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 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 . 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