Diff of /src/MB04DD.f [000000] .. [6a0e2a]  Maximize  Restore

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+      SUBROUTINE MB04DD( JOB, N, A, LDA, QG, LDQG, ILO, SCALE, 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 balance a real Hamiltonian matrix,
+C
+C                   [  A   G  ]
+C              H =  [       T ] ,
+C                   [  Q  -A  ]
+C
+C     where A is an N-by-N matrix and G, Q are N-by-N symmetric
+C     matrices. This involves, first, permuting H by a symplectic
+C     similarity transformation to isolate eigenvalues in the first
+C     1:ILO-1 elements on the diagonal of A; and second, applying a
+C     diagonal similarity transformation to rows and columns
+C     ILO:2*N-ILO+1 to make the rows and columns as close in 1-norm
+C     as possible. Both steps are optional.
+C
+C     ARGUMENTS
+C
+C     Mode Parameters
+C
+C     JOB     CHARACTER*1
+C             Specifies the operations to be performed on H:
+C             = 'N':  none, set ILO = 1, SCALE(I) = 1.0, I = 1 .. N;
+C             = 'P':  permute only;
+C             = 'S':  scale only;
+C             = 'B':  both permute and scale.
+C
+C     Input/Output Parameters
+C
+C     N       (input) INTEGER
+C             The order of the matrix A. N >= 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.
+C             On exit, the leading N-by-N part of this array contains
+C             the matrix A of the balanced Hamiltonian. In particular,
+C             the lower triangular part of the first ILO-1 columns of A
+C             is zero.
+C
+C     LDA     INTEGER
+C             The leading dimension of the array A.  LDA >= MAX(1,N).
+C
+C     QG      (input/output) DOUBLE PRECISION array, dimension
+C                            (LDQG,N+1)
+C             On entry, the leading N-by-N+1 part of this array must
+C             contain the lower triangular part of the matrix Q and
+C             the upper triangular part of the matrix G.
+C             On exit, the leading N-by-N+1 part of this array contains
+C             the lower and upper triangular parts of the matrices Q and
+C             G, respectively, of the balanced Hamiltonian. In
+C             particular, the lower triangular and diagonal part of the
+C             first ILO-1 columns of QG is zero.
+C
+C     LDQG    INTEGER
+C             The leading dimension of the array QG.  LDQG >= MAX(1,N).
+C
+C     ILO     (output) INTEGER
+C             ILO-1 is the number of deflated eigenvalues in the
+C             balanced Hamiltonian matrix.
+C
+C     SCALE   (output) DOUBLE PRECISION array of dimension (N)
+C             Details of the permutations and scaling factors applied to
+C             H.  For j = 1,...,ILO-1 let P(j) = SCALE(j). If P(j) <= N,
+C             then rows and columns P(j) and P(j)+N are interchanged
+C             with rows and columns j and j+N, respectively. If
+C             P(j) > N, then row and column P(j)-N are interchanged with
+C             row and column j+N by a generalized symplectic
+C             permutation. For j = ILO,...,N the j-th element of SCALE
+C             contains the factor of the scaling applied to row and
+C             column j.
+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
+C     REFERENCES
+C
+C     [1] Benner, P.
+C         Symplectic balancing of Hamiltonian matrices.
+C         SIAM J. Sci. Comput., 22 (5), pp. 1885-1904, 2000.
+C
+C     CONTRIBUTORS
+C
+C     D. Kressner, Technical Univ. Berlin, Germany, and
+C     P. Benner, Technical Univ. Chemnitz, Germany, December 2003.
+C
+C     REVISIONS
+C
+C     V. Sima, June 2008 (SLICOT version of the HAPACK routine DHABAL).
+C
+C     KEYWORDS
+C
+C     Balancing, Hamiltonian matrix.
+C
+C     ******************************************************************
+C
+C     .. Parameters ..
+      DOUBLE PRECISION  ZERO, ONE
+      PARAMETER         ( ZERO = 0.0D0, ONE = 1.0D0 )
+C     .. Scalar Arguments ..
+      CHARACTER         JOB
+      INTEGER           ILO, INFO, LDA, LDQG, N
+C     .. Array Arguments ..
+      DOUBLE PRECISION  A(LDA,*), QG(LDQG,*), SCALE(*)
+C     .. Local Scalars ..
+      LOGICAL           CONV, LPERM, LSCAL
+      INTEGER           I, IC, ILOOLD, J
+      DOUBLE PRECISION  C, F, GII, MAXC, MAXR, QII, R, SCLFAC,
+     $                  SFMAX1, SFMAX2, SFMIN1, SFMIN2, TEMP
+C     .. External Functions ..
+      LOGICAL           LSAME
+      INTEGER           IDAMAX
+      DOUBLE PRECISION  DASUM, DLAMCH
+      EXTERNAL          DASUM, DLAMCH, IDAMAX, LSAME
+C     .. External Subroutines ..
+      EXTERNAL          DRSCL, DSCAL, DSWAP, XERBLA
+C     .. Intrinsic Functions ..
+      INTRINSIC         ABS, DBLE, MAX, MIN
+C
+C     .. Executable Statements ..
+C
+C     Check the scalar input parameters.
+C
+      INFO = 0
+      LPERM = LSAME( JOB, 'P' ) .OR. LSAME( JOB, 'B' )
+      LSCAL = LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' )
+C
+      IF ( .NOT.LPERM .AND. .NOT.LSCAL
+     $     .AND. .NOT.LSAME( JOB, 'N' ) ) THEN
+         INFO = -1
+      ELSE IF ( N.LT.0 ) THEN
+         INFO = -2
+      ELSE IF ( LDA.LT.MAX( 1, N ) ) THEN
+         INFO = -4
+      ELSE IF ( LDQG.LT.MAX( 1, N ) ) THEN
+         INFO = -6
+      END IF
+C
+C     Return if there were illegal values.
+C
+      IF( INFO.NE.0 ) THEN
+         CALL XERBLA( 'MB04DD', -INFO )
+         RETURN
+      END IF
+C
+      ILO = 1
+C
+C     Quick return if possible.
+C
+      IF ( N.EQ.0 )
+     $   RETURN
+      IF ( .NOT.LPERM .AND. .NOT.LSCAL ) THEN
+         DO 10  I = 1, N
+            SCALE(I) = ONE
+   10    CONTINUE
+         RETURN
+      END IF
+C
+C     Permutations to isolate eigenvalues if possible.
+C
+      IF ( LPERM ) THEN
+         ILOOLD = 0
+C        WHILE ( ILO.NE.ILOOLD )
+   20    IF ( ILO.NE.ILOOLD ) THEN
+            ILOOLD = ILO
+C
+C           Scan columns ILO .. N.
+C
+            I = ILO
+C           WHILE ( I.LE.N .AND. ILO.EQ.ILOOLD )
+   30       IF ( I.LE.N .AND. ILO.EQ.ILOOLD ) THEN
+               DO 40  J = ILO, I-1
+                  IF ( A(J,I).NE.ZERO ) THEN
+                     I = I + 1
+                     GOTO 30
+                  END IF
+   40          CONTINUE
+               DO 50  J = I+1, N
+                  IF ( A(J,I).NE.ZERO ) THEN
+                     I = I + 1
+                     GOTO 30
+                  END IF
+   50          CONTINUE
+               DO 60  J = ILO, I
+                  IF ( QG(I,J).NE.ZERO ) THEN
+                     I = I + 1
+                     GOTO 30
+                  END IF
+   60          CONTINUE
+               DO 70  J = I+1, N
+                  IF ( QG(J,I).NE.ZERO ) THEN
+                     I = I + 1
+                     GOTO 30
+                  END IF
+   70          CONTINUE
+C
+C              Exchange columns/rows ILO <-> I.
+C
+               SCALE( ILO ) = DBLE( I )
+               IF ( ILO.NE.I ) THEN
+C
+                  CALL DSWAP( N, A(1,ILO), 1, A(1,I), 1 )
+                  CALL DSWAP( N-ILO+1, A(ILO,ILO), LDA, A(I,ILO), LDA )
+C
+                  CALL DSWAP( 1, QG(I,ILO), LDQG, QG(ILO,ILO), LDQG )
+                  CALL DSWAP( N-I+1, QG(I,I), 1, QG(I,ILO), 1 )
+                  CALL DSWAP( I-ILO, QG(ILO,ILO), 1, QG(I,ILO), LDQG )
+C
+                  CALL DSWAP( ILO, QG(1,I+1), 1, QG(1,ILO+1), 1 )
+                  CALL DSWAP( N-I+1, QG(I,I+1), LDQG, QG(ILO,I+1),
+     $                        LDQG )
+                  CALL DSWAP( I-ILO, QG(ILO,ILO+1), LDQG, QG(ILO,I+1),
+     $                        1 )
+               END IF
+               ILO = ILO + 1
+            END IF
+C           END WHILE 30
+C
+C           Scan columns N+ILO .. 2*N.
+C
+            I = ILO
+C           WHILE ( I.LE.N .AND. ILO.EQ.ILOOLD )
+   80       IF ( I.LE.N .AND. ILO.EQ.ILOOLD ) THEN
+               DO 90  J = ILO, I-1
+                  IF ( A(I,J).NE.ZERO ) THEN
+                     I = I + 1
+                     GOTO 80
+                  END IF
+   90          CONTINUE
+               DO 100  J = I+1, N
+                  IF ( A(I,J).NE.ZERO ) THEN
+                     I = I + 1
+                     GOTO 80
+                  END IF
+  100          CONTINUE
+               DO 110  J = ILO, I
+                  IF ( QG(J,I+1).NE.ZERO ) THEN
+                     I = I + 1
+                     GOTO 80
+                  END IF
+  110          CONTINUE
+               DO 120  J = I+1, N
+                  IF ( QG(I,J+1).NE.ZERO ) THEN
+                     I = I + 1
+                     GOTO 80
+                  END IF
+  120          CONTINUE
+               SCALE( ILO ) = DBLE( N+I )
+C
+C              Exchange columns/rows I <-> I+N with a symplectic
+C              generalized permutation.
+C
+               CALL DSWAP( I-ILO, A(I,ILO), LDA, QG(I,ILO), LDQG )
+               CALL DSCAL( I-ILO, -ONE, A(I,ILO), LDA )
+               CALL DSWAP( N-I, A(I,I+1), LDA, QG(I+1,I), 1 )
+               CALL DSCAL( N-I, -ONE, A(I,I+1), LDA )
+               CALL DSWAP( I-1, A(1,I), 1, QG(1,I+1), 1 )
+               CALL DSCAL( I-1, -ONE, A(1,I), 1 )
+               CALL DSWAP( N-I, A(I+1,I), 1, QG(I,I+2), LDQG )
+               CALL DSCAL( N-I, -ONE, A(I+1,I), 1 )
+               A(I,I) = -A(I,I)
+               TEMP   = QG(I,I)
+               QG(I,I) = -QG(I,I+1)
+               QG(I,I+1) = -TEMP
+C
+C              Exchange columns/rows ILO <-> I.
+C
+               IF ( ILO.NE.I ) THEN
+C
+                  CALL DSWAP( N, A(1,ILO), 1, A(1,I), 1 )
+                  CALL DSWAP( N-ILO+1, A(ILO,ILO), LDA, A(I,ILO), LDA )
+C
+                  CALL DSWAP( 1, QG(I,ILO), LDQG, QG(ILO,ILO), LDQG )
+                  CALL DSWAP( N-I+1, QG(I,I), 1, QG(I,ILO), 1 )
+                  CALL DSWAP( I-ILO, QG(ILO,ILO), 1, QG(I,ILO), LDQG )
+C
+                  CALL DSWAP( ILO, QG(1,I+1), 1, QG(1,ILO+1), 1 )
+                  CALL DSWAP( N-I+1, QG(I,I+1), LDQG, QG(ILO,I+1),
+     $                        LDQG )
+                  CALL DSWAP( I-ILO, QG(ILO,ILO+1), LDQG, QG(ILO,I+1),
+     $                        1 )
+               END IF
+               ILO = ILO + 1
+            END IF
+C           END WHILE 80
+            GOTO 20
+         END IF
+C        END WHILE 20
+      END IF
+C
+      DO 130  I = ILO, N
+         SCALE(I) = ONE
+  130 CONTINUE
+C
+C     Scale to reduce the 1-norm of the remaining blocks.
+C
+      IF ( LSCAL ) THEN
+         SCLFAC = DLAMCH( 'B' )
+         SFMIN1 = DLAMCH( 'S' ) / DLAMCH( 'P' )
+         SFMAX1 = ONE / SFMIN1
+         SFMIN2 = SFMIN1*SCLFAC
+         SFMAX2 = ONE / SFMIN2
+C
+C        Scale the rows and columns one at a time to minimize the
+C        1-norm of the remaining Hamiltonian submatrix.
+C        Stop when the 1-norm is very roughly minimal.
+C
+  140    CONTINUE
+            CONV = .TRUE.
+            DO 170 I = ILO, N
+C
+C              Compute 1-norm of row and column I without diagonal
+C              elements.
+C
+               R = DASUM( I-ILO, A(I,ILO), LDA ) +
+     $             DASUM( N-I,   A(I,I+1), LDA ) +
+     $             DASUM( I-ILO, QG(ILO,I+1), 1 ) +
+     $             DASUM( N-I,   QG(I,I+2), LDQG )
+               C = DASUM( I-ILO, A(ILO,I), 1 ) +
+     $             DASUM( N-I,   A(I+1,I), 1 ) +
+     $             DASUM( I-ILO, QG(I,ILO), LDQG ) +
+     $             DASUM( N-I,   QG(I+1,I), 1 )
+               QII = ABS( QG(I,I) )
+               GII = ABS( QG(I,I+1) )
+C
+C              Compute inf-norms of row and column I.
+C
+               IC = IDAMAX( N-ILO+1, A(I,ILO), LDA )
+               MAXR = ABS( A(I,IC+ILO-1) )
+               IF ( I.GT.1 ) THEN
+                  IC = IDAMAX( I-1, QG(1,I+1), 1 )
+                  MAXR = MAX( MAXR, ABS( QG(IC,I+1) ) )
+               END IF
+               IF ( N.GT.I ) THEN
+                  IC = IDAMAX( N-I, QG(I,I+2), LDQG )
+                  MAXR = MAX( MAXR, ABS( QG(I,IC+I+1) ) )
+               END IF
+               IC = IDAMAX( N, A(1,I), 1 )
+               MAXC = ABS( A(IC,I) )
+               IF ( I.GT.ILO ) THEN
+                  IC = IDAMAX( I-ILO, QG(I,ILO), LDQG )
+                  MAXC = MAX( MAXC, ABS( QG(I,IC+ILO-1) ) )
+               END IF
+               IF ( N.GT.I ) THEN
+                  IC = IDAMAX( N-I, QG(I+1,I), 1 )
+                  MAXC = MAX( MAXC, ABS( QG(IC+I,I) ) )
+               END IF
+               IF ( ( C + QII ).EQ.ZERO .OR. ( R + GII ).EQ.ZERO )
+     $            GO TO 170
+C
+               F = ONE
+  150          CONTINUE
+               IF ( ( ( R + GII/SCLFAC )/SCLFAC ).GE.
+     $              ( ( C + QII*SCLFAC )*SCLFAC ) .AND.
+     $              MAX( F*SCLFAC, C*SCLFAC, MAXC*SCLFAC,
+     $                   QII*SCLFAC*SCLFAC ).LT.SFMAX2 .AND.
+     $              MIN( ( R + GII/SCLFAC )/SCLFAC, MAX( MAXR/SCLFAC,
+     $                   GII/SCLFAC/SCLFAC ) ).GT.SFMIN2 ) THEN
+                  F = F*SCLFAC
+                  C = C*SCLFAC
+                  QII = QII*SCLFAC*SCLFAC
+                  R = R / SCLFAC
+                  GII = GII/SCLFAC/SCLFAC
+                  MAXC = MAXC*SCLFAC
+                  MAXR = MAXR / SCLFAC
+                  GO TO 150
+               END IF
+C
+  160          CONTINUE
+               IF ( ( ( R + GII*SCLFAC )*SCLFAC ).LE.
+     $              ( ( C + QII/SCLFAC )/SCLFAC ) .AND.
+     $              MAX( R*SCLFAC, MAXR*SCLFAC,
+     $                   GII*SCLFAC*SCLFAC ).LT.SFMAX2 .AND.
+     $              MIN( F/SCLFAC, ( C + QII/SCLFAC )/SCLFAC,
+     $                   MAX( MAXC/SCLFAC, QII/SCLFAC/SCLFAC ) )
+     $                   .GT.SFMIN2 ) THEN
+                  F = F / SCLFAC
+                  C = C / SCLFAC
+                  QII = QII/SCLFAC/SCLFAC
+                  R = R*SCLFAC
+                  GII = GII*SCLFAC*SCLFAC
+                  MAXC = MAXC/SCLFAC
+                  MAXR = MAXR*SCLFAC
+                  GO TO 160
+               END IF
+C
+C              Now balance if necessary.
+C
+               IF ( F.NE.ONE ) THEN
+                  IF ( F.LT.ONE .AND. SCALE(I).LT.ONE ) THEN
+                     IF ( F*SCALE(I).LE.SFMIN1 )
+     $                  GO TO 170
+                  END IF
+                  IF ( F.GT.ONE .AND. SCALE(I).GT.ONE ) THEN
+                     IF ( SCALE(I).GE.SFMAX1 / F )
+     $                  GO TO 170
+                  END IF
+                  CONV = .FALSE.
+                  SCALE(I) = SCALE(I)*F
+                  CALL DRSCL( I-ILO, F, A(I,ILO), LDA )
+                  CALL DRSCL( N-I, F, A(I,I+1), LDA )
+                  CALL DSCAL( I-1, F, A(1,I), 1 )
+                  CALL DSCAL( N-I, F, A(I+1,I), 1 )
+                  CALL DRSCL( I-1, F, QG(1,I+1), 1 )
+                  QG(I,I+1) = QG(I,I+1) / F / F
+                  CALL DRSCL( N-I, F, QG(I,I+1+1), LDQG )
+                  CALL DSCAL( I-ILO, F, QG(I,ILO), LDQG )
+                  QG(I,I)   = QG(I,I) * F * F
+                  CALL DSCAL( N-I, F, QG(I+1,I), 1 )
+               END IF
+  170       CONTINUE
+         IF ( .NOT.CONV ) GO TO 140
+      END IF
+      RETURN
+C *** Last line of MB04DD ***
+      END