Purpose
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.Specification
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, *)
Arguments
Mode Parameters
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'.
Input/Output Parameters
NOBR (input) INTEGER
The number of block rows, s, in the input and output
block Hankel matrices to be processed. NOBR > 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 >= 0.
When M = 0, no system inputs are processed.
L (input) INTEGER
The number of system outputs. L > 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 >= 2*(M+L+1)*NOBR - 1, for non-sequential
processing;
NSMP >= 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 >= NSMP, if M > 0;
LDU >= 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 >= 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 >= MAX( 2*(M+L)*NOBR, 3*M*NOBR ),
for METH = 'M' and JOBD = 'M';
LDR >= 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.
Tolerances
RCOND DOUBLE PRECISION
The tolerance to be used for estimating the rank of
matrices. If the user sets RCOND > 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 <= 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 >= 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 < 0, the estimate is indicated by the
index of the singular value that has the largest
logarithmic gap to its successor.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK >= (M+L)*NOBR, if METH = 'N';
LIWORK >= M+L, if METH = 'M' and ALG = 'F';
LIWORK >= 0, if METH = 'M' and ALG = 'C' or 'Q'.
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 >= (4*NOBR-2)*(M+L), if ALG = 'C', BATCH = 'F' or
'I' and CONCT = 'C';
LDWORK >= 1, if ALG = 'C', BATCH = 'F' or 'I' and
CONCT = 'N';
LDWORK >= max((4*NOBR-2)*(M+L), 5*L*NOBR), if METH = 'M',
ALG = 'C', BATCH = 'L' and CONCT = 'C';
LDWORK >= 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 >= 5*L*NOBR, if METH = 'M', JOBD = 'N', ALG = 'C',
BATCH = 'O', or
(BATCH = 'L' and CONCT = 'N');
LDWORK >= 5*(M+L)*NOBR+1, if METH = 'N', ALG = 'C', and
BATCH = 'L' or 'O';
LDWORK >= (M+L)*2*NOBR*(M+L+3), if ALG = 'F',
BATCH <> 'O' and CONCT = 'C';
LDWORK >= (M+L)*2*NOBR*(M+L+1), if ALG = 'F',
BATCH = 'F', 'I' and CONCT = 'N';
LDWORK >= (M+L)*4*NOBR*(M+L+1)+(M+L)*2*NOBR, if ALG = 'F',
BATCH = 'L' and CONCT = 'N', or
BATCH = 'O';
LDWORK >= 4*(M+L)*NOBR, if ALG = 'Q', BATCH = 'F', and
LDR >= NS = NSMP - 2*NOBR + 1;
LDWORK >= max(4*(M+L)*NOBR, 5*L*NOBR), if METH = 'M',
ALG = 'Q', BATCH = 'O', and LDR >= NS;
LDWORK >= 5*(M+L)*NOBR+1, if METH = 'N', ALG = 'Q',
BATCH = 'O', and LDR >= NS;
LDWORK >= 6*(M+L)*NOBR, if ALG = 'Q', (BATCH = 'F' or 'O',
and LDR < NS), or (BATCH = 'I' or
'L' and CONCT = 'N');
LDWORK >= 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.
Warning Indicator
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.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 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.
Method
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.
References
[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.
Numerical Aspects
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.
Further Comments
For ALG = 'Q', BATCH = 'O' and LDR < NS, or BATCH <> '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.Example
Program Text
* IB01AD EXAMPLE PROGRAM TEXT
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. 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
Program Data
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 6.41 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 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 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 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 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 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 3.41 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 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 3.41 3.41 3.41 3.41 3.41 6.41 6.41 6.41 6.41 3.41 3.41 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 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 6.41 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 6.040829 6.096999 6.123859 6.162929 6.040829 5.845469 5.772209 5.799069 5.923609 5.928499 6.001759 6.001759 6.060369 5.882099 5.510909 5.322879 5.371719 5.454749 5.437649 5.159269 4.902859 4.587839 4.502369 4.595159 4.824709 5.064029 5.271599 5.466959 5.615919 5.528009 5.254499 4.883319 4.517019 4.197119 4.001759 3.806399 3.904079 3.923609 3.869889 3.806399 3.720929 3.818609 4.140949 4.529229 4.805179 5.086009 5.339969 5.532889 5.576849 5.667199 5.791739 5.850349 5.923609 5.921169 5.977339 5.740459 5.388809 5.000539 4.849129 4.944369 5.173919 5.369279 5.447419 5.603709 5.730689 5.850349 5.979779 5.991989 6.084789 5.940709 5.803949 5.791739 5.603709 5.264269 4.946809 4.619579 4.514579 4.433989 4.285029 4.121419 3.945589 3.984659 4.219099 4.546319 4.873549 5.154379 5.388809 5.613479 5.835699 5.884539 5.955359 5.762439 5.459629 5.061589 4.707499 4.458409 4.267939 4.053039 3.943149 3.825929 3.967569 4.280149 4.480389 4.492599 4.390039 4.197119 4.111649 3.982219 3.867449 3.767319 3.872329 4.236189 4.663539 4.971229 5.066469 4.902859 4.675749 4.392479 4.099439 4.114089 4.326539 4.643999 4.971229 5.159269 5.388809 5.576849 5.652549 5.803949 5.913839 5.886979 5.799069 5.730689 5.762439 5.813719 5.821049 5.928499 6.013969 5.764879 5.413229 5.098219 4.678189 4.372939 4.392479 4.590279 4.919949 5.017629 4.858899 4.675749 4.619579 4.834479 5.090889 5.376599 5.681849 5.823489 5.952919 6.062809 6.089669 6.075019 6.026179 5.994429 6.077459 5.857679 5.701389 5.730689 5.784419 5.823489 5.894309 5.762439 5.415679 4.961459 4.595159 4.331429 4.297239 4.582949 4.861339 5.173919 5.166589 4.919949 4.607369 4.370499 4.182469 4.038389 4.145839 4.431549 4.556089 4.480389 4.375379 4.370499 4.558529 4.858899 4.895529 4.741679 4.744129 4.875989 5.105539 5.239849 5.518239 5.652549 5.723369 5.855239 5.962679 5.984659 5.984659 6.055479 6.062809 6.055479 6.070129 5.784419 5.440099 5.056699 4.941929 5.010299 5.134849 5.313109 5.479169 5.623249 5.562199 5.330209 5.010299 4.665979 4.414459 4.201999 4.048159 4.079899 4.189789 4.131179 4.004199 3.916289 3.960239 4.199559 4.624469 4.883319 5.137289 5.379049 5.623249 5.762439 5.833259 5.686739 5.366839 5.225199 5.239849 5.354629 5.508469 5.596379 5.752669 5.874769 5.906519 5.894309 5.742899 5.447419 5.024959 4.883319 4.885759 4.893089 4.714819 4.451089 4.233749 4.043269 3.864999 3.757559 3.669639 3.593939 3.547539 3.506029 3.454749 3.398579 3.361949 3.339969 3.374159 3.520679 3.713599 3.757559 3.779529 3.696509 3.777089 3.886979 3.904079 3.850349 3.965129 4.282589 4.521899 4.714819 4.971229 5.220319 5.532889 5.652549 5.781979 5.955359 6.035939 6.118969 6.133629 6.153159 6.192229 6.143389 6.167809 5.991989 5.652549 5.459629 5.437649 5.339969 5.098219 4.785639 4.492599 4.236189 4.067689 3.933379 3.823489 3.730689 3.611039 3.564639 3.549989 3.557309 3.513359 3.515799 3.694059 4.072579 4.480389 4.705049 4.612259 4.385149 4.201999 4.026179 3.904079 3.774649 3.691619 3.845469 4.201999 4.585399 4.902859 5.256949 5.510909 5.640339 5.843029 5.974889 5.935819 5.821049 5.528009 5.171479 4.810059 4.453529 4.380269 4.565859 4.805179 5.125079 5.354629 5.589059 5.764879 5.923609 5.940709 5.857679 5.694059 5.486489 5.149499 4.844249 4.541439 4.267939 4.060369 3.960239 3.789299 3.642779 3.525569 3.498699 3.454749 3.408349 3.379049 3.376599 3.361949 3.359509 3.369279 3.398579 3.579289 3.948029 4.412009 4.585399 4.514579 4.343639 4.155599 3.984659 4.043269 4.307009 4.421779 4.353409 4.223979 4.053039 3.940709 3.838139 3.730689 3.652549 3.611039 3.564639 3.496259 3.462069 3.454749 3.425439 3.379049 3.432769 3.623249 3.974889 4.380269 4.714819 5.073799 5.369279 5.603709 5.745349 5.652549 5.401019 5.015189 4.709939 4.416899 4.236189 4.236189 4.248399 4.221539 4.297239 4.590279 4.893089 5.134849 5.427889 5.379049 5.364389 5.452309 5.567079 5.672089 5.769769 5.830819 5.923609 5.965129 6.057919 6.050599 6.072579 6.111649 6.070129 5.896749 5.755109 5.718479 5.821049 6.001759 6.001759 5.901629 5.557309 5.173919 4.800289 4.431549 4.194679 4.006639 3.850349 3.747789 3.642779 3.591499 3.569519 3.528009 3.537779 3.554869 3.493819 3.447419 3.440099 3.408349 3.410789 3.452309 3.681849 4.060369 4.441319 4.854019 5.154379 5.425439 5.596379 5.586619 5.354629 5.027399 4.863779 4.761219 4.570739 4.368059 4.397359 4.573189 4.841809 5.203219 5.452309 5.652549 5.855239 5.906519 5.952919 5.828369 5.791739 5.799069 5.813719 5.877209 5.955359 5.781979 5.518239 5.127519 4.763659 4.492599 4.233749 4.011519 3.855239 3.691619 3.635459 3.818609 4.155599 4.590279 4.988329 5.076239 4.907739 4.648889 4.377829 4.216649 4.287469 4.590279 4.846689 5.139729 5.388809 5.689179 5.884539 6.043269 6.170259 6.211769 6.250839 6.209329 6.013969 5.701389 5.469399 5.479169 5.557309 5.728249 5.882099 5.984659 5.901629 5.581729 5.371719 5.418119 5.510909 5.667199 5.791739 5.698949 5.484049 5.154379 4.980999 5.061589 5.195899 5.359509 5.615919 5.762439 5.857679 5.948029 5.835699 5.706269 5.498699 5.188569 5.117749 5.191009 5.315549 5.532889 5.444979 5.396139 5.274039 5.027399 4.744129 4.668419 4.651329 4.514579 4.267939 4.260609 4.263049 4.189789 4.277699 4.600049 4.932159 5.283809 5.528009 5.740459 5.874769 5.955359 5.991989 5.845469 5.528009 5.061589 4.734359 4.534109 4.534109 4.697729 4.744129 4.619579 4.643999 4.832039 5.132399 5.410789 5.625689 5.603709 5.315549 4.961459 4.619579 4.358289 4.155599 4.033499 3.886979 3.772209 3.640339 3.532889 3.435209 3.427889 3.422999 3.398579 3.603709 4.023729 4.451089 4.792969 4.902859 4.780759 4.590279 4.336309 4.145839 4.216649 4.433989 4.714819 5.098219 5.359509 5.569519 5.772209 5.921169 6.055479 5.962679 5.642779 5.435209 5.388809 5.537779 5.681849 5.701389 5.615919 5.667199 5.740459 5.803949 5.882099 5.950469 6.072579 6.148279 6.116529 6.177579 6.201999 6.206889 5.991989 5.564639 5.178799 4.998089 5.051819 5.232529 5.484049 5.686739 5.899189 5.869889 5.977339 6.053039 6.079899 6.128739 6.079899 6.167809 6.194679 6.236189 6.053039 5.652549 5.274039 4.858899 4.534109 4.455969 4.619579 4.866229 5.117749 5.166589 5.056699 5.002979 5.098219 5.325319 5.567079 5.466959 5.252059 4.946809 4.880879 4.980999 5.225199 5.459629 5.723369 5.791739 5.906519 5.991989 5.835699 5.528009 5.142169 4.775869 4.490159 4.236189 4.023729 3.886979 3.752669 3.681849 3.806399 4.145839 4.600049 5.002979 5.303339 5.552429 5.615919 5.523119 5.611039 5.713599 5.845469 5.899189 5.994429 6.092109 6.092109 6.143389 6.153159 6.233749 6.187349 6.013969 5.835699 5.774649 5.686739 5.537779 5.327759 5.054259 4.700169 4.394919 4.180019 4.043269 3.877209 3.752669 3.728249 3.869889 4.206889 4.355849 4.426669 4.453529 4.521899 4.392479 4.155599 3.965129 3.877209 3.970009 4.258169 4.421779 4.336309 4.299679 4.392479 4.675749 4.761219 4.658659 4.490159 4.307009 4.126299 3.972449 4.077459 4.372939 4.741679 5.088449 5.186129 5.037169 4.785639 4.563419 4.534109 4.705049 4.741679 4.648889 4.431549 4.238629 4.065249 3.943149 3.811279 3.691619 3.652549 3.825929 4.223979 4.424219 4.429109 4.319219 4.138509 3.965129 3.886979 3.801509 3.701389 3.640339 3.767319 4.150719 4.648889 4.990769 5.088449 5.022509 4.783199 4.685519 4.665979 4.707499 4.912619 5.195899 5.415679 5.623249 5.740459 5.899189 5.928499 6.050599 6.153159 5.965129 5.586619 5.381489 5.371719 5.486489 5.567079 5.821049 5.913839 5.994429 6.011519 5.999309 6.018849 5.821049 5.728249 5.740459 5.764879 5.882099 5.926049 5.750229 5.415679 4.995649 4.861339 4.902859 5.103099 5.364389 5.596379 5.752669 5.845469 5.928499 6.006639 5.840579 5.518239 5.173919 4.739239 4.458409 4.426669 4.602489 4.822269 5.183689 5.430329 5.652549 5.821049 5.706269 5.369279 5.027399 4.705049 4.414459 4.145839 3.965129 4.033499 4.372939 4.683079Program Results
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