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|
SUBROUTINE IB01PD( METH, JOB, JOBCV, NOBR, N, M, L, NSMPL, R,
$ LDR, A, LDA, C, LDC, B, LDB, D, LDD, Q, LDQ,
$ RY, LDRY, S, LDS, O, LDO, TOL, IWORK, DWORK,
$ LDWORK, IWARN, INFO )
C
C SLICOT RELEASE 5.0.
C
C Copyright (c) 2002-2009 NICONET e.V.
C
C This program is free software: you can redistribute it and/or
C modify it under the terms of the GNU General Public License as
C published by the Free Software Foundation, either version 2 of
C the License, or (at your option) any later version.
C
C This program is distributed in the hope that it will be useful,
C but WITHOUT ANY WARRANTY; without even the implied warranty of
C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
C GNU General Public License for more details.
C
C You should have received a copy of the GNU General Public License
C along with this program. If not, see
C <http://www.gnu.org/licenses/>.
C
C PURPOSE
C
C To estimate the matrices A, C, B, and D of a linear time-invariant
C (LTI) state space model, using the singular value decomposition
C information provided by other routines. Optionally, the system and
C noise covariance matrices, needed for the Kalman gain, are also
C determined.
C
C ARGUMENTS
C
C Mode Parameters
C
C METH CHARACTER*1
C Specifies the subspace identification method to be used,
C as follows:
C = 'M': MOESP algorithm with past inputs and outputs;
C = 'N': N4SID algorithm.
C
C JOB CHARACTER*1
C Specifies which matrices should be computed, as follows:
C = 'A': compute all system matrices, A, B, C, and D;
C = 'C': compute the matrices A and C only;
C = 'B': compute the matrix B only;
C = 'D': compute the matrices B and D only.
C
C JOBCV CHARACTER*1
C Specifies whether or not the covariance matrices are to
C be computed, as follows:
C = 'C': the covariance matrices should be computed;
C = 'N': the covariance matrices should not be computed.
C
C Input/Output Parameters
C
C NOBR (input) INTEGER
C The number of block rows, s, in the input and output
C Hankel matrices processed by other routines. NOBR > 1.
C
C N (input) INTEGER
C The order of the system. NOBR > N > 0.
C
C M (input) INTEGER
C The number of system inputs. M >= 0.
C
C L (input) INTEGER
C The number of system outputs. L > 0.
C
C NSMPL (input) INTEGER
C If JOBCV = 'C', the total number of samples used for
C calculating the covariance matrices.
C NSMPL >= 2*(M+L)*NOBR.
C This parameter is not meaningful if JOBCV = 'N'.
C
C R (input/workspace) DOUBLE PRECISION array, dimension
C ( LDR,2*(M+L)*NOBR )
C On entry, the leading 2*(M+L)*NOBR-by-2*(M+L)*NOBR part
C of this array must contain the relevant data for the MOESP
C or N4SID algorithms, as constructed by SLICOT Library
C routines IB01AD or IB01ND. Let R_ij, i,j = 1:4, be the
C ij submatrix of R (denoted S in IB01AD and IB01ND),
C partitioned by M*NOBR, L*NOBR, M*NOBR, and L*NOBR
C rows and columns. The submatrix R_22 contains the matrix
C of left singular vectors used. Also needed, for
C METH = 'N' or JOBCV = 'C', are the submatrices R_11,
C R_14 : R_44, and, for METH = 'M' and JOB <> 'C', the
C submatrices R_31 and R_12, containing the processed
C matrices R_1c and R_2c, respectively, as returned by
C SLICOT Library routines IB01AD or IB01ND.
C Moreover, if METH = 'N' and JOB = 'A' or 'C', the
C block-row R_41 : R_43 must contain the transpose of the
C block-column R_14 : R_34 as returned by SLICOT Library
C routines IB01AD or IB01ND.
C The remaining part of R is used as workspace.
C On exit, part of this array is overwritten. Specifically,
C if METH = 'M', R_22 and R_31 are overwritten if
C JOB = 'B' or 'D', and R_12, R_22, R_14 : R_34,
C and possibly R_11 are overwritten if JOBCV = 'C';
C if METH = 'N', all needed submatrices are overwritten.
C
C LDR INTEGER
C The leading dimension of the array R.
C LDR >= 2*(M+L)*NOBR.
C
C A (input or output) DOUBLE PRECISION array, dimension
C (LDA,N)
C On entry, if METH = 'N' and JOB = 'B' or 'D', the
C leading N-by-N part of this array must contain the system
C state matrix.
C If METH = 'M' or (METH = 'N' and JOB = 'A' or 'C'),
C this array need not be set on input.
C On exit, if JOB = 'A' or 'C' and INFO = 0, the
C leading N-by-N part of this array contains the system
C state matrix.
C
C LDA INTEGER
C The leading dimension of the array A.
C LDA >= N, if JOB = 'A' or 'C', or METH = 'N' and
C JOB = 'B' or 'D';
C LDA >= 1, otherwise.
C
C C (input or output) DOUBLE PRECISION array, dimension
C (LDC,N)
C On entry, if METH = 'N' and JOB = 'B' or 'D', the
C leading L-by-N part of this array must contain the system
C output matrix.
C If METH = 'M' or (METH = 'N' and JOB = 'A' or 'C'),
C this array need not be set on input.
C On exit, if JOB = 'A' or 'C' and INFO = 0, or
C INFO = 3 (or INFO >= 0, for METH = 'M'), the leading
C L-by-N part of this array contains the system output
C matrix.
C
C LDC INTEGER
C The leading dimension of the array C.
C LDC >= L, if JOB = 'A' or 'C', or METH = 'N' and
C JOB = 'B' or 'D';
C LDC >= 1, otherwise.
C
C B (output) DOUBLE PRECISION array, dimension (LDB,M)
C If M > 0, JOB = 'A', 'B', or 'D' and INFO = 0, the
C leading N-by-M part of this array contains the system
C input matrix. If M = 0 or JOB = 'C', this array is
C not referenced.
C
C LDB INTEGER
C The leading dimension of the array B.
C LDB >= N, if M > 0 and JOB = 'A', 'B', or 'D';
C LDB >= 1, if M = 0 or JOB = 'C'.
C
C D (output) DOUBLE PRECISION array, dimension (LDD,M)
C If M > 0, JOB = 'A' or 'D' and INFO = 0, the leading
C L-by-M part of this array contains the system input-output
C matrix. If M = 0 or JOB = 'C' or 'B', this array is
C not referenced.
C
C LDD INTEGER
C The leading dimension of the array D.
C LDD >= L, if M > 0 and JOB = 'A' or 'D';
C LDD >= 1, if M = 0 or JOB = 'C' or 'B'.
C
C Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
C If JOBCV = 'C', the leading N-by-N part of this array
C contains the positive semidefinite state covariance matrix
C to be used as state weighting matrix when computing the
C Kalman gain.
C This parameter is not referenced if JOBCV = 'N'.
C
C LDQ INTEGER
C The leading dimension of the array Q.
C LDQ >= N, if JOBCV = 'C';
C LDQ >= 1, if JOBCV = 'N'.
C
C RY (output) DOUBLE PRECISION array, dimension (LDRY,L)
C If JOBCV = 'C', the leading L-by-L part of this array
C contains the positive (semi)definite output covariance
C matrix to be used as output weighting matrix when
C computing the Kalman gain.
C This parameter is not referenced if JOBCV = 'N'.
C
C LDRY INTEGER
C The leading dimension of the array RY.
C LDRY >= L, if JOBCV = 'C';
C LDRY >= 1, if JOBCV = 'N'.
C
C S (output) DOUBLE PRECISION array, dimension (LDS,L)
C If JOBCV = 'C', the leading N-by-L part of this array
C contains the state-output cross-covariance matrix to be
C used as cross-weighting matrix when computing the Kalman
C gain.
C This parameter is not referenced if JOBCV = 'N'.
C
C LDS INTEGER
C The leading dimension of the array S.
C LDS >= N, if JOBCV = 'C';
C LDS >= 1, if JOBCV = 'N'.
C
C O (output) DOUBLE PRECISION array, dimension ( LDO,N )
C If METH = 'M' and JOBCV = 'C', or METH = 'N',
C the leading L*NOBR-by-N part of this array contains
C the estimated extended observability matrix, i.e., the
C first N columns of the relevant singular vectors.
C If METH = 'M' and JOBCV = 'N', this array is not
C referenced.
C
C LDO INTEGER
C The leading dimension of the array O.
C LDO >= L*NOBR, if JOBCV = 'C' or METH = 'N';
C LDO >= 1, otherwise.
C
C Tolerances
C
C TOL DOUBLE PRECISION
C The tolerance to be used for estimating the rank of
C matrices. If the user sets TOL > 0, then the given value
C of TOL is used as a lower bound for the reciprocal
C condition number; an m-by-n matrix whose estimated
C condition number is less than 1/TOL is considered to
C be of full rank. If the user sets TOL <= 0, then an
C implicitly computed, default tolerance, defined by
C TOLDEF = m*n*EPS, is used instead, where EPS is the
C relative machine precision (see LAPACK Library routine
C DLAMCH).
C
C Workspace
C
C IWORK INTEGER array, dimension (LIWORK)
C LIWORK = N, if METH = 'M' and M = 0
C or JOB = 'C' and JOBCV = 'N';
C LIWORK = M*NOBR+N, if METH = 'M', JOB = 'C',
C and JOBCV = 'C';
C LIWORK = max(L*NOBR,M*NOBR), if METH = 'M', JOB <> 'C',
C and JOBCV = 'N';
C LIWORK = max(L*NOBR,M*NOBR+N), if METH = 'M', JOB <> 'C',
C and JOBCV = 'C';
C LIWORK = max(M*NOBR+N,M*(N+L)), if METH = 'N'.
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) returns the optimal value
C of LDWORK, and DWORK(2), DWORK(3), DWORK(4), and
C DWORK(5) contain the reciprocal condition numbers of the
C triangular factors of the matrices, defined in the code,
C GaL (GaL = Un(1:(s-1)*L,1:n)), R_1c (if METH = 'M'),
C M (if JOBCV = 'C' or METH = 'N'), and Q or T (see
C SLICOT Library routines IB01PY or IB01PX), respectively.
C If METH = 'N', DWORK(3) is set to one without any
C calculations. Similarly, if METH = 'M' and JOBCV = 'N',
C DWORK(4) is set to one. If M = 0 or JOB = 'C',
C DWORK(3) and DWORK(5) are set to one.
C On exit, if INFO = -30, DWORK(1) returns the minimum
C value of LDWORK.
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= max( LDW1,LDW2 ), where, if METH = 'M',
C LDW1 >= max( 2*(L*NOBR-L)*N+2*N, (L*NOBR-L)*N+N*N+7*N ),
C if JOB = 'C' or JOB = 'A' and M = 0;
C LDW1 >= max( 2*(L*NOBR-L)*N+N*N+7*N,
C (L*NOBR-L)*N+N+6*M*NOBR, (L*NOBR-L)*N+N+
C max( L+M*NOBR, L*NOBR +
C max( 3*L*NOBR+1, M ) ) )
C if M > 0 and JOB = 'A', 'B', or 'D';
C LDW2 >= 0, if JOBCV = 'N';
C LDW2 >= max( (L*NOBR-L)*N+Aw+2*N+max(5*N,(2*M+L)*NOBR+L),
C 4*(M*NOBR+N)+1, M*NOBR+2*N+L ),
C if JOBCV = 'C',
C where Aw = N+N*N, if M = 0 or JOB = 'C';
C Aw = 0, otherwise;
C and, if METH = 'N',
C LDW1 >= max( (L*NOBR-L)*N+2*N+(2*M+L)*NOBR+L,
C 2*(L*NOBR-L)*N+N*N+8*N, N+4*(M*NOBR+N)+1,
C M*NOBR+3*N+L );
C LDW2 >= 0, if M = 0 or JOB = 'C';
C LDW2 >= M*NOBR*(N+L)*(M*(N+L)+1)+
C max( (N+L)**2, 4*M*(N+L)+1 ),
C if M > 0 and JOB = 'A', 'B', or 'D'.
C For good performance, LDWORK should be larger.
C
C Warning Indicator
C
C IWARN INTEGER
C = 0: no warning;
C = 4: a least squares problem to be solved has a
C rank-deficient coefficient matrix;
C = 5: the computed covariance matrices are too small.
C The problem seems to be a deterministic one.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value;
C = 2: the singular value decomposition (SVD) algorithm did
C not converge;
C = 3: a singular upper triangular matrix was found.
C
C METHOD
C
C In the MOESP approach, the matrices A and C are first
C computed from an estimated extended observability matrix [1],
C and then, the matrices B and D are obtained by solving an
C extended linear system in a least squares sense.
C In the N4SID approach, besides the estimated extended
C observability matrix, the solutions of two least squares problems
C are used to build another least squares problem, whose solution
C is needed to compute the system matrices A, C, B, and D. The
C solutions of the two least squares problems are also optionally
C used by both approaches to find the covariance matrices.
C
C REFERENCES
C
C [1] Verhaegen M., and Dewilde, P.
C Subspace Model Identification. Part 1: The output-error state-
C space model identification class of algorithms.
C Int. J. Control, 56, pp. 1187-1210, 1992.
C
C [2] Van Overschee, P., and De Moor, B.
C N4SID: Two Subspace Algorithms for the Identification
C of Combined Deterministic-Stochastic Systems.
C Automatica, Vol.30, No.1, pp. 75-93, 1994.
C
C [3] Van Overschee, P.
C Subspace Identification : Theory - Implementation -
C Applications.
C Ph. D. Thesis, Department of Electrical Engineering,
C Katholieke Universiteit Leuven, Belgium, Feb. 1995.
C
C [4] Sima, V.
C Subspace-based Algorithms for Multivariable System
C Identification.
C Studies in Informatics and Control, 5, pp. 335-344, 1996.
C
C NUMERICAL ASPECTS
C
C The implemented method is numerically stable.
C
C FURTHER COMMENTS
C
C In some applications, it is useful to compute the system matrices
C using two calls to this routine, the first one with JOB = 'C',
C and the second one with JOB = 'B' or 'D'. This is slightly less
C efficient than using a single call with JOB = 'A', because some
C calculations are repeated. If METH = 'N', all the calculations
C at the first call are performed again at the second call;
C moreover, it is required to save the needed submatrices of R
C before the first call and restore them before the second call.
C If the covariance matrices are desired, JOBCV should be set
C to 'C' at the second call. If B and D are both needed, they
C should be computed at once.
C It is possible to compute the matrices A and C using the MOESP
C algorithm (METH = 'M'), and the matrices B and D using the N4SID
C algorithm (METH = 'N'). This combination could be slightly more
C efficient than N4SID algorithm alone and it could be more accurate
C than MOESP algorithm. No saving/restoring is needed in such a
C combination, provided JOBCV is set to 'N' at the first call.
C Recommended usage: either one call with JOB = 'A', or
C first call with METH = 'M', JOB = 'C', JOBCV = 'N',
C second call with METH = 'M', JOB = 'D', JOBCV = 'C', or
C first call with METH = 'M', JOB = 'C', JOBCV = 'N',
C second call with METH = 'N', JOB = 'D', JOBCV = 'C'.
C
C CONTRIBUTOR
C
C V. Sima, Research Institute for Informatics, Bucharest, Dec. 1999.
C
C REVISIONS
C
C March 2000, Feb. 2001, Sep. 2001, March 2005.
C
C KEYWORDS
C
C Identification methods; least squares solutions; multivariable
C systems; QR decomposition; singular value decomposition.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, THREE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ THREE = 3.0D0 )
C .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, IWARN, L, LDA, LDB, LDC, LDD, LDO, LDQ,
$ LDR, LDRY, LDS, LDWORK, M, N, NOBR, NSMPL
CHARACTER JOB, JOBCV, METH
C .. Array Arguments ..
DOUBLE PRECISION A(LDA, *), B(LDB, *), C(LDC, *), D(LDD, *),
$ DWORK(*), O(LDO, *), Q(LDQ, *), R(LDR, *),
$ RY(LDRY, *), S(LDS, *)
INTEGER IWORK( * )
C .. Local Scalars ..
DOUBLE PRECISION EPS, RCOND1, RCOND2, RCOND3, RCOND4, RNRM,
$ SVLMAX, THRESH, TOLL, TOLL1
INTEGER I, IAW, ID, IERR, IGAL, IHOUS, ISV, ITAU,
$ ITAU1, ITAU2, IU, IUN2, IWARNL, IX, JWORK,
$ LDUN2, LDUNN, LDW, LMMNOB, LMMNOL, LMNOBR,
$ LNOBR, LNOBRN, MAXWRK, MINWRK, MNOBR, MNOBRN,
$ N2, NCOL, NN, NPL, NR, NR2, NR3, NR4, NR4MN,
$ NR4PL, NROW, RANK, RANK11, RANKM
CHARACTER FACT, JOBP, JOBPY
LOGICAL FULLR, MOESP, N4SID, SHIFT, WITHAL, WITHB,
$ WITHC, WITHCO, WITHD
C .. Local Array ..
DOUBLE PRECISION SVAL(3)
C .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL DLAMCH, DLANGE, ILAENV, LSAME
C .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DGEQRF, DLACPY, DLASET, DORMQR,
$ DSYRK, DTRCON, DTRSM, DTRTRS, IB01PX, IB01PY,
$ MA02AD, MA02ED, MB02QY, MB02UD, MB03OD, XERBLA
C .. Intrinsic Functions ..
INTRINSIC DBLE, MAX
C .. Executable Statements ..
C
C Decode the scalar input parameters.
C
MOESP = LSAME( METH, 'M' )
N4SID = LSAME( METH, 'N' )
WITHAL = LSAME( JOB, 'A' )
WITHC = LSAME( JOB, 'C' ) .OR. WITHAL
WITHD = LSAME( JOB, 'D' ) .OR. WITHAL
WITHB = LSAME( JOB, 'B' ) .OR. WITHD
WITHCO = LSAME( JOBCV, 'C' )
MNOBR = M*NOBR
LNOBR = L*NOBR
LMNOBR = LNOBR + MNOBR
LMMNOB = LNOBR + 2*MNOBR
MNOBRN = MNOBR + N
LNOBRN = LNOBR - N
LDUN2 = LNOBR - L
LDUNN = LDUN2*N
LMMNOL = LMMNOB + L
NR = LMNOBR + LMNOBR
NPL = N + L
N2 = N + N
NN = N*N
MINWRK = 1
IWARN = 0
INFO = 0
C
C Check the scalar input parameters.
C
IF( .NOT.( MOESP .OR. N4SID ) ) THEN
INFO = -1
ELSE IF( .NOT.( WITHB .OR. WITHC ) ) THEN
INFO = -2
ELSE IF( .NOT.( WITHCO .OR. LSAME( JOBCV, 'N' ) ) ) THEN
INFO = -3
ELSE IF( NOBR.LE.1 ) THEN
INFO = -4
ELSE IF( N.LE.0 .OR. N.GE.NOBR ) THEN
INFO = -5
ELSE IF( M.LT.0 ) THEN
INFO = -6
ELSE IF( L.LE.0 ) THEN
INFO = -7
ELSE IF( WITHCO .AND. NSMPL.LT.NR ) THEN
INFO = -8
ELSE IF( LDR.LT.NR ) THEN
INFO = -10
ELSE IF( LDA.LT.1 .OR. ( ( WITHC .OR. ( WITHB .AND. N4SID ) )
$ .AND. LDA.LT.N ) ) THEN
INFO = -12
ELSE IF( LDC.LT.1 .OR. ( ( WITHC .OR. ( WITHB .AND. N4SID ) )
$ .AND. LDC.LT.L ) ) THEN
INFO = -14
ELSE IF( LDB.LT.1 .OR. ( WITHB .AND. LDB.LT.N .AND. M.GT.0 ) )
$ THEN
INFO = -16
ELSE IF( LDD.LT.1 .OR. ( WITHD .AND. LDD.LT.L .AND. M.GT.0 ) )
$ THEN
INFO = -18
ELSE IF( LDQ.LT.1 .OR. ( WITHCO .AND. LDQ.LT.N ) ) THEN
INFO = -20
ELSE IF( LDRY.LT.1 .OR. ( WITHCO .AND. LDRY.LT.L ) ) THEN
INFO = -22
ELSE IF( LDS.LT.1 .OR. ( WITHCO .AND. LDS.LT.N ) ) THEN
INFO = -24
ELSE IF( LDO.LT.1 .OR. ( ( WITHCO .OR. N4SID ) .AND.
$ LDO.LT.LNOBR ) ) THEN
INFO = -26
ELSE
C
C Compute workspace.
C (Note: Comments in the code beginning "Workspace:" describe the
C minimal amount of workspace needed at that point in the code,
C as well as the preferred amount for good performance.
C NB refers to the optimal block size for the immediately
C following subroutine, as returned by ILAENV.)
C
IAW = 0
MINWRK = LDUNN + 4*N
MAXWRK = LDUNN + N + N*ILAENV( 1, 'DGEQRF', ' ', LDUN2, N, -1,
$ -1 )
IF( MOESP ) THEN
ID = 0
IF( WITHC ) THEN
MINWRK = MAX( MINWRK, 2*LDUNN + N2, LDUNN + NN + 7*N )
MAXWRK = MAX( MAXWRK, 2*LDUNN + N + N*ILAENV( 1,
$ 'DORMQR', 'LT', LDUN2, N, N, -1 ) )
END IF
ELSE
ID = N
END IF
C
IF( ( M.GT.0 .AND. WITHB ) .OR. N4SID ) THEN
MINWRK = MAX( MINWRK, 2*LDUNN + NN + ID + 7*N )
IF ( MOESP )
$ MINWRK = MAX( MINWRK, LDUNN + N + 6*MNOBR, LDUNN + N +
$ MAX( L + MNOBR, LNOBR +
$ MAX( 3*LNOBR + 1, M ) ) )
ELSE
IF( MOESP )
$ IAW = N + NN
END IF
C
IF( N4SID .OR. WITHCO ) THEN
MINWRK = MAX( MINWRK, LDUNN + IAW + N2 + MAX( 5*N, LMMNOL ),
$ ID + 4*MNOBRN+1, ID + MNOBRN + NPL )
MAXWRK = MAX( MAXWRK, LDUNN + IAW + N2 +
$ MAX( N*ILAENV( 1, 'DGEQRF', ' ', LNOBR, N, -1,
$ -1 ), LMMNOB*
$ ILAENV( 1, 'DORMQR', 'LT', LNOBR,
$ LMMNOB, N, -1 ), LMMNOL*
$ ILAENV( 1, 'DORMQR', 'LT', LDUN2,
$ LMMNOL, N, -1 ) ),
$ ID + N + N*ILAENV( 1, 'DGEQRF', ' ', LMNOBR,
$ N, -1, -1 ),
$ ID + N + NPL*ILAENV( 1, 'DORMQR', 'LT',
$ LMNOBR, NPL, N, -1 ) )
IF( N4SID .AND. ( M.GT.0 .AND. WITHB ) )
$ MINWRK = MAX( MINWRK, MNOBR*NPL*( M*NPL + 1 ) +
$ MAX( NPL**2, 4*M*NPL + 1 ) )
END IF
MAXWRK = MAX( MINWRK, MAXWRK )
C
IF ( LDWORK.LT.MINWRK ) THEN
INFO = -30
DWORK( 1 ) = MINWRK
END IF
END IF
C
C Return if there are illegal arguments.
C
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'IB01PD', -INFO )
RETURN
END IF
C
NR2 = MNOBR + 1
NR3 = LMNOBR + 1
NR4 = LMMNOB + 1
C
C Set the precision parameters. A threshold value EPS**(2/3) is
C used for deciding to use pivoting or not, where EPS is the
C relative machine precision (see LAPACK Library routine DLAMCH).
C
EPS = DLAMCH( 'Precision' )
THRESH = EPS**( TWO/THREE )
SVLMAX = ZERO
RCOND4 = ONE
C
C Let Un be the matrix of left singular vectors (stored in R_22).
C Copy un1 = GaL = Un(1:(s-1)*L,1:n) in the workspace.
C
IGAL = 1
CALL DLACPY( 'Full', LDUN2, N, R(NR2,NR2), LDR, DWORK(IGAL),
$ LDUN2 )
C
C Factor un1 = Q1*[r1' 0]' (' means transposition).
C Workspace: need L*(NOBR-1)*N+2*N,
C prefer L*(NOBR-1)*N+N+N*NB.
C
ITAU1 = IGAL + LDUNN
JWORK = ITAU1 + N
LDW = JWORK
CALL DGEQRF( LDUN2, N, DWORK(IGAL), LDUN2, DWORK(ITAU1),
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
C
C Compute the reciprocal of the condition number of r1.
C Workspace: need L*(NOBR-1)*N+4*N.
C
CALL DTRCON( '1-norm', 'Upper', 'NonUnit', N, DWORK(IGAL), LDUN2,
$ RCOND1, DWORK(JWORK), IWORK, INFO )
C
TOLL1 = TOL
IF( TOLL1.LE.ZERO )
$ TOLL1 = NN*EPS
C
IF ( ( M.GT.0 .AND. WITHB ) .OR. N4SID ) THEN
JOBP = 'P'
IF ( WITHAL ) THEN
JOBPY = 'D'
ELSE
JOBPY = JOB
END IF
ELSE
JOBP = 'N'
END IF
C
IF ( MOESP ) THEN
NCOL = 0
IUN2 = JWORK
IF ( WITHC ) THEN
C
C Set C = Un(1:L,1:n) and then compute the system matrix A.
C
C Set un2 = Un(L+1:L*s,1:n) in DWORK(IUN2).
C Workspace: need 2*L*(NOBR-1)*N+N.
C
CALL DLACPY( 'Full', L, N, R(NR2,NR2), LDR, C, LDC )
CALL DLACPY( 'Full', LDUN2, N, R(NR2+L,NR2), LDR,
$ DWORK(IUN2), LDUN2 )
C
C Note that un1 has already been factored as
C un1 = Q1*[r1' 0]' and usually (generically, assuming
C observability) has full column rank.
C Update un2 <-- Q1'*un2 in DWORK(IUN2) and save its
C first n rows in A.
C Workspace: need 2*L*(NOBR-1)*N+2*N;
C prefer 2*L*(NOBR-1)*N+N+N*NB.
C
JWORK = IUN2 + LDUNN
CALL DORMQR( 'Left', 'Transpose', LDUN2, N, N, DWORK(IGAL),
$ LDUN2, DWORK(ITAU1), DWORK(IUN2), LDUN2,
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
CALL DLACPY( 'Full', N, N, DWORK(IUN2), LDUN2, A, LDA )
NCOL = N
JWORK = IUN2
END IF
C
IF ( RCOND1.GT.MAX( TOLL1, THRESH ) ) THEN
C
C The triangular factor r1 is considered to be of full rank.
C Solve for A (if requested), r1*A = un2(1:n,:) in A.
C
IF ( WITHC ) THEN
CALL DTRTRS( 'Upper', 'NoTranspose', 'NonUnit', N, N,
$ DWORK(IGAL), LDUN2, A, LDA, IERR )
IF ( IERR.GT.0 ) THEN
INFO = 3
RETURN
END IF
END IF
RANK = N
ELSE
C
C Rank-deficient triangular factor r1. Use SVD of r1,
C r1 = U*S*V', also for computing A (if requested) from
C r1*A = un2(1:n,:). Matrix U is computed in DWORK(IU),
C and V' overwrites r1. If B is requested, the
C pseudoinverse of r1 and then of GaL are also computed
C in R(NR3,NR2).
C Workspace: need c*L*(NOBR-1)*N+N*N+7*N,
C where c = 1 if B and D are not needed,
C c = 2 if B and D are needed;
C prefer larger.
C
IU = IUN2
ISV = IU + NN
JWORK = ISV + N
IF ( M.GT.0 .AND. WITHB ) THEN
C
C Save the elementary reflectors used for computing r1,
C if B, D are needed.
C Workspace: need 2*L*(NOBR-1)*N+2*N+N*N.
C
IHOUS = JWORK
JWORK = IHOUS + LDUNN
CALL DLACPY( 'Lower', LDUN2, N, DWORK(IGAL), LDUN2,
$ DWORK(IHOUS), LDUN2 )
ELSE
IHOUS = IGAL
END IF
C
CALL MB02UD( 'Not factored', 'Left', 'NoTranspose', JOBP, N,
$ NCOL, ONE, TOLL1, RANK, DWORK(IGAL), LDUN2,
$ DWORK(IU), N, DWORK(ISV), A, LDA, R(NR3,NR2),
$ LDR, DWORK(JWORK), LDWORK-JWORK+1, IERR )
IF ( IERR.NE.0 ) THEN
INFO = 2
RETURN
END IF
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
C
IF ( RANK.EQ.0 ) THEN
JOBP = 'N'
ELSE IF ( M.GT.0 .AND. WITHB ) THEN
C
C Compute pinv(GaL) in R(NR3,NR2) if B, D are needed.
C Workspace: need 2*L*(NOBR-1)*N+N*N+3*N;
C prefer 2*L*(NOBR-1)*N+N*N+2*N+N*NB.
C
CALL DLASET( 'Full', N, LDUN2-N, ZERO, ZERO,
$ R(NR3,NR2+N), LDR )
CALL DORMQR( 'Right', 'Transpose', N, LDUN2, N,
$ DWORK(IHOUS), LDUN2, DWORK(ITAU1),
$ R(NR3,NR2), LDR, DWORK(JWORK),
$ LDWORK-JWORK+1, IERR )
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
IF ( WITHCO ) THEN
C
C Save pinv(GaL) in DWORK(IGAL).
C
CALL DLACPY( 'Full', N, LDUN2, R(NR3,NR2), LDR,
$ DWORK(IGAL), N )
END IF
JWORK = IUN2
END IF
LDW = JWORK
END IF
C
IF ( M.GT.0 .AND. WITHB ) THEN
C
C Computation of B and D.
C
C Compute the reciprocal of the condition number of R_1c.
C Workspace: need L*(NOBR-1)*N+N+3*M*NOBR.
C
CALL DTRCON( '1-norm', 'Upper', 'NonUnit', MNOBR, R(NR3,1),
$ LDR, RCOND2, DWORK(JWORK), IWORK, IERR )
C
TOLL = TOL
IF( TOLL.LE.ZERO )
$ TOLL = MNOBR*MNOBR*EPS
C
C Compute the right hand side and solve for K (in R_23),
C K*R_1c' = u2'*R_2c',
C where u2 = Un(:,n+1:L*s), and K is (Ls-n) x ms.
C
CALL DGEMM( 'Transpose', 'Transpose', LNOBRN, MNOBR, LNOBR,
$ ONE, R(NR2,NR2+N), LDR, R(1,NR2), LDR, ZERO,
$ R(NR2,NR3), LDR )
C
IF ( RCOND2.GT.MAX( TOLL, THRESH ) ) THEN
C
C The triangular factor R_1c is considered to be of full
C rank. Solve for K, K*R_1c' = u2'*R_2c'.
C
CALL DTRSM( 'Right', 'Upper', 'Transpose', 'Non-unit',
$ LNOBRN, MNOBR, ONE, R(NR3,1), LDR,
$ R(NR2,NR3), LDR )
ELSE
C
C Rank-deficient triangular factor R_1c. Use SVD of R_1c
C for computing K from K*R_1c' = u2'*R_2c', where
C R_1c = U1*S1*V1'. Matrix U1 is computed in R_33,
C and V1' overwrites R_1c.
C Workspace: need L*(NOBR-1)*N+N+6*M*NOBR;
C prefer larger.
C
ISV = LDW
JWORK = ISV + MNOBR
CALL MB02UD( 'Not factored', 'Right', 'Transpose',
$ 'No pinv', LNOBRN, MNOBR, ONE, TOLL, RANK11,
$ R(NR3,1), LDR, R(NR3,NR3), LDR, DWORK(ISV),
$ R(NR2,NR3), LDR, DWORK(JWORK), 1,
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
IF ( IERR.NE.0 ) THEN
INFO = 2
RETURN
END IF
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
JWORK = LDW
END IF
C
C Compute the triangular factor of the structured matrix Q
C and apply the transformations to the matrix Kexpand, where
C Q and Kexpand are defined in SLICOT Library routine
C IB01PY. Compute also the matrices B, D.
C Workspace: need L*(NOBR-1)*N+N+max(L+M*NOBR,L*NOBR+
C max(3*L*NOBR+1,M));
C prefer larger.
C
IF ( WITHCO )
$ CALL DLACPY( 'Full', LNOBR, N, R(NR2,NR2), LDR, O, LDO )
CALL IB01PY( METH, JOBPY, NOBR, N, M, L, RANK, R(NR2,NR2),
$ LDR, DWORK(IGAL), LDUN2, DWORK(ITAU1),
$ R(NR3,NR2), LDR, R(NR2,NR3), LDR, R(NR4,NR2),
$ LDR, R(NR4,NR3), LDR, B, LDB, D, LDD, TOL,
$ IWORK, DWORK(JWORK), LDWORK-JWORK+1, IWARN,
$ INFO )
IF ( INFO.NE.0 )
$ RETURN
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
RCOND4 = DWORK(JWORK+1)
IF ( WITHCO )
$ CALL DLACPY( 'Full', LNOBR, N, O, LDO, R(NR2,1), LDR )
C
ELSE
RCOND2 = ONE
END IF
C
IF ( .NOT.WITHCO ) THEN
RCOND3 = ONE
GO TO 30
END IF
ELSE
C
C For N4SID, set RCOND2 to one.
C
RCOND2 = ONE
END IF
C
C If needed, save the first n columns, representing Gam, of the
C matrix of left singular vectors, Un, in R_21 and in O.
C
IF ( N4SID .OR. ( WITHC .AND. .NOT.WITHAL ) ) THEN
IF ( M.GT.0 )
$ CALL DLACPY( 'Full', LNOBR, N, R(NR2,NR2), LDR, R(NR2,1),
$ LDR )
CALL DLACPY( 'Full', LNOBR, N, R(NR2,NR2), LDR, O, LDO )
END IF
C
C Computations for covariance matrices, and system matrices (N4SID).
C Solve the least squares problems Gam*Y = R4(1:L*s,1:(2*m+L)*s),
C GaL*X = R4(L+1:L*s,:), where
C GaL = Gam(1:L*(s-1),:), Gam has full column rank, and
C R4 = [ R_14' R_24' R_34' R_44L' ], R_44L = R_44(1:L,:), as
C returned by SLICOT Library routine IB01ND.
C First, find the QR factorization of Gam, Gam = Q*R.
C Workspace: need L*(NOBR-1)*N+Aw+3*N;
C prefer L*(NOBR-1)*N+Aw+2*N+N*NB, where
C Aw = N+N*N, if (M = 0 or JOB = 'C'), rank(r1) < N,
C and METH = 'M';
C Aw = 0, otherwise.
C
ITAU2 = LDW
JWORK = ITAU2 + N
CALL DGEQRF( LNOBR, N, R(NR2,1), LDR, DWORK(ITAU2),
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
C
C For METH = 'M' or when JOB = 'B' or 'D', transpose
C [ R_14' R_24' R_34' ]' in the last block-row of R, obtaining Z,
C and for METH = 'N' and JOB = 'A' or 'C', use the matrix Z
C already available in the last block-row of R, and then apply
C the transformations, Z <-- Q'*Z.
C Workspace: need L*(NOBR-1)*N+Aw+2*N+(2*M+L)*NOBR;
C prefer L*(NOBR-1)*N+Aw+2*N+(2*M+L)*NOBR*NB.
C
IF ( MOESP .OR. ( WITHB .AND. .NOT. WITHAL ) )
$ CALL MA02AD( 'Full', LMMNOB, LNOBR, R(1,NR4), LDR, R(NR4,1),
$ LDR )
CALL DORMQR( 'Left', 'Transpose', LNOBR, LMMNOB, N, R(NR2,1), LDR,
$ DWORK(ITAU2), R(NR4,1), LDR, DWORK(JWORK),
$ LDWORK-JWORK+1, IERR )
C
C Solve for Y, RY = Z in Z and save the transpose of the
C solution Y in the second block-column of R.
C
CALL DTRTRS( 'Upper', 'NoTranspose', 'NonUnit', N, LMMNOB,
$ R(NR2,1), LDR, R(NR4,1), LDR, IERR )
IF ( IERR.GT.0 ) THEN
INFO = 3
RETURN
END IF
CALL MA02AD( 'Full', N, LMMNOB, R(NR4,1), LDR, R(1,NR2), LDR )
NR4MN = NR4 - N
NR4PL = NR4 + L
NROW = LMMNOL
C
C SHIFT is .TRUE. if some columns of R_14 : R_44L should be
C shifted to the right, to avoid overwriting useful information.
C
SHIFT = M.EQ.0 .AND. LNOBR.LT.N2
C
IF ( RCOND1.GT.MAX( TOLL1, THRESH ) ) THEN
C
C The triangular factor r1 of GaL (GaL = Q1*r1) is
C considered to be of full rank.
C
C Transpose [ R_14' R_24' R_34' R_44L' ]'(:,L+1:L*s) in the
C last block-row of R (beginning with the (L+1)-th row),
C obtaining Z1, and then apply the transformations,
C Z1 <-- Q1'*Z1.
C Workspace: need L*(NOBR-1)*N+Aw+2*N+ (2*M+L)*NOBR + L;
C prefer L*(NOBR-1)*N+Aw+2*N+((2*M+L)*NOBR + L)*NB.
C
CALL MA02AD( 'Full', LMMNOL, LDUN2, R(1,NR4PL), LDR,
$ R(NR4PL,1), LDR )
CALL DORMQR( 'Left', 'Transpose', LDUN2, LMMNOL, N,
$ DWORK(IGAL), LDUN2, DWORK(ITAU1), R(NR4PL,1), LDR,
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
C
C Solve for X, r1*X = Z1 in Z1, and copy the transpose of X
C into the last part of the third block-column of R.
C
CALL DTRTRS( 'Upper', 'NoTranspose', 'NonUnit', N, LMMNOL,
$ DWORK(IGAL), LDUN2, R(NR4PL,1), LDR, IERR )
IF ( IERR.GT.0 ) THEN
INFO = 3
RETURN
END IF
C
IF ( SHIFT ) THEN
NR4MN = NR4
C
DO 10 I = L - 1, 0, -1
CALL DCOPY( LMMNOL, R(1,NR4+I), 1, R(1,NR4+N+I), 1 )
10 CONTINUE
C
END IF
CALL MA02AD( 'Full', N, LMMNOL, R(NR4PL,1), LDR, R(1,NR4MN),
$ LDR )
NROW = 0
END IF
C
IF ( N4SID .OR. NROW.GT.0 ) THEN
C
C METH = 'N' or rank-deficient triangular factor r1.
C For METH = 'N', use SVD of r1, r1 = U*S*V', for computing
C X' from X'*GaL' = Z1', if rank(r1) < N. Matrix U is
C computed in DWORK(IU) and V' overwrites r1. Then, the
C pseudoinverse of GaL is determined in R(NR4+L,NR2).
C For METH = 'M', the pseudoinverse of GaL is already available
C if M > 0 and B is requested; otherwise, the SVD of r1 is
C available in DWORK(IU), DWORK(ISV), and DWORK(IGAL).
C Workspace for N4SID: need 2*L*(NOBR-1)*N+N*N+8*N;
C prefer larger.
C
IF ( MOESP ) THEN
FACT = 'F'
IF ( M.GT.0 .AND. WITHB )
$ CALL DLACPY( 'Full', N, LDUN2, DWORK(IGAL), N,
$ R(NR4PL,NR2), LDR )
ELSE
C
C Save the elementary reflectors used for computing r1.
C
IHOUS = JWORK
CALL DLACPY( 'Lower', LDUN2, N, DWORK(IGAL), LDUN2,
$ DWORK(IHOUS), LDUN2 )
FACT = 'N'
IU = IHOUS + LDUNN
ISV = IU + NN
JWORK = ISV + N
END IF
C
CALL MB02UD( FACT, 'Right', 'Transpose', JOBP, NROW, N, ONE,
$ TOLL1, RANK, DWORK(IGAL), LDUN2, DWORK(IU), N,
$ DWORK(ISV), R(1,NR4PL), LDR, R(NR4PL,NR2), LDR,
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
IF ( NROW.GT.0 ) THEN
IF ( SHIFT ) THEN
NR4MN = NR4
CALL DLACPY( 'Full', LMMNOL, L, R(1,NR4), LDR,
$ R(1,NR4-L), LDR )
CALL DLACPY( 'Full', LMMNOL, N, R(1,NR4PL), LDR,
$ R(1,NR4MN), LDR )
CALL DLACPY( 'Full', LMMNOL, L, R(1,NR4-L), LDR,
$ R(1,NR4+N), LDR )
ELSE
CALL DLACPY( 'Full', LMMNOL, N, R(1,NR4PL), LDR,
$ R(1,NR4MN), LDR )
END IF
END IF
C
IF ( N4SID ) THEN
IF ( IERR.NE.0 ) THEN
INFO = 2
RETURN
END IF
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
C
C Compute pinv(GaL) in R(NR4+L,NR2).
C Workspace: need 2*L*(NOBR-1)*N+3*N;
C prefer 2*L*(NOBR-1)*N+2*N+N*NB.
C
JWORK = IU
CALL DLASET( 'Full', N, LDUN2-N, ZERO, ZERO, R(NR4PL,NR2+N),
$ LDR )
CALL DORMQR( 'Right', 'Transpose', N, LDUN2, N,
$ DWORK(IHOUS), LDUN2, DWORK(ITAU1),
$ R(NR4PL,NR2), LDR, DWORK(JWORK),
$ LDWORK-JWORK+1, IERR )
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
END IF
END IF
C
C For METH = 'N', find part of the solution (corresponding to A
C and C) and, optionally, for both METH = 'M', or METH = 'N',
C find the residual of the least squares problem that gives the
C covariances, M*V = N, where
C ( R_11 )
C M = ( Y' ), N = ( X' R4'(:,1:L) ), V = V(n+m*s, n+L),
C ( 0 0 )
C with M((2*m+L)*s+L, n+m*s), N((2*m+L)*s+L, n+L), R4' being
C stored in the last block-column of R. The last L rows of M
C are not explicitly considered. Note that, for efficiency, the
C last m*s columns of M are in the first positions of arrray R.
C This permutation does not affect the residual, only the
C solution. (The solution is not needed for METH = 'M'.)
C Note that R_11 corresponds to the future outputs for both
C METH = 'M', or METH = 'N' approaches. (For METH = 'N', the
C first two block-columns have been interchanged.)
C For METH = 'N', A and C are obtained as follows:
C [ A' C' ] = V(m*s+1:m*s+n,:).
C
C First, find the QR factorization of Y'(m*s+1:(2*m+L)*s,:)
C and apply the transformations to the corresponding part of N.
C Compress the workspace for N4SID by moving the scalar reflectors
C corresponding to Q.
C Workspace: need d*N+2*N;
C prefer d*N+N+N*NB;
C where d = 0, for MOESP, and d = 1, for N4SID.
C
IF ( MOESP ) THEN
ITAU = 1
ELSE
CALL DCOPY( N, DWORK(ITAU2), 1, DWORK, 1 )
ITAU = N + 1
END IF
C
JWORK = ITAU + N
CALL DGEQRF( LMNOBR, N, R(NR2,NR2), LDR, DWORK(ITAU),
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
C
C Workspace: need d*N+N+(N+L);
C prefer d*N+N+(N+L)*NB.
C
CALL DORMQR( 'Left', 'Transpose', LMNOBR, NPL, N, R(NR2,NR2), LDR,
$ DWORK(ITAU), R(NR2,NR4MN), LDR, DWORK(JWORK),
$ LDWORK-JWORK+1, IERR )
C
CALL DLASET( 'Lower', L-1, L-1, ZERO, ZERO, R(NR4+1,NR4), LDR )
C
C Now, matrix M with permuted block-columns has been
C triangularized.
C Compute the reciprocal of the condition number of its
C triangular factor in R(1:m*s+n,1:m*s+n).
C Workspace: need d*N+3*(M*NOBR+N).
C
JWORK = ITAU
CALL DTRCON( '1-norm', 'Upper', 'NonUnit', MNOBRN, R, LDR, RCOND3,
$ DWORK(JWORK), IWORK, INFO )
C
TOLL = TOL
IF( TOLL.LE.ZERO )
$ TOLL = MNOBRN*MNOBRN*EPS
IF ( RCOND3.GT.MAX( TOLL, THRESH ) ) THEN
C
C The triangular factor is considered to be of full rank.
C Solve for V(m*s+1:m*s+n,:), giving [ A' C' ].
C
FULLR = .TRUE.
RANKM = MNOBRN
IF ( N4SID )
$ CALL DTRSM( 'Left', 'Upper', 'NoTranspose', 'NonUnit', N,
$ NPL, ONE, R(NR2,NR2), LDR, R(NR2,NR4MN), LDR )
ELSE
FULLR = .FALSE.
C
C Use QR factorization (with pivoting). For METH = 'N', save
C (and then restore) information about the QR factorization of
C Gam, for later use. Note that R_11 could be modified by
C MB03OD, but the corresponding part of N is also modified
C accordingly.
C Workspace: need d*N+4*(M*NOBR+N)+1;
C prefer d*N+3*(M*NOBR+N)+(M*NOBR+N+1)*NB.
C
DO 20 I = 1, MNOBRN
IWORK(I) = 0
20 CONTINUE
C
IF ( N4SID .AND. ( M.GT.0 .AND. WITHB ) )
$ CALL DLACPY( 'Full', LNOBR, N, R(NR2,1), LDR, R(NR4,1),
$ LDR )
JWORK = ITAU + MNOBRN
CALL DLASET( 'Lower', MNOBRN-1, MNOBRN, ZERO, ZERO, R(2,1),
$ LDR )
CALL MB03OD( 'QR', MNOBRN, MNOBRN, R, LDR, IWORK, TOLL,
$ SVLMAX, DWORK(ITAU), RANKM, SVAL, DWORK(JWORK),
$ LDWORK-JWORK+1, IERR )
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
C
C Workspace: need d*N+M*NOBR+N+N+L;
C prefer d*N+M*NOBR+N+(N+L)*NB.
C
CALL DORMQR( 'Left', 'Transpose', MNOBRN, NPL, MNOBRN,
$ R, LDR, DWORK(ITAU), R(1,NR4MN), LDR,
$ DWORK(JWORK), LDWORK-JWORK+1, IERR )
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
END IF
C
IF ( WITHCO ) THEN
C
C The residual (transposed) of the least squares solution
C (multiplied by a matrix with orthogonal columns) is stored
C in the rows RANKM+1:(2*m+L)*s+L of V, and it should be
C squared-up for getting the covariance matrices. (Generically,
C RANKM = m*s+n.)
C
RNRM = ONE/DBLE( NSMPL )
IF ( MOESP ) THEN
CALL DSYRK( 'Upper', 'Transpose', NPL, LMMNOL-RANKM, RNRM,
$ R(RANKM+1,NR4MN), LDR, ZERO, R, LDR )
CALL DLACPY( 'Upper', N, N, R, LDR, Q, LDQ )
CALL DLACPY( 'Full', N, L, R(1,N+1), LDR, S, LDS )
CALL DLACPY( 'Upper', L, L, R(N+1,N+1), LDR, RY, LDRY )
ELSE
CALL DSYRK( 'Upper', 'Transpose', NPL, LMMNOL-RANKM, RNRM,
$ R(RANKM+1,NR4MN), LDR, ZERO, DWORK(JWORK), NPL )
CALL DLACPY( 'Upper', N, N, DWORK(JWORK), NPL, Q, LDQ )
CALL DLACPY( 'Full', N, L, DWORK(JWORK+N*NPL), NPL, S,
$ LDS )
CALL DLACPY( 'Upper', L, L, DWORK(JWORK+N*(NPL+1)), NPL, RY,
$ LDRY )
END IF
CALL MA02ED( 'Upper', N, Q, LDQ )
CALL MA02ED( 'Upper', L, RY, LDRY )
C
C Check the magnitude of the residual.
C
RNRM = DLANGE( '1-norm', LMMNOL-RANKM, NPL, R(RANKM+1,NR4MN),
$ LDR, DWORK(JWORK) )
IF ( RNRM.LT.THRESH )
$ IWARN = 5
END IF
C
IF ( N4SID ) THEN
IF ( .NOT.FULLR ) THEN
IWARN = 4
C
C Compute part of the solution of the least squares problem,
C M*V = N, for the rank-deficient problem.
C Remark: this computation should not be performed before the
C symmetric updating operation above.
C Workspace: need M*NOBR+3*N+L;
C prefer larger.
C
CALL MB03OD( 'No QR', N, N, R(NR2,NR2), LDR, IWORK, TOLL1,
$ SVLMAX, DWORK(ITAU), RANKM, SVAL, DWORK(JWORK),
$ LDWORK-JWORK+1, IERR )
CALL MB02QY( N, N, NPL, RANKM, R(NR2,NR2), LDR, IWORK,
$ R(NR2,NR4MN), LDR, DWORK(ITAU+MNOBR),
$ DWORK(JWORK), LDWORK-JWORK+1, INFO )
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
JWORK = ITAU
IF ( M.GT.0 .AND. WITHB )
$ CALL DLACPY( 'Full', LNOBR, N, R(NR4,1), LDR, R(NR2,1),
$ LDR )
END IF
C
IF ( WITHC ) THEN
C
C Obtain A and C, noting that block-permutations have been
C implicitly used.
C
CALL MA02AD( 'Full', N, N, R(NR2,NR4MN), LDR, A, LDA )
CALL MA02AD( 'Full', N, L, R(NR2,NR4MN+N), LDR, C, LDC )
ELSE
C
C Use the given A and C.
C
CALL MA02AD( 'Full', N, N, A, LDA, R(NR2,NR4MN), LDR )
CALL MA02AD( 'Full', L, N, C, LDC, R(NR2,NR4MN+N), LDR )
END IF
C
IF ( M.GT.0 .AND. WITHB ) THEN
C
C Obtain B and D.
C First, compute the transpose of the matrix K as
C N(1:m*s,:) - M(1:m*s,m*s+1:m*s+n)*[A' C'], in the first
C m*s rows of R(1,NR4MN).
C
CALL DGEMM ( 'NoTranspose', 'NoTranspose', MNOBR, NPL, N,
$ -ONE, R(1,NR2), LDR, R(NR2,NR4MN), LDR, ONE,
$ R(1,NR4MN), LDR )
C
C Denote M = pinv(GaL) and construct
C
C [ [ A ] -1 ] [ R ]
C and L = [ [ ] R 0 ] Q', where Gam = Q * [ ].
C [ [ C ] ] [ 0 ]
C
C Then, solve the least squares problem.
C
CALL DLACPY( 'Full', N, N, A, LDA, R(NR2,NR4), LDR )
CALL DLACPY( 'Full', L, N, C, LDC, R(NR2+N,NR4), LDR )
CALL DTRSM( 'Right', 'Upper', 'NoTranspose', 'NonUnit',
$ NPL, N, ONE, R(NR2,1), LDR, R(NR2,NR4), LDR )
CALL DLASET( 'Full', NPL, LNOBRN, ZERO, ZERO, R(NR2,NR4+N),
$ LDR )
C
C Workspace: need 2*N+L; prefer N + (N+L)*NB.
C
CALL DORMQR( 'Right', 'Transpose', NPL, LNOBR, N, R(NR2,1),
$ LDR, DWORK, R(NR2,NR4), LDR, DWORK(JWORK),
$ LDWORK-JWORK+1, IERR )
C
C Obtain the matrix K by transposition, and find B and D.
C Workspace: need NOBR*(M*(N+L))**2+M*NOBR*(N+L)+
C max((N+L)**2,4*M*(N+L)+1);
C prefer larger.
C
CALL MA02AD( 'Full', MNOBR, NPL, R(1,NR4MN), LDR,
$ R(NR2,NR3), LDR )
IX = MNOBR*NPL**2*M + 1
JWORK = IX + MNOBR*NPL
CALL IB01PX( JOBPY, NOBR, N, M, L, R, LDR, O, LDO,
$ R(NR2,NR4), LDR, R(NR4PL,NR2), LDR, R(NR2,NR3),
$ LDR, DWORK, MNOBR*NPL, DWORK(IX), B, LDB, D,
$ LDD, TOL, IWORK, DWORK(JWORK), LDWORK-JWORK+1,
$ IWARNL, INFO )
IF ( INFO.NE.0 )
$ RETURN
IWARN = MAX( IWARN, IWARNL )
MAXWRK = MAX( MAXWRK, INT( DWORK(JWORK) ) + JWORK - 1 )
RCOND4 = DWORK(JWORK+1)
C
END IF
END IF
C
30 CONTINUE
C
C Return optimal workspace in DWORK(1) and reciprocal condition
C numbers in the next locations.
C
DWORK(1) = MAXWRK
DWORK(2) = RCOND1
DWORK(3) = RCOND2
DWORK(4) = RCOND3
DWORK(5) = RCOND4
RETURN
C
C *** Last line of IB01PD ***
END
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