SUBROUTINE MB04DY( JOBSCL, N, A, LDA, QG, LDQG, D, DWORK, INFO )
C
C PURPOSE
C
C To perform a symplectic scaling on the Hamiltonian matrix
C
C ( A G )
C H = ( T ), (1)
C ( Q -A )
C
C i.e., perform either the symplectic scaling transformation
C
C -1
C ( A' G' ) ( D 0 ) ( A G ) ( D 0 )
C H' <-- ( T ) = ( ) ( T ) ( -1 ), (2)
C ( Q' -A' ) ( 0 D ) ( Q -A ) ( 0 D )
C
C where D is a diagonal scaling matrix, or the symplectic norm
C scaling transformation
C
C ( A'' G'' ) 1 ( A G/tau )
C H'' <-- ( T ) = --- ( T ), (3)
C ( Q'' -A'' ) tau ( tau Q -A )
C
C where tau is a real scalar. Note that if tau is not equal to 1,
C then (3) is NOT a similarity transformation. The eigenvalues
C of H are then tau times the eigenvalues of H''.
C
C For symplectic scaling (2), D is chosen to give the rows and
C columns of A' approximately equal 1-norms and to give Q' and G'
C approximately equal norms. (See METHOD below for details.) For
C norm scaling, tau = MAX(1, ||A||, ||G||, ||Q||) where ||.||
C denotes the 1-norm (column sum norm).
C
C ARGUMENTS
C
C Mode Parameters
C
C JOBSCL CHARACTER*1
C Indicates which scaling strategy is used, as follows:
C = 'S' : do the symplectic scaling (2);
C = '1' or 'O': do the 1-norm scaling (3);
C = 'N' : do nothing; set INFO and return.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the matrices A, G, and Q. N >= 0.
C
C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
C On input, if JOBSCL <> 'N', the leading N-by-N part of
C this array must contain the upper left block A of the
C Hamiltonian matrix H in (1).
C On output, if JOBSCL <> 'N', the leading N-by-N part of
C this array contains the leading N-by-N part of the scaled
C Hamiltonian matrix H' in (2) or H'' in (3), depending on
C the setting of JOBSCL.
C If JOBSCL = 'N', this array is not referenced.
C
C LDA INTEGER
C The leading dimension of the array A.
C LDA >= MAX(1,N), if JOBSCL <> 'N';
C LDA >= 1, if JOBSCL = 'N'.
C
C QG (input/output) DOUBLE PRECISION array, dimension
C (LDQG,N+1)
C On input, if JOBSCL <> 'N', the leading N-by-N lower
C triangular part of this array must contain the lower
C triangle of the lower left symmetric block Q of the
C Hamiltonian matrix H in (1), and the N-by-N upper
C triangular part of the submatrix in the columns 2 to N+1
C of this array must contain the upper triangle of the upper
C right symmetric block G of H in (1).
C So, if i >= j, then Q(i,j) = Q(j,i) is stored in QG(i,j)
C and G(i,j) = G(j,i) is stored in QG(j,i+1).
C On output, if JOBSCL <> 'N', the leading N-by-N lower
C triangular part of this array contains the lower triangle
C of the lower left symmetric block Q' or Q'', and the
C N-by-N upper triangular part of the submatrix in the
C columns 2 to N+1 of this array contains the upper triangle
C of the upper right symmetric block G' or G'' of the scaled
C Hamiltonian matrix H' in (2) or H'' in (3), depending on
C the setting of JOBSCL.
C If JOBSCL = 'N', this array is not referenced.
C
C LDQG INTEGER
C The leading dimension of the array QG.
C LDQG >= MAX(1,N), if JOBSCL <> 'N';
C LDQG >= 1, if JOBSCL = 'N'.
C
C D (output) DOUBLE PRECISION array, dimension (nd)
C If JOBSCL = 'S', then nd = N and D contains the diagonal
C elements of the diagonal scaling matrix in (2).
C If JOBSCL = '1' or 'O', then nd = 1 and D(1) is set to tau
C from (3). In this case, no other elements of D are
C referenced.
C If JOBSCL = 'N', this array is not referenced.
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (N)
C If JOBSCL = 'N', this array is not referenced.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, then the i-th argument had an illegal
C value.
C
C METHOD
C
C 1. Symplectic scaling (JOBSCL = 'S'):
C
C First, LAPACK subroutine DGEBAL is used to equilibrate the 1-norms
C of the rows and columns of A using a diagonal scaling matrix D_A.
C Then, H is similarily transformed by the symplectic diagonal
C matrix D1 = diag(D_A,D_A**(-1)). Next, the off-diagonal blocks of
C the resulting Hamiltonian matrix are equilibrated in the 1-norm
C using the symplectic diagonal matrix D2 of the form
C
C ( I/rho 0 )
C D2 = ( )
C ( 0 rho*I )
C
C where rho is a real scalar. Thus, in (2), D = D1*D2.
C
C 2. Norm scaling (JOBSCL = '1' or 'O'):
C
C The norm of the matrices A and G of (1) is reduced by setting
C A := A/tau and G := G/(tau**2) where tau is the power of the
C base of the arithmetic closest to MAX(1, ||A||, ||G||, ||Q||) and
C ||.|| denotes the 1-norm.
C
C REFERENCES
C
C [1] Benner, P., Byers, R., and Barth, E.
C Fortran 77 Subroutines for Computing the Eigenvalues of
C Hamiltonian Matrices. I: The Square-Reduced Method.
C ACM Trans. Math. Software, 26, 1, pp. 49-77, 2000.
C
C NUMERICAL ASPECTS
C
C For symplectic scaling, the complexity of the used algorithms is
C hard to estimate and depends upon how well the rows and columns of
C A in (1) are equilibrated. In one sweep, each row/column of A is
C scaled once, i.e., the cost of one sweep is N**2 multiplications.
C Usually, 3-6 sweeps are enough to equilibrate the norms of the
C rows and columns of a matrix. Roundoff errors are possible as
C LAPACK routine DGEBAL does NOT use powers of the machine base for
C scaling. The second stage (equilibrating ||G|| and ||Q||) requires
C N**2 multiplications.
C For norm scaling, 3*N**2 + O(N) multiplications are required and
C NO rounding errors occur as all multiplications are performed with
C powers of the machine base.
C
C CONTRIBUTOR
C
C P. Benner, Universitaet Bremen, Germany, and
C R. Byers, University of Kansas, Lawrence, USA.
C Aug. 1998, routine DHABL.
C V. Sima, Research Institute for Informatics, Bucharest, Romania,
C Oct. 1998, SLICOT Library version.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, May 2009.
C
C KEYWORDS
C
C Balancing, Hamiltonian matrix, norms, symplectic similarity
C transformation.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDQG, N
CHARACTER JOBSCL
C ..
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), D(*), DWORK(*), QG(LDQG,*)
C ..
C .. Local Scalars ..
DOUBLE PRECISION ANRM, BASE, EPS, GNRM, OFL, QNRM,
$ RHO, SFMAX, SFMIN, TAU, UFL, Y
INTEGER I, IERR, IHI, ILO, J
LOGICAL NONE, NORM, SYMP
C ..
C .. External Functions ..
DOUBLE PRECISION DLAMCH, DLANGE, DLANSY
LOGICAL LSAME
EXTERNAL DLAMCH, DLANGE, DLANSY, LSAME
C ..
C .. External Subroutines ..
EXTERNAL DGEBAL, DLABAD, DLASCL, DRSCL, XERBLA
C ..
C .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
C ..
C .. Executable Statements ..
C
INFO = 0
SYMP = LSAME( JOBSCL, 'S' )
NORM = LSAME( JOBSCL, '1' ) .OR. LSAME( JOBSCL, 'O' )
NONE = LSAME( JOBSCL, 'N' )
C
IF( .NOT.SYMP .AND. .NOT.NORM .AND. .NOT.NONE ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.1 .OR. ( .NOT.NONE .AND. LDA.LT.N ) ) THEN
INFO = -4
ELSE IF( LDQG.LT.1 .OR. ( .NOT.NONE .AND. LDQG.LT.N ) ) THEN
INFO = -6
END IF
C
IF ( INFO.NE.0 ) THEN
CALL XERBLA( 'MB04DY', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( N.EQ.0 .OR. NONE )
$ RETURN
C
C Set some machine dependant constants.
C
BASE = DLAMCH( 'Base' )
EPS = DLAMCH( 'Precision' )
UFL = DLAMCH( 'Safe minimum' )
OFL = ONE/UFL
CALL DLABAD( UFL, OFL )
SFMAX = ( EPS/BASE )/UFL
SFMIN = ONE/SFMAX
C
IF ( NORM ) THEN
C
C Compute norms.
C
ANRM = DLANGE( '1-norm', N, N, A, LDA, DWORK )
GNRM = DLANSY( '1-norm', 'Upper', N, QG(1,2), LDQG, DWORK )
QNRM = DLANSY( '1-norm', 'Lower', N, QG, LDQG, DWORK )
Y = MAX( ONE, ANRM, GNRM, QNRM )
TAU = ONE
C
C WHILE ( TAU < Y ) DO
10 CONTINUE
IF ( ( TAU.LT.Y ) .AND. ( TAU.LT.SQRT( SFMAX ) ) ) THEN
TAU = TAU*BASE
GO TO 10
END IF
C END WHILE 10
IF ( TAU.GT.ONE ) THEN
IF ( ABS( TAU/BASE - Y ).LT.ABS( TAU - Y ) )
$ TAU = TAU/BASE
CALL DLASCL( 'General', 0, 0, TAU, ONE, N, N, A, LDA, IERR )
CALL DLASCL( 'Upper', 0, 0, TAU, ONE, N, N, QG(1,2), LDQG,
$ IERR )
CALL DLASCL( 'Upper', 0, 0, TAU, ONE, N, N, QG(1,2), LDQG,
$ IERR )
END IF
C
D(1) = TAU
C
ELSE
CALL DGEBAL( 'Scale', N, A, LDA, ILO, IHI, D, IERR )
C
DO 30 J = 1, N
C
DO 20 I = J, N
QG(I,J) = QG(I,J)*D(J)*D(I)
20 CONTINUE
C
30 CONTINUE
C
DO 50 J = 2, N + 1
C
DO 40 I = 1, J - 1
QG(I,J) = QG(I,J)/D(J-1)/D(I)
40 CONTINUE
C
50 CONTINUE
C
GNRM = DLANSY( '1-norm', 'Upper', N, QG(1,2), LDQG, DWORK )
QNRM = DLANSY( '1-norm', 'Lower', N, QG, LDQG, DWORK )
IF ( GNRM.EQ.ZERO ) THEN
IF ( QNRM.EQ.ZERO ) THEN
RHO = ONE
ELSE
RHO = SFMAX
END IF
ELSE IF ( QNRM.EQ.ZERO ) THEN
RHO = SFMIN
ELSE
RHO = SQRT( QNRM )/SQRT( GNRM )
END IF
C
CALL DLASCL( 'Lower', 0, 0, RHO, ONE, N, N, QG, LDQG, IERR )
CALL DLASCL( 'Upper', 0, 0, ONE, RHO, N, N, QG(1,2), LDQG,
$ IERR )
CALL DRSCL( N, SQRT( RHO ), D, 1 )
END IF
C
RETURN
C *** Last line of MB04DY ***
END