METHOD). (If L > 1, DWORK(8)
contains the maximum of the Levenberg factors for all
outputs.) If INIT = 'L' or INIT = 'B', and INFO = 0,
DWORK(9) to DWORK(8+IWORK(3)) contain reciprocal condition
number estimates set by SLICOT Library routines IB01AD,
IB01BD, and IB01CD.
On exit, if  INFO = -21,  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 + BSN +
MAX( 4, NSMP +
MAX( NSMP*BSN + MAX( 2*NN, 5*BSN + 1 ),
BSN**2 + BSN +
MAX( NSMP + 2*NN, 5*BSN ) ) );
LW3 = MAX( LDW7, NSMP*L + (N+L)*(2*N+M) + 2*N );
LDW7 = NSMP*L + (N+L)*(N+M) + 3*N+M+L,  if M > 0;
LDW7 = NSMP*L + (N+L)*N + 2*N+L,        if M = 0;
LW4 = NSMP*L + NX +
MAX( 4, NSMP*L +
MAX( NSMP*L*( BSN + LTHS ) +
MAX( NSMP*L + L1, L2 + NX ),
NX*( BSN + LTHS ) + NX +
MAX( NSMP*L + L1, NX + L3 ) ) ),
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 = 4*NX + 1,  if L <= 1 or BSN = 0; otherwise,
L2 = BSN + MAX(3*BSN+1,LTHS);
L2 = MAX(L2,4*LTHS+1),         if NSMP > BSN;
L2 = MAX(L2,(NSMP-BSN)*(L-1)), if BSN < NSMP < 2*BSN;
L3 = 4*NX,                     if L <= 1 or BSN = 0;
L3 = LTHS*BSN + 2*NX + 2*MAX(BSN,LTHS),
if L > 1 and BSN > 0,
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:  the user set IFLAG = IWARN in (one of) the
subroutine(s) FCN, i.e., NF01BE, if INIT = 'S'
or 'B', and/or NF01BF; 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:  both actual and predicted relative reductions in
the sum of squares are at most TOL*;
= 2:  relative error between two consecutive iterates is
at most TOL*;
= 3:  conditions for i or j = 1 and i or j = 2 both hold;
= 4:  the cosine of the angle between the vector of error
function values and any column of the Jacobian is at
most EPS in absolute value;
= 5:  the number of iterations has reached ITMAX* without
satisfying any convergence condition;
= 6:  TOL* is too small: no further reduction in the sum
of squares is possible;
= 7:  TOL* is too small: no further improvement in the
approximate solution X is possible;
= 8:  the vector of function values e is orthogonal to the
columns of the Jacobian to machine precision.
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(1:4),
DWORK(5:8) (if INIT = 'S' or 'B'), and DWORK(9:8+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:  the routine QRFACT returned with INFO <> 0;
= 4:  the routine LMPARM returned with INFO <> 0.
In addition, if INIT = 'L' or 'B', i could also be
= 5:  if a Lyapunov equation could not be solved;
= 6:  if the identified linear system is unstable;
= 7:  if the QR algorithm failed on the state matrix
of the identified linear system.
QRFACT and LMPARM are generic names for SLICOT Library
routines NF01BS and NF01BP, respectively, for the whole
optimization process, and MD03BA and MD03BB, respectively,
for the initialization step (if INIT = 'S' or 'B').
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

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 [1,3].
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] 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.

[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.

[3] More, J.J.
The Levenberg-Marquardt algorithm: implementation and theory.
In Watson, G.A. (Ed.), Numerical Analysis, Lecture Notes in
Mathematics, vol. 630, Springer-Verlag, Berlin, Heidelberg
and New York, pp. 105-116, 1978.

```
Numerical Aspects
```  The Levenberg-Marquardt algorithm described in [3] is scaling
invariant and globally convergent to (maybe local) minima.
The convergence rate near a local minimum is quadratic, if the
Jacobian is computed analytically, and linear, if the Jacobian
is computed numerically.

```
```  None
```
Example

Program Text

```*     IB03BD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER           NIN, NOUT
PARAMETER         ( NIN = 5, NOUT = 6 )
INTEGER           BSNM, LDU, LDY, LIWORK, LMAX, LTHS, LXM, MMAX,
\$                  NMAX, NNMAX, NOBRMX, NSMPMX
PARAMETER         ( LMAX = 2, MMAX = 3, NOBRMX = 10, NNMAX = 12,
\$                    NMAX = 4, NSMPMX = 1024,
\$                    BSNM = NNMAX*( LMAX + 2 ) + 1,
\$                    LTHS = NMAX*( LMAX + MMAX + 1 ) + LMAX*MMAX,
\$                    LDU  = NSMPMX, LDY = NSMPMX,
\$                    LXM  = BSNM*LMAX + LTHS,
\$                    LIWORK = MAX( MMAX + LMAX, MMAX*NOBRMX + NMAX,
\$                                  MMAX*( NMAX + LMAX ), 3 +
\$                                  MAX( BSNM + 1, LXM + LMAX ) ) )
INTEGER           L0, L1M, L2M, L3M, LDW1, LDW2, LDW3, LDW4, LDW5,
\$                  LDW6, LDW7, LDWORK, LW1, LW2, LW3, LW4
PARAMETER         ( L0   = MAX( NMAX*( NMAX + LMAX ),
\$                                NMAX + MMAX + LMAX ),
\$                    L1M  = NSMPMX*LMAX +
\$                           MAX( 2*NNMAX,
\$                                ( NMAX + LMAX )*( NMAX + MMAX ) +
\$                                2*NMAX + L0 ),
\$                    L2M  = MAX( 4*LXM + 1, BSNM +
\$                                MAX( 3*BSNM + 1, LTHS ),
\$                                     NSMPMX*( LMAX - 1 ) ),
\$                    L3M  = MAX( 4*LXM, LTHS*BSNM + 2*LXM +
\$                                2*MAX( BSNM, LTHS ) ),
\$                    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 = 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 + BSNM +
\$                           MAX( 4, NSMPMX +
\$                                MAX( NSMPMX*BSNM +
\$                                     MAX( 2*NNMAX, 5*BSNM + 1 ),
\$                                     BSNM**2 + BSNM +
\$                                     MAX( NSMPMX + 2*NNMAX,
\$                                          5*BSNM ) ) ),
\$                    LW3  = MAX( LDW7, NSMPMX*LMAX +
\$                                ( NMAX + LMAX )*( 2*NMAX + MMAX )+
\$                                2*NMAX ),
\$                    LW4  = NSMPMX*LMAX + LXM +
\$                           MAX( 4, NSMPMX*LMAX +
\$                                MAX( NSMPMX*LMAX*( BSNM + LTHS ) +
\$                                     MAX( NSMPMX*LMAX + L1M,
\$                                          L2M + LXM ),
\$                                          LXM*( BSNM + LTHS ) +
\$                                          LXM +
\$                                          MAX( NSMPMX*LMAX + L1M,
\$                                               LXM + L3M ) ) ),
\$                    LDWORK = MAX( LW1, LW2, LW3, LW4 ) )
*     .. Local Scalars ..
LOGICAL           INIT1, INITB, INITL, INITN, INITS
CHARACTER*1       INIT
INTEGER           BSN, I, INFO, INI, ITER, 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 ..
EXTERNAL          IB03BD
*     .. 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
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 IB03BD( INIT, 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)
WRITE ( NOUT, FMT = 99997 ) DWORK(2)
WRITE ( NOUT, FMT = 99996 ) ITER, 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 (' IB03BD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from IB03BD = ',I4)
99997 FORMAT (/' Final 2-norm of the residuals = ',D15.7)
99996 FORMAT (/' Number of iterations                     = ', I7,
\$        /' Number of function evaluations           = ', I7,
\$        /' Number of Jacobian evaluations           = ', I7)
99995 FORMAT (10(1X,F9.4))
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 IB03BD = ',I4)
END
```
Program Data
``` IB03BD EXAMPLE PROGRAM DATA
10     1     1  1024    4   12   500  1000     0  .00001  .00001   B
2.2183165e-01
3.9027807e-02
-5.0295887e-02
8.5386224e-03
7.2431159e-02
-1.7082198e-03
-1.7176287e-01
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1.8566868e-02
1.5625362e-01
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2.8779248e-02
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2.1525013e-02
1.4363602e-01
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2.0263451e-01
2.1983417e-01
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1.2023747e-01
1.8863385e-02
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1.0326675e-02
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1998.   1999.   2000.   2001.
```
Program Results
``` IB03BD EXAMPLE PROGRAM RESULTS

IWARN on exit from IB03BD =   12

Final 2-norm of the residuals =   0.2995840D+00

Number of iterations                     =      42
Number of function evaluations           =     898
Number of Jacobian evaluations           =     295

Final approximate solution is
14.1294    1.1232    6.4322  -11.2418    7.6380  -33.4730  -64.7203  747.1515   -0.4623  -92.6092
6.1682   -0.7672    0.1194    0.3558    0.9091    0.2948    1.3465    0.0093    0.0560   -0.0035
-0.4179   -0.0455   -2.0871   -0.9196    1.0777    0.9213    0.5373    1.0412   -0.3978    7.6832
-6.8614  -31.6119   -0.1092   -9.8984    0.1257    0.4056    0.0472    7.5819  -13.3969    2.4869
-66.0727   -0.8411   -0.7040    1.9641    1.3059   -0.2046   -0.9326    0.0040    0.4032    0.1479
```