SUBROUTINE SB10ZD( N, M, NP, A, LDA, B, LDB, C, LDC, D, LDD,
$ FACTOR, AK, LDAK, BK, LDBK, CK, LDCK, DK,
$ LDDK, RCOND, TOL, IWORK, DWORK, LDWORK, BWORK,
$ INFO )
C
C PURPOSE
C
C To compute the matrices of the positive feedback controller
C
C | Ak | Bk |
C K = |----|----|
C | Ck | Dk |
C
C for the shaped plant
C
C | A | B |
C G = |---|---|
C | C | D |
C
C in the Discrete-Time Loop Shaping Design Procedure.
C
C ARGUMENTS
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the plant. N >= 0.
C
C M (input) INTEGER
C The column size of the matrix B. M >= 0.
C
C NP (input) INTEGER
C The row size of the matrix C. NP >= 0.
C
C A (input) DOUBLE PRECISION array, dimension (LDA,N)
C The leading N-by-N part of this array must contain the
C system state matrix A of the shaped plant.
C
C LDA INTEGER
C The leading dimension of the array A. LDA >= max(1,N).
C
C B (input) DOUBLE PRECISION array, dimension (LDB,M)
C The leading N-by-M part of this array must contain the
C system input matrix B of the shaped plant.
C
C LDB INTEGER
C The leading dimension of the array B. LDB >= max(1,N).
C
C C (input) DOUBLE PRECISION array, dimension (LDC,N)
C The leading NP-by-N part of this array must contain the
C system output matrix C of the shaped plant.
C
C LDC INTEGER
C The leading dimension of the array C. LDC >= max(1,NP).
C
C D (input) DOUBLE PRECISION array, dimension (LDD,M)
C The leading NP-by-M part of this array must contain the
C system input/output matrix D of the shaped plant.
C
C LDD INTEGER
C The leading dimension of the array D. LDD >= max(1,NP).
C
C FACTOR (input) DOUBLE PRECISION
C = 1 implies that an optimal controller is required
C (not recommended);
C > 1 implies that a suboptimal controller is required
C achieving a performance FACTOR less than optimal.
C FACTOR >= 1.
C
C AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
C The leading N-by-N part of this array contains the
C controller state matrix Ak.
C
C LDAK INTEGER
C The leading dimension of the array AK. LDAK >= max(1,N).
C
C BK (output) DOUBLE PRECISION array, dimension (LDBK,NP)
C The leading N-by-NP part of this array contains the
C controller input matrix Bk.
C
C LDBK INTEGER
C The leading dimension of the array BK. LDBK >= max(1,N).
C
C CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
C The leading M-by-N part of this array contains the
C controller output matrix Ck.
C
C LDCK INTEGER
C The leading dimension of the array CK. LDCK >= max(1,M).
C
C DK (output) DOUBLE PRECISION array, dimension (LDDK,NP)
C The leading M-by-NP part of this array contains the
C controller matrix Dk.
C
C LDDK INTEGER
C The leading dimension of the array DK. LDDK >= max(1,M).
C
C RCOND (output) DOUBLE PRECISION array, dimension (6)
C RCOND(1) contains an estimate of the reciprocal condition
C number of the linear system of equations from
C which the solution of the P-Riccati equation is
C obtained;
C RCOND(2) contains an estimate of the reciprocal condition
C number of the linear system of equations from
C which the solution of the Q-Riccati equation is
C obtained;
C RCOND(3) contains an estimate of the reciprocal condition
C number of the matrix (gamma^2-1)*In - P*Q;
C RCOND(4) contains an estimate of the reciprocal condition
C number of the matrix Rx + Bx'*X*Bx;
C RCOND(5) contains an estimate of the reciprocal condition
C ^
C number of the matrix Ip + D*Dk;
C RCOND(6) contains an estimate of the reciprocal condition
C ^
C number of the matrix Im + Dk*D.
C
C Tolerances
C
C TOL DOUBLE PRECISION
C Tolerance used for checking the nonsingularity of the
C matrices to be inverted. If TOL <= 0, then a default value
C equal to sqrt(EPS) is used, where EPS is the relative
C machine precision. TOL < 1.
C
C Workspace
C
C IWORK INTEGER array, dimension (2*max(N,M+NP))
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) contains the optimal value
C of LDWORK.
C
C LDWORK INTEGER
C The dimension of the array DWORK.
C LDWORK >= 16*N*N + 5*M*M + 7*NP*NP + 6*M*N + 7*M*NP +
C 7*N*NP + 6*N + 2*(M + NP) +
C max(14*N+23,16*N,2*M-1,2*NP-1).
C For good performance, LDWORK must generally be larger.
C
C BWORK LOGICAL array, dimension (2*N)
C
C Error Indicator
C
C INFO (output) INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value;
C = 1: the P-Riccati equation is not solved successfully;
C = 2: the Q-Riccati equation is not solved successfully;
C = 3: the iteration to compute eigenvalues or singular
C values failed to converge;
C = 4: the matrix (gamma^2-1)*In - P*Q is singular;
C = 5: the matrix Rx + Bx'*X*Bx is singular;
C ^
C = 6: the matrix Ip + D*Dk is singular;
C ^
C = 7: the matrix Im + Dk*D is singular;
C = 8: the matrix Ip - D*Dk is singular;
C = 9: the matrix Im - Dk*D is singular;
C = 10: the closed-loop system is unstable.
C
C METHOD
C
C The routine implements the formulas given in [1].
C
C REFERENCES
C
C [1] Gu, D.-W., Petkov, P.H., and Konstantinov, M.M.
C On discrete H-infinity loop shaping design procedure routines.
C Technical Report 00-6, Dept. of Engineering, Univ. of
C Leicester, UK, 2000.
C
C NUMERICAL ASPECTS
C
C The accuracy of the results depends on the conditioning of the
C two Riccati equations solved in the controller design. For
C better conditioning it is advised to take FACTOR > 1.
C
C CONTRIBUTORS
C
C P.Hr. Petkov, D.W. Gu and M.M. Konstantinov, May 2001.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, July 2001.
C
C KEYWORDS
C
C H_infinity control, Loop-shaping design, Robust control.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
C ..
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDWORK, M, N, NP
DOUBLE PRECISION FACTOR, TOL
C ..
C .. Array Arguments ..
INTEGER IWORK( * )
LOGICAL BWORK( * )
DOUBLE PRECISION A ( LDA, * ), AK( LDAK, * ), B ( LDB, * ),
$ BK( LDBK, * ), C ( LDC, * ), CK( LDCK, * ),
$ D ( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ RCOND( 6 )
C ..
C .. Local Scalars ..
INTEGER I, I1, I2, I3, I4, I5, I6, I7, I8, I9, I10,
$ I11, I12, I13, I14, I15, I16, I17, I18, I19,
$ I20, I21, I22, I23, I24, I25, I26, INFO2, IWRK,
$ J, LWAMAX, MINWRK, N2, NS, SDIM
DOUBLE PRECISION ANORM, GAMMA, TOLL
C ..
C .. External Functions ..
LOGICAL SELECT
DOUBLE PRECISION DLAMCH, DLANGE, DLANSY, DLAPY2
EXTERNAL DLAMCH, DLANGE, DLANSY, DLAPY2, SELECT
C ..
C .. External Subroutines ..
EXTERNAL DCOPY, DGECON, DGEES, DGEMM, DGETRF, DGETRS,
$ DLACPY, DLASCL, DLASET, DPOTRF, DPOTRS, DSWAP,
$ DSYCON, DSYEV, DSYRK, DSYTRF, DSYTRS, DTRSM,
$ DTRTRS, MA02AD, MB01RX, MB02VD, SB02OD, XERBLA
C ..
C .. Intrinsic Functions ..
INTRINSIC INT, MAX, SQRT
C ..
C .. Executable Statements ..
C
C Decode and Test input parameters.
C
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( NP.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, NP ) ) THEN
INFO = -9
ELSE IF( LDD.LT.MAX( 1, NP ) ) THEN
INFO = -11
ELSE IF( FACTOR.LT.ONE ) THEN
INFO = -12
ELSE IF( LDAK.LT.MAX( 1, N ) ) THEN
INFO = -14
ELSE IF( LDBK.LT.MAX( 1, N ) ) THEN
INFO = -16
ELSE IF( LDCK.LT.MAX( 1, M ) ) THEN
INFO = -18
ELSE IF( LDDK.LT.MAX( 1, M ) ) THEN
INFO = -20
ELSE IF( TOL.GE.ONE ) THEN
INFO = -22
END IF
C
C Compute workspace.
C
MINWRK = 16*N*N + 5*M*M + 7*NP*NP + 6*M*N + 7*M*NP + 7*N*NP +
$ 6*N + 2*(M + NP) + MAX( 14*N+23, 16*N, 2*M-1, 2*NP-1 )
IF( LDWORK.LT.MINWRK ) THEN
INFO = -25
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SB10ZD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C Note that some computation could be made if one or two of the
C dimension parameters N, M, and P are zero, but the results are
C not so meaningful.
C
IF( N.EQ.0 .OR. M.EQ.0 .OR. NP.EQ.0 ) THEN
RCOND( 1 ) = ONE
RCOND( 2 ) = ONE
RCOND( 3 ) = ONE
RCOND( 4 ) = ONE
RCOND( 5 ) = ONE
RCOND( 6 ) = ONE
DWORK( 1 ) = ONE
RETURN
END IF
C
C Set the default tolerance, if needed.
C
IF( TOL.LE.ZERO ) THEN
TOLL = SQRT( DLAMCH( 'Epsilon' ) )
ELSE
TOLL = TOL
END IF
C
C Workspace usage.
C
N2 = 2*N
I1 = 1 + N*N
I2 = I1 + N*N
I3 = I2 + NP*NP
I4 = I3 + M*M
I5 = I4 + NP*NP
I6 = I5 + M*M
I7 = I6 + M*N
I8 = I7 + M*N
I9 = I8 + N*N
I10 = I9 + N*N
I11 = I10 + N2
I12 = I11 + N2
I13 = I12 + N2
I14 = I13 + N2*N2
I15 = I14 + N2*N2
C
IWRK = I15 + N2*N2
LWAMAX = 0
C
C Compute R1 = Ip + D*D' .
C
CALL DLASET( 'U', NP, NP, ZERO, ONE, DWORK( I2 ), NP )
CALL DSYRK( 'U', 'N', NP, M, ONE, D, LDD, ONE, DWORK( I2 ), NP )
CALL DLACPY( 'U', NP, NP, DWORK( I2 ), NP, DWORK( I4 ), NP )
C
C Factorize R1 = R'*R .
C
CALL DPOTRF( 'U', NP, DWORK( I4 ), NP, INFO2 )
C -1
C Compute C'*R in BK .
C
CALL MA02AD( 'F', NP, N, C, LDC, BK, LDBK )
CALL DTRSM( 'R', 'U', 'N', 'N', N, NP, ONE, DWORK( I4 ), NP, BK,
$ LDBK )
C
C Compute R2 = Im + D'*D .
C
CALL DLASET( 'U', M, M, ZERO, ONE, DWORK( I3 ), M )
CALL DSYRK( 'U', 'T', M, NP, ONE, D, LDD, ONE, DWORK( I3 ), M )
CALL DLACPY( 'U', M, M, DWORK( I3 ), M, DWORK( I5 ), M )
C
C Factorize R2 = U'*U .
C
CALL DPOTRF( 'U', M, DWORK( I5 ), M, INFO2 )
C -1
C Compute (U )'*B' .
C
CALL MA02AD( 'F', N, M, B, LDB, DWORK( I6 ), M )
CALL DTRTRS( 'U', 'T', 'N', M, N, DWORK( I5 ), M, DWORK( I6 ), M,
$ INFO2 )
C
C Compute D'*C .
C
CALL DGEMM( 'T', 'N', M, N, NP, ONE, D, LDD, C, LDC, ZERO,
$ DWORK( I7 ), M )
C -1
C Compute (U )'*D'*C .
C
CALL DTRTRS( 'U', 'T', 'N', M, N, DWORK( I5 ), M, DWORK( I7 ), M,
$ INFO2 )
C -1
C Compute Ar = A - B*R2 D'*C .
C
CALL DLACPY( 'F', N, N, A, LDA, DWORK( I8 ), N )
CALL DGEMM( 'T', 'N', N, N, M, -ONE, DWORK( I6 ), M, DWORK( I7 ),
$ M, ONE, DWORK( I8 ), N )
C -1
C Compute Cr = C'*R1 *C .
C
CALL DSYRK( 'U', 'N', N, NP, ONE, BK, LDBK, ZERO, DWORK( I9 ), N )
C -1
C Compute Dr = B*R2 B' in AK .
C
CALL DSYRK( 'U', 'T', N, M, ONE, DWORK( I6 ), M, ZERO, AK, LDAK )
C -1
C Solution of the Riccati equation Ar'*P*(In + Dr*P) Ar - P +
C Cr = 0 .
CALL SB02OD( 'D', 'G', 'N', 'U', 'Z', 'S', N, M, NP, DWORK( I8 ),
$ N, AK, LDAK, DWORK( I9 ), N, DWORK, M, DWORK, N,
$ RCOND( 1 ), DWORK, N, DWORK( I10 ), DWORK( I11 ),
$ DWORK( I12 ), DWORK( I13 ), N2, DWORK( I14 ), N2,
$ DWORK( I15 ), N2, -ONE, IWORK, DWORK( IWRK ),
$ LDWORK-IWRK+1, BWORK, INFO2 )
IF( INFO2.NE.0 ) THEN
INFO = 1
RETURN
END IF
LWAMAX = MAX( LWAMAX, INT( DWORK( IWRK ) ) + IWRK - 1 )
C
C Transpose Ar .
C
DO 10 J = 1, N - 1
CALL DSWAP( J, DWORK( I8+J ), N, DWORK( I8+J*N ), 1 )
10 CONTINUE
C -1
C Solution of the Riccati equation Ar*Q*(In + Cr*Q) *Ar' - Q +
C Dr = 0 .
CALL SB02OD( 'D', 'G', 'N', 'U', 'Z', 'S', N, M, NP, DWORK( I8 ),
$ N, DWORK( I9 ), N, AK, LDAK, DWORK, M, DWORK, N,
$ RCOND( 2 ), DWORK( I1 ), N, DWORK( I10 ),
$ DWORK( I11 ), DWORK( I12 ), DWORK( I13 ), N2,
$ DWORK( I14 ), N2, DWORK( I15 ), N2, -ONE, IWORK,
$ DWORK( IWRK ), LDWORK-IWRK+1, BWORK, INFO2 )
IF( INFO2.NE.0 ) THEN
INFO = 2
RETURN
END IF
LWAMAX = MAX( LWAMAX, INT( DWORK( IWRK ) ) + IWRK - 1 )
C
C Compute gamma.
C
CALL DGEMM( 'N', 'N', N, N, N, ONE, DWORK( I1 ), N, DWORK, N,
$ ZERO, DWORK( I8 ), N )
CALL DGEES( 'N', 'N', SELECT, N, DWORK( I8 ), N, SDIM,
$ DWORK( I10 ), DWORK( I11 ), DWORK( IWRK ), N,
$ DWORK( IWRK ), LDWORK-IWRK+1, BWORK, INFO2 )
IF( INFO2.NE.0 ) THEN
INFO = 3
RETURN
END IF
LWAMAX = MAX( LWAMAX, INT( DWORK( IWRK ) ) + IWRK - 1 )
GAMMA = ZERO
C
DO 20 I = 0, N - 1
GAMMA = MAX( GAMMA, DWORK( I10+I ) )
20 CONTINUE
C
GAMMA = FACTOR*SQRT( ONE + GAMMA )
C
C Workspace usage.
C
I5 = I4 + NP*NP
I6 = I5 + M*M
I7 = I6 + NP*NP
I8 = I7 + NP*NP
I9 = I8 + NP*NP
I10 = I9 + NP
I11 = I10 + NP*NP
I12 = I11 + M*M
I13 = I12 + M
C
IWRK = I13 + M*M
C
C Compute the eigenvalues and eigenvectors of R1 .
C
CALL DLACPY( 'U', NP, NP, DWORK( I2 ), NP, DWORK( I8 ), NP )
CALL DSYEV( 'V', 'U', NP, DWORK( I8 ), NP, DWORK( I9 ),
$ DWORK( IWRK ), LDWORK-IWRK+1, INFO2 )
IF( INFO2.NE.0 ) THEN
INFO = 3
RETURN
END IF
LWAMAX = MAX( LWAMAX, INT( DWORK( IWRK ) ) + IWRK - 1 )
C -1/2
C Compute R1 .
C
DO 40 J = 1, NP
DO 30 I = 1, NP
DWORK( I10-1+I+(J-1)*NP ) = DWORK( I8-1+J+(I-1)*NP ) /
$ SQRT( DWORK( I9+I-1 ) )
30 CONTINUE
40 CONTINUE
C
CALL DGEMM( 'N', 'N', NP, NP, NP, ONE, DWORK( I8 ), NP,
$ DWORK( I10 ), NP, ZERO, DWORK( I4 ), NP )
C
C Compute the eigenvalues and eigenvectors of R2 .
C
CALL DLACPY( 'U', M, M, DWORK( I3 ), M, DWORK( I11 ), M )
CALL DSYEV( 'V', 'U', M, DWORK( I11 ), M, DWORK( I12 ),
$ DWORK( IWRK ), LDWORK-IWRK+1, INFO2 )
IF( INFO2.NE.0 ) THEN
INFO = 3
RETURN
END IF
LWAMAX = MAX( LWAMAX, INT( DWORK( IWRK ) ) + IWRK - 1 )
C -1/2
C Compute R2 .
C
DO 60 J = 1, M
DO 50 I = 1, M
DWORK( I13-1+I+(J-1)*M ) = DWORK( I11-1+J+(I-1)*M ) /
$ SQRT( DWORK( I12+I-1 ) )
50 CONTINUE
60 CONTINUE
C
CALL DGEMM( 'N', 'N', M, M, M, ONE, DWORK( I11 ), M, DWORK( I13 ),
$ M, ZERO, DWORK( I5 ), M )
C
C Compute R1 + C*Q*C' .
C
CALL DGEMM( 'N', 'T', N, NP, N, ONE, DWORK( I1 ), N, C, LDC,
$ ZERO, BK, LDBK )
CALL MB01RX( 'L', 'U', 'N', NP, N, ONE, ONE, DWORK( I2 ), NP,
$ C, LDC, BK, LDBK, INFO2 )
CALL DLACPY( 'U', NP, NP, DWORK( I2 ), NP, DWORK( I8 ), NP )
C
C Compute the eigenvalues and eigenvectors of R1 + C*Q*C' .
C
CALL DSYEV( 'V', 'U', NP, DWORK( I8 ), NP, DWORK( I9 ),
$ DWORK( IWRK ), LDWORK-IWRK+1, INFO2 )
IF( INFO2.NE.0 ) THEN
INFO = 3
RETURN
END IF
LWAMAX = MAX( LWAMAX, INT( DWORK( IWRK ) ) + IWRK - 1 )
C -1
C Compute ( R1 + C*Q*C' ) .
C
DO 80 J = 1, NP
DO 70 I = 1, NP
DWORK( I10-1+I+(J-1)*NP ) = DWORK( I8-1+J+(I-1)*NP ) /
$ DWORK( I9+I-1 )
70 CONTINUE
80 CONTINUE
C
CALL DGEMM( 'N', 'N', NP, NP, NP, ONE, DWORK( I8 ), NP,
$ DWORK( I10 ), NP, ZERO, DWORK( I6 ), NP )
C -1
C Compute Z2 .
C
DO 100 J = 1, NP
DO 90 I = 1, NP
DWORK( I10-1+I+(J-1)*NP ) = DWORK( I8-1+J+(I-1)*NP )*
$ SQRT( DWORK( I9+I-1 ) )
90 CONTINUE
100 CONTINUE
C
CALL DGEMM( 'N', 'N', NP, NP, NP, ONE, DWORK( I8 ), NP,
$ DWORK( I10 ), NP, ZERO, DWORK( I7 ), NP )
C
C Workspace usage.
C
I9 = I8 + N*NP
I10 = I9 + N*NP
I11 = I10 + NP*M
I12 = I11 + ( NP + M )*( NP + M )
I13 = I12 + N*( NP + M )
I14 = I13 + N*( NP + M )
I15 = I14 + N*N
I16 = I15 + N*N
I17 = I16 + ( NP + M )*N
I18 = I17 + ( NP + M )*( NP + M )
I19 = I18 + ( NP + M )*N
I20 = I19 + M*N
I21 = I20 + M*NP
I22 = I21 + NP*N
I23 = I22 + N*N
I24 = I23 + N*NP
I25 = I24 + NP*NP
I26 = I25 + M*M
C
IWRK = I26 + N*M
C
C Compute A*Q*C' + B*D' .
C
CALL DGEMM( 'N', 'T', N, NP, M, ONE, B, LDB, D, LDD, ZERO,
$ DWORK( I8 ), N )
CALL DGEMM( 'N', 'N', N, NP, N, ONE, A, LDA, BK, LDBK,
$ ONE, DWORK( I8 ), N )
C -1
C Compute H = -( A*Q*C'+B*D' )*( R1 + C*Q*C' ) .
C
CALL DGEMM( 'N', 'N', N, NP, NP, -ONE, DWORK( I8 ), N,
$ DWORK( I6 ), NP, ZERO, DWORK( I9 ), N )
C -1/2
C Compute R1 D .
C
CALL DGEMM( 'N', 'N', NP, M, NP, ONE, DWORK( I4 ), NP, D, LDD,
$ ZERO, DWORK( I10 ), NP )
C
C Compute Rx .
C
DO 110 J = 1, NP
CALL DCOPY( J, DWORK( I2+(J-1)*NP ), 1,
$ DWORK( I11+(J-1)*(NP+M) ), 1 )
DWORK( I11-1+J+(J-1)*(NP+M) ) = DWORK( I2-1+J+(J-1)*NP ) -
$ GAMMA*GAMMA
110 CONTINUE
C
CALL DGEMM( 'N', 'N', NP, M, NP, ONE, DWORK( I7 ), NP,
$ DWORK( I10 ), NP, ZERO, DWORK( I11+(NP+M)*NP ),
$ NP+M )
CALL DLASET( 'U', M, M, ZERO, ONE, DWORK( I11+(NP+M)*NP+NP ),
$ NP+M )
C
C Compute Bx .
C
CALL DGEMM( 'N', 'N', N, NP, NP, -ONE, DWORK( I9 ), N,
$ DWORK( I7 ), NP, ZERO, DWORK( I12 ), N )
CALL DGEMM( 'N', 'N', N, M, M, ONE, B, LDB, DWORK( I5 ), M,
$ ZERO, DWORK( I12+N*NP ), N )
C
C Compute Sx .
C
CALL DGEMM( 'T', 'N', N, NP, NP, ONE, C, LDC, DWORK( I7 ), NP,
$ ZERO, DWORK( I13 ), N )
CALL DGEMM( 'T', 'N', N, M, NP, ONE, C, LDC, DWORK( I10 ), NP,
$ ZERO, DWORK( I13+N*NP ), N )
C
C Compute (gamma^2 - 1)*In - P*Q .
C
CALL DLASET( 'F', N, N, ZERO, GAMMA*GAMMA-ONE, DWORK( I14 ), N )
CALL DGEMM( 'N', 'N', N, N, N, -ONE, DWORK, N, DWORK( I1 ), N,
$ ONE, DWORK( I14 ), N )
C -1
C Compute X = ((gamma^2 - 1)*In - P*Q) *gamma^2*P .
C
CALL DLACPY( 'F', N, N, DWORK, N, DWORK( I15 ), N )
CALL DLASCL( 'G', 0, 0, ONE, GAMMA*GAMMA, N, N, DWORK( I15 ), N,
$ INFO )
ANORM = DLANGE( '1', N, N, DWORK( I14 ), N, DWORK( IWRK ) )
CALL DGETRF( N, N, DWORK( I14 ), N, IWORK, INFO2 )
IF( INFO2.GT.0 ) THEN
INFO = 4
RETURN
END IF
CALL DGECON( '1', N, DWORK( I14 ), N, ANORM, RCOND( 3 ),
$ DWORK( IWRK ), IWORK( N+1 ), INFO2 )
C
C Return if the matrix is singular to working precision.
C
IF( RCOND( 3 ).LT.TOLL ) THEN
INFO = 4
RETURN
END IF
CALL DGETRS( 'N', N, N, DWORK( I14 ), N, IWORK, DWORK( I15 ),
$ N, INFO2 )
C
C Compute Bx'*X .
C
CALL DGEMM( 'T', 'N', NP+M, N, N, ONE, DWORK( I12 ), N,
$ DWORK( I15 ), N, ZERO, DWORK( I16 ), NP+M )
C
C Compute Rx + Bx'*X*Bx .
C
CALL DLACPY( 'U', NP+M, NP+M, DWORK( I11 ), NP+M, DWORK( I17 ),
$ NP+M )
CALL MB01RX( 'L', 'U', 'N', NP+M, N, ONE, ONE, DWORK( I17 ), NP+M,
$ DWORK( I16 ), NP+M, DWORK( I12 ), N, INFO2 )
C
C Compute -( Sx' + Bx'*X*A ) .
C
CALL MA02AD( 'F', N, NP+M, DWORK( I13 ), N, DWORK( I18 ), NP+M )
CALL DGEMM( 'N', 'N', NP+M, N, N, -ONE, DWORK( I16 ), NP+M,
$ A, LDA, -ONE, DWORK( I18 ), NP+M )
C
C Factorize Rx + Bx'*X*Bx .
C
ANORM = DLANSY( '1', 'U', NP+M, DWORK( I17 ), NP+M,
$ DWORK( IWRK ) )
CALL DSYTRF( 'U', NP+M, DWORK( I17 ), NP+M, IWORK,
$ DWORK( IWRK ), LDWORK-IWRK, INFO2 )
IF( INFO2.NE.0 ) THEN
INFO = 5
RETURN
END IF
CALL DSYCON( 'U', NP+M, DWORK( I17 ), NP+M, IWORK, ANORM,
$ RCOND( 4 ), DWORK( IWRK ), IWORK( NP+M+1), INFO2 )
C
C Return if the matrix is singular to working precision.
C
IF( RCOND( 4 ).LT.TOLL ) THEN
INFO = 5
RETURN
END IF
C -1
C Compute F = -( Rx + Bx'*X*Bx ) ( Sx' + Bx'*X*A ) .
C
CALL DSYTRS( 'U', NP+M, N, DWORK( I17 ), NP+M, IWORK,
$ DWORK( I18 ), NP+M, INFO2 )
C
C Compute B'*X .
C
CALL DGEMM( 'T', 'N', M, N, N, ONE, B, LDB, DWORK( I15 ), N,
$ ZERO, DWORK( I19 ), M )
C
C Compute -( D' - B'*X*H ) .
C
DO 130 J = 1, NP
DO 120 I = 1, M
DWORK( I20-1+I+(J-1)*M ) = -D( J, I )
120 CONTINUE
130 CONTINUE
C
CALL DGEMM( 'N', 'N', M, NP, N, ONE, DWORK( I19 ), M,
$ DWORK( I9 ), N, ONE, DWORK( I20 ), M )
C -1
C Compute C + Z2 *F1 .
C
CALL DLACPY( 'F', NP, N, C, LDC, DWORK( I21 ), NP )
CALL DGEMM( 'N', 'N', NP, N, NP, ONE, DWORK( I7 ), NP,
$ DWORK( I18 ), NP+M, ONE, DWORK( I21 ), NP )
C
C Compute R2 + B'*X*B .
C
CALL MB01RX( 'L', 'U', 'N', M, N, ONE, ONE, DWORK( I3 ), M,
$ DWORK( I19 ), M, B, LDB, INFO2 )
C
C Factorize R2 + B'*X*B .
C
CALL DPOTRF( 'U', M, DWORK( I3 ), M, INFO2 )
C ^ -1
C Compute Dk = -( R2 + B'*X*B ) (D' - B'*X*H) .
C
CALL DLACPY( 'F', M, NP, DWORK( I20 ), M, DK, LDDK )
CALL DPOTRS( 'U', M, NP, DWORK( I3 ), M, DK, LDDK, INFO2 )
C ^ ^
C Compute Bk = -H + B*Dk .
C
CALL DLACPY( 'F', N, NP, DWORK( I9 ), N, DWORK( I23 ), N )
CALL DGEMM( 'N', 'N', N, NP, M, ONE, B, LDB, DK, LDDK,
$ -ONE, DWORK( I23 ), N )
C -1/2
C Compute R2 *F2 .
C
CALL DGEMM( 'N', 'N', M, N, M, ONE, DWORK( I5 ), M,
$ DWORK( I18+NP ), NP+M, ZERO, CK, LDCK )
C ^ -1/2 ^ -1
C Compute Ck = R2 *F2 - Dk*( C + Z2 *F1 ) .
C
CALL DGEMM( 'N', 'N', M, N, NP, -ONE, DK, LDDK,
$ DWORK( I21 ), NP, ONE, CK, LDCK )
C ^ ^
C Compute Ak = A + H*C + B*Ck .
C
CALL DLACPY( 'F', N, N, A, LDA, AK, LDAK )
CALL DGEMM( 'N', 'N', N, N, NP, ONE, DWORK( I9 ), N, C, LDC,
$ ONE, AK, LDAK )
CALL DGEMM( 'N', 'N', N, N, M, ONE, B, LDB, CK, LDCK,
$ ONE, AK, LDAK )
C ^
C Compute Ip + D*Dk .
C
CALL DLASET( 'Full', NP, NP, ZERO, ONE, DWORK( I24 ), NP )
CALL DGEMM( 'N', 'N', NP, NP, M, ONE, D, LDD, DK, LDDK,
$ ONE, DWORK( I24 ), NP )
C ^
C Compute Im + Dk*D .
C
CALL DLASET( 'Full', M, M, ZERO, ONE, DWORK( I25 ), M )
CALL DGEMM( 'N', 'N', M, M, NP, ONE, DK, LDDK, D, LDD,
$ ONE, DWORK( I25 ), M )
C ^ ^ ^ ^ -1
C Compute Ck = M*Ck, M = (Im + Dk*D) .
C
ANORM = DLANGE( '1', M, M, DWORK( I25 ), M, DWORK( IWRK ) )
CALL DGETRF( M, M, DWORK( I25 ), M, IWORK, INFO2 )
IF( INFO2.NE.0 ) THEN
INFO = 7
RETURN
END IF
CALL DGECON( '1', M, DWORK( I25 ), M, ANORM, RCOND( 6 ),
$ DWORK( IWRK ), IWORK( M+1 ), INFO2 )
C
C Return if the matrix is singular to working precision.
C
IF( RCOND( 6 ).LT.TOLL ) THEN
INFO = 7
RETURN
END IF
CALL DGETRS( 'N', M, N, DWORK( I25 ), M, IWORK, CK, LDCK, INFO2 )
C ^ ^
C Compute Dk = M*Dk .
C
CALL DGETRS( 'N', M, NP, DWORK( I25 ), M, IWORK, DK, LDDK, INFO2 )
C ^
C Compute Bk*D .
C
CALL DGEMM( 'N', 'N', N, M, NP, ONE, DWORK( I23 ), N, D, LDD,
$ ZERO, DWORK( I26 ), N )
C ^ ^
C Compute Ak = Ak - Bk*D*Ck.
C
CALL DGEMM( 'N', 'N', N, N, M, -ONE, DWORK( I26 ), N, CK, LDCK,
$ ONE, AK, LDAK )
C ^ ^ -1
C Compute Bk = Bk*(Ip + D*Dk) .
C
ANORM = DLANGE( '1', NP, NP, DWORK( I24 ), NP, DWORK( IWRK ) )
CALL DLACPY( 'Full', N, NP, DWORK( I23 ), N, BK, LDBK )
CALL MB02VD( 'N', N, NP, DWORK( I24 ), NP, IWORK, BK, LDBK,
$ INFO2 )
IF( INFO2.NE.0 ) THEN
INFO = 6
RETURN
END IF
CALL DGECON( '1', NP, DWORK( I24 ), NP, ANORM, RCOND( 5 ),
$ DWORK( IWRK ), IWORK( NP+1 ), INFO2 )
C
C Return if the matrix is singular to working precision.
C
IF( RCOND( 5 ).LT.TOLL ) THEN
INFO = 6
RETURN
END IF
C
C Workspace usage.
C
I2 = 1 + NP*NP
I3 = I2 + N*NP
I4 = I3 + M*M
I5 = I4 + N*M
I6 = I5 + NP*N
I7 = I6 + M*N
I8 = I7 + N2*N2
I9 = I8 + N2
C
IWRK = I9 + N2
C
C Compute Ip - D*Dk .
C
CALL DLASET( 'Full', NP, NP, ZERO, ONE, DWORK, NP )
CALL DGEMM( 'N', 'N', NP, NP, M, -ONE, D, LDD, DK, LDDK, ONE,
$ DWORK, NP )
C -1
C Compute Bk*(Ip-D*Dk) .
C
CALL DLACPY( 'Full', N, NP, BK, LDBK, DWORK( I2 ), N )
CALL MB02VD( 'N', N, NP, DWORK, NP, IWORK, DWORK( I2 ), N, INFO2 )
IF( INFO2.NE.0 ) THEN
INFO = 8
RETURN
END IF
C
C Compute Im - Dk*D .
C
CALL DLASET( 'Full', M, M, ZERO, ONE, DWORK( I3 ), M )
CALL DGEMM( 'N', 'N', M, M, NP, -ONE, DK, LDDK, D, LDD, ONE,
$ DWORK( I3 ), M )
C -1
C Compute B*(Im-Dk*D) .
C
CALL DLACPY( 'Full', N, M, B, LDB, DWORK( I4 ), N )
CALL MB02VD( 'N', N, M, DWORK( I3 ), M, IWORK, DWORK( I4 ), N,
$ INFO2 )
IF( INFO2.NE.0 ) THEN
INFO = 9
RETURN
END IF
C
C Compute D*Ck .
C
CALL DGEMM( 'N', 'N', NP, N, M, ONE, D, LDD, CK, LDCK, ZERO,
$ DWORK( I5 ), NP )
C
C Compute Dk*C .
C
CALL DGEMM( 'N', 'N', M, N, NP, ONE, DK, LDDK, C, LDC, ZERO,
$ DWORK( I6 ), M )
C
C Compute the closed-loop state matrix.
C
CALL DLACPY( 'F', N, N, A, LDA, DWORK( I7 ), N2 )
CALL DGEMM( 'N', 'N', N, N, M, ONE, DWORK( I4 ), N,
$ DWORK( I6 ), M, ONE, DWORK( I7 ), N2 )
CALL DGEMM( 'N', 'N', N, N, M, ONE, DWORK( I4 ), N, CK, LDCK,
$ ZERO, DWORK( I7+N2*N ), N2 )
CALL DGEMM( 'N', 'N', N, N, NP, ONE, DWORK( I2 ), N, C, LDC,
$ ZERO, DWORK( I7+N ), N2 )
CALL DLACPY( 'F', N, N, AK, LDAK, DWORK( I7+N2*N+N ), N2 )
CALL DGEMM( 'N', 'N', N, N, NP, ONE, DWORK( I2 ), N,
$ DWORK( I5 ), NP, ONE, DWORK( I7+N2*N+N ), N2 )
C
C Compute the closed-loop poles.
C
CALL DGEES( 'N', 'N', SELECT, N2, DWORK( I7 ), N2, SDIM,
$ DWORK( I8 ), DWORK( I9 ), DWORK( IWRK ), N,
$ DWORK( IWRK ), LDWORK-IWRK+1, BWORK, INFO2 )
IF( INFO2.NE.0 ) THEN
INFO = 3
RETURN
END IF
LWAMAX = MAX( LWAMAX, INT( DWORK( IWRK ) ) + IWRK - 1 )
C
C Check the stability of the closed-loop system.
C
NS = 0
C
DO 140 I = 0, N2 - 1
IF( DLAPY2( DWORK( I8+I ), DWORK( I9+I ) ).GT.ONE )
$ NS = NS + 1
140 CONTINUE
C
IF( NS.GT.0 ) THEN
INFO = 10
RETURN
END IF
C
DWORK( 1 ) = DBLE( LWAMAX )
RETURN
C *** Last line of SB10ZD ***
END