Purpose
To perform a symplectic scaling on the Hamiltonian matrix
( A G )
H = ( T ), (1)
( Q -A )
i.e., perform either the symplectic scaling transformation
-1
( A' G' ) ( D 0 ) ( A G ) ( D 0 )
H' <-- ( T ) = ( ) ( T ) ( -1 ), (2)
( Q' -A' ) ( 0 D ) ( Q -A ) ( 0 D )
where D is a diagonal scaling matrix, or the symplectic norm
scaling transformation
( A'' G'' ) 1 ( A G/tau )
H'' <-- ( T ) = --- ( T ), (3)
( Q'' -A'' ) tau ( tau Q -A )
where tau is a real scalar. Note that if tau is not equal to 1,
then (3) is NOT a similarity transformation. The eigenvalues
of H are then tau times the eigenvalues of H''.
For symplectic scaling (2), D is chosen to give the rows and
columns of A' approximately equal 1-norms and to give Q' and G'
approximately equal norms. (See METHOD below for details.) For
norm scaling, tau = MAX(1, ||A||, ||G||, ||Q||) where ||.||
denotes the 1-norm (column sum norm).
Specification
SUBROUTINE MB04DY( JOBSCL, N, A, LDA, QG, LDQG, D, DWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDQG, N
CHARACTER JOBSCL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), D(*), DWORK(*), QG(LDQG,*)
Arguments
Mode Parameters
JOBSCL CHARACTER*1
Indicates which scaling strategy is used, as follows:
= 'S' : do the symplectic scaling (2);
= '1' or 'O': do the 1-norm scaling (3);
= 'N' : do nothing; set INFO and return.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, G, and Q. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On input, if JOBSCL <> 'N', the leading N-by-N part of
this array must contain the upper left block A of the
Hamiltonian matrix H in (1).
On output, if JOBSCL <> 'N', the leading N-by-N part of
this array contains the leading N-by-N part of the scaled
Hamiltonian matrix H' in (2) or H'' in (3), depending on
the setting of JOBSCL.
If JOBSCL = 'N', this array is not referenced.
LDA INTEGER
The leading dimension of the array A.
LDA >= MAX(1,N), if JOBSCL <> 'N';
LDA >= 1, if JOBSCL = 'N'.
QG (input/output) DOUBLE PRECISION array, dimension
(LDQG,N+1)
On input, if JOBSCL <> 'N', the leading N-by-N lower
triangular part of this array must contain the lower
triangle of the lower left symmetric block Q of the
Hamiltonian matrix H in (1), and the N-by-N upper
triangular part of the submatrix in the columns 2 to N+1
of this array must contain the upper triangle of the upper
right symmetric block G of H in (1).
So, if i >= j, then Q(i,j) = Q(j,i) is stored in QG(i,j)
and G(i,j) = G(j,i) is stored in QG(j,i+1).
On output, if JOBSCL <> 'N', the leading N-by-N lower
triangular part of this array contains the lower triangle
of the lower left symmetric block Q' or Q'', and the
N-by-N upper triangular part of the submatrix in the
columns 2 to N+1 of this array contains the upper triangle
of the upper right symmetric block G' or G'' of the scaled
Hamiltonian matrix H' in (2) or H'' in (3), depending on
the setting of JOBSCL.
If JOBSCL = 'N', this array is not referenced.
LDQG INTEGER
The leading dimension of the array QG.
LDQG >= MAX(1,N), if JOBSCL <> 'N';
LDQG >= 1, if JOBSCL = 'N'.
D (output) DOUBLE PRECISION array, dimension (nd)
If JOBSCL = 'S', then nd = N and D contains the diagonal
elements of the diagonal scaling matrix in (2).
If JOBSCL = '1' or 'O', then nd = 1 and D(1) is set to tau
from (3). In this case, no other elements of D are
referenced.
If JOBSCL = 'N', this array is not referenced.
Workspace
DWORK DOUBLE PRECISION array, dimension (N)
If JOBSCL = 'N', this array is not referenced.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, then the i-th argument had an illegal
value.
Method
1. Symplectic scaling (JOBSCL = 'S'):
First, LAPACK subroutine DGEBAL is used to equilibrate the 1-norms
of the rows and columns of A using a diagonal scaling matrix D_A.
Then, H is similarily transformed by the symplectic diagonal
matrix D1 = diag(D_A,D_A**(-1)). Next, the off-diagonal blocks of
the resulting Hamiltonian matrix are equilibrated in the 1-norm
using the symplectic diagonal matrix D2 of the form
( I/rho 0 )
D2 = ( )
( 0 rho*I )
where rho is a real scalar. Thus, in (2), D = D1*D2.
2. Norm scaling (JOBSCL = '1' or 'O'):
The norm of the matrices A and G of (1) is reduced by setting
A := A/tau and G := G/(tau**2) where tau is the power of the
base of the arithmetic closest to MAX(1, ||A||, ||G||, ||Q||) and
||.|| denotes the 1-norm.
References
[1] Benner, P., Byers, R., and Barth, E.
Fortran 77 Subroutines for Computing the Eigenvalues of
Hamiltonian Matrices. I: The Square-Reduced Method.
ACM Trans. Math. Software, 26, 1, pp. 49-77, 2000.
Numerical Aspects
For symplectic scaling, the complexity of the used algorithms is hard to estimate and depends upon how well the rows and columns of A in (1) are equilibrated. In one sweep, each row/column of A is scaled once, i.e., the cost of one sweep is N**2 multiplications. Usually, 3-6 sweeps are enough to equilibrate the norms of the rows and columns of a matrix. Roundoff errors are possible as LAPACK routine DGEBAL does NOT use powers of the machine base for scaling. The second stage (equilibrating ||G|| and ||Q||) requires N**2 multiplications. For norm scaling, 3*N**2 + O(N) multiplications are required and NO rounding errors occur as all multiplications are performed with powers of the machine base.Further Comments
NoneExample
Program Text
* MB04DY EXAMPLE PROGRAM TEXT.
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDQG
PARAMETER ( LDA = NMAX, LDQG = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, N
CHARACTER*1 JOBSCL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), D(NMAX), DWORK(LDWORK),
$ QG(LDQG,NMAX+1)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB04DY
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOBSCL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99998 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( QG(J,I+1), I = J,N ), J = 1,N )
READ ( NIN, FMT = * ) ( ( QG(I,J), I = J,N ), J = 1,N )
* Scale the Hamiltonian matrix.
CALL MB04DY( JOBSCL, N, A, LDA, QG, LDQG, D, DWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
* Show the scaled Hamiltonian matrix.
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( A(I,J), J = 1,N ),
$ ( QG(J,I+1), J = 1,I-1 ), ( QG(I,J+1), J = I,N )
10 CONTINUE
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( QG(I,J), J = 1,I-1 ),
$ ( QG(J,I), J = I,N ), ( -A(J,I), J = 1,N )
20 CONTINUE
* Show the scaling factors.
IF ( LSAME( JOBSCL, 'S' ) ) THEN
WRITE ( NOUT, FMT = 99995 )
WRITE ( NOUT, FMT = 99993 ) ( D(I), I = 1,N )
ELSE IF ( LSAME( JOBSCL, '1' ) .OR. LSAME( JOBSCL, 'O' ) )
$ THEN
WRITE ( NOUT, FMT = 99994 )
WRITE ( NOUT, FMT = 99993 ) D(1)
END IF
ENDIF
END IF
STOP
*
99999 FORMAT (' MB04DY EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' N is out of range.',/' N = ',I5)
99997 FORMAT (' INFO on exit from MB04DY = ',I2)
99996 FORMAT (/' The scaled Hamiltonian is ')
99995 format (/' The scaling factors are ')
99994 format (/' The scaling factor tau is ')
99993 FORMAT (1X,8(F10.4))
END
Program Data
MB04DY EXAMPLE PROGRAM DATA 3 S -0.4 0.05 0.0007 -4.7 0.8 0.025 81.0 29.0 -0.9 0.0034 0.0014 0.00077 -0.005 0.0004 0.003 -18.0 -12.0 43.0 99.0 420.0 -200.0Program Results
MB04DY EXAMPLE PROGRAM RESULTS
The scaled Hamiltonian is
-0.4000 0.4000 0.3584 418.4403 21.5374 0.1851
-0.5875 0.8000 1.6000 21.5374 -9.6149 0.0120
0.1582 0.4531 -0.9000 0.1851 0.0120 0.0014
-0.0001 -0.0008 0.1789 0.4000 0.5875 -0.1582
-0.0008 0.0515 13.9783 -0.4000 -0.8000 -0.4531
0.1789 13.9783 -426.0056 -0.3584 -1.6000 0.9000
The scaling factors are
0.0029 0.0228 1.4595