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SUBROUTINE AB09IX( DICO, JOB, FACT, ORDSEL, N, M, P, NR,
$ SCALEC, SCALEO, A, LDA, B, LDB, C, LDC, D, LDD,
$ TI, LDTI, T, LDT, NMINR, HSV, TOL1, TOL2,
$ IWORK, DWORK, LDWORK, IWARN, INFO )
C
C PURPOSE
C
C To compute a reduced order model (Ar,Br,Cr,Dr) for an original
C state-space representation (A,B,C,D) by using the square-root or
C balancing-free square-root Balance & Truncate (B&T) or
C Singular Perturbation Approximation (SPA) model reduction methods.
C The computation of truncation matrices TI and T is based on
C the Cholesky factor S of a controllability Grammian P = S*S'
C and the Cholesky factor R of an observability Grammian Q = R'*R,
C where S and R are given upper triangular matrices.
C
C For the B&T approach, the matrices of the reduced order system
C are computed using the truncation formulas:
C
C Ar = TI * A * T , Br = TI * B , Cr = C * T . (1)
C
C For the SPA approach, the matrices of a minimal realization
C (Am,Bm,Cm) are computed using the truncation formulas:
C
C Am = TI * A * T , Bm = TI * B , Cm = C * T . (2)
C
C Am, Bm, Cm and D serve further for computing the SPA of the given
C system.
C
C ARGUMENTS
C
C Mode Parameters
C
C DICO CHARACTER*1
C Specifies the type of the original system as follows:
C = 'C': continuous-time system;
C = 'D': discrete-time system.
C
C JOB CHARACTER*1
C Specifies the model reduction approach to be used
C as follows:
C = 'B': use the square-root B&T method;
C = 'F': use the balancing-free square-root B&T method;
C = 'S': use the square-root SPA method;
C = 'P': use the balancing-free square-root SPA method.
C
C FACT CHARACTER*1
C Specifies whether or not, on entry, the matrix A is in a
C real Schur form, as follows:
C = 'S': A is in a real Schur form;
C = 'N': A is a general dense square matrix.
C
C ORDSEL CHARACTER*1
C Specifies the order selection method as follows:
C = 'F': the resulting order NR is fixed;
C = 'A': the resulting order NR is automatically determined
C on basis of the given tolerance TOL1.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the original state-space representation,
C i.e., the order of the matrix A. N >= 0.
C
C M (input) INTEGER
C The number of system inputs. M >= 0.
C
C P (input) INTEGER
C The number of system outputs. P >= 0.
C
C NR (input/output) INTEGER
C On entry with ORDSEL = 'F', NR is the desired order of
C the resulting reduced order system. 0 <= NR <= N.
C On exit, if INFO = 0, NR is the order of the resulting
C reduced order model. NR is set as follows:
C if ORDSEL = 'F', NR is equal to MIN(NR,NMINR), where NR
C is the desired order on entry and NMINR is the number of
C the Hankel singular values greater than N*EPS*S1, where
C EPS is the machine precision (see LAPACK Library Routine
C DLAMCH) and S1 is the largest Hankel singular value
C (computed in HSV(1));
C NR can be further reduced to ensure HSV(NR) > HSV(NR+1);
C if ORDSEL = 'A', NR is equal to the number of Hankel
C singular values greater than MAX(TOL1,N*EPS*S1).
C
C SCALEC (input) DOUBLE PRECISION
C Scaling factor for the Cholesky factor S of the
C controllability Grammian, i.e., S/SCALEC is used to
C compute the Hankel singular values. SCALEC > 0.
C
C SCALEO (input) DOUBLE PRECISION
C Scaling factor for the Cholesky factor R of the
C observability Grammian, i.e., R/SCALEO is used to
C compute the Hankel singular values. SCALEO > 0.
C
C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
C On entry, the leading N-by-N part of this array must
C contain the state dynamics matrix A. If FACT = 'S',
C A is in a real Schur form.
C On exit, if INFO = 0, the leading NR-by-NR part of this
C array contains the state dynamics matrix Ar of the
C reduced order system.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,N).
C
C B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
C On entry, the leading N-by-M part of this array must
C contain the original input/state matrix B.
C On exit, if INFO = 0, the leading NR-by-M part of this
C array contains the input/state matrix Br of the reduced
C order system.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= MAX(1,N).
C
C C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
C On entry, the leading P-by-N part of this array must
C contain the original state/output matrix C.
C On exit, if INFO = 0, the leading P-by-NR part of this
C array contains the state/output matrix Cr of the reduced
C order system.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,P).
C
C D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
C On entry, if JOB = 'S' or JOB = 'P', the leading P-by-M
C part of this array must contain the original input/output
C matrix D.
C On exit, if INFO = 0 and JOB = 'S' or JOB = 'P', the
C leading P-by-M part of this array contains the
C input/output matrix Dr of the reduced order system.
C If JOB = 'B' or JOB = 'F', this array is not referenced.
C
C LDD INTEGER
C The leading dimension of array D.
C LDD >= 1, if JOB = 'B' or JOB = 'F';
C LDD >= MAX(1,P), if JOB = 'S' or JOB = 'P'.
C
C TI (input/output) DOUBLE PRECISION array, dimension (LDTI,N)
C On entry, the leading N-by-N upper triangular part of
C this array must contain the Cholesky factor S of a
C controllability Grammian P = S*S'.
C On exit, if INFO = 0, and NR > 0, the leading NMINR-by-N
C part of this array contains the left truncation matrix
C TI in (1), for the B&T approach, or in (2), for the
C SPA approach.
C
C LDTI INTEGER
C The leading dimension of array TI. LDTI >= MAX(1,N).
C
C T (input/output) DOUBLE PRECISION array, dimension (LDT,N)
C On entry, the leading N-by-N upper triangular part of
C this array must contain the Cholesky factor R of an
C observability Grammian Q = R'*R.
C On exit, if INFO = 0, and NR > 0, the leading N-by-NMINR
C part of this array contains the right truncation matrix
C T in (1), for the B&T approach, or in (2), for the
C SPA approach.
C
C LDT INTEGER
C The leading dimension of array T. LDT >= MAX(1,N).
C
C NMINR (output) INTEGER
C The number of Hankel singular values greater than
C MAX(TOL2,N*EPS*S1).
C Note: If S and R are the Cholesky factors of the
C controllability and observability Grammians of the
C original system (A,B,C,D), respectively, then NMINR is
C the order of a minimal realization of the original system.
C
C HSV (output) DOUBLE PRECISION array, dimension (N)
C If INFO = 0, it contains the Hankel singular values,
C ordered decreasingly. The Hankel singular values are
C singular values of the product R*S.
C
C Tolerances
C
C TOL1 DOUBLE PRECISION
C If ORDSEL = 'A', TOL1 contains the tolerance for
C determining the order of the reduced system.
C For model reduction, the recommended value lies in the
C interval [0.00001,0.001].
C If TOL1 <= 0 on entry, the used default value is
C TOL1 = N*EPS*S1, where EPS is the machine precision
C (see LAPACK Library Routine DLAMCH) and S1 is the largest
C Hankel singular value (computed in HSV(1)).
C If ORDSEL = 'F', the value of TOL1 is ignored.
C
C TOL2 DOUBLE PRECISION
C The tolerance for determining the order of a minimal
C realization of the system.
C The recommended value is TOL2 = N*EPS*S1.
C This value is used by default if TOL2 <= 0 on entry.
C If TOL2 > 0, and ORDSEL = 'A', then TOL2 <= TOL1.
C
C Workspace
C
C IWORK INTEGER array, dimension (LIWORK), where
C LIWORK = 0, if JOB = 'B';
C LIWORK = N, if JOB = 'F';
C LIWORK = 2*N, if JOB = 'S' or 'P'.
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) returns the optimal value
C of LDWORK.
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= MAX( 1, 2*N*N + 5*N, N*MAX(M,P) ).
C For optimum performance LDWORK should be larger.
C
C Warning Indicator
C
C IWARN INTEGER
C = 0: no warning;
C = 1: with ORDSEL = 'F', the selected order NR is greater
C than NMINR, the order of a minimal realization of
C the given system; in this case, the resulting NR is
C set automatically to NMINR;
C = 2: with ORDSEL = 'F', the selected order NR corresponds
C to repeated singular values, which are neither all
C included nor all excluded from the reduced model;
C in this case, the resulting NR is set automatically
C to the largest value such that HSV(NR) > HSV(NR+1).
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 = 1: the computation of Hankel singular values failed.
C
C METHOD
C
C Let be the stable linear system
C
C d[x(t)] = Ax(t) + Bu(t)
C y(t) = Cx(t) + Du(t), (3)
C
C where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
C for a discrete-time system. The subroutine AB09IX determines for
C the given system (3), the matrices of a reduced NR order system
C
C d[z(t)] = Ar*z(t) + Br*u(t)
C yr(t) = Cr*z(t) + Dr*u(t), (4)
C
C by using the square-root or balancing-free square-root
C Balance & Truncate (B&T) or Singular Perturbation Approximation
C (SPA) model reduction methods.
C
C The projection matrices TI and T are determined using the
C Cholesky factors S and R of a controllability Grammian P and an
C observability Grammian Q.
C The Hankel singular values HSV(1), ...., HSV(N) are computed as
C singular values of the product R*S.
C
C If JOB = 'B', the square-root Balance & Truncate technique
C of [1] is used.
C
C If JOB = 'F', the balancing-free square-root version of the
C Balance & Truncate technique [2] is used.
C
C If JOB = 'S', the square-root version of the Singular Perturbation
C Approximation method [3,4] is used.
C
C If JOB = 'P', the balancing-free square-root version of the
C Singular Perturbation Approximation method [3,4] is used.
C
C REFERENCES
C
C [1] Tombs M.S. and Postlethwaite I.
C Truncated balanced realization of stable, non-minimal
C state-space systems.
C Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
C
C [2] Varga A.
C Efficient minimal realization procedure based on balancing.
C Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991,
C A. El Moudni, P. Borne, S. G. Tzafestas (Eds.),
C Vol. 2, pp. 42-46.
C
C [3] Liu Y. and Anderson B.D.O.
C Singular Perturbation Approximation of balanced systems.
C Int. J. Control, Vol. 50, pp. 1379-1405, 1989.
C
C [4] Varga A.
C Balancing-free square-root algorithm for computing singular
C perturbation approximations.
C Proc. 30-th CDC, Brighton, Dec. 11-13, 1991,
C Vol. 2, pp. 1062-1065.
C
C NUMERICAL ASPECTS
C
C The implemented method relies on accuracy enhancing square-root
C or balancing-free square-root methods.
C
C CONTRIBUTORS
C
C A. Varga, German Aerospace Center, Oberpfaffenhofen, August 2000.
C D. Sima, University of Bucharest, August 2000.
C V. Sima, Research Institute for Informatics, Bucharest, Aug. 2000.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, Dec. 2000,
C Sep. 2001.
C
C KEYWORDS
C
C Balance and truncate, minimal state-space representation,
C model reduction, multivariable system,
C singular perturbation approximation, state-space model.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
C .. Scalar Arguments ..
CHARACTER DICO, FACT, JOB, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDT, LDTI,
$ LDWORK, M, N, NMINR, NR, P
DOUBLE PRECISION SCALEC, SCALEO, TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), HSV(*), T(LDT,*), TI(LDTI,*)
C .. Local Scalars ..
LOGICAL BAL, BTA, DISCR, FIXORD, RSF, SPA
INTEGER IERR, IJ, J, K, KTAU, KU, KV, KW, LDW, LW,
$ NRED, NR1, NS, WRKOPT
DOUBLE PRECISION ATOL, RCOND, SKP, TEMP, TOLDEF
C .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH, LSAME
C .. External Subroutines ..
EXTERNAL AB09DD, DGEMM, DGEMV, DGEQRF, DGETRF, DGETRS,
$ DLACPY, DORGQR, DSCAL, DTRMM, DTRMV, MA02AD,
$ MB03UD, XERBLA
C .. Intrinsic Functions ..
INTRINSIC DBLE, INT, MAX, MIN, SQRT
C .. Executable Statements ..
C
INFO = 0
IWARN = 0
DISCR = LSAME( DICO, 'D' )
BTA = LSAME( JOB, 'B' ) .OR. LSAME( JOB, 'F' )
SPA = LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'P' )
BAL = LSAME( JOB, 'B' ) .OR. LSAME( JOB, 'S' )
RSF = LSAME( FACT, 'S' )
FIXORD = LSAME( ORDSEL, 'F' )
C
LW = MAX( 1, 2*N*N + 5*N, N*MAX( M, P ) )
C
C Test the input scalar arguments.
C
IF( .NOT. ( LSAME( DICO, 'C' ) .OR. DISCR ) ) THEN
INFO = -1
ELSE IF( .NOT. ( BTA .OR. SPA ) ) THEN
INFO = -2
ELSE IF( .NOT. ( RSF .OR. LSAME( FACT, 'N' ) ) ) THEN
INFO = -3
ELSE IF( .NOT. ( FIXORD .OR. LSAME( ORDSEL, 'A' ) ) ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( M.LT.0 ) THEN
INFO = -6
ELSE IF( P.LT.0 ) THEN
INFO = -7
ELSE IF( FIXORD .AND. ( NR.LT.0 .OR. NR.GT.N ) ) THEN
INFO = -8
ELSE IF( SCALEC.LE.ZERO ) THEN
INFO = -9
ELSE IF( SCALEO.LE.ZERO ) THEN
INFO = -10
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -12
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -14
ELSE IF( LDC.LT.MAX( 1, P ) ) THEN
INFO = -16
ELSE IF( LDD.LT.1 .OR. ( SPA .AND. LDD.LT.P ) ) THEN
INFO = -18
ELSE IF( LDTI.LT.MAX( 1, N ) ) THEN
INFO = -20
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
INFO = -22
ELSE IF( TOL2.GT.ZERO .AND. .NOT.FIXORD .AND. TOL2.GT.TOL1 ) THEN
INFO = -26
ELSE IF( LDWORK.LT.LW ) THEN
INFO = -29
END IF
C
IF( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'AB09IX', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( MIN( N, M, P ).EQ.0 ) THEN
NR = 0
NMINR = 0
DWORK(1) = ONE
RETURN
END IF
C
C Save S in DWORK(KV).
C
KV = 1
KU = KV + N*N
KW = KU + N*N
CALL DLACPY( 'Upper', N, N, TI, LDTI, DWORK(KV), N )
C | x x |
C Compute R*S in the form | 0 x | in TI.
C
DO 10 J = 1, N
CALL DTRMV( 'Upper', 'NoTranspose', 'NonUnit', J, T, LDT,
$ TI(1,J), 1 )
10 CONTINUE
C
C Compute the singular value decomposition R*S = V*Sigma*UT of the
C upper triangular matrix R*S, with UT in TI and V in DWORK(KU).
C
C Workspace: need 2*N*N + 5*N;
C prefer larger.
C
CALL MB03UD( 'Vectors', 'Vectors', N, TI, LDTI, DWORK(KU), N, HSV,
$ DWORK(KW), LDWORK-KW+1, IERR )
IF( IERR.NE.0 ) THEN
INFO = 1
RETURN
ENDIF
WRKOPT = INT( DWORK(KW) ) + KW - 1
C
C Scale the singular values.
C
CALL DSCAL( N, ONE / SCALEC / SCALEO, HSV, 1 )
C
C Partition Sigma, U and V conformally as:
C
C Sigma = diag(Sigma1,Sigma2,Sigma3), U = [U1,U2,U3] (U' in TI) and
C V = [V1,V2,V3] (in DWORK(KU)).
C
C Compute NMINR, the order of a minimal realization, as the order
C of [Sigma1 Sigma2].
C
TOLDEF = DBLE( N )*DLAMCH( 'Epsilon' )
ATOL = MAX( TOL2, TOLDEF*HSV(1) )
NMINR = N
20 IF( NMINR.GT.0 ) THEN
IF( HSV(NMINR).LE.ATOL ) THEN
NMINR = NMINR - 1
GO TO 20
END IF
END IF
C
C Compute the order NR of reduced system, as the order of Sigma1.
C
IF( FIXORD ) THEN
C
C Check if the desired order is less than the order of a minimal
C realization.
C
IF( NR.GT.NMINR ) THEN
C
C Reduce the order to NMINR.
C
NR = NMINR
IWARN = 1
END IF
C
C Check for singular value multiplicity at cut-off point.
C
IF( NR.GT.0 .AND. NR.LT.NMINR ) THEN
SKP = HSV(NR)
IF( SKP-HSV(NR+1).LE.TOLDEF*SKP ) THEN
IWARN = 2
C
C Reduce the order such that HSV(NR) > HSV(NR+1).
C
30 NR = NR - 1
IF( NR.GT.0 ) THEN
IF( HSV(NR)-SKP.LE.TOLDEF*SKP ) GO TO 30
END IF
END IF
END IF
ELSE
C
C The order is given as the number of singular values
C exceeding MAX( TOL1, N*EPS*HSV(1) ).
C
ATOL = MAX( TOL1, ATOL )
NR = 0
DO 40 J = 1, NMINR
IF( HSV(J).LE.ATOL ) GO TO 50
NR = NR + 1
40 CONTINUE
50 CONTINUE
ENDIF
C
C Finish if the order is zero.
C
IF( NR.EQ.0 ) THEN
IF( SPA )
$ CALL AB09DD( DICO, N, M, P, NR, A, LDA, B, LDB, C, LDC,
$ D, LDD, RCOND, IWORK, DWORK, IERR )
DWORK(1) = WRKOPT
RETURN
END IF
C
C Compute NS, the order of Sigma2. For BTA, NS = 0.
C
IF( SPA ) THEN
NRED = NMINR
ELSE
NRED = NR
END IF
NS = NRED - NR
C
C Compute the truncation matrices.
C
C Compute TI' = | TI1' TI2' | = R'*| V1 V2 | in DWORK(KU).
C
CALL DTRMM( 'Left', 'Upper', 'Transpose', 'NonUnit', N, NRED,
$ ONE, T, LDT, DWORK(KU), N )
C
C Compute T = | T1 T2 | = S*| U1 U2 | .
C
CALL MA02AD( 'Full', NRED, N, TI, LDTI, T, LDT )
CALL DTRMM( 'Left', 'Upper', 'NoTranspose', 'NonUnit', N,
$ NRED, ONE, DWORK(KV), N, T, LDT )
C
KTAU = KW
IF( BAL ) THEN
IJ = KU
C
C Square-Root B&T/SPA method.
C
C Compute the truncation matrices for balancing
C -1/2 -1/2
C T1*Sigma1 and TI1'*Sigma1 .
C
DO 60 J = 1, NR
TEMP = ONE/SQRT( HSV(J) )
CALL DSCAL( N, TEMP, T(1,J), 1 )
CALL DSCAL( N, TEMP, DWORK(IJ), 1 )
IJ = IJ + N
60 CONTINUE
C
ELSE
C
C Balancing-Free B&T/SPA method.
C
C Compute orthogonal bases for the images of matrices T1 and
C TI1'.
C
C Workspace: need 2*N*N + 2*N;
C prefer larger.
C
KW = KTAU + NR
LDW = LDWORK - KW + 1
CALL DGEQRF( N, NR, T, LDT, DWORK(KTAU), DWORK(KW), LDW, IERR )
CALL DORGQR( N, NR, NR, T, LDT, DWORK(KTAU), DWORK(KW), LDW,
$ IERR )
CALL DGEQRF( N, NR, DWORK(KU), N, DWORK(KTAU), DWORK(KW), LDW,
$ IERR )
WRKOPT = MAX( WRKOPT, INT( DWORK(KW) ) + KW - 1 )
CALL DORGQR( N, NR, NR, DWORK(KU), N, DWORK(KTAU), DWORK(KW),
$ LDW, IERR )
WRKOPT = MAX( WRKOPT, INT( DWORK(KW) ) + KW - 1 )
ENDIF
C
IF( NS.GT.0 ) THEN
C
C Compute orthogonal bases for the images of matrices T2 and
C TI2'.
C
C Workspace: need 2*N*N + 2*N;
C prefer larger.
C
NR1 = NR + 1
KW = KTAU + NS
LDW = LDWORK - KW + 1
CALL DGEQRF( N, NS, T(1,NR1), LDT, DWORK(KTAU), DWORK(KW), LDW,
$ IERR )
CALL DORGQR( N, NS, NS, T(1,NR1), LDT, DWORK(KTAU), DWORK(KW),
$ LDW, IERR )
CALL DGEQRF( N, NS, DWORK(KU+N*NR), N, DWORK(KTAU), DWORK(KW),
$ LDW, IERR )
WRKOPT = MAX( WRKOPT, INT( DWORK(KW) ) + KW - 1 )
CALL DORGQR( N, NS, NS, DWORK(KU+N*NR), N, DWORK(KTAU),
$ DWORK(KW), LDW, IERR )
WRKOPT = MAX( WRKOPT, INT( DWORK(KW) ) + KW - 1 )
ENDIF
C
C Transpose TI' in TI.
C
CALL MA02AD( 'Full', N, NRED, DWORK(KU), N, TI, LDTI )
C
IF( .NOT.BAL ) THEN
C -1
C Compute (TI1*T1) *TI1 in TI.
C
CALL DGEMM( 'NoTranspose', 'NoTranspose', NR, NR, N, ONE, TI,
$ LDTI, T, LDT, ZERO, DWORK(KU), N )
CALL DGETRF( NR, NR, DWORK(KU), N, IWORK, IERR )
CALL DGETRS( 'NoTranspose', NR, N, DWORK(KU), N, IWORK, TI,
$ LDTI, IERR )
C
IF( NS.GT.0 ) THEN
C -1
C Compute (TI2*T2) *TI2 in TI2.
C
CALL DGEMM( 'NoTranspose', 'NoTranspose', NS, NS, N, ONE,
$ TI(NR1,1), LDTI, T(1,NR1), LDT, ZERO, DWORK(KU),
$ N )
CALL DGETRF( NS, NS, DWORK(KU), N, IWORK, IERR )
CALL DGETRS( 'NoTranspose', NS, N, DWORK(KU), N, IWORK,
$ TI(NR1,1), LDTI, IERR )
END IF
END IF
C
C Compute TI*A*T. Exploit RSF of A if possible.
C Workspace: need N*N.
C
IF( RSF ) THEN
IJ = 1
DO 80 J = 1, N
K = MIN( J+1, N )
CALL DGEMV( 'NoTranspose', NRED, K, ONE, TI, LDTI,
$ A(1,J), 1, ZERO, DWORK(IJ), 1 )
IJ = IJ + N
80 CONTINUE
ELSE
CALL DGEMM( 'NoTranspose', 'NoTranspose', NRED, N, N, ONE,
$ TI, LDTI, A, LDA, ZERO, DWORK, N )
END IF
CALL DGEMM( 'NoTranspose', 'NoTranspose', NRED, NRED, N, ONE,
$ DWORK, N, T, LDT, ZERO, A, LDA )
C
C Compute TI*B and C*T.
C Workspace: need N*MAX(M,P).
C
CALL DLACPY( 'Full', N, M, B, LDB, DWORK, N )
CALL DGEMM( 'NoTranspose', 'NoTranspose', NRED, M, N, ONE, TI,
$ LDTI, DWORK, N, ZERO, B, LDB )
C
CALL DLACPY( 'Full', P, N, C, LDC, DWORK, P )
CALL DGEMM( 'NoTranspose', 'NoTranspose', P, NRED, N, ONE,
$ DWORK, P, T, LDT, ZERO, C, LDC )
C
C Compute the singular perturbation approximation if possible.
C Note that IERR = 1 on exit from AB09DD cannot appear here.
C
C Workspace: need real 4*(NMINR-NR);
C need integer 2*(NMINR-NR).
C
IF( SPA) THEN
CALL AB09DD( DICO, NRED, M, P, NR, A, LDA, B, LDB,
$ C, LDC, D, LDD, RCOND, IWORK, DWORK, IERR )
ELSE
NMINR = NR
END IF
DWORK(1) = WRKOPT
C
RETURN
C *** Last line of AB09IX ***
END