SUBROUTINE AB09HD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR, ALPHA, $ BETA, A, LDA, B, LDB, C, LDC, D, LDD, NS, HSV, $ TOL1, TOL2, IWORK, DWORK, LDWORK, BWORK, 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 stochastic C balancing approach in conjunction with the square-root or C the balancing-free square-root Balance & Truncate (B&T) C or Singular Perturbation Approximation (SPA) model reduction C methods for the ALPHA-stable part of the 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 Balance & Truncate method; C = 'F': use the balancing-free square-root C Balance & Truncate method; C = 'S': use the square-root Singular Perturbation C Approximation method; C = 'P': use the balancing-free square-root C Singular Perturbation Approximation method. C C EQUIL CHARACTER*1 C Specifies whether the user wishes to preliminarily C equilibrate the triplet (A,B,C) as follows: C = 'S': perform equilibration (scaling); C = 'N': do not perform equilibration. 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 P <= M if BETA = 0. C C NR (input/output) INTEGER C On entry with ORDSEL = 'F', NR is the desired order of the C resulting reduced order system. 0 <= NR <= N. C On exit, if INFO = 0, NR is the order of the resulting C reduced order model. For a system with NU ALPHA-unstable C eigenvalues and NS ALPHA-stable eigenvalues (NU+NS = N), C NR is set as follows: if ORDSEL = 'F', NR is equal to C NU+MIN(MAX(0,NR-NU),NMIN), where NR is the desired order C on entry, and NMIN is the order of a minimal realization C of the ALPHA-stable part of the given system; NMIN is C determined as the number of Hankel singular values greater C than NS*EPS, where EPS is the machine precision C (see LAPACK Library Routine DLAMCH); C if ORDSEL = 'A', NR is the sum of NU and the number of C Hankel singular values greater than MAX(TOL1,NS*EPS); C NR can be further reduced to ensure that C HSV(NR-NU) > HSV(NR+1-NU). C C ALPHA (input) DOUBLE PRECISION C Specifies the ALPHA-stability boundary for the eigenvalues C of the state dynamics matrix A. For a continuous-time C system (DICO = 'C'), ALPHA <= 0 is the boundary value for C the real parts of eigenvalues, while for a discrete-time C system (DICO = 'D'), 0 <= ALPHA <= 1 represents the C boundary value for the moduli of eigenvalues. C The ALPHA-stability domain does not include the boundary. C C BETA (input) DOUBLE PRECISION C BETA > 0 specifies the absolute/relative error weighting C parameter. A large positive value of BETA favours the C minimization of the absolute approximation error, while a C small value of BETA is appropriate for the minimization C of the relative error. C BETA = 0 means a pure relative error method and can be C used only if rank(D) = P. 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. C On exit, if INFO = 0, the leading NR-by-NR part of this C array contains the state dynamics matrix Ar of the reduced C order system. C The resulting A has a block-diagonal form with two blocks. C For a system with NU ALPHA-unstable eigenvalues and C NS ALPHA-stable eigenvalues (NU+NS = N), the leading C NU-by-NU block contains the unreduced part of A C corresponding to ALPHA-unstable eigenvalues in an C upper real Schur form. C The trailing (NR+NS-N)-by-(NR+NS-N) block contains C the reduced part of A corresponding to ALPHA-stable C eigenvalues. 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, the leading P-by-M part of this array must C contain the original input/output matrix D. C On exit, if INFO = 0, the leading P-by-M part of this C array contains the input/output matrix Dr of the reduced C order system. C C LDD INTEGER C The leading dimension of array D. LDD >= MAX(1,P). C C NS (output) INTEGER C The dimension of the ALPHA-stable subsystem. C C HSV (output) DOUBLE PRECISION array, dimension (N) C If INFO = 0, the leading NS elements of HSV contain the C Hankel singular values of the phase system corresponding C to the ALPHA-stable part of the original system. C The Hankel singular values are ordered decreasingly. C C Tolerances C C TOL1 DOUBLE PRECISION C If ORDSEL = 'A', TOL1 contains the tolerance for C determining the order of reduced system. C For model reduction, the recommended value of TOL1 lies C in the interval [0.00001,0.001]. C If TOL1 <= 0 on entry, the used default value is C TOL1 = NS*EPS, where NS is the number of C ALPHA-stable eigenvalues of A and EPS is the machine C precision (see LAPACK Library Routine DLAMCH). C If ORDSEL = 'F', the value of TOL1 is ignored. C TOL1 < 1. C C TOL2 DOUBLE PRECISION C The tolerance for determining the order of a minimal C realization of the phase system (see METHOD) corresponding C to the ALPHA-stable part of the given system. C The recommended value is TOL2 = NS*EPS. C This value is used by default if TOL2 <= 0 on entry. C If TOL2 > 0 and ORDSEL = 'A', then TOL2 <= TOL1. C TOL2 < 1. C C Workspace C C IWORK INTEGER array, dimension MAX(1,2*N) C On exit with INFO = 0, IWORK(1) contains the order of the C minimal realization of the system. 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) contains RCOND, the reciprocal C condition number of the U11 matrix from the expression C used to compute the solution X = U21*inv(U11) of the C Riccati equation for spectral factorization. C A small value RCOND indicates possible ill-conditioning C of the respective Riccati equation. C C LDWORK INTEGER C The length of the array DWORK. C LDWORK >= 2*N*N + MB*(N+P) + MAX( 2, N*(MAX(N,MB,P)+5), C 2*N*P+MAX(P*(MB+2),10*N*(N+1) ) ), C where MB = M if BETA = 0 and MB = M+P if BETA > 0. C For optimum performance LDWORK should be larger. C C BWORK LOGICAL array, dimension 2*N 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 NSMIN, the sum of the order of the C ALPHA-unstable part and the order of a minimal C realization of the ALPHA-stable part of the given C system; in this case, the resulting NR is set equal C to NSMIN; C = 2: with ORDSEL = 'F', the selected order NR corresponds C to repeated singular values for the ALPHA-stable C part, which are neither all included nor all C excluded from the reduced model; in this case, the C resulting NR is automatically decreased to exclude C all repeated singular values; C = 3: with ORDSEL = 'F', the selected order NR is less C than the order of the ALPHA-unstable part of the C given system; in this case NR is set equal to the C order of the ALPHA-unstable part. 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 the ordered real Schur form of A C failed; C = 2: the reduction of the Hamiltonian matrix to real C Schur form failed; C = 3: the reordering of the real Schur form of the C Hamiltonian matrix failed; C = 4: the Hamiltonian matrix has less than N stable C eigenvalues; C = 5: the coefficient matrix U11 in the linear system C X*U11 = U21 to determine X is singular to working C precision; C = 6: BETA = 0 and D has not a maximal row rank; C = 7: the computation of Hankel singular values failed; C = 8: the separation of the ALPHA-stable/unstable diagonal C blocks failed because of very close eigenvalues; C = 9: the resulting order of reduced stable part is less C than the number of unstable zeros of the stable C part. C METHOD C C Let be the following linear system C C d[x(t)] = Ax(t) + Bu(t) C y(t) = Cx(t) + Du(t), (1) 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 AB09HD determines for C the given system (1), the matrices of a reduced order system C C d[z(t)] = Ar*z(t) + Br*u(t) C yr(t) = Cr*z(t) + Dr*u(t), (2) C C such that C C INFNORM[inv(conj(W))*(G-Gr)] <= C (1+HSV(NR+NS-N+1)) / (1-HSV(NR+NS-N+1)) + ... C + (1+HSV(NS)) / (1-HSV(NS)) - 1, C C where G and Gr are transfer-function matrices of the systems C (A,B,C,D) and (Ar,Br,Cr,Dr), respectively, W is the right, minimum C phase spectral factor satisfying C C G1*conj(G1) = conj(W)* W, (3) C C G1 is the NS-order ALPHA-stable part of G, and INFNORM(G) is the C infinity-norm of G. HSV(1), ... , HSV(NS) are the Hankel-singular C values of the stable part of the phase system (Ap,Bp,Cp) C with the transfer-function matrix C C P = inv(conj(W))*G1. C C If BETA > 0, then the model reduction is performed on [G BETA*I] C instead of G. This is the recommended approach to be used when D C has not a maximal row rank or when a certain balance between C relative and absolute approximation errors is desired. For C increasingly large values of BETA, the obtained reduced system C assymptotically approaches that computed by using the C Balance & Truncate or Singular Perturbation Approximation methods. C C Note: conj(G) denotes either G'(-s) for a continuous-time system C or G'(1/z) for a discrete-time system. C inv(G) is the inverse of G. C C The following procedure is used to reduce a given G: C C 1) Decompose additively G as C C G = G1 + G2, C C such that G1 = (As,Bs,Cs,D) has only ALPHA-stable poles and C G2 = (Au,Bu,Cu) has only ALPHA-unstable poles. C C 2) Determine G1r, a reduced order approximation of the C ALPHA-stable part G1 using the balancing stochastic method C in conjunction with either the B&T [1,2] or SPA methods [3]. C C 3) Assemble the reduced model Gr as C C Gr = G1r + G2. C C Note: The employed stochastic truncation algorithm [2,3] has the C property that right half plane zeros of G1 remain as right half C plane zeros of G1r. Thus, the order can not be chosen smaller than C the sum of the number of unstable poles of G and the number of C unstable zeros of G1. C C The reduction of the ALPHA-stable part G1 is done as follows. C C If JOB = 'B', the square-root stochastic Balance & Truncate C method of [1] is used. C For an ALPHA-stable continuous-time system (DICO = 'C'), C the resulting reduced model is stochastically balanced. C C If JOB = 'F', the balancing-free square-root version of the C stochastic Balance & Truncate method [1] is used to reduce C the ALPHA-stable part G1. C C If JOB = 'S', the stochastic balancing method is used to reduce C the ALPHA-stable part G1, in conjunction with the square-root C version of the Singular Perturbation Approximation method [3,4]. C C If JOB = 'P', the stochastic balancing method is used to reduce C the ALPHA-stable part G1, in conjunction with the balancing-free C square-root version of the Singular Perturbation Approximation C method [3,4]. C C REFERENCES C C [1] Varga A. and Fasol K.H. C A new square-root balancing-free stochastic truncation model C reduction algorithm. C Proc. 12th IFAC World Congress, Sydney, 1993. C C [2] Safonov M. G. and Chiang R. Y. C Model reduction for robust control: a Schur relative error C method. C Int. J. Adapt. Contr. Sign. Proc., vol. 2, pp. 259-272, 1988. C C [3] Green M. and Anderson B. D. O. C Generalized balanced stochastic truncation. C Proc. 29-th CDC, Honolulu, Hawaii, pp. 476-481, 1990. C C [4] Varga A. C Balancing-free square-root algorithm for computing C singular perturbation approximations. C Proc. 30-th IEEE CDC, Brighton, Dec. 11-13, 1991, C Vol. 2, pp. 1062-1065. C C NUMERICAL ASPECTS C C The implemented methods rely on accuracy enhancing square-root or C balancing-free square-root techniques. The effectiveness of the C accuracy enhancing technique depends on the accuracy of the C solution of a Riccati equation. An ill-conditioned Riccati C solution typically results when [D BETA*I] is nearly C rank deficient. C 3 C The algorithm requires about 100N floating point operations. C C CONTRIBUTORS C C A. Varga, German Aerospace Center, Oberpfaffenhofen, May 2000. C D. Sima, University of Bucharest, May 2000. C V. Sima, Research Institute for Informatics, Bucharest, May 2000. C Partly based on the RASP routine SRBFS, by A. Varga, 1992. C C REVISIONS C C A. Varga, Australian National University, Canberra, November 2000. C V. Sima, Research Institute for Informatics, Bucharest, Dec. 2000. C Oct. 2001. C C KEYWORDS C C Minimal realization, model reduction, multivariable system, C state-space model, state-space representation, C stochastic balancing. C C ****************************************************************** C C .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO, TWOBY3, C100 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0, $ TWOBY3 = TWO/3.0D0, C100 = 100.0D0 ) C .. Scalar Arguments .. CHARACTER DICO, EQUIL, JOB, ORDSEL INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK, $ M, N, NR, NS, P DOUBLE PRECISION ALPHA, BETA, TOL1, TOL2 C .. Array Arguments .. INTEGER IWORK(*) DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), $ DWORK(*), HSV(*) LOGICAL BWORK(*) C .. Local Scalars .. LOGICAL BTA, DISCR, FIXORD, LEQUIL, SPA INTEGER IERR, IWARNL, KB, KD, KT, KTI, KU, KW, KWI, KWR, $ LW, LWR, MB, N2, NMR, NN, NRA, NU, NU1, WRKOPT DOUBLE PRECISION EPSM, MAXRED, RICOND, SCALEC, SCALEO C .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH, LSAME C .. External Subroutines .. EXTERNAL AB04MD, AB09HY, AB09IX, DLACPY, DLASET, TB01ID, $ TB01KD, XERBLA C .. Intrinsic Functions .. INTRINSIC INT, MAX, MIN C .. Executable Statements .. C INFO = 0 IWARN = 0 DISCR = LSAME( DICO, 'D' ) FIXORD = LSAME( ORDSEL, 'F' ) LEQUIL = LSAME( EQUIL, 'S' ) BTA = LSAME( JOB, 'B' ) .OR. LSAME( JOB, 'F' ) SPA = LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'P' ) MB = M IF( BETA.GT.ZERO ) MB = M + P LW = 2*N*N + MB*(N+P) + MAX( 2, N*(MAX( N, MB, P )+5), $ 2*N*P+MAX( P*(MB+2), 10*N*(N+1) ) ) 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. ( LEQUIL .OR. LSAME( EQUIL, '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 .OR. ( BETA.EQ.ZERO .AND. P.GT.M ) ) THEN INFO = -7 ELSE IF( FIXORD .AND. ( NR.LT.0 .OR. NR.GT.N ) ) THEN INFO = -8 ELSE IF( ( DISCR .AND. ( ALPHA.LT.ZERO .OR. ALPHA.GT.ONE ) ) .OR. $ ( .NOT.DISCR .AND. ALPHA.GT.ZERO ) ) THEN INFO = -9 ELSE IF( BETA.LT.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.MAX( 1, P ) ) THEN INFO = -18 ELSE IF( TOL1.GE.ONE ) THEN INFO = -21 ELSE IF( ( TOL2.GT.ZERO .AND. .NOT.FIXORD .AND. TOL2.GT.TOL1 ) $ .OR. TOL2.GE.ONE ) THEN INFO = -22 ELSE IF( LDWORK.LT.LW ) THEN INFO = -25 END IF C IF( INFO.NE.0 ) THEN C C Error return. C CALL XERBLA( 'AB09HD', -INFO ) RETURN END IF C C Quick return if possible. C IF( MIN( N, M, P ).EQ.0 .OR. $ ( BTA .AND. FIXORD .AND. NR.EQ.0 ) ) THEN NR = 0 NS = 0 IWORK(1) = 0 DWORK(1) = TWO DWORK(2) = ONE RETURN END IF C IF( LEQUIL ) THEN C C Scale simultaneously the matrices A, B and C: C A <- inv(D)*A*D, B <- inv(D)*B and C <- C*D, where D is a C diagonal matrix. C Workspace: N. C MAXRED = C100 CALL TB01ID( 'All', N, M, P, MAXRED, A, LDA, B, LDB, C, LDC, $ DWORK, INFO ) END IF C C Allocate working storage. C NN = N*N KU = 1 KWR = KU + NN KWI = KWR + N KW = KWI + N LWR = LDWORK - KW + 1 C C Reduce A to a block-diagonal real Schur form, with the C ALPHA-unstable part in the leading diagonal position, using a C non-orthogonal similarity transformation A <- inv(T)*A*T and C apply the transformation to B and C: B <- inv(T)*B and C <- C*T. C C Workspace needed: N*(N+2); C Additional workspace: need 3*N; C prefer larger. C CALL TB01KD( DICO, 'Unstable', 'General', N, M, P, ALPHA, A, LDA, $ B, LDB, C, LDC, NU, DWORK(KU), N, DWORK(KWR), $ DWORK(KWI), DWORK(KW), LWR, IERR ) C IF( IERR.NE.0 ) THEN IF( IERR.NE.3 ) THEN INFO = 1 ELSE INFO = 8 END IF RETURN END IF C WRKOPT = INT( DWORK(KW) ) + KW - 1 C IWARNL = 0 NS = N - NU IF( FIXORD ) THEN NRA = MAX( 0, NR-NU ) IF( NR.LT.NU ) $ IWARNL = 3 ELSE NRA = 0 END IF C C Finish if the system is completely unstable. C IF( NS.EQ.0 ) THEN NR = NU IWORK(1) = NS DWORK(1) = WRKOPT DWORK(2) = ONE RETURN END IF C NU1 = NU + 1 C C Allocate working storage. C N2 = N + N KB = 1 KD = KB + N*MB KT = KD + P*MB KTI = KT + N*N KW = KTI + N*N C C Form [B 0] and [D BETA*I]. C CALL DLACPY( 'F', NS, M, B(NU1,1), LDB, DWORK(KB), N ) CALL DLACPY( 'F', P, M, D, LDD, DWORK(KD), P ) IF( BETA.GT.ZERO ) THEN CALL DLASET( 'F', NS, P, ZERO, ZERO, DWORK(KB+N*M), N ) CALL DLASET( 'F', P, P, ZERO, BETA, DWORK(KD+P*M), P ) END IF C C For discrete-time case, apply the discrete-to-continuous bilinear C transformation to the stable part. C IF( DISCR ) THEN C C Real workspace: need N, prefer larger; C Integer workspace: need N. C CALL AB04MD( 'Discrete', NS, MB, P, ONE, ONE, A(NU1,NU1), LDA, $ DWORK(KB), N, C(1,NU1), LDC, DWORK(KD), P, $ IWORK, DWORK(KT), LDWORK-KT+1, IERR ) WRKOPT = MAX( WRKOPT, INT( DWORK(KT) ) + KT - 1 ) END IF C C Compute in DWORK(KTI) and DWORK(KT) the Cholesky factors S and R C of the controllability and observability Grammians, respectively. C Real workspace: need 2*N*N + MB*(N+P)+ C MAX( 2, N*(MAX(N,MB,P)+5), C 2*N*P+MAX(P*(MB+2), 10*N*(N+1) ) ); C prefer larger. C Integer workspace: need 2*N. C CALL AB09HY( NS, MB, P, A(NU1,NU1), LDA, DWORK(KB), N, $ C(1,NU1), LDC, DWORK(KD), P, SCALEC, SCALEO, $ DWORK(KTI), N, DWORK(KT), N, IWORK, DWORK(KW), $ LDWORK-KW+1, BWORK, INFO ) IF( INFO.NE.0 ) $ RETURN WRKOPT = MAX( WRKOPT, INT( DWORK(KW) ) + KW - 1 ) RICOND = DWORK(KW+1) C C Compute a BTA or SPA of the stable part. C Real workspace: need 2*N*N + MB*(N+P)+ C MAX( 1, 2*N*N+5*N, N*MAX(MB,P) ). C EPSM = DLAMCH( 'Epsilon' ) CALL AB09IX( 'C', JOB, 'Schur', ORDSEL, NS, MB, P, NRA, SCALEC, $ SCALEO, A(NU1,NU1), LDA, DWORK(KB), N, C(1,NU1), LDC, $ DWORK(KD), P, DWORK(KTI), N, DWORK(KT), N, NMR, HSV, $ MAX( TOL1, N*EPSM ), TOL2, IWORK, DWORK(KW), $ LDWORK-KW+1, IWARN, IERR ) IWARN = MAX( IWARN, IWARNL ) IF( IERR.NE.0 ) THEN INFO = 7 RETURN END IF WRKOPT = MAX( WRKOPT, INT( DWORK(KW) ) + KW - 1 ) C C Check if the resulting order is greater than the number of C unstable zeros (this check is implicit by looking at Hankel C singular values equal to 1). C IF( NRA.LT.NS .AND. HSV(NRA+1).GE.ONE-EPSM**TWOBY3 ) THEN INFO = 9 RETURN END IF C C For discrete-time case, apply the continuous-to-discrete C bilinear transformation. C IF( DISCR ) THEN CALL AB04MD( 'Continuous', NRA, MB, P, ONE, ONE, $ A(NU1,NU1), LDA, DWORK(KB), N, C(1,NU1), LDC, $ DWORK(KD), P, IWORK, DWORK, LDWORK, IERR ) WRKOPT = MAX( WRKOPT, INT( DWORK(1) ) ) END IF C CALL DLACPY( 'F', NRA, M, DWORK(KB), N, B(NU1,1), LDB ) CALL DLACPY( 'F', P, M, DWORK(KD), P, D, LDD ) C NR = NRA + NU C IWORK(1) = NMR DWORK(1) = WRKOPT DWORK(2) = RICOND C RETURN C *** Last line of AB09HD *** END