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SUBROUTINE SB03OD( DICO, FACT, TRANS, N, M, A, LDA, Q, LDQ, B,
$ LDB, SCALE, WR, WI, DWORK, LDWORK, 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 <http://www.gnu.org/licenses/>.
C
C PURPOSE
C
C To solve for X = op(U)'*op(U) either the stable non-negative
C definite continuous-time Lyapunov equation
C 2
C op(A)'*X + X*op(A) = -scale *op(B)'*op(B) (1)
C
C or the convergent non-negative definite discrete-time Lyapunov
C equation
C 2
C op(A)'*X*op(A) - X = -scale *op(B)'*op(B) (2)
C
C where op(K) = K or K' (i.e., the transpose of the matrix K), A is
C an N-by-N matrix, op(B) is an M-by-N matrix, U is an upper
C triangular matrix containing the Cholesky factor of the solution
C matrix X, X = op(U)'*op(U), and scale is an output scale factor,
C set less than or equal to 1 to avoid overflow in X. If matrix B
C has full rank then the solution matrix X will be positive-definite
C and hence the Cholesky factor U will be nonsingular, but if B is
C rank deficient then X may be only positive semi-definite and U
C will be singular.
C
C In the case of equation (1) the matrix A must be stable (that
C is, all the eigenvalues of A must have negative real parts),
C and for equation (2) the matrix A must be convergent (that is,
C all the eigenvalues of A must lie inside the unit circle).
C
C ARGUMENTS
C
C Mode Parameters
C
C DICO CHARACTER*1
C Specifies the type of Lyapunov equation to be solved as
C follows:
C = 'C': Equation (1), continuous-time case;
C = 'D': Equation (2), discrete-time case.
C
C FACT CHARACTER*1
C Specifies whether or not the real Schur factorization
C of the matrix A is supplied on entry, as follows:
C = 'F': On entry, A and Q contain the factors from the
C real Schur factorization of the matrix A;
C = 'N': The Schur factorization of A will be computed
C and the factors will be stored in A and Q.
C
C TRANS CHARACTER*1
C Specifies the form of op(K) to be used, as follows:
C = 'N': op(K) = K (No transpose);
C = 'T': op(K) = K**T (Transpose).
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the matrix A and the number of columns in
C matrix op(B). N >= 0.
C
C M (input) INTEGER
C The number of rows in matrix op(B). M >= 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 matrix A. If FACT = 'F', then A contains
C an upper quasi-triangular matrix S in Schur canonical
C form; the elements below the upper Hessenberg part of the
C array A are not referenced.
C On exit, the leading N-by-N upper Hessenberg part of this
C array contains the upper quasi-triangular matrix S in
C Schur canonical form from the Shur factorization of A.
C The contents of array A is not modified if FACT = 'F'.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,N).
C
C Q (input or output) DOUBLE PRECISION array, dimension
C (LDQ,N)
C On entry, if FACT = 'F', then the leading N-by-N part of
C this array must contain the orthogonal matrix Q of the
C Schur factorization of A.
C Otherwise, Q need not be set on entry.
C On exit, the leading N-by-N part of this array contains
C the orthogonal matrix Q of the Schur factorization of A.
C The contents of array Q is not modified if FACT = 'F'.
C
C LDQ INTEGER
C The leading dimension of array Q. LDQ >= MAX(1,N).
C
C B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
C if TRANS = 'N', and dimension (LDB,max(M,N)), if
C TRANS = 'T'.
C On entry, if TRANS = 'N', the leading M-by-N part of this
C array must contain the coefficient matrix B of the
C equation.
C On entry, if TRANS = 'T', the leading N-by-M part of this
C array must contain the coefficient matrix B of the
C equation.
C On exit, the leading N-by-N part of this array contains
C the upper triangular Cholesky factor U of the solution
C matrix X of the problem, X = op(U)'*op(U).
C If M = 0 and N > 0, then U is set to zero.
C
C LDB INTEGER
C The leading dimension of array B.
C LDB >= MAX(1,N,M), if TRANS = 'N';
C LDB >= MAX(1,N), if TRANS = 'T'.
C
C SCALE (output) DOUBLE PRECISION
C The scale factor, scale, set less than or equal to 1 to
C prevent the solution overflowing.
C
C WR (output) DOUBLE PRECISION array, dimension (N)
C WI (output) DOUBLE PRECISION array, dimension (N)
C If FACT = 'N', and INFO >= 0 and INFO <= 2, WR and WI
C contain the real and imaginary parts, respectively, of
C the eigenvalues of A.
C If FACT = 'F', WR and WI are not referenced.
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, or INFO = 1, DWORK(1) returns the
C optimal value of LDWORK.
C
C LDWORK INTEGER
C The length of the array DWORK.
C If M > 0, LDWORK >= MAX(1,4*N + MIN(M,N));
C If M = 0, LDWORK >= 1.
C For optimum performance LDWORK should sometimes be larger.
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 Lyapunov equation is (nearly) singular
C (warning indicator);
C if DICO = 'C' this means that while the matrix A
C (or the factor S) has computed eigenvalues with
C negative real parts, it is only just stable in the
C sense that small perturbations in A can make one or
C more of the eigenvalues have a non-negative real
C part;
C if DICO = 'D' this means that while the matrix A
C (or the factor S) has computed eigenvalues inside
C the unit circle, it is nevertheless only just
C convergent, in the sense that small perturbations
C in A can make one or more of the eigenvalues lie
C outside the unit circle;
C perturbed values were used to solve the equation;
C = 2: if FACT = 'N' and DICO = 'C', but the matrix A is
C not stable (that is, one or more of the eigenvalues
C of A has a non-negative real part), or DICO = 'D',
C but the matrix A is not convergent (that is, one or
C more of the eigenvalues of A lies outside the unit
C circle); however, A will still have been factored
C and the eigenvalues of A returned in WR and WI.
C = 3: if FACT = 'F' and DICO = 'C', but the Schur factor S
C supplied in the array A is not stable (that is, one
C or more of the eigenvalues of S has a non-negative
C real part), or DICO = 'D', but the Schur factor S
C supplied in the array A is not convergent (that is,
C one or more of the eigenvalues of S lies outside the
C unit circle);
C = 4: if FACT = 'F' and the Schur factor S supplied in
C the array A has two or more consecutive non-zero
C elements on the first sub-diagonal, so that there is
C a block larger than 2-by-2 on the diagonal;
C = 5: if FACT = 'F' and the Schur factor S supplied in
C the array A has a 2-by-2 diagonal block with real
C eigenvalues instead of a complex conjugate pair;
C = 6: if FACT = 'N' and the LAPACK Library routine DGEES
C has failed to converge. This failure is not likely
C to occur. The matrix B will be unaltered but A will
C be destroyed.
C
C METHOD
C
C The method used by the routine is based on the Bartels and Stewart
C method [1], except that it finds the upper triangular matrix U
C directly without first finding X and without the need to form the
C normal matrix op(B)'*op(B).
C
C The Schur factorization of a square matrix A is given by
C
C A = QSQ',
C
C where Q is orthogonal and S is an N-by-N block upper triangular
C matrix with 1-by-1 and 2-by-2 blocks on its diagonal (which
C correspond to the eigenvalues of A). If A has already been
C factored prior to calling the routine however, then the factors
C Q and S may be supplied and the initial factorization omitted.
C
C If TRANS = 'N', the matrix B is factored as (QR factorization)
C _ _ _ _ _
C B = P ( R ), M >= N, B = P ( R Z ), M < N,
C ( 0 )
C _ _
C where P is an M-by-M orthogonal matrix and R is a square upper
C _ _ _ _ _
C triangular matrix. Then, the matrix B = RQ, or B = ( R Z )Q (if
C M < N) is factored as
C _ _
C B = P ( R ), M >= N, B = P ( R Z ), M < N.
C
C If TRANS = 'T', the matrix B is factored as (RQ factorization)
C _
C _ _ ( Z ) _
C B = ( 0 R ) P, M >= N, B = ( _ ) P, M < N,
C ( R )
C _ _
C where P is an M-by-M orthogonal matrix and R is a square upper
C _ _ _ _ _
C triangular matrix. Then, the matrix B = Q'R, or B = Q'( Z' R' )'
C (if M < N) is factored as
C _ _
C B = ( R ) P, M >= N, B = ( Z ) P, M < N.
C ( R )
C
C These factorizations are utilised to either transform the
C continuous-time Lyapunov equation to the canonical form
C 2
C op(S)'*op(V)'*op(V) + op(V)'*op(V)*op(S) = -scale *op(F)'*op(F),
C
C or the discrete-time Lyapunov equation to the canonical form
C 2
C op(S)'*op(V)'*op(V)*op(S) - op(V)'*op(V) = -scale *op(F)'*op(F),
C
C where V and F are upper triangular, and
C
C F = R, M >= N, F = ( R Z ), M < N, if TRANS = 'N';
C ( 0 0 )
C
C F = R, M >= N, F = ( 0 Z ), M < N, if TRANS = 'T'.
C ( 0 R )
C
C The transformed equation is then solved for V, from which U is
C obtained via the QR factorization of V*Q', if TRANS = 'N', or
C via the RQ factorization of Q*V, if TRANS = 'T'.
C
C REFERENCES
C
C [1] Bartels, R.H. and Stewart, G.W.
C Solution of the matrix equation A'X + XB = C.
C Comm. A.C.M., 15, pp. 820-826, 1972.
C
C [2] Hammarling, S.J.
C Numerical solution of the stable, non-negative definite
C Lyapunov equation.
C IMA J. Num. Anal., 2, pp. 303-325, 1982.
C
C NUMERICAL ASPECTS
C 3
C The algorithm requires 0(N ) operations and is backward stable.
C
C FURTHER COMMENTS
C
C The Lyapunov equation may be very ill-conditioned. In particular,
C if A is only just stable (or convergent) then the Lyapunov
C equation will be ill-conditioned. A symptom of ill-conditioning
C is "large" elements in U relative to those of A and B, or a
C "small" value for scale. A condition estimate can be computed
C using SLICOT Library routine SB03MD.
C
C SB03OD routine can be also used for solving "unstable" Lyapunov
C equations, i.e., when matrix A has all eigenvalues with positive
C real parts, if DICO = 'C', or with moduli greater than one,
C if DICO = 'D'. Specifically, one may solve for X = op(U)'*op(U)
C either the continuous-time Lyapunov equation
C 2
C op(A)'*X + X*op(A) = scale *op(B)'*op(B), (3)
C
C or the discrete-time Lyapunov equation
C 2
C op(A)'*X*op(A) - X = scale *op(B)'*op(B), (4)
C
C provided, for equation (3), the given matrix A is replaced by -A,
C or, for equation (4), the given matrices A and B are replaced by
C inv(A) and B*inv(A), if TRANS = 'N' (or inv(A)*B, if TRANS = 'T'),
C respectively. Although the inversion generally can rise numerical
C problems, in case of equation (4) it is expected that the matrix A
C is enough well-conditioned, having only eigenvalues with moduli
C greater than 1. However, if A is ill-conditioned, it could be
C preferable to use the more general SLICOT Lyapunov solver SB03MD.
C
C CONTRIBUTOR
C
C Release 3.0: V. Sima, Katholieke Univ. Leuven, Belgium, Aug. 1997.
C Supersedes Release 2.0 routine SB03CD by Sven Hammarling,
C NAG Ltd, United Kingdom.
C
C REVISIONS
C
C Dec. 1997, April 1998, May 1998, May 1999, Oct. 2001 (V. Sima).
C March 2002 (A. Varga).
C
C KEYWORDS
C
C Lyapunov equation, orthogonal transformation, real Schur form,
C Sylvester equation.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
CHARACTER DICO, FACT, TRANS
INTEGER INFO, LDA, LDB, LDQ, LDWORK, M, N
DOUBLE PRECISION SCALE
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), Q(LDQ,*), WI(*),
$ WR(*)
C .. Local Scalars ..
LOGICAL CONT, LTRANS, NOFACT
INTEGER I, IFAIL, INFORM, ITAU, J, JWORK, K, L, MINMN,
$ NE, SDIM, WRKOPT
DOUBLE PRECISION EMAX, TEMP
C .. Local Arrays ..
LOGICAL BWORK(1)
C .. External Functions ..
LOGICAL LSAME, SELECT
DOUBLE PRECISION DLAPY2
EXTERNAL DLAPY2, LSAME, SELECT
C .. External Subroutines ..
EXTERNAL DCOPY, DGEES, DGEMM, DGEMV, DGEQRF, DGERQF,
$ DLACPY, DLASET, DTRMM, SB03OU, XERBLA
C .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
C .. Executable Statements ..
C
C Test the input scalar arguments.
C
CONT = LSAME( DICO, 'C' )
NOFACT = LSAME( FACT, 'N' )
LTRANS = LSAME( TRANS, 'T' )
MINMN = MIN( M, N )
C
INFO = 0
IF( .NOT.CONT .AND. .NOT.LSAME( DICO, 'D' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
INFO = -2
ELSE IF( .NOT.LTRANS .AND. .NOT.LSAME( TRANS, 'N' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( M.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( ( LDB.LT.MAX( 1, N ) ) .OR.
$ ( LDB.LT.MAX( 1, N, M ) .AND. .NOT.LTRANS ) ) THEN
INFO = -11
ELSE IF( LDWORK.LT.1 .OR. ( M.GT.0 .AND. LDWORK.LT.4*N + MINMN ) )
$ THEN
INFO = -16
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'SB03OD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( MINMN.EQ.0 ) THEN
IF( M.EQ.0 )
$ CALL DLASET( 'Full', N, N, ZERO, ZERO, B, LDB )
SCALE = ONE
DWORK(1) = ONE
RETURN
END IF
C
C Start the solution.
C
C (Note: Comments in the code beginning "Workspace:" describe the
C minimal amount of real workspace needed at that point in the
C code, as well as the preferred amount for good performance.
C NB refers to the optimal block size for the immediately
C following subroutine, as returned by ILAENV.)
C
IF ( NOFACT ) THEN
C
C Find the Schur factorization of A, A = Q*S*Q'.
C Workspace: need 3*N;
C prefer larger.
C
CALL DGEES( 'Vectors', 'Not ordered', SELECT, N, A, LDA, SDIM,
$ WR, WI, Q, LDQ, DWORK, LDWORK, BWORK, INFORM )
IF ( INFORM.NE.0 ) THEN
INFO = 6
RETURN
END IF
WRKOPT = DWORK(1)
C
C Check the eigenvalues for stability.
C
IF ( CONT ) THEN
EMAX = WR(1)
C
DO 20 J = 2, N
IF ( WR(J).GT.EMAX )
$ EMAX = WR(J)
20 CONTINUE
C
ELSE
EMAX = DLAPY2( WR(1), WI(1) )
C
DO 40 J = 2, N
TEMP = DLAPY2( WR(J), WI(J) )
IF ( TEMP.GT.EMAX )
$ EMAX = TEMP
40 CONTINUE
C
END IF
C
IF ( ( CONT ) .AND. ( EMAX.GE.ZERO ) .OR.
$ ( .NOT.CONT ) .AND. ( EMAX.GE.ONE ) ) THEN
INFO = 2
RETURN
END IF
ELSE
WRKOPT = 0
END IF
C
C Perform the QR or RQ factorization of B,
C _ _ _ _ _
C B = P ( R ), or B = P ( R Z ), if TRANS = 'N', or
C ( 0 )
C _
C _ _ ( Z ) _
C B = ( 0 R ) P, or B = ( _ ) P, if TRANS = 'T'.
C ( R )
C Workspace: need MIN(M,N) + N;
C prefer MIN(M,N) + N*NB.
C
ITAU = 1
JWORK = ITAU + MINMN
IF ( LTRANS ) THEN
CALL DGERQF( N, M, B, LDB, DWORK(ITAU), DWORK(JWORK),
$ LDWORK-JWORK+1, IFAIL )
WRKOPT = MAX( WRKOPT, INT( DWORK(JWORK) ) + JWORK - 1, MINMN*N)
JWORK = ITAU
C
C Form in B
C _ _ _ _ _ _
C B := Q'R, m >= n, B := Q'*( Z' R' )', m < n, with B an
C n-by-min(m,n) matrix.
C Use a BLAS 3 operation if enough workspace, and BLAS 2,
C _
C otherwise: B is formed column by column.
C
IF ( LDWORK.GE.JWORK+MINMN*N-1 ) THEN
K = JWORK
C
DO 60 I = 1, MINMN
CALL DCOPY( N, Q(N-MINMN+I,1), LDQ, DWORK(K), 1 )
K = K + N
60 CONTINUE
C
CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Non-unit',
$ N, MINMN, ONE, B(N-MINMN+1,M-MINMN+1), LDB,
$ DWORK(JWORK), N )
IF ( M.LT.N )
$ CALL DGEMM( 'Transpose', 'No transpose', N, M, N-M,
$ ONE, Q, LDQ, B, LDB, ONE, DWORK(JWORK), N )
CALL DLACPY( 'Full', N, MINMN, DWORK(JWORK), N, B, LDB )
ELSE
NE = N - MINMN
C
DO 80 J = 1, MINMN
NE = NE + 1
CALL DCOPY( NE, B(1,M-MINMN+J), 1, DWORK(JWORK), 1 )
CALL DGEMV( 'Transpose', NE, N, ONE, Q, LDQ,
$ DWORK(JWORK), 1, ZERO, B(1,J), 1 )
80 CONTINUE
C
END IF
ELSE
CALL DGEQRF( M, N, B, LDB, DWORK(ITAU), DWORK(JWORK),
$ LDWORK-JWORK+1, IFAIL )
WRKOPT = MAX( WRKOPT, INT( DWORK(JWORK) ) + JWORK - 1, MINMN*N)
JWORK = ITAU
C
C Form in B
C _ _ _ _ _ _
C B := RQ, m >= n, B := ( R Z )*Q, m < n, with B an
C min(m,n)-by-n matrix.
C Use a BLAS 3 operation if enough workspace, and BLAS 2,
C _
C otherwise: B is formed row by row.
C
IF ( LDWORK.GE.JWORK+MINMN*N-1 ) THEN
CALL DLACPY( 'Full', MINMN, N, Q, LDQ, DWORK(JWORK), MINMN )
CALL DTRMM( 'Left', 'Upper', 'No transpose', 'Non-unit',
$ MINMN, N, ONE, B, LDB, DWORK(JWORK), MINMN )
IF ( M.LT.N )
$ CALL DGEMM( 'No transpose', 'No transpose', M, N, N-M,
$ ONE, B(1,M+1), LDB, Q(M+1,1), LDQ, ONE,
$ DWORK(JWORK), MINMN )
CALL DLACPY( 'Full', MINMN, N, DWORK(JWORK), MINMN, B, LDB )
ELSE
NE = MINMN + MAX( 0, N-M )
C
DO 100 J = 1, MINMN
CALL DCOPY( NE, B(J,J), LDB, DWORK(JWORK), 1 )
CALL DGEMV( 'Transpose', NE, N, ONE, Q(J,1), LDQ,
$ DWORK(JWORK), 1, ZERO, B(J,1), LDB )
NE = NE - 1
100 CONTINUE
C
END IF
END IF
JWORK = ITAU + MINMN
C
C Solve for U the transformed Lyapunov equation
C 2 _ _
C op(S)'*op(U)'*op(U) + op(U)'*op(U)*op(S) = -scale *op(B)'*op(B),
C
C or
C 2 _ _
C op(S)'*op(U)'*op(U)*op(S) - op(U)'*op(U) = -scale *op(B)'*op(B)
C
C Workspace: need MIN(M,N) + 4*N;
C prefer larger.
C
CALL SB03OU( .NOT.CONT, LTRANS, N, MINMN, A, LDA, B, LDB,
$ DWORK(ITAU), B, LDB, SCALE, DWORK(JWORK),
$ LDWORK-JWORK+1, INFO )
IF ( INFO.GT.1 ) THEN
INFO = INFO + 1
RETURN
END IF
WRKOPT = MAX( WRKOPT, INT( DWORK(JWORK) ) + JWORK - 1 )
JWORK = ITAU
C
C Form U := U*Q' or U := Q*U in the array B.
C Use a BLAS 3 operation if enough workspace, and BLAS 2, otherwise.
C Workspace: need N;
C prefer N*N;
C
IF ( LDWORK.GE.JWORK+N*N-1 ) THEN
IF ( LTRANS ) THEN
CALL DLACPY( 'Full', N, N, Q, LDQ, DWORK(JWORK), N )
CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Non-unit', N,
$ N, ONE, B, LDB, DWORK(JWORK), N )
ELSE
K = JWORK
C
DO 120 I = 1, N
CALL DCOPY( N, Q(1,I), 1, DWORK(K), N )
K = K + 1
120 CONTINUE
C
CALL DTRMM( 'Left', 'Upper', 'No transpose', 'Non-unit', N,
$ N, ONE, B, LDB, DWORK(JWORK), N )
END IF
CALL DLACPY( 'Full', N, N, DWORK(JWORK), N, B, LDB )
WRKOPT = MAX( WRKOPT, JWORK + N*N - 1 )
ELSE
IF ( LTRANS ) THEN
C
C U is formed column by column ( U := Q*U ).
C
DO 140 I = 1, N
CALL DCOPY( I, B(1,I), 1, DWORK(JWORK), 1 )
CALL DGEMV( 'No transpose', N, I, ONE, Q, LDQ,
$ DWORK(JWORK), 1, ZERO, B(1,I), 1 )
140 CONTINUE
ELSE
C
C U is formed row by row ( U' := Q*U' ).
C
DO 160 I = 1, N
CALL DCOPY( N-I+1, B(I,I), LDB, DWORK(JWORK), 1 )
CALL DGEMV( 'No transpose', N, N-I+1, ONE, Q(1,I), LDQ,
$ DWORK(JWORK), 1, ZERO, B(I,1), LDB )
160 CONTINUE
END IF
END IF
C
C Lastly find the QR or RQ factorization of U, overwriting on B,
C to give the required Cholesky factor.
C Workspace: need 2*N;
C prefer N + N*NB;
C
JWORK = ITAU + N
IF ( LTRANS ) THEN
CALL DGERQF( N, N, B, LDB, DWORK(ITAU), DWORK(JWORK),
$ LDWORK-JWORK+1, IFAIL )
ELSE
CALL DGEQRF( N, N, B, LDB, DWORK(ITAU), DWORK(JWORK),
$ LDWORK-JWORK+1, IFAIL )
END IF
WRKOPT = MAX( WRKOPT, INT( DWORK(JWORK) ) + JWORK - 1 )
C
C Make the diagonal elements of U non-negative.
C
IF ( LTRANS ) THEN
C
DO 200 J = 1, N
IF ( B(J,J).LT.ZERO ) THEN
C
DO 180 I = 1, J
B(I,J) = -B(I,J)
180 CONTINUE
C
END IF
200 CONTINUE
C
ELSE
K = JWORK
C
DO 240 J = 1, N
DWORK(K) = B(J,J)
L = JWORK
C
DO 220 I = 1, J
IF ( DWORK(L).LT.ZERO ) B(I,J) = -B(I,J)
L = L + 1
220 CONTINUE
C
K = K + 1
240 CONTINUE
END IF
C
IF( N.GT.1 )
$ CALL DLASET( 'Lower', N-1, N-1, ZERO, ZERO, B(2,1), LDB )
C
C Set the optimal workspace.
C
DWORK(1) = WRKOPT
C
RETURN
C *** Last line of SB03OD ***
END
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