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<HEAD><TITLE>IB03AD - SLICOT Library Routine Documentation</TITLE>
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<H2><A Name="IB03AD">IB03AD</A></H2>
<H3>
Estimating parameters of a Wiener system using Levenberg-Marquardt algorithm
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>

<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
  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).

</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
      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(*)

</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>

<B>Mode Parameters</B>
<PRE>
  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'.

</PRE>
<B>Input/Output Parameters</B>
<PRE>
  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 &gt; 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 &gt;= 0.

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

  NSMP    (input) INTEGER
          The number of input and output samples, t.  NSMP &gt;= 0, and
          NSMP &gt;= 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 &lt; 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 &gt;= 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 &gt;= 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 &gt;= 0.

  ITMAX2  (input) INTEGER
          The maximum number of iterations.  ITMAX2 &gt;= 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 &lt;= 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 &gt;= 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 &gt;= 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 &gt;= 0, then LX &gt;= NX := lin2 (see parameter X).
          If N is unknown (N &lt; 0 on entry), a large enough estimate
          of N should be used in the formula of lin2.
          On exit, if N &lt; 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.

</PRE>
<B>Tolerances</B>
<PRE>
  TOL1    DOUBLE PRECISION
          If INIT = 'S' or 'B' and TOL1 &gt;= 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 &lt; 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 &gt;= 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 &lt; 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.

</PRE>
<B>Workspace</B>
<PRE>
  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 &gt; 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 &lt; 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 &gt;= 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 &gt;= 0,                                  if M = 0;
              LDW2 &gt;= L*NOBR*N + M*NOBR*(N+L)*(M*(N+L)+1) +
                      MAX( (N+L)**2, 4*M*(N+L)+1 ),       if M &gt; 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 &gt; 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 &gt; 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.

</PRE>
<B>Warning Indicator</B>
<PRE>
  IWARN   INTEGER
          = 0:  no warning;
          &lt; 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.

</PRE>
<B>Error Indicator</B>
<PRE>
  INFO    INTEGER
          = 0:  successful exit;
          &lt; 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 &lt;&gt; 0 for
                IFLAG = 1;
          = 2:  the routine FCN returned with INFO &lt;&gt; 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 &lt;&gt; 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).

</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
  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.

</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
  [1] Kelley, C.T.
      Iterative Methods for Optimization.
      Society for Industrial and Applied Mathematics (SIAM),
      Philadelphia (Pa.), 1999.

  [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.

</PRE>

<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
  None
</PRE>

<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
*     IB03AD EXAMPLE PROGRAM TEXT
*
*     .. 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 ..
      EXTERNAL          IB03AD
*     .. 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
</PRE>
<B>Program Data</B>
<PRE>
 IB03AD EXAMPLE PROGRAM DATA
 10     1     1  1024    4   12   500  1000     0  .00001  .00001   B   D   F
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  -1.2043322e-01
  -2.1083864e-01
  -1.4415945e-01
   4.3538937e-02
   2.3203364e-01
   2.9044234e-01
   1.6171416e-01
  -9.5674666e-02
  -3.3749265e-01
  -4.1795872e-01
  -2.7746809e-01
   2.0648626e-02
   3.2603206e-01
   4.8410918e-01
   4.1672303e-01
   1.5905611e-01
  -1.6318595e-01
  -3.9931562e-01
  -4.4568803e-01
  -2.9169291e-01
  -2.0960934e-02
   2.3175866e-01
   3.4693819e-01
   2.7877641e-01
   7.7125945e-02
  -1.4069530e-01
  -2.5367798e-01
  -2.0150506e-01
  -1.6778161e-02
   1.9116819e-01
   2.9409556e-01
   2.1593628e-01
  -1.9610708e-02
  -2.9401135e-01
  -4.5512990e-01
  -4.0311941e-01
  -1.5075705e-01
   1.7921653e-01
   4.2153577e-01
   4.6143206e-01
   2.9688389e-01
   3.5275834e-02
  -1.7206796e-01
  -2.2040717e-01
  -1.1280250e-01
   4.6014479e-02
   1.2005000e-01
   3.5297082e-02
  -1.6459920e-01
  -3.4121448e-01
  -3.5130088e-01
  -1.4787707e-01
   1.7615712e-01
   4.3972643e-01
   4.8949447e-01
   2.9899548e-01
  -1.6059656e-02
  -2.7414987e-01
  -3.4124596e-01
  -2.0476598e-01
   3.1287353e-02
   2.1535118e-01
   2.3693813e-01
   8.7039128e-02
  -1.3914592e-01
  -2.9731202e-01
  -2.8057123e-01
  -8.9244625e-02
   1.6445576e-01
   3.2621002e-01
   2.9949560e-01
   1.0678193e-01
  -1.3016725e-01
  -2.7225661e-01
  -2.4687907e-01
  -8.3173776e-02
   1.1381888e-01
   2.2819642e-01
   1.9830143e-01
   4.8505476e-02
  -1.2763594e-01
  -2.2560309e-01
  -1.9560311e-01
  -7.1212054e-02
   6.0380807e-02
   1.2445307e-01
   1.0835168e-01
   5.5609724e-02
   1.7269294e-02
   9.3997346e-03
   1.1223045e-02
  -4.3543819e-03
  -4.2668837e-02
  -8.5657964e-02
  -1.0909342e-01
  -9.7154374e-02
  -4.6781850e-02
   3.1101930e-02
   1.0973840e-01
   1.5122945e-01
   1.2531404e-01
   3.3620966e-02
  -8.3194568e-02
  -1.6716420e-01
 1998.   1999.   2000.   2001.
</PRE>
<B>Program Results</B>
<PRE>
 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
</PRE>

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