SUBROUTINE SB10SD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
$ D, LDD, AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK,
$ X, LDX, Y, LDY, RCOND, TOL, IWORK, DWORK,
$ LDWORK, BWORK, INFO )
C
C PURPOSE
C
C To compute the matrices of the H2 optimal controller
C
C | AK | BK |
C K = |----|----|,
C | CK | DK |
C
C for the normalized discrete-time system
C
C | A | B1 B2 | | A | B |
C P = |----|---------| = |---|---|
C | C1 | D11 D12 | | C | D |
C | C2 | D21 0 |
C
C where B2 has as column size the number of control inputs (NCON)
C and C2 has as row size the number of measurements (NMEAS) being
C provided to the controller.
C
C It is assumed that
C
C (A1) (A,B2) is stabilizable and (C2,A) is detectable,
C
C (A2) D12 is full column rank with D12 = | 0 | and D21 is
C | I |
C full row rank with D21 = | 0 I | as obtained by the
C SLICOT Library routine SB10PD,
C
C j*Theta
C (A3) | A-e *I B2 | has full column rank for all
C | C1 D12 |
C
C 0 <= Theta < 2*Pi ,
C
C
C j*Theta
C (A4) | A-e *I B1 | has full row rank for all
C | C2 D21 |
C
C 0 <= Theta < 2*Pi .
C
C ARGUMENTS
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the system. 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 NCON (input) INTEGER
C The number of control inputs (M2). M >= NCON >= 0,
C NP-NMEAS >= NCON.
C
C NMEAS (input) INTEGER
C The number of measurements (NP2). NP >= NMEAS >= 0,
C M-NCON >= NMEAS.
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.
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.
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.
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. Only the leading
C (NP-NP2)-by-(M-M2) submatrix D11 is used.
C
C LDD INTEGER
C The leading dimension of the array D. LDD >= max(1,NP).
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,NMEAS)
C The leading N-by-NMEAS 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 NCON-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.
C LDCK >= max(1,NCON).
C
C DK (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
C The leading NCON-by-NMEAS part of this array contains the
C controller input/output matrix DK.
C
C LDDK INTEGER
C The leading dimension of the array DK.
C LDDK >= max(1,NCON).
C
C X (output) DOUBLE PRECISION array, dimension (LDX,N)
C The leading N-by-N part of this array contains the matrix
C X, solution of the X-Riccati equation.
C
C LDX INTEGER
C The leading dimension of the array X. LDX >= max(1,N).
C
C Y (output) DOUBLE PRECISION array, dimension (LDY,N)
C The leading N-by-N part of this array contains the matrix
C Y, solution of the Y-Riccati equation.
C
C LDY INTEGER
C The leading dimension of the array Y. LDY >= max(1,N).
C
C RCOND (output) DOUBLE PRECISION array, dimension (4)
C RCOND contains estimates of the reciprocal condition
C numbers of the matrices which are to be inverted and the
C reciprocal condition numbers of the Riccati equations
C which have to be solved during the computation of the
C controller. (See the description of the algorithm in [2].)
C RCOND(1) contains the reciprocal condition number of the
C matrix Im2 + B2'*X2*B2;
C RCOND(2) contains the reciprocal condition number of the
C matrix Ip2 + C2*Y2*C2';
C RCOND(3) contains the reciprocal condition number of the
C X-Riccati equation;
C RCOND(4) contains the reciprocal condition number of the
C Y-Riccati equation.
C
C Tolerances
C
C TOL DOUBLE PRECISION
C Tolerance used in determining the nonsingularity of the
C matrices which must be inverted. If TOL <= 0, then a
C default value equal to sqrt(EPS) is used, where EPS is the
C relative machine precision.
C
C Workspace
C
C IWORK INTEGER array, dimension (max(M2,2*N,N*N,NP2))
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) contains the optimal
C LDWORK.
C
C LDWORK INTEGER
C The dimension of the array DWORK.
C LDWORK >= max(1, 14*N*N+6*N+max(14*N+23,16*N),
C M2*(N+M2+max(3,M1)), NP2*(N+NP2+3)),
C where M1 = M - M2.
C For good performance, LDWORK must generally be larger.
C
C BWORK LOGICAL array, dimension (2*N)
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: if the X-Riccati equation was not solved
C successfully;
C = 2: if the matrix Im2 + B2'*X2*B2 is not positive
C definite, or it is numerically singular (with
C respect to the tolerance TOL);
C = 3: if the Y-Riccati equation was not solved
C successfully;
C = 4: if the matrix Ip2 + C2*Y2*C2' is not positive
C definite, or it is numerically singular (with
C respect to the tolerance TOL).
C
C METHOD
C
C The routine implements the formulas given in [1]. The X- and
C Y-Riccati equations are solved with condition estimates.
C
C REFERENCES
C
C [1] Zhou, K., Doyle, J.C., and Glover, K.
C Robust and Optimal Control.
C Prentice-Hall, Upper Saddle River, NJ, 1996.
C
C [2] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
C Fortran 77 routines for Hinf and H2 design of linear
C discrete-time control systems.
C Report 99-8, Department of Engineering, Leicester University,
C April 1999.
C
C NUMERICAL ASPECTS
C
C The accuracy of the result depends on the condition numbers of the
C matrices which are to be inverted and on the condition numbers of
C the matrix Riccati equations which are to be solved in the
C computation of the controller. (The corresponding reciprocal
C condition numbers are given in the output array RCOND.)
C
C CONTRIBUTORS
C
C P.Hr. Petkov, D.W. Gu and M.M. Konstantinov, April 1999.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, May 1999,
C January 2003.
C
C KEYWORDS
C
C Algebraic Riccati equation, H2 optimal control, LQG, LQR, optimal
C regulator, 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, LDX, LDY, M, N, NCON, NMEAS, NP
DOUBLE PRECISION TOL
C ..
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ RCOND( * ), X( LDX, * ), Y( LDY, * )
LOGICAL BWORK( * )
C ..
C .. Local Scalars ..
INTEGER INFO2, IW2, IWB, IWC, IWG, IWI, IWQ, IWR, IWRK,
$ IWS, IWT, IWU, IWV, J, LWAMAX, M1, M2, MINWRK,
$ ND1, ND2, NP1, NP2
DOUBLE PRECISION ANORM, FERR, RCOND2, SEPD, TOLL
C ..
C .. External functions ..
DOUBLE PRECISION DLAMCH, DLANSY
EXTERNAL DLAMCH, DLANSY
C ..
C .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLASET, DPOCON, DPOTRF, DPOTRS,
$ DSWAP, DSYRK, DTRSM, MB01RX, SB02OD, SB02SD,
$ XERBLA
C ..
C .. Intrinsic Functions ..
INTRINSIC DBLE, INT, MAX
C ..
C .. Executable Statements ..
C
C Decode and Test input parameters.
C
M1 = M - NCON
M2 = NCON
NP1 = NP - NMEAS
NP2 = NMEAS
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( NCON.LT.0 .OR. M1.LT.0 .OR. M2.GT.NP1 ) THEN
INFO = -4
ELSE IF( NMEAS.LT.0 .OR. NP1.LT.0 .OR. NP2.GT.M1 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDC.LT.MAX( 1, NP ) ) THEN
INFO = -11
ELSE IF( LDD.LT.MAX( 1, NP ) ) THEN
INFO = -13
ELSE IF( LDAK.LT.MAX( 1, N ) ) THEN
INFO = -15
ELSE IF( LDBK.LT.MAX( 1, N ) ) THEN
INFO = -17
ELSE IF( LDCK.LT.MAX( 1, M2 ) ) THEN
INFO = -19
ELSE IF( LDDK.LT.MAX( 1, M2 ) ) THEN
INFO = -21
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -23
ELSE IF( LDY.LT.MAX( 1, N ) ) THEN
INFO = -25
ELSE
C
C Compute workspace.
C
MINWRK = MAX( 1, 14*N*N + 6*N + MAX( 14*N + 23, 16*N ),
$ M2*( N + M2 + MAX( 3, M1 ) ), NP2*( N + NP2 + 3 ) )
IF( LDWORK.LT.MINWRK )
$ INFO = -30
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SB10SD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( N.EQ.0 .OR. M.EQ.0 .OR. NP.EQ.0 .OR. M1.EQ.0 .OR. M2.EQ.0
$ .OR. NP1.EQ.0 .OR. NP2.EQ.0 ) THEN
RCOND( 1 ) = ONE
RCOND( 2 ) = ONE
RCOND( 3 ) = ONE
RCOND( 4 ) = ONE
DWORK( 1 ) = ONE
RETURN
END IF
C
ND1 = NP1 - M2
ND2 = M1 - NP2
TOLL = TOL
IF( TOLL.LE.ZERO ) THEN
C
C Set the default value of the tolerance for nonsingularity test.
C
TOLL = SQRT( DLAMCH( 'Epsilon' ) )
END IF
C
C Workspace usage.
C
IWQ = 1
IWG = IWQ + N*N
IWR = IWG + N*N
IWI = IWR + 2*N
IWB = IWI + 2*N
IWS = IWB + 2*N
IWT = IWS + 4*N*N
IWU = IWT + 4*N*N
IWRK = IWU + 4*N*N
IWC = IWR
IWV = IWC + N*N
C
C Compute Ax = A - B2*D12'*C1 in AK .
C
CALL DLACPY( 'Full', N, N, A, LDA, AK, LDAK )
CALL DGEMM( 'N', 'N', N, N, M2, -ONE, B( 1, M1+1 ), LDB,
$ C( ND1+1, 1), LDC, ONE, AK, LDAK )
C
C Compute Cx = C1'*C1 - C1'*D12*D12'*C1 .
C
IF( ND1.GT.0 ) THEN
CALL DSYRK( 'L', 'T', N, ND1, ONE, C, LDC, ZERO, DWORK( IWQ ),
$ N )
ELSE
CALL DLASET( 'L', N, N, ZERO, ZERO, DWORK( IWQ ), N )
END IF
C
C Compute Dx = B2*B2' .
C
CALL DSYRK( 'L', 'N', N, M2, ONE, B( 1, M1+1 ), LDB, ZERO,
$ DWORK( IWG ), N )
C
C Solution of the discrete-time Riccati equation
C Ax'*inv(In + X2*Dx)*X2*Ax - X2 + Cx = 0 .
C Workspace: need 14*N*N + 6*N + max(14*N+23,16*N);
C prefer larger.
C
CALL SB02OD( 'D', 'G', 'N', 'L', 'Z', 'S', N, M2, NP1, AK, LDAK,
$ DWORK( IWG ), N, DWORK( IWQ ), N, DWORK( IWRK ), M,
$ DWORK( IWRK ), N, RCOND2, X, LDX, DWORK( IWR ),
$ DWORK( IWI ), DWORK( IWB ), DWORK( IWS ), 2*N,
$ DWORK( IWT ), 2*N, DWORK( IWU ), 2*N, TOLL, IWORK,
$ DWORK( IWRK ), LDWORK-IWRK+1, BWORK, INFO2 )
IF( INFO2.GT.0 ) THEN
INFO = 1
RETURN
END IF
LWAMAX = INT( DWORK( IWRK ) ) + IWRK - 1
C
C Condition estimation.
C Workspace: need 4*N*N + max(N*N+5*N,max(3,2*N*N)+N*N);
C prefer larger.
C
IWRK = IWV + N*N
CALL SB02SD( 'C', 'N', 'N', 'L', 'O', N, AK, LDAK, DWORK( IWC ),
$ N, DWORK( IWV ), N, DWORK( IWG ), N, DWORK( IWQ ), N,
$ X, LDX, SEPD, RCOND( 3 ), FERR, IWORK, DWORK( IWRK ),
$ LDWORK-IWRK+1, INFO2 )
IF( INFO2.GT.0 ) RCOND( 3 ) = ZERO
LWAMAX = MAX( INT( DWORK( IWRK ) ) + IWRK - 1, LWAMAX )
C
C Workspace usage.
C
IW2 = M2*N + 1
IWRK = IW2 + M2*M2
C
C Compute B2'*X2 .
C
CALL DGEMM( 'T', 'N', M2, N, N, ONE, B( 1, M1+1 ), LDB, X, LDX,
$ ZERO, DWORK, M2 )
C
C Compute Im2 + B2'*X2*B2 .
C
CALL DLASET( 'L', M2, M2, ZERO, ONE, DWORK( IW2 ), M2 )
CALL MB01RX( 'Left', 'Lower', 'N', M2, N, ONE, ONE, DWORK( IW2 ),
$ M2, DWORK, M2, B( 1, M1+1 ), LDB, INFO2 )
C
C Compute the Cholesky factorization of Im2 + B2'*X2*B2 .
C Workspace: need M2*N + M2*M2 + max(3*M2,M2*M1);
C prefer larger.
C
ANORM = DLANSY( 'I', 'L', M2, DWORK( IW2 ), M2, DWORK( IWRK ) )
CALL DPOTRF( 'L', M2, DWORK( IW2 ), M2, INFO2 )
IF( INFO2.GT.0 ) THEN
INFO = 2
RETURN
END IF
CALL DPOCON( 'L', M2, DWORK( IW2 ), M2, ANORM, RCOND( 1 ),
$ DWORK( IWRK ), IWORK, INFO2 )
C
C Return if the matrix is singular to working precision.
C
IF( RCOND( 1 ).LT.TOLL ) THEN
INFO = 2
RETURN
END IF
C
C Compute -( B2'*X2*A + D12'*C1 ) in CK .
C
CALL DLACPY( 'Full', M2, N, C( ND1+1, 1 ), LDC, CK, LDCK )
CALL DGEMM( 'N', 'N', M2, N, N, -ONE, DWORK, M2, A, LDA, -ONE, CK,
$ LDCK )
C
C Compute F2 = -inv( Im2 + B2'*X2*B2 )*( B2'*X2*A + D12'*C1 ) .
C
CALL DPOTRS( 'L', M2, N, DWORK( IW2 ), M2, CK, LDCK, INFO2 )
C
C Compute -( B2'*X2*B1 + D12'*D11 ) .
C
CALL DLACPY( 'Full', M2, M1, D( ND1+1, 1 ), LDD, DWORK( IWRK ),
$ M2 )
CALL DGEMM( 'N', 'N', M2, M1, N, -ONE, DWORK, M2, B, LDB, -ONE,
$ DWORK( IWRK ), M2 )
C
C Compute F0 = -inv( Im2 + B2'*X2*B2 )*( B2'*X2*B1 + D12'*D11 ) .
C
CALL DPOTRS( 'L', M2, M1, DWORK( IW2 ), M2, DWORK( IWRK ), M2,
$ INFO2 )
C
C Save F0*D21' in DK .
C
CALL DLACPY( 'Full', M2, NP2, DWORK( IWRK+ND2*M2 ), M2, DK,
$ LDDK )
C
C Workspace usage.
C
IWRK = IWU + 4*N*N
C
C Compute Ay = A - B1*D21'*C2 in AK .
C
CALL DLACPY( 'Full', N, N, A, LDA, AK, LDAK )
CALL DGEMM( 'N', 'N', N, N, NP2, -ONE, B( 1, ND2+1 ), LDB,
$ C( NP1+1, 1 ), LDC, ONE, AK, LDAK )
C
C Transpose Ay in-situ.
C
DO 20 J = 1, N - 1
CALL DSWAP( J, AK( J+1, 1 ), LDAK, AK( 1, J+1 ), 1 )
20 CONTINUE
C
C Compute Cy = B1*B1' - B1*D21'*D21*B1' .
C
IF( ND2.GT.0 ) THEN
CALL DSYRK( 'U', 'N', N, ND2, ONE, B, LDB, ZERO, DWORK( IWQ ),
$ N )
ELSE
CALL DLASET( 'U', N, N, ZERO, ZERO, DWORK( IWQ ), N )
END IF
C
C Compute Dy = C2'*C2 .
C
CALL DSYRK( 'U', 'T', N, NP2, ONE, C( NP1+1, 1 ), LDC, ZERO,
$ DWORK( IWG ), N )
C
C Solution of the discrete-time Riccati equation
C Ay*inv( In + Y2*Dy )*Y2*Ay' - Y2 + Cy = 0 .
C
CALL SB02OD( 'D', 'G', 'N', 'U', 'Z', 'S', N, NP2, M1, AK, LDAK,
$ DWORK( IWG ), N, DWORK( IWQ ), N, DWORK( IWRK ), M,
$ DWORK( IWRK ), N, RCOND2, Y, LDY, DWORK( IWR ),
$ DWORK( IWI ), DWORK( IWB ), DWORK( IWS ), 2*N,
$ DWORK( IWT ), 2*N, DWORK( IWU ), 2*N, TOLL, IWORK,
$ DWORK( IWRK ), LDWORK-IWRK+1, BWORK, INFO2 )
IF( INFO2.GT.0 ) THEN
INFO = 3
RETURN
END IF
LWAMAX = MAX( INT( DWORK( IWRK ) ) + IWRK - 1, LWAMAX )
C
C Condition estimation.
C
IWRK = IWV + N*N
CALL SB02SD( 'C', 'N', 'N', 'U', 'O', N, AK, LDAK, DWORK( IWC ),
$ N, DWORK( IWV ), N, DWORK( IWG ), N, DWORK( IWQ ), N,
$ Y, LDY, SEPD, RCOND( 4 ), FERR, IWORK, DWORK( IWRK ),
$ LDWORK-IWRK+1, INFO2 )
IF( INFO2.GT.0 ) RCOND( 4 ) = ZERO
LWAMAX = MAX( INT( DWORK( IWRK ) ) + IWRK - 1, LWAMAX )
C
C Workspace usage.
C
IW2 = N*NP2 + 1
IWRK = IW2 + NP2*NP2
C
C Compute Y2*C2' .
C
CALL DGEMM( 'N', 'T', N, NP2, N, ONE, Y, LDY, C( NP1+1, 1 ), LDC,
$ ZERO, DWORK, N )
C
C Compute Ip2 + C2*Y2*C2' .
C
CALL DLASET( 'U', NP2, NP2, ZERO, ONE, DWORK( IW2 ), NP2 )
CALL MB01RX( 'Left', 'Upper', 'N', NP2, N, ONE, ONE, DWORK( IW2 ),
$ NP2, C( NP1+1, 1 ), LDC, DWORK, N, INFO2 )
C
C Compute the Cholesky factorization of Ip2 + C2*Y2*C2' .
C
ANORM = DLANSY( 'I', 'U', NP2, DWORK( IW2 ), NP2, DWORK( IWRK ) )
CALL DPOTRF( 'U', NP2, DWORK( IW2 ), NP2, INFO2 )
IF( INFO2.GT.0 ) THEN
INFO = 4
RETURN
END IF
CALL DPOCON( 'U', NP2, DWORK( IW2 ), NP2, ANORM, RCOND( 2 ),
$ DWORK( IWRK ), IWORK, INFO2 )
C
C Return if the matrix is singular to working precision.
C
IF( RCOND( 2 ).LT.TOLL ) THEN
INFO = 4
RETURN
END IF
C
C Compute A*Y2*C2' + B1*D21' in BK .
C
CALL DLACPY ( 'Full', N, NP2, B( 1, ND2+1 ), LDB, BK, LDBK )
CALL DGEMM( 'N', 'N', N, NP2, N, ONE, A, LDA, DWORK, N, ONE,
$ BK, LDBK )
C
C Compute L2 = -( A*Y2*C2' + B1*D21' )*inv( Ip2 + C2*Y2*C2' ) .
C
CALL DTRSM( 'R', 'U', 'N', 'N', N, NP2, -ONE, DWORK( IW2 ), NP2,
$ BK, LDBK )
CALL DTRSM( 'R', 'U', 'T', 'N', N, NP2, ONE, DWORK( IW2 ), NP2,
$ BK, LDBK )
C
C Compute F2*Y2*C2' + F0*D21' .
C
CALL DGEMM( 'N', 'N', M2, NP2, N, ONE, CK, LDCK, DWORK, N, ONE,
$ DK, LDDK )
C
C Compute DK = L0 = ( F2*Y2*C2' + F0*D21' )*inv( Ip2 + C2*Y2*C2' ) .
C
CALL DTRSM( 'R', 'U', 'N', 'N', M2, NP2, ONE, DWORK( IW2 ), NP2,
$ DK, LDDK )
CALL DTRSM( 'R', 'U', 'T', 'N', M2, NP2, ONE, DWORK( IW2 ), NP2,
$ DK, LDDK )
C
C Compute CK = F2 - L0*C2 .
C
CALL DGEMM( 'N', 'N', M2, N, NP2, -ONE, DK, LDDK, C( NP1+1, 1),
$ LDC, ONE, CK, LDCK )
C
C Find AK = A + B2*( F2 - L0*C2 ) + L2*C2 .
C
CALL DLACPY( 'Full', N, N, A, LDA, AK, LDAK )
CALL DGEMM( 'N', 'N', N, N, M2, ONE, B(1, M1+1 ), LDB, CK, LDCK,
$ ONE, AK, LDAK )
CALL DGEMM( 'N', 'N', N, N, NP2, ONE, BK, LDBK, C( NP1+1, 1),
$ LDC, ONE, AK, LDAK )
C
C Find BK = -L2 + B2*L0 .
C
CALL DGEMM( 'N', 'N', N, NP2, M2, ONE, B( 1, M1+1 ), LDB, DK,
$ LDDK, -ONE, BK, LDBK )
C
DWORK( 1 ) = DBLE( LWAMAX )
RETURN
C *** Last line of SB10SD ***
END