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SUBROUTINE SG02AD( DICO, JOBB, FACT, UPLO, JOBL, SCAL, SORT, ACC,
$ N, M, P, A, LDA, E, LDE, B, LDB, Q, LDQ, R,
$ LDR, L, LDL, RCONDU, X, LDX, ALFAR, ALFAI,
$ BETA, S, LDS, T, LDT, U, LDU, TOL, 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 <http://www.gnu.org/licenses/>.
C
C PURPOSE
C
C To solve for X either the continuous-time algebraic Riccati
C equation
C -1
C Q + A'XE + E'XA - (L+E'XB)R (L+E'XB)' = 0 , (1)
C
C or the discrete-time algebraic Riccati equation
C -1
C E'XE = A'XA - (L+A'XB)(R + B'XB) (L+A'XB)' + Q , (2)
C
C where A, E, B, Q, R, and L are N-by-N, N-by-N, N-by-M, N-by-N,
C M-by-M and N-by-M matrices, respectively, such that Q = C'C,
C R = D'D and L = C'D; X is an N-by-N symmetric matrix.
C The routine also returns the computed values of the closed-loop
C spectrum of the system, i.e., the stable eigenvalues
C lambda(1),...,lambda(N) of the pencil (A - BF,E), where F is
C the optimal gain matrix,
C -1
C F = R (L+E'XB)' , for (1),
C
C and
C -1
C F = (R+B'XB) (L+A'XB)' , for (2).
C -1
C Optionally, matrix G = BR B' may be given instead of B and R.
C Other options include the case with Q and/or R given in a
C factored form, Q = C'C, R = D'D, and with L a zero matrix.
C
C The routine uses the method of deflating subspaces, based on
C reordering the eigenvalues in a generalized Schur matrix pair.
C
C It is assumed that E is nonsingular, but this condition is not
C checked. Note that the definition (1) of the continuous-time
C algebraic Riccati equation, and the formula for the corresponding
C optimal gain matrix, require R to be nonsingular, but the
C associated linear quadratic optimal problem could have a unique
C solution even when matrix R is singular, under mild assumptions
C (see METHOD). The routine SG02AD works accordingly in this case.
C
C ARGUMENTS
C
C Mode Parameters
C
C DICO CHARACTER*1
C Specifies the type of Riccati equation to be solved as
C follows:
C = 'C': Equation (1), continuous-time case;
C = 'D': Equation (2), discrete-time case.
C
C JOBB CHARACTER*1
C Specifies whether or not the matrix G is given, instead
C of the matrices B and R, as follows:
C = 'B': B and R are given;
C = 'G': G is given.
C
C FACT CHARACTER*1
C Specifies whether or not the matrices Q and/or R (if
C JOBB = 'B') are factored, as follows:
C = 'N': Not factored, Q and R are given;
C = 'C': C is given, and Q = C'C;
C = 'D': D is given, and R = D'D;
C = 'B': Both factors C and D are given, Q = C'C, R = D'D.
C
C UPLO CHARACTER*1
C If JOBB = 'G', or FACT = 'N', specifies which triangle of
C the matrices G, or Q and R, is stored, as follows:
C = 'U': Upper triangle is stored;
C = 'L': Lower triangle is stored.
C
C JOBL CHARACTER*1
C Specifies whether or not the matrix L is zero, as follows:
C = 'Z': L is zero;
C = 'N': L is nonzero.
C JOBL is not used if JOBB = 'G' and JOBL = 'Z' is assumed.
C SLICOT Library routine SB02MT should be called just before
C SG02AD, for obtaining the results when JOBB = 'G' and
C JOBL = 'N'.
C
C SCAL CHARACTER*1
C If JOBB = 'B', specifies whether or not a scaling strategy
C should be used to scale Q, R, and L, as follows:
C = 'G': General scaling should be used;
C = 'N': No scaling should be used.
C SCAL is not used if JOBB = 'G'.
C
C SORT CHARACTER*1
C Specifies which eigenvalues should be obtained in the top
C of the generalized Schur form, as follows:
C = 'S': Stable eigenvalues come first;
C = 'U': Unstable eigenvalues come first.
C
C ACC CHARACTER*1
C Specifies whether or not iterative refinement should be
C used to solve the system of algebraic equations giving
C the solution matrix X, as follows:
C = 'R': Use iterative refinement;
C = 'N': Do not use iterative refinement.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The actual state dimension, i.e., the order of the
C matrices A, E, Q, and X, and the number of rows of the
C matrices B and L. N >= 0.
C
C M (input) INTEGER
C The number of system inputs. If JOBB = 'B', M is the
C order of the matrix R, and the number of columns of the
C matrix B. M >= 0.
C M is not used if JOBB = 'G'.
C
C P (input) INTEGER
C The number of system outputs. If FACT = 'C' or 'D' or 'B',
C P is the number of rows of the matrices C and/or D.
C P >= 0.
C Otherwise, P is not used.
C
C A (input) DOUBLE PRECISION array, dimension (LDA,N)
C The leading N-by-N part of this array must contain the
C state matrix A of the descriptor system.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,N).
C
C E (input) DOUBLE PRECISION array, dimension (LDE,N)
C The leading N-by-N part of this array must contain the
C matrix E of the descriptor system.
C
C LDE INTEGER
C The leading dimension of array E. LDE >= MAX(1,N).
C
C B (input) DOUBLE PRECISION array, dimension (LDB,*)
C If JOBB = 'B', the leading N-by-M part of this array must
C contain the input matrix B of the system.
C If JOBB = 'G', the leading N-by-N upper triangular part
C (if UPLO = 'U') or lower triangular part (if UPLO = 'L')
C of this array must contain the upper triangular part or
C lower triangular part, respectively, of the matrix
C -1
C G = BR B'. The stricly lower triangular part (if
C UPLO = 'U') or stricly upper triangular part (if
C UPLO = 'L') is not referenced.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= MAX(1,N).
C
C Q (input) DOUBLE PRECISION array, dimension (LDQ,N)
C If FACT = 'N' or 'D', the leading N-by-N upper triangular
C part (if UPLO = 'U') or lower triangular part (if UPLO =
C 'L') of this array must contain the upper triangular part
C or lower triangular part, respectively, of the symmetric
C state weighting matrix Q. The stricly lower triangular
C part (if UPLO = 'U') or stricly upper triangular part (if
C UPLO = 'L') is not referenced.
C If FACT = 'C' or 'B', the leading P-by-N part of this
C array must contain the output matrix C of the system.
C If JOBB = 'B' and SCAL = 'G', then Q is modified
C internally, but is restored on exit.
C
C LDQ INTEGER
C The leading dimension of array Q.
C LDQ >= MAX(1,N) if FACT = 'N' or 'D';
C LDQ >= MAX(1,P) if FACT = 'C' or 'B'.
C
C R (input) DOUBLE PRECISION array, dimension (LDR,*)
C If FACT = 'N' or 'C', the leading M-by-M upper triangular
C part (if UPLO = 'U') or lower triangular part (if UPLO =
C 'L') of this array must contain the upper triangular part
C or lower triangular part, respectively, of the symmetric
C input weighting matrix R. The stricly lower triangular
C part (if UPLO = 'U') or stricly upper triangular part (if
C UPLO = 'L') is not referenced.
C If FACT = 'D' or 'B', the leading P-by-M part of this
C array must contain the direct transmission matrix D of the
C system.
C If JOBB = 'B' and SCAL = 'G', then R is modified
C internally, but is restored on exit.
C If JOBB = 'G', this array is not referenced.
C
C LDR INTEGER
C The leading dimension of array R.
C LDR >= MAX(1,M) if JOBB = 'B' and FACT = 'N' or 'C';
C LDR >= MAX(1,P) if JOBB = 'B' and FACT = 'D' or 'B';
C LDR >= 1 if JOBB = 'G'.
C
C L (input) DOUBLE PRECISION array, dimension (LDL,*)
C If JOBL = 'N' and JOBB = 'B', the leading N-by-M part of
C this array must contain the cross weighting matrix L.
C If JOBB = 'B' and SCAL = 'G', then L is modified
C internally, but is restored on exit.
C If JOBL = 'Z' or JOBB = 'G', this array is not referenced.
C
C LDL INTEGER
C The leading dimension of array L.
C LDL >= MAX(1,N) if JOBL = 'N' and JOBB = 'B';
C LDL >= 1 if JOBL = 'Z' or JOBB = 'G'.
C
C RCONDU (output) DOUBLE PRECISION
C If N > 0 and INFO = 0 or INFO = 7, an estimate of the
C reciprocal of the condition number (in the 1-norm) of
C the N-th order system of algebraic equations from which
C the solution matrix X is obtained.
C
C X (output) DOUBLE PRECISION array, dimension (LDX,N)
C If INFO = 0, the leading N-by-N part of this array
C contains the solution matrix X of the problem.
C
C LDX INTEGER
C The leading dimension of array X. LDX >= MAX(1,N).
C
C ALFAR (output) DOUBLE PRECISION array, dimension (2*N)
C ALFAI (output) DOUBLE PRECISION array, dimension (2*N)
C BETA (output) DOUBLE PRECISION array, dimension (2*N)
C The generalized eigenvalues of the 2N-by-2N matrix pair,
C ordered as specified by SORT (if INFO = 0, or INFO >= 5).
C For instance, if SORT = 'S', the leading N elements of
C these arrays contain the closed-loop spectrum of the
C system. Specifically,
C lambda(k) = [ALFAR(k)+j*ALFAI(k)]/BETA(k) for
C k = 1,2,...,N.
C
C S (output) DOUBLE PRECISION array, dimension (LDS,*)
C The leading 2N-by-2N part of this array contains the
C ordered real Schur form S of the first matrix in the
C reduced matrix pencil associated to the optimal problem,
C corresponding to the scaled Q, R, and L, if JOBB = 'B'
C and SCAL = 'G'. That is,
C
C (S S )
C ( 11 12)
C S = ( ),
C (0 S )
C ( 22)
C
C where S , S and S are N-by-N matrices.
C 11 12 22
C Array S must have 2*N+M columns if JOBB = 'B', and 2*N
C columns, otherwise.
C
C LDS INTEGER
C The leading dimension of array S.
C LDS >= MAX(1,2*N+M) if JOBB = 'B';
C LDS >= MAX(1,2*N) if JOBB = 'G'.
C
C T (output) DOUBLE PRECISION array, dimension (LDT,2*N)
C The leading 2N-by-2N part of this array contains the
C ordered upper triangular form T of the second matrix in
C the reduced matrix pencil associated to the optimal
C problem, corresponding to the scaled Q, R, and L, if
C JOBB = 'B' and SCAL = 'G'. That is,
C
C (T T )
C ( 11 12)
C T = ( ),
C (0 T )
C ( 22)
C
C where T , T and T are N-by-N matrices.
C 11 12 22
C
C LDT INTEGER
C The leading dimension of array T.
C LDT >= MAX(1,2*N+M) if JOBB = 'B';
C LDT >= MAX(1,2*N) if JOBB = 'G'.
C
C U (output) DOUBLE PRECISION array, dimension (LDU,2*N)
C The leading 2N-by-2N part of this array contains the right
C transformation matrix U which reduces the 2N-by-2N matrix
C pencil to the ordered generalized real Schur form (S,T).
C That is,
C
C (U U )
C ( 11 12)
C U = ( ),
C (U U )
C ( 21 22)
C
C where U , U , U and U are N-by-N matrices.
C 11 12 21 22
C If JOBB = 'B' and SCAL = 'G', then U corresponds to the
C scaled pencil. If a basis for the stable deflating
C subspace of the original problem is needed, then the
C submatrix U must be multiplied by the scaling factor
C 21
C contained in DWORK(4).
C
C LDU INTEGER
C The leading dimension of array U. LDU >= MAX(1,2*N).
C
C Tolerances
C
C TOL DOUBLE PRECISION
C The tolerance to be used to test for near singularity of
C the original matrix pencil, specifically of the triangular
C M-by-M factor obtained during the reduction process. If
C the user sets TOL > 0, then the given value of TOL is used
C as a lower bound for the reciprocal condition number of
C that matrix; a matrix whose estimated condition number is
C less than 1/TOL is considered to be nonsingular. If the
C user sets TOL <= 0, then a default tolerance, defined by
C TOLDEF = EPS, is used instead, where EPS is the machine
C precision (see LAPACK Library routine DLAMCH).
C This parameter is not referenced if JOBB = 'G'.
C
C Workspace
C
C IWORK INTEGER array, dimension (LIWORK)
C LIWORK >= MAX(1,M,2*N) if JOBB = 'B';
C LIWORK >= MAX(1,2*N) if JOBB = 'G'.
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) returns the optimal value
C of LDWORK. If JOBB = 'B' and N > 0, DWORK(2) returns the
C reciprocal of the condition number of the M-by-M bottom
C right lower triangular matrix obtained while compressing
C the matrix pencil of order 2N+M to obtain a pencil of
C order 2N. If ACC = 'R', and INFO = 0 or INFO = 7, DWORK(3)
C returns the reciprocal pivot growth factor (see SLICOT
C Library routine MB02PD) for the LU factorization of the
C coefficient matrix of the system of algebraic equations
C giving the solution matrix X; if DWORK(3) is much
C less than 1, then the computed X and RCONDU could be
C unreliable. If INFO = 0 or INFO = 7, DWORK(4) returns the
C scaling factor used to scale Q, R, and L. DWORK(4) is set
C to 1 if JOBB = 'G' or SCAL = 'N'.
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= MAX(7*(2*N+1)+16,16*N), if JOBB = 'G';
C LDWORK >= MAX(7*(2*N+1)+16,16*N,2*N+M,3*M), if JOBB = 'B'.
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: the computed solution may be inaccurate due to poor
C scaling or eigenvalues too close to the boundary of
C the stability domain (the imaginary axis, if
C DICO = 'C', or the unit circle, if DICO = 'D').
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 computed extended matrix pencil is singular,
C possibly due to rounding errors;
C = 2: if the QZ algorithm failed;
C = 3: if reordering of the generalized eigenvalues failed;
C = 4: if after reordering, roundoff changed values of
C some complex eigenvalues so that leading eigenvalues
C in the generalized Schur form no longer satisfy the
C stability condition; this could also be caused due
C to scaling;
C = 5: if the computed dimension of the solution does not
C equal N;
C = 6: if the spectrum is too close to the boundary of
C the stability domain;
C = 7: if a singular matrix was encountered during the
C computation of the solution matrix X.
C
C METHOD
C
C The routine uses a variant of the method of deflating subspaces
C proposed by van Dooren [1]. See also [2], [3], [4].
C It is assumed that E is nonsingular, the triple (E,A,B) is
C strongly stabilizable and detectable (see [3]); if, in addition,
C
C - [ Q L ]
C R := [ ] >= 0 ,
C [ L' R ]
C
C then the pencils
C
C discrete-time continuous-time
C
C |A 0 B| |E 0 0| |A 0 B| |E 0 0|
C |Q -E' L| - z |0 -A' 0| , |Q A' L| - s |0 -E' 0| , (3)
C |L' 0 R| |0 -B' 0| |L' B' R| |0 0 0|
C
C are dichotomic, i.e., they have no eigenvalues on the boundary of
C the stability domain. The above conditions are sufficient for
C regularity of these pencils. A necessary condition is that
C rank([ B' L' R']') = m.
C
C Under these assumptions the algebraic Riccati equation is known to
C have a unique non-negative definite solution.
C The first step in the method of deflating subspaces is to form the
C extended matrices in (3), of order 2N + M. Next, these pencils are
C compressed to a form of order 2N (see [1])
C
C lambda x A - B .
C f f
C
C This generalized eigenvalue problem is then solved using the QZ
C algorithm and the stable deflating subspace Ys is determined.
C If [Y1'|Y2']' is a basis for Ys, then the required solution is
C -1
C X = Y2 x Y1 .
C
C REFERENCES
C
C [1] Van Dooren, P.
C A Generalized Eigenvalue Approach for Solving Riccati
C Equations.
C SIAM J. Sci. Stat. Comp., 2, pp. 121-135, 1981.
C
C [2] Arnold, III, W.F. and Laub, A.J.
C Generalized Eigenproblem Algorithms and Software for
C Algebraic Riccati Equations.
C Proc. IEEE, 72, 1746-1754, 1984.
C
C [3] Mehrmann, V.
C The Autonomous Linear Quadratic Control Problem. Theory and
C Numerical Solution.
C Lect. Notes in Control and Information Sciences, vol. 163,
C Springer-Verlag, Berlin, 1991.
C
C [4] Sima, V.
C Algorithms for Linear-Quadratic Optimization.
C Pure and Applied Mathematics: A Series of Monographs and
C Textbooks, vol. 200, Marcel Dekker, Inc., New York, 1996.
C
C NUMERICAL ASPECTS
C
C This routine is particularly suited for systems where the matrix R
C is ill-conditioned, or even singular.
C
C FURTHER COMMENTS
C
C To obtain a stabilizing solution of the algebraic Riccati
C equations set SORT = 'S'.
C
C The routine can also compute the anti-stabilizing solutions of
C the algebraic Riccati equations, by specifying SORT = 'U'.
C
C CONTRIBUTOR
C
C V. Sima, Katholieke Univ. Leuven, Belgium, June 2002.
C
C REVISIONS
C
C V. Sima, Katholieke Univ. Leuven, Belgium, September 2002,
C December 2002.
C
C KEYWORDS
C
C Algebraic Riccati equation, closed loop system, continuous-time
C system, discrete-time system, optimal regulator, Schur form.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, HALF, ONE, P1, FOUR
PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
$ P1 = 0.1D0, FOUR = 4.0D0 )
C .. Scalar Arguments ..
CHARACTER ACC, DICO, FACT, JOBB, JOBL, SCAL, SORT, UPLO
INTEGER INFO, IWARN, LDA, LDB, LDE, LDL, LDQ, LDR, LDS,
$ LDT, LDU, LDWORK, LDX, M, N, P
DOUBLE PRECISION RCONDU, TOL
C .. Array Arguments ..
LOGICAL BWORK(*)
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), ALFAI(*), ALFAR(*), B(LDB,*), BETA(*),
$ DWORK(*), E(LDE,*), L(LDL,*), Q(LDQ,*),
$ R(LDR,*), S(LDS,*), T(LDT,*), U(LDU,*), X(LDX,*)
C .. Local Scalars ..
CHARACTER EQUED, QTYPE, RTYPE
LOGICAL COLEQU, DISCR, LFACB, LFACN, LFACQ, LFACR,
$ LJOBB, LJOBL, LJOBLN, LSCAL, LSORT, LUPLO,
$ REFINE, ROWEQU
INTEGER I, INFO1, IW, IWB, IWC, IWF, IWR, J, LDW, MP,
$ NDIM, NN, NNM, NP, NP1, WRKOPT
DOUBLE PRECISION ASYM, EPS, PIVOTU, RCONDL, RNORM, SCALE, SEPS,
$ U12M, UNORM
C .. External Functions ..
LOGICAL LSAME, SB02OU, SB02OV, SB02OW
DOUBLE PRECISION DLAMCH, DLANGE, DLANSY
EXTERNAL DLAMCH, DLANGE, DLANSY, LSAME, SB02OU, SB02OV,
$ SB02OW
C .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGECON, DGEMM, DGEQRF, DGGES,
$ DLACPY, DLASCL, DLASET, DORGQR, DSCAL, DSWAP,
$ MB01SD, MB02PD, MB02VD, SB02OY, XERBLA
C .. Intrinsic Functions ..
INTRINSIC ABS, INT, MAX, SQRT
C .. Executable Statements ..
C
IWARN = 0
INFO = 0
DISCR = LSAME( DICO, 'D' )
LJOBB = LSAME( JOBB, 'B' )
LFACN = LSAME( FACT, 'N' )
LFACQ = LSAME( FACT, 'C' )
LFACR = LSAME( FACT, 'D' )
LFACB = LSAME( FACT, 'B' )
LUPLO = LSAME( UPLO, 'U' )
LSORT = LSAME( SORT, 'S' )
REFINE = LSAME( ACC, 'R' )
NN = 2*N
IF ( LJOBB ) THEN
LJOBL = LSAME( JOBL, 'Z' )
LJOBLN = LSAME( JOBL, 'N' )
LSCAL = LSAME( SCAL, 'G' )
NNM = NN + M
LDW = MAX( NNM, 3*M )
ELSE
LSCAL = .FALSE.
NNM = NN
LDW = 1
END IF
NP1 = N + 1
C
C Test the input scalar arguments.
C
IF( .NOT.DISCR .AND. .NOT.LSAME( DICO, 'C' ) ) THEN
INFO = -1
ELSE IF( .NOT.LJOBB .AND. .NOT.LSAME( JOBB, 'G' ) ) THEN
INFO = -2
ELSE IF( .NOT.LFACQ .AND. .NOT.LFACR .AND. .NOT.LFACB
$ .AND. .NOT.LFACN ) THEN
INFO = -3
ELSE IF( .NOT.LJOBB .OR. LFACN ) THEN
IF( .NOT.LUPLO .AND. .NOT.LSAME( UPLO, 'L' ) )
$ INFO = -4
END IF
IF( INFO.EQ.0 .AND. LJOBB ) THEN
IF( .NOT.LJOBL .AND. .NOT.LJOBLN ) THEN
INFO = -5
ELSE IF( .NOT.LSCAL .AND. .NOT. LSAME( SCAL, 'N' ) ) THEN
INFO = -6
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( .NOT.LSORT .AND. .NOT.LSAME( SORT, 'U' ) ) THEN
INFO = -7
ELSE IF( .NOT.REFINE .AND. .NOT.LSAME( ACC, 'N' ) ) THEN
INFO = -8
ELSE IF( N.LT.0 ) THEN
INFO = -9
ELSE IF( LJOBB ) THEN
IF( M.LT.0 )
$ INFO = -10
END IF
END IF
IF( INFO.EQ.0 .AND. .NOT.LFACN ) THEN
IF( P.LT.0 )
$ INFO = -11
END IF
IF( INFO.EQ.0 ) THEN
IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -13
ELSE IF( LDE.LT.MAX( 1, N ) ) THEN
INFO = -15
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -17
ELSE IF( ( ( LFACN.OR.LFACR ) .AND. LDQ.LT.MAX( 1, N ) ) .OR.
$ ( ( LFACQ.OR.LFACB ) .AND. LDQ.LT.MAX( 1, P ) ) ) THEN
INFO = -19
ELSE IF( LJOBB ) THEN
IF ( ( LFACN.OR.LFACQ ) .AND. LDR.LT.MAX( 1, M ) .OR.
$ ( LFACR.OR.LFACB ) .AND. LDR.LT.MAX( 1, P ) ) THEN
INFO = -21
ELSE IF( ( LJOBLN .AND. LDL.LT.MAX( 1, N ) ) .OR.
$ ( LJOBL .AND. LDL.LT.1 ) ) THEN
INFO = -23
END IF
ELSE
IF( LDR.LT.1 ) THEN
INFO = -21
ELSE IF( LDL.LT.1 ) THEN
INFO = -23
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -26
ELSE IF( LDS.LT.MAX( 1, NNM ) ) THEN
INFO = -31
ELSE IF( LDT.LT.MAX( 1, NNM ) ) THEN
INFO = -33
ELSE IF( LDU.LT.MAX( 1, NN ) ) THEN
INFO = -35
ELSE IF( LDWORK.LT.MAX( 14*N + 23, 16*N, LDW ) ) THEN
INFO = -39
END IF
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'SG02AD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF ( N.EQ.0 ) THEN
DWORK(1) = FOUR
DWORK(4) = ONE
RETURN
END IF
C
C Start computations.
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
LSCAL = LSCAL .AND. LJOBB
IF ( LSCAL ) THEN
C
C Scale the matrices Q, R (or G), and L so that
C norm(Q) + norm(R) + norm(L) = 1,
C using the 1-norm. If Q and/or R are factored, the norms of
C the factors are used.
C Workspace: need max(N,M), if FACT = 'N';
C N, if FACT = 'D';
C M, if FACT = 'C'.
C
IF ( LFACN .OR. LFACR ) THEN
SCALE = DLANSY( '1-norm', UPLO, N, Q, LDQ, DWORK )
QTYPE = UPLO
NP = N
ELSE
SCALE = DLANGE( '1-norm', P, N, Q, LDQ, DWORK )
QTYPE = 'G'
NP = P
END IF
C
IF ( LFACN .OR. LFACQ ) THEN
RNORM = DLANSY( '1-norm', UPLO, M, R, LDR, DWORK )
RTYPE = UPLO
MP = M
ELSE
RNORM = DLANGE( '1-norm', P, M, R, LDR, DWORK )
RTYPE = 'G'
MP = P
END IF
SCALE = SCALE + RNORM
C
IF ( LJOBLN )
$ SCALE = SCALE + DLANGE( '1-norm', N, M, L, LDL, DWORK )
IF ( SCALE.EQ.ZERO )
$ SCALE = ONE
C
CALL DLASCL( QTYPE, 0, 0, SCALE, ONE, NP, N, Q, LDQ, INFO1 )
CALL DLASCL( RTYPE, 0, 0, SCALE, ONE, MP, M, R, LDR, INFO1 )
IF ( LJOBLN )
$ CALL DLASCL( 'G', 0, 0, SCALE, ONE, N, M, L, LDL, INFO1 )
ELSE
SCALE = ONE
END IF
C
C Construct the extended matrix pair.
C Workspace: need 1, if JOBB = 'G',
C max(1,2*N+M,3*M), if JOBB = 'B';
C prefer larger.
C
CALL SB02OY( 'Optimal control', DICO, JOBB, FACT, UPLO, JOBL,
$ 'Not identity E', N, M, P, A, LDA, B, LDB, Q, LDQ, R,
$ LDR, L, LDL, E, LDE, S, LDS, T, LDT, TOL, IWORK,
$ DWORK, LDWORK, INFO )
C
IF ( LSCAL ) THEN
C
C Undo scaling of the data arrays.
C
CALL DLASCL( QTYPE, 0, 0, ONE, SCALE, NP, N, Q, LDQ, INFO1 )
CALL DLASCL( RTYPE, 0, 0, ONE, SCALE, MP, M, R, LDR, INFO1 )
IF ( LJOBLN )
$ CALL DLASCL( 'G', 0, 0, ONE, SCALE, N, M, L, LDL, INFO1 )
END IF
C
IF ( INFO.NE.0 )
$ RETURN
WRKOPT = DWORK(1)
IF ( LJOBB )
$ RCONDL = DWORK(2)
C
C Workspace: need max(7*(2*N+1)+16,16*N);
C prefer larger.
C
IF ( DISCR ) THEN
IF ( LSORT ) THEN
C
C The natural tendency of the QZ algorithm to get the largest
C eigenvalues in the leading part of the matrix pair is
C exploited, by computing the unstable eigenvalues of the
C permuted matrix pair.
C
CALL DGGES( 'No vectors', 'Vectors', 'Sort', SB02OV, NN, T,
$ LDT, S, LDS, NDIM, ALFAR, ALFAI, BETA, U, LDU,
$ U, LDU, DWORK, LDWORK, BWORK, INFO1 )
CALL DSWAP( N, ALFAR(NP1), 1, ALFAR, 1 )
CALL DSWAP( N, ALFAI(NP1), 1, ALFAI, 1 )
CALL DSWAP( N, BETA (NP1), 1, BETA , 1 )
ELSE
CALL DGGES( 'No vectors', 'Vectors', 'Sort', SB02OV, NN, S,
$ LDS, T, LDT, NDIM, ALFAR, ALFAI, BETA, U, LDU,
$ U, LDU, DWORK, LDWORK, BWORK, INFO1 )
END IF
ELSE
IF ( LSORT ) THEN
CALL DGGES( 'No vectors', 'Vectors', 'Sort', SB02OW, NN, S,
$ LDS, T, LDT, NDIM, ALFAR, ALFAI, BETA, U, LDU,
$ U, LDU, DWORK, LDWORK, BWORK, INFO1 )
ELSE
CALL DGGES( 'No vectors', 'Vectors', 'Sort', SB02OU, NN, S,
$ LDS, T, LDT, NDIM, ALFAR, ALFAI, BETA, U, LDU,
$ U, LDU, DWORK, LDWORK, BWORK, INFO1 )
END IF
END IF
IF ( INFO1.GT.0 .AND. INFO1.LE.NN+1 ) THEN
INFO = 2
ELSE IF ( INFO1.EQ.NN+2 ) THEN
INFO = 4
ELSE IF ( INFO1.EQ.NN+3 ) THEN
INFO = 3
ELSE IF ( NDIM.NE.N ) THEN
INFO = 5
END IF
IF ( INFO.NE.0 )
$ RETURN
WRKOPT = MAX( WRKOPT, INT( DWORK(1) ) )
C
C Take the non-identity matrix E into account and orthogonalize the
C basis. Use the array X as workspace.
C Workspace: need N;
C prefer N*NB.
C
CALL DGEMM( 'No transpose', 'No transpose', N, N, N, ONE, E, LDE,
$ U, LDU, ZERO, X, LDX )
CALL DLACPY( 'Full', N, N, X, LDX, U, LDU )
CALL DGEQRF( NN, N, U, LDU, X, DWORK, LDWORK, INFO1 )
WRKOPT = MAX( WRKOPT, INT( DWORK(1) ) )
CALL DORGQR( NN, N, N, U, LDU, X, DWORK, LDWORK, INFO1 )
WRKOPT = MAX( WRKOPT, INT( DWORK(1) ) )
C
C Check for the symmetry of the solution. The array X is again used
C as workspace.
C
CALL DGEMM( 'Transpose', 'No transpose', N, N, N, ONE, U, LDU,
$ U(NP1,1), LDU, ZERO, X, LDX )
U12M = ZERO
ASYM = ZERO
C
DO 20 J = 1, N
C
DO 10 I = 1, N
U12M = MAX( U12M, ABS( X(I,J) ) )
ASYM = MAX( ASYM, ABS( X(I,J) - X(J,I) ) )
10 CONTINUE
C
20 CONTINUE
C
EPS = DLAMCH( 'Epsilon' )
SEPS = SQRT( EPS )
ASYM = ASYM - SEPS
IF ( ASYM.GT.P1*U12M ) THEN
INFO = 6
RETURN
ELSE IF ( ASYM.GT.SEPS ) THEN
IWARN = 1
END IF
C
C Compute the solution of X*U(1,1) = U(2,1). Use the (2,1) block
C of S as a workspace for factoring U(1,1).
C
IF ( REFINE ) THEN
C
C Use LU factorization and iterative refinement for finding X.
C Workspace: need 8*N.
C
C First transpose U(2,1) in-situ.
C
DO 30 I = 1, N - 1
CALL DSWAP( N-I, U(N+I,I+1), LDU, U(N+I+1,I), 1 )
30 CONTINUE
C
IWR = 1
IWC = IWR + N
IWF = IWC + N
IWB = IWF + N
IW = IWB + N
C
CALL MB02PD( 'Equilibrate', 'Transpose', N, N, U, LDU,
$ S(NP1,1), LDS, IWORK, EQUED, DWORK(IWR),
$ DWORK(IWC), U(NP1,1), LDU, X, LDX, RCONDU,
$ DWORK(IWF), DWORK(IWB), IWORK(NP1), DWORK(IW),
$ INFO1 )
C
C Transpose U(2,1) back in-situ.
C
DO 40 I = 1, N - 1
CALL DSWAP( N-I, U(N+I,I+1), LDU, U(N+I+1,I), 1 )
40 CONTINUE
C
IF( .NOT.LSAME( EQUED, 'N' ) ) THEN
C
C Undo the equilibration of U(1,1) and U(2,1).
C
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
C
IF( ROWEQU ) THEN
C
DO 50 I = 0, N - 1
DWORK(IWR+I) = ONE / DWORK(IWR+I)
50 CONTINUE
C
CALL MB01SD( 'Row scaling', N, N, U, LDU, DWORK(IWR),
$ DWORK(IWC) )
END IF
C
IF( COLEQU ) THEN
C
DO 60 I = 0, N - 1
DWORK(IWC+I) = ONE / DWORK(IWC+I)
60 CONTINUE
C
CALL MB01SD( 'Column scaling', NN, N, U, LDU, DWORK(IWR),
$ DWORK(IWC) )
END IF
END IF
C
PIVOTU = DWORK(IW)
C
IF ( INFO1.GT.0 ) THEN
C
C Singular matrix. Set INFO and DWORK for error return.
C
INFO = 7
GO TO 80
END IF
C
ELSE
C
C Use LU factorization and a standard solution algorithm.
C
CALL DLACPY( 'Full', N, N, U, LDU, S(NP1,1), LDS )
CALL DLACPY( 'Full', N, N, U(NP1,1), LDU, X, LDX )
C
C Solve the system X*U(1,1) = U(2,1).
C
CALL MB02VD( 'No Transpose', N, N, S(NP1,1), LDS, IWORK, X,
$ LDX, INFO1 )
C
IF ( INFO1.NE.0 ) THEN
INFO = 7
RCONDU = ZERO
GO TO 80
ELSE
C
C Compute the norm of U(1,1).
C
UNORM = DLANGE( '1-norm', N, N, U, LDU, DWORK )
C
C Estimate the reciprocal condition of U(1,1).
C Workspace: need 4*N.
C
CALL DGECON( '1-norm', N, S(NP1,1), LDS, UNORM, RCONDU,
$ DWORK, IWORK(NP1), INFO )
C
IF ( RCONDU.LT.EPS ) THEN
C
C Nearly singular matrix. Set IWARN for warning indication.
C
IWARN = 1
END IF
WRKOPT = MAX( WRKOPT, 4*N )
END IF
END IF
C
C Set S(2,1) to zero.
C
CALL DLASET( 'Full', N, N, ZERO, ZERO, S(NP1,1), LDS )
C
C Make sure the solution matrix X is symmetric.
C
DO 70 I = 1, N - 1
CALL DAXPY( N-I, ONE, X(I,I+1), LDX, X(I+1,I), 1 )
CALL DSCAL( N-I, HALF, X(I+1,I), 1 )
CALL DCOPY( N-I, X(I+1,I), 1, X(I,I+1), LDX )
70 CONTINUE
C
IF ( LSCAL ) THEN
C
C Undo scaling for the solution X.
C
CALL DLASCL( 'G', 0, 0, ONE, SCALE, N, N, X, LDX, INFO1 )
END IF
C
DWORK(1) = WRKOPT
C
80 CONTINUE
IF ( LJOBB )
$ DWORK(2) = RCONDL
IF ( REFINE )
$ DWORK(3) = PIVOTU
DWORK(4) = SCALE
C
RETURN
C *** Last line of SG02AD ***
END
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