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<HEAD><TITLE>IB01AD - SLICOT Library Routine Documentation</TITLE>
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<BODY>

<H2><A Name="IB01AD">IB01AD</A></H2>
<H3>
Upper triangular factor in the QR factorization of the block-Hankel matrix (driver)
</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 preprocess the input-output data for estimating the matrices
  of a linear time-invariant dynamical system and to find an
  estimate of the system order. The input-output data can,
  optionally, be processed sequentially.

</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
      SUBROUTINE IB01AD( METH, ALG, JOBD, BATCH, CONCT, CTRL, NOBR, M,
     $                   L, NSMP, U, LDU, Y, LDY, N, R, LDR, SV, RCOND,
     $                   TOL, IWORK, DWORK, LDWORK, IWARN, INFO )
C     .. Scalar Arguments ..
      DOUBLE PRECISION   RCOND, TOL
      INTEGER            INFO, IWARN, L, LDR, LDU, LDWORK, LDY, M, N,
     $                   NOBR, NSMP
      CHARACTER          ALG, BATCH, CONCT, CTRL, JOBD, METH
C     .. Array Arguments ..
      INTEGER            IWORK(*)
      DOUBLE PRECISION   DWORK(*), R(LDR, *), SV(*), U(LDU, *),
     $                   Y(LDY, *)

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

<B>Mode Parameters</B>
<PRE>
  METH    CHARACTER*1
          Specifies the subspace identification method to be used,
          as follows:
          = 'M':  MOESP  algorithm with past inputs and outputs;
          = 'N':  N4SID  algorithm.

  ALG     CHARACTER*1
          Specifies the algorithm for computing the triangular
          factor R, as follows:
          = 'C':  Cholesky algorithm applied to the correlation
                  matrix of the input-output data;
          = 'F':  Fast QR algorithm;
          = 'Q':  QR algorithm applied to the concatenated block
                  Hankel matrices.

  JOBD    CHARACTER*1
          Specifies whether or not the matrices B and D should later
          be computed using the MOESP approach, as follows:
          = 'M':  the matrices B and D should later be computed
                  using the MOESP approach;
          = 'N':  the matrices B and D should not be computed using
                  the MOESP approach.
          This parameter is not relevant for METH = 'N'.

  BATCH   CHARACTER*1
          Specifies whether or not sequential data processing is to
          be used, and, for sequential processing, whether or not
          the current data block is the first block, an intermediate
          block, or the last block, as follows:
          = 'F':  the first block in sequential data processing;
          = 'I':  an intermediate block in sequential data
                  processing;
          = 'L':  the last block in sequential data processing;
          = 'O':  one block only (non-sequential data processing).
          NOTE that when  100  cycles of sequential data processing
               are completed for  BATCH = 'I',  a warning is
               issued, to prevent for an infinite loop.

  CONCT   CHARACTER*1
          Specifies whether or not the successive data blocks in
          sequential data processing belong to a single experiment,
          as follows:
          = 'C':  the current data block is a continuation of the
                  previous data block and/or it will be continued
                  by the next data block;
          = 'N':  there is no connection between the current data
                  block and the previous and/or the next ones.
          This parameter is not used if BATCH = 'O'.

  CTRL    CHARACTER*1
          Specifies whether or not the user's confirmation of the
          system order estimate is desired, as follows:
          = 'C':  user's confirmation;
          = 'N':  no confirmation.
          If  CTRL = 'C',  a reverse communication routine,  IB01OY,
          is indirectly called (by SLICOT Library routine IB01OD),
          and, after inspecting the singular values and system order
          estimate,  n,  the user may accept  n  or set a new value.
          IB01OY  is not called if CTRL = 'N'.

</PRE>
<B>Input/Output Parameters</B>
<PRE>
  NOBR    (input) INTEGER
          The number of block rows,  s,  in the input and output
          block Hankel matrices to be processed.  NOBR &gt; 0.
          (In the MOESP theory,  NOBR  should be larger than  n,
          the estimated dimension of state vector.)

  M       (input) INTEGER
          The number of system inputs.  M &gt;= 0.
          When M = 0, no system inputs are processed.

  L       (input) INTEGER
          The number of system outputs.  L &gt; 0.

  NSMP    (input) INTEGER
          The number of rows of matrices  U  and  Y  (number of
          samples,  t). (When sequential data processing is used,
          NSMP  is the number of samples of the current data
          block.)
          NSMP &gt;= 2*(M+L+1)*NOBR - 1,  for non-sequential
                                       processing;
          NSMP &gt;= 2*NOBR,  for sequential processing.
          The total number of samples when calling the routine with
          BATCH = 'L'  should be at least  2*(M+L+1)*NOBR - 1.
          The  NSMP  argument may vary from a cycle to another in
          sequential data processing, but  NOBR, M,  and  L  should
          be kept constant. For efficiency, it is advisable to use
          NSMP  as large as possible.

  U       (input) DOUBLE PRECISION array, dimension (LDU,M)
          The leading NSMP-by-M part of this array must contain the
          t-by-m input-data sequence matrix  U,
          U = [u_1 u_2 ... u_m].  Column  j  of  U  contains the
          NSMP  values of the j-th input component for consecutive
          time increments.
          If M = 0, this array is not referenced.

  LDU     INTEGER
          The leading dimension of the array U.
          LDU &gt;= NSMP, if M &gt; 0;
          LDU &gt;= 1,    if M = 0.

  Y       (input) DOUBLE PRECISION array, dimension (LDY,L)
          The leading NSMP-by-L part of this array must contain the
          t-by-l output-data sequence matrix  Y,
          Y = [y_1 y_2 ... y_l].  Column  j  of  Y  contains the
          NSMP  values of the j-th output component for consecutive
          time increments.

  LDY     INTEGER
          The leading dimension of the array Y.  LDY &gt;= NSMP.

  N       (output) INTEGER
          The estimated order of the system.
          If  CTRL = 'C',  the estimated order has been reset to a
          value specified by the user.

  R       (output or input/output) DOUBLE PRECISION array, dimension
          ( LDR,2*(M+L)*NOBR )
          On exit, if ALG = 'C' and BATCH = 'F' or 'I', the leading
          2*(M+L)*NOBR-by-2*(M+L)*NOBR upper triangular part of this
          array contains the current upper triangular part of the
          correlation matrix in sequential data processing.
          If ALG = 'F' and BATCH = 'F' or 'I', the array R is not
          referenced.
          On exit, if INFO = 0, ALG = 'Q', and BATCH = 'F' or 'I',
          the leading 2*(M+L)*NOBR-by-2*(M+L)*NOBR upper triangular
          part of this array contains the current upper triangular
          factor R from the QR factorization of the concatenated
          block Hankel matrices. Denote  R_ij, i,j = 1:4,  the
          ij submatrix of  R,  partitioned by M*NOBR,  M*NOBR,
          L*NOBR,  and  L*NOBR  rows and columns.
          On exit, if INFO = 0 and BATCH = 'L' or 'O', the leading
          2*(M+L)*NOBR-by-2*(M+L)*NOBR upper triangular part of
          this array contains the matrix S, the processed upper
          triangular factor R from the QR factorization of the
          concatenated block Hankel matrices, as required by other
          subroutines. Specifically, let  S_ij, i,j = 1:4,  be the
          ij submatrix of  S,  partitioned by M*NOBR,  L*NOBR,
          M*NOBR,  and  L*NOBR  rows and columns. The submatrix
          S_22  contains the matrix of left singular vectors needed
          subsequently. Useful information is stored in  S_11  and
          in the block-column  S_14 : S_44.  For METH = 'M' and
          JOBD = 'M', the upper triangular part of  S_31  contains
          the upper triangular factor in the QR factorization of the
          matrix  R_1c = [ R_12'  R_22'  R_11' ]',  and  S_12
          contains the corresponding leading part of the transformed
          matrix  R_2c = [ R_13'  R_23'  R_14' ]'.  For  METH = 'N',
          the subarray  S_41 : S_43  contains the transpose of the
          matrix contained in  S_14 : S_34.
          The details of the contents of R need not be known if this
          routine is followed by SLICOT Library routine IB01BD.
          On entry, if ALG = 'C', or ALG = 'Q', and BATCH = 'I' or
          'L', the leading  2*(M+L)*NOBR-by-2*(M+L)*NOBR  upper
          triangular part of this array must contain the upper
          triangular matrix R computed at the previous call of this
          routine in sequential data processing. The array R need
          not be set on entry if ALG = 'F' or if BATCH = 'F' or 'O'.

  LDR     INTEGER
          The leading dimension of the array  R.
          LDR &gt;= MAX( 2*(M+L)*NOBR, 3*M*NOBR ),
                               for METH = 'M' and JOBD = 'M';
          LDR &gt;= 2*(M+L)*NOBR, for METH = 'M' and JOBD = 'N' or
                               for METH = 'N'.

  SV      (output) DOUBLE PRECISION array, dimension ( L*NOBR )
          The singular values used to estimate the system order.

</PRE>
<B>Tolerances</B>
<PRE>
  RCOND   DOUBLE PRECISION
          The tolerance to be used for estimating the rank of
          matrices. If the user sets  RCOND &gt; 0,  the given value
          of  RCOND  is used as a lower bound for the reciprocal
          condition number;  an m-by-n matrix whose estimated
          condition number is less than  1/RCOND  is considered to
          be of full rank.  If the user sets  RCOND &lt;= 0,  then an
          implicitly computed, default tolerance, defined by
          RCONDEF = m*n*EPS,  is used instead, where  EPS  is the
          relative machine precision (see LAPACK Library routine
          DLAMCH).
          This parameter is not used for  METH = 'M'.

  TOL     DOUBLE PRECISION
          Absolute tolerance used for determining an estimate of
          the system order. If  TOL &gt;= 0,  the estimate is
          indicated by the index of the last singular value greater
          than or equal to  TOL.  (Singular values less than  TOL
          are considered as zero.) When  TOL = 0,  an internally
          computed default value,  TOL = NOBR*EPS*SV(1),  is used,
          where  SV(1)  is the maximal singular value, and  EPS  is
          the relative machine precision (see LAPACK Library routine
          DLAMCH). When  TOL &lt; 0,  the estimate is indicated by the
          index of the singular value that has the largest
          logarithmic gap to its successor.

</PRE>
<B>Workspace</B>
<PRE>
  IWORK   INTEGER array, dimension (LIWORK)
          LIWORK &gt;= MAX(3,(M+L)*NOBR), if METH = 'N';
          LIWORK &gt;= MAX(3,M+L), if METH = 'M' and ALG = 'F';
          LIWORK &gt;= 3,   if METH = 'M' and ALG = 'C' or 'Q'.
          On entry with  BATCH = 'I'  or  BATCH = 'L',  IWORK(1:3)
          must contain the values of ICYCLE, MAXWRK, and NSMPSM
          set by the previous call of this routine.
          On exit with  BATCH = 'F'  or  BATCH = 'I',  IWORK(1:3)
          contains the values of ICYCLE, MAXWRK, and NSMPSM to be
          used by the next call of the routine.
          ICYCLE  counts the cycles for  BATCH = 'I'.
          MAXWRK  stores the current optimal workspace.
          NSMPSM  sums up the  NSMP  values for  BATCH &lt;&gt; 'O'.
          The first three elements of  IWORK  should be preserved
          during successive calls of the routine with  BATCH = 'F'
          or  BATCH = 'I',  till the final call with   BATCH = 'L'.

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if  INFO = 0,  DWORK(1) returns the optimal value
          of LDWORK,  and, for  METH = 'N',  and  BATCH = 'L'  or
          'O',  DWORK(2)  and  DWORK(3)  contain the reciprocal
          condition numbers of the triangular factors of the
          matrices  U_f  and  r_1  [6].
          On exit, if  INFO = -23,  DWORK(1)  returns the minimum
          value of LDWORK.
          Let
          k = 0,               if CONCT = 'N' and ALG = 'C' or 'Q';
          k = 2*NOBR-1,        if CONCT = 'C' and ALG = 'C' or 'Q';
          k = 2*NOBR*(M+L+1),  if CONCT = 'N' and ALG = 'F';
          k = 2*NOBR*(M+L+2),  if CONCT = 'C' and ALG = 'F'.
          The first (M+L)*k elements of  DWORK  should be preserved
          during successive calls of the routine with  BATCH = 'F'
          or  'I',  till the final call with  BATCH = 'L'.

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK &gt;= (4*NOBR-2)*(M+L), if ALG = 'C', BATCH = 'F' or
                          'I' and CONCT = 'C';
          LDWORK &gt;= 1, if ALG = 'C', BATCH = 'F' or 'I' and
                          CONCT = 'N';
          LDWORK &gt;= max((4*NOBR-2)*(M+L), 5*L*NOBR), if METH = 'M',
                          ALG = 'C', BATCH = 'L' and CONCT = 'C';
          LDWORK &gt;= max((2*M-1)*NOBR, (M+L)*NOBR, 5*L*NOBR),
                          if METH = 'M', JOBD = 'M', ALG = 'C',
                           BATCH = 'O', or
                          (BATCH = 'L' and CONCT = 'N');
          LDWORK &gt;= 5*L*NOBR, if METH = 'M', JOBD = 'N', ALG = 'C',
                           BATCH = 'O', or
                          (BATCH = 'L' and CONCT = 'N');
          LDWORK &gt;= 5*(M+L)*NOBR+1, if METH = 'N', ALG = 'C', and
                          BATCH = 'L' or 'O';
          LDWORK &gt;= (M+L)*2*NOBR*(M+L+3), if ALG = 'F',
                          BATCH &lt;&gt; 'O' and CONCT = 'C';
          LDWORK &gt;= (M+L)*2*NOBR*(M+L+1), if ALG = 'F',
                          BATCH = 'F', 'I' and CONCT = 'N';
          LDWORK &gt;= (M+L)*4*NOBR*(M+L+1)+(M+L)*2*NOBR, if ALG = 'F',
                          BATCH = 'L' and CONCT = 'N', or
                          BATCH = 'O';
          LDWORK &gt;= 4*(M+L)*NOBR, if ALG = 'Q', BATCH = 'F', and
                          LDR &gt;= NS = NSMP - 2*NOBR + 1;
          LDWORK &gt;= max(4*(M+L)*NOBR, 5*L*NOBR), if METH = 'M',
                          ALG = 'Q', BATCH = 'O', and LDR &gt;= NS;
          LDWORK &gt;= 5*(M+L)*NOBR+1, if METH = 'N', ALG = 'Q',
                          BATCH = 'O', and LDR &gt;= NS;
          LDWORK &gt;= 6*(M+L)*NOBR, if ALG = 'Q', (BATCH = 'F' or 'O',
                          and LDR &lt; NS), or (BATCH = 'I' or
                          'L' and CONCT = 'N');
          LDWORK &gt;= 4*(NOBR+1)*(M+L)*NOBR, if ALG = 'Q', BATCH = 'I'
                          or 'L' and CONCT = 'C'.
          The workspace used for ALG = 'Q' is
                    LDRWRK*2*(M+L)*NOBR + 4*(M+L)*NOBR,
          where LDRWRK = LDWORK/(2*(M+L)*NOBR) - 2; recommended
          value LDRWRK = NS, assuming a large enough cache size.
          For good performance,  LDWORK  should be larger.

</PRE>
<B>Warning Indicator</B>
<PRE>
  IWARN   INTEGER
          = 0:  no warning;
          = 1:  the number of 100 cycles in sequential data
                processing has been exhausted without signaling
                that the last block of data was get; the cycle
                counter was reinitialized;
          = 2:  a fast algorithm was requested (ALG = 'C' or 'F'),
                but it failed, and the QR algorithm was then used
                (non-sequential data processing);
          = 3:  all singular values were exactly zero, hence  N = 0
                (both input and output were identically zero);
          = 4:  the least squares problems with coefficient matrix
                U_f,  used for computing the weighted oblique
                projection (for METH = 'N'), have a rank-deficient
                coefficient matrix;
          = 5:  the least squares problem with coefficient matrix
                r_1  [6], used for computing the weighted oblique
                projection (for METH = 'N'), has a rank-deficient
                coefficient matrix.
          NOTE: the values 4 and 5 of IWARN have no significance
                for the identification problem.

</PRE>
<B>Error Indicator</B>
<PRE>
  INFO    INTEGER
          = 0:  successful exit;
          &lt; 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  a fast algorithm was requested (ALG = 'C', or 'F')
                in sequential data processing, but it failed; the
                routine can be repeatedly called again using the
                standard QR algorithm;
          = 2:  the singular value decomposition (SVD) algorithm did
                not converge.

</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
  The procedure consists in three main steps, the first step being
  performed by one of the three algorithms included.

  1.a) For non-sequential data processing using QR algorithm, a
  t x 2(m+l)s  matrix H is constructed, where

       H = [ Uf'         Up'      Y'      ],  for METH = 'M',
               s+1,2s,t    1,s,t   1,2s,t

       H = [ U'       Y'      ],              for METH = 'N',
              1,2s,t   1,2s,t

  and  Up     , Uf        , U      , and  Y        are block Hankel
         1,s,t    s+1,2s,t   1,2s,t        1,2s,t
  matrices defined in terms of the input and output data [3].
  A QR factorization is used to compress the data.
  The fast QR algorithm uses a QR factorization which exploits
  the block-Hankel structure. Actually, the Cholesky factor of H'*H
  is computed.

  1.b) For sequential data processing using QR algorithm, the QR
  decomposition is done sequentially, by updating the upper
  triangular factor  R.  This is also performed internally if the
  workspace is not large enough to accommodate an entire batch.

  1.c) For non-sequential or sequential data processing using
  Cholesky algorithm, the correlation matrix of input-output data is
  computed (sequentially, if requested), taking advantage of the
  block Hankel structure [7].  Then, the Cholesky factor of the
  correlation matrix is found, if possible.

  2) A singular value decomposition (SVD) of a certain matrix is
  then computed, which reveals the order  n  of the system as the
  number of "non-zero" singular values. For the MOESP approach, this
  matrix is  [ R_24'  R_34' ]' := R(ms+1:(2m+l)s,(2m+l)s+1:2(m+l)s),
  where  R  is the upper triangular factor  R  constructed by SLICOT
  Library routine  IB01MD.  For the N4SID approach, a weighted
  oblique projection is computed from the upper triangular factor  R
  and its SVD is then found.

  3) The singular values are compared to the given, or default TOL,
  and the estimated order  n  is returned, possibly after user's
  confirmation.

</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
  [1] Verhaegen M., and Dewilde, P.
      Subspace Model Identification. Part 1: The output-error
      state-space model identification class of algorithms.
      Int. J. Control, 56, pp. 1187-1210, 1992.

  [2] Verhaegen M.
      Subspace Model Identification. Part 3: Analysis of the
      ordinary output-error state-space model identification
      algorithm.
      Int. J. Control, 58, pp. 555-586, 1993.

  [3] Verhaegen M.
      Identification of the deterministic part of MIMO state space
      models given in innovations form from input-output data.
      Automatica, Vol.30, No.1, pp.61-74, 1994.

  [4] Van Overschee, P., and De Moor, B.
      N4SID: Subspace Algorithms for the Identification of
      Combined Deterministic-Stochastic Systems.
      Automatica, Vol.30, No.1, pp. 75-93, 1994.

  [5] Peternell, K., Scherrer, W. and Deistler, M.
      Statistical Analysis of Novel Subspace Identification Methods.
      Signal Processing, 52, pp. 161-177, 1996.

  [6] Sima, V.
      Subspace-based Algorithms for Multivariable System
      Identification.
      Studies in Informatics and Control, 5, pp. 335-344, 1996.

  [7] Sima, V.
      Cholesky or QR Factorization for Data Compression in
      Subspace-based Identification ?
      Proceedings of the Second NICONET Workshop on ``Numerical
      Control Software: SLICOT, a Useful Tool in Industry'',
      December 3, 1999, INRIA Rocquencourt, France, pp. 75-80, 1999.

</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
  The implemented method is numerically stable (when QR algorithm is
  used), reliable and efficient. The fast Cholesky or QR algorithms
  are more efficient, but the accuracy could diminish by forming the
  correlation matrix.
  The most time-consuming computational step is step 1:
                                     2
  The QR algorithm needs 0(t(2(m+l)s) ) floating point operations.
                                        2              3
  The Cholesky algorithm needs 0(2t(m+l) s)+0((2(m+l)s) ) floating
  point operations.
                                       2           3 2
  The fast QR algorithm needs 0(2t(m+l) s)+0(4(m+l) s ) floating
  point operations.
                                             3
  Step 2 of the algorithm requires 0(((m+l)s) ) floating point
  operations.

</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
  For ALG = 'Q', BATCH = 'O' and LDR &lt; NS, or BATCH &lt;&gt; 'O', the
  calculations could be rather inefficient if only minimal workspace
  (see argument LDWORK) is provided. It is advisable to provide as
  much workspace as possible. Almost optimal efficiency can be
  obtained for  LDWORK = (NS+2)*(2*(M+L)*NOBR),  assuming that the
  cache size is large enough to accommodate R, U, Y, and DWORK.

</PRE>

<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
*     IB01AD EXAMPLE PROGRAM TEXT
*
*     .. Parameters ..
      INTEGER          NIN, NOUT
      PARAMETER        ( NIN = 5, NOUT = 6 )
      INTEGER          LDR, LDU, LDWORK, LDY, LIWORK, LMAX, MMAX,
     $                 NOBRMX, NSMPMX
      PARAMETER        ( LMAX = 5, MMAX = 5, NOBRMX = 20, NSMPMX = 2000,
     $                   LDR = MAX( 2*( MMAX + LMAX )*NOBRMX,
     $                              3*MMAX*NOBRMX ), LDU = NSMPMX,
     $                   LDWORK = MAX( 6*( MMAX + LMAX )*NOBRMX,
     $                                 ( MMAX + LMAX )*( 4*NOBRMX*
     $                                 ( MMAX + LMAX + 1 ) + 2*NOBRMX ),
     $                                 ( MMAX + LMAX )*4*NOBRMX*
     $                                 ( NOBRMX + 1 ) ),
     $                   LDY = NSMPMX, LIWORK = ( MMAX + LMAX )*NOBRMX )
*     .. Local Scalars ..
      LOGICAL          NGIVEN
      CHARACTER        ALG, BATCH, CONCT, CTRL, JOBD, METH
      INTEGER          I, ICYCLE, II, INFO, IWARN, J, L, M, N, NCYCLE,
     $                 NGIV, NOBR, NSAMPL, NSMP
      DOUBLE PRECISION RCOND, TOL
*     .. Local Arrays ..
      DOUBLE PRECISION DWORK(LDWORK), R(LDR, 2*(MMAX+LMAX)*NOBRMX),
     $                 SV(LMAX*NOBRMX), U(LDU, MMAX), Y(LDY, LMAX)
      INTEGER          IWORK(LIWORK)
*     .. External Functions ..
      LOGICAL          LSAME
      EXTERNAL         LSAME
*     .. External Subroutines ..
      EXTERNAL         IB01AD
*     .. Intrinsic Functions ..
      INTRINSIC        MAX
*     .. Executable Statements ..
*
      WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
*     If the value of N is positive, it will be taken as system order.
      READ ( NIN, FMT = '()' )
      READ ( NIN, FMT = * ) NOBR, N, M, L, NSMP, RCOND, TOL, METH, ALG,
     $                      JOBD, BATCH, CONCT, CTRL
      IF ( LSAME( BATCH, 'F' ) ) THEN
         READ ( NIN, FMT = * ) NCYCLE
      ELSE
         NCYCLE = 1
      END IF
      NSAMPL = NCYCLE*NSMP
*
      NGIVEN = N.GT.0
      IF( NGIVEN )
     $   NGIV = N
      IF ( NOBR.LE.0 .OR. NOBR.GT.NOBRMX ) THEN
         WRITE ( NOUT, FMT = 99997 ) NOBR
      ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
         WRITE ( NOUT, FMT = 99996 ) M
      ELSE IF ( L.LE.0 .OR. L.GT.LMAX ) THEN
         WRITE ( NOUT, FMT = 99995 ) L
      ELSE IF ( NSMP.LT.0 .OR. NSMP.GT.NSMPMX .OR.
     $        ( NSMP.LT.2*( M + L + 1 )*NOBR - 1 .AND.
     $          LSAME( BATCH, 'O' ) ) .OR.
     $        ( NSAMPL.LT.2*( M + L + 1 )*NOBR - 1 .AND.
     $          LSAME( BATCH, 'L' ) ) .OR.
     $          NSMP.LT.2*NOBR .AND. ( LSAME( BATCH, 'F' ) .OR.
     $                                 LSAME( BATCH, 'I' ) ) ) THEN
         WRITE ( NOUT, FMT = 99994 ) NSMP
      ELSE IF ( NCYCLE.LE.0 .OR. NSAMPL.GT.NSMPMX ) THEN
         WRITE ( NOUT, FMT = 99993 ) NCYCLE
      ELSE
*        Read the matrices U and Y from the input file.
         IF ( M.GT.0 )
     $      READ ( NIN, FMT = * )
     $                         ( ( U(I,J), J = 1, M ), I = 1, NSAMPL )
         READ ( NIN, FMT = * ) ( ( Y(I,J), J = 1, L ), I = 1, NSAMPL )
*        Compute the  R  factor from a QR (or Cholesky) factorization
*        of the Hankel-like matrix (or correlation matrix).
         DO 10 ICYCLE = 1, NCYCLE
            II = ( ICYCLE - 1 )*NSMP + 1
            IF ( NCYCLE.GT.1 ) THEN
               IF ( ICYCLE.GT.1 )      BATCH = 'I'
               IF ( ICYCLE.EQ.NCYCLE ) BATCH = 'L'
            END IF
            CALL IB01AD( METH, ALG, JOBD, BATCH, CONCT, CTRL, NOBR, M,
     $                   L, NSMP, U(II,1), LDU, Y(II,1), LDY, N, R, LDR,
     $                   SV, RCOND, TOL, IWORK, DWORK, LDWORK, IWARN,
     $                   INFO )
   10    CONTINUE
         IF ( INFO.NE.0 ) THEN
            WRITE ( NOUT, FMT = 99998 ) INFO
         ELSE
            IF ( IWARN.NE.0 )
     $         WRITE ( NOUT, FMT = 99990 ) IWARN
            IF( NGIVEN )
     $         N = NGIV
            WRITE ( NOUT, FMT = 99992 ) N
            WRITE ( NOUT, FMT = 99991 ) ( SV(I), I = 1,L*NOBR )
         END IF
      END IF
      STOP
99999 FORMAT ( ' IB01AD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT ( ' INFO on exit from IB01AD = ',I2)
99997 FORMAT (/' NOBR is out of range.',/' NOBR = ', I5)
99996 FORMAT (/' M is out of range.',/' M = ', I5)
99995 FORMAT (/' L is out of range.',/' L = ', I5)
99994 FORMAT (/' NSMP is out of range.',/' NSMP = ', I5)
99993 FORMAT (/' NCYCLE is out of range.',/' NCYCLE = ', I5)
99992 FORMAT ( ' The order of the system is ', I5)
99991 FORMAT ( ' The singular values are ',/ (8(1X,F8.4)))
99990 FORMAT ( ' IWARN on exit from IB01AD = ',I2)
      END
</PRE>
<B>Program Data</B>
<PRE>
 IB01AD EXAMPLE PROGRAM DATA
  15     0     1     1  1000    0.0   -1.0     M     C     N     O     N     N
   6.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   6.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   3.41
   3.41
   3.41
   3.41
   6.41
   3.41
   3.41
   3.41
   6.41
   6.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   3.41
   3.41
   6.41
   6.41
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   6.41
   3.41
   6.41
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   6.41
   6.41
   6.41
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   6.41
   6.41
   3.41
   6.41
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   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   6.41
   3.41
   3.41
   3.41
   6.41
   6.41
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   6.41
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   6.41
   6.41
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   6.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   6.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   3.41
   6.41
   3.41
   3.41
   3.41
   6.41
   3.41
   3.41
   3.41
   6.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
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   3.41
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   3.41
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   3.41
   6.41
   6.41
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   6.41
   3.41
   3.41
   6.41
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   3.41
   6.41
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   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
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   6.41
   6.41
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   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   6.41
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   3.41
   3.41
   3.41
   6.41
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   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
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   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   6.41
   6.41
   3.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   3.41
   3.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   3.41
   3.41
   3.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   6.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   3.41
   6.41
   6.41
   6.41
   6.41
   6.41
   6.41
   4.766099
   4.763659
   4.839359
   5.002979
   5.017629
   5.056699
   5.154379
   5.361949
   5.425439
   5.569519
   5.681849
   5.742899
   5.803949
   5.918729
   5.821049
   5.447419
   5.061589
   4.629349
   4.267939
   4.011519
   3.850349
   3.711159
   3.569519
   3.518239
   3.652549
   3.818609
   3.862559
   4.011519
   4.353409
   4.705049
   5.083559
   5.344859
   5.274039
   5.127519
   4.761219
   4.451089
   4.221539
   4.045709
   3.874769
   3.730689
   3.662319
   3.576849
   3.542659
   3.479169
   3.454749
   3.359509
   3.298459
   3.225199
   3.200779
   3.225199
   3.227639
   3.274039
   3.457189
   3.867449
   4.321659
   4.492599
   4.431549
   4.243519
   4.050599
   3.857679
   3.730689
   3.791739
   3.921169
   3.955359
   3.847909
   3.725809
   3.611039
   3.716039
   4.092109
   4.480389
   4.814939
   5.054259
   5.303339
   5.486489
   5.672089
   5.779529
   5.799069
   5.664759
   5.291129
   4.880879
   4.558529
   4.184909
   3.889419
   3.708719
   3.623249
   3.569519
   3.718479
   4.033499
   4.412009
   4.629349
   4.558529
   4.394919
   4.180019
   4.197119
   4.431549
   4.714819
   4.961459
   5.300899
   5.567079
   5.681849
   5.545099
   5.188569
   4.883319
   4.600049
   4.270379
   4.038389
   3.838139
   3.711159
   3.591499
   3.535329
   3.486489
   3.476729
   3.425439
   3.381489
   3.369279
   3.364389
   3.347299
   3.381489
   3.420559
   3.413229
   3.452309
   3.635459
   4.038389
   4.375379
   4.727029
   5.056699
   5.298459
   5.532889
   5.466959
   5.195899
   4.885759
   4.763659
   4.875989
   5.042049
   5.283809
   5.491379
   5.596379
   5.672089
   5.772209
   5.830819
   5.933379
   5.899189
   5.935819
   5.894309
   5.918729
   5.994429
   5.957799
   6.031059
   6.062809
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   4.426669
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   5.183689
   5.430329
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   5.706269
   5.369279
   5.027399
   4.705049
   4.414459
   4.145839
   3.965129
   4.033499
   4.372939
   4.683079
</PRE>
<B>Program Results</B>
<PRE>
 IB01AD EXAMPLE PROGRAM RESULTS

 The order of the system is     4
 The singular values are 
  69.8841  14.9963   3.6675   1.9677   0.3000   0.2078   0.1651   0.1373
   0.1133   0.1059   0.0856   0.0784   0.0733   0.0678   0.0571
</PRE>

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