Purpose
To balance the 2*N-by-2*N complex skew-Hamiltonian/Hamiltonian
pencil aS - bH, with
( A D ) ( C V )
S = ( ) and H = ( ), A, C N-by-N, (1)
( E A' ) ( W -C' )
where D and E are skew-Hermitian, V and W are Hermitian matrices,
and ' denotes conjugate transpose. This involves, first, permuting
aS - bH by a symplectic equivalence transformation to isolate
eigenvalues in the first 1:ILO-1 elements on the diagonal of A
and C; and second, applying a diagonal equivalence transformation
to make the pairs of rows and columns ILO:N and N+ILO:2*N as close
in 1-norm as possible. Both steps are optional. Balancing may
reduce the 1-norms of the matrices S and H.
Specification
SUBROUTINE MB4DPZ( JOB, N, THRESH, A, LDA, DE, LDDE, C, LDC, VW,
$ LDVW, ILO, LSCALE, RSCALE, DWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER JOB
INTEGER ILO, INFO, IWARN, LDA, LDC, LDDE, LDVW, N
DOUBLE PRECISION THRESH
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), LSCALE(*), RSCALE(*)
COMPLEX*16 A(LDA,*), C(LDC,*), DE(LDDE,*), VW(LDVW,*)
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the operations to be performed on S and H:
= 'N': none: simply set ILO = 1, LSCALE(I) = 1.0 and
RSCALE(I) = 1.0 for i = 1,...,N.
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale.
Input/Output Parameters
N (input) INTEGER
The order of matrices A, D, E, C, V, and W. N >= 0.
THRESH (input) DOUBLE PRECISION
If JOB = 'S' or JOB = 'B', and THRESH >= 0, threshold
value for magnitude of the elements to be considered in
the scaling process: elements with magnitude less than or
equal to THRESH*MXNORM are ignored for scaling, where
MXNORM is the maximum of the 1-norms of the original
submatrices S(s,s) and H(s,s), with s = [ILO:N,N+ILO:2*N].
If THRESH < 0, the subroutine finds the scaling factors
for which some conditions, detailed below, are fulfilled.
A sequence of increasing strictly positive threshold
values is used.
If THRESH = -1, the condition is that
max( norm(H(s,s),1)/norm(S(s,s),1),
norm(S(s,s),1)/norm(H(s,s),1) ) (1)
has the smallest value, for the threshold values used,
where S(s,s) and H(s,s) are the scaled submatrices.
If THRESH = -2, the norm ratio reduction (1) is tried, but
the subroutine may return IWARN = 1 and reset the scaling
factors to 1, if this seems suitable. See the description
of the argument IWARN and FURTHER COMMENTS.
If THRESH = -3, the condition is that
norm(H(s,s),1)*norm(S(s,s),1) (2)
has the smallest value for the scaled submatrices.
If THRESH = -4, the norm reduction in (2) is tried, but
the subroutine may return IWARN = 1 and reset the scaling
factors to 1, as for THRESH = -2 above.
If THRESH = -VALUE, with VALUE >= 10, the condition
numbers of the left and right scaling transformations will
be bounded by VALUE, i.e., the ratios between the largest
and smallest entries in [LSCALE(ILO:N); RSCALE(ILO:N)]
will be at most VALUE. VALUE should be a power of 10.
If JOB = 'N' or JOB = 'P', the value of THRESH is
irrelevant.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A.
On exit, the leading N-by-N part of this array contains
the matrix A of the balanced skew-Hamiltonian matrix S.
In particular, the strictly lower triangular part of the
first ILO-1 columns of A is zero.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
DE (input/output) COMPLEX*16 array, dimension (LDDE, N+1)
On entry, the leading N-by-N lower triangular part of
this array must contain the lower triangular part of the
skew-Hermitian matrix E, 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 triangular part of the
skew-Hermitian matrix D. The real parts of the entries on
the diagonal and the first superdiagonal of this array
should be zero.
On exit, the leading N-by-N lower triangular part of this
array contains the lower triangular part of the balanced
matrix E, and the N-by-N upper triangular part of the
submatrix in the columns 2 to N+1 of this array contains
the upper triangular part of the balanced matrix D.
In particular, the lower triangular part of the first
ILO-1 columns of DE is zero.
LDDE INTEGER
The leading dimension of the array DE. LDDE >= MAX(1, N).
C (input/output) COMPLEX*16 array, dimension (LDC, N)
On entry, the leading N-by-N part of this array must
contain the matrix C.
On exit, the leading N-by-N part of this array contains
the matrix C of the balanced Hamiltonian matrix H.
In particular, the strictly lower triangular part of the
first ILO-1 columns of C is zero.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1, N).
VW (input/output) COMPLEX*16 array, dimension (LDVW, N+1)
On entry, the leading N-by-N lower triangular part of
this array must contain the lower triangular part of the
Hermitian matrix W, 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 triangular part of the
Hermitian matrix V. The imaginary parts of the entries on
the diagonal and the first superdiagonal of this array
should be zero.
On exit, the leading N-by-N lower triangular part of this
array contains the lower triangular part of the balanced
matrix W, and the N-by-N upper triangular part of the
submatrix in the columns 2 to N+1 of this array contains
the upper triangular part of the balanced matrix V. In
particular, the lower triangular part of the first ILO-1
columns of VW is zero.
LDVW INTEGER
The leading dimension of the array VW. LDVW >= MAX(1, N).
ILO (output) INTEGER
ILO-1 is the number of deflated eigenvalues in the
balanced skew-Hamiltonian/Hamiltonian matrix pencil.
ILO is set to 1 if JOB = 'N' or JOB = 'S'.
LSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations of S and H and scaling applied
to A, D, C, and V from the left. For j = 1,...,ILO-1 let
P(j) = LSCALE(j). If P(j) <= N, then rows and columns P(j)
and P(j)+N are interchanged with rows and columns j and
j+N, respectively. If P(j) > N, then row and column P(j)-N
are interchanged with row and column j+N by a generalized
symplectic permutation. For j = ILO,...,N the j-th element
of LSCALE contains the factor of the scaling applied to
row j of the matrices A, D, C, and V.
RSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations of S and H and scaling applied
to A, E, C, and W from the right. For j = 1,...,ILO-1 let
P(j) = RSCALE(j). If P(j) <= N, then rows and columns P(j)
and P(j)+N are interchanged with rows and columns j and
j+N, respectively. If P(j) > N, then row and column P(j)-N
are interchanged with row and column j+N by a generalized
symplectic permutation. For j = ILO,...,N the j-th element
of RSCALE contains the factor of the scaling applied to
column j of the matrices A, E, C, and W.
Workspace
DWORK DOUBLE PRECISION array, dimension (LDWORK) where
LDWORK = 0, if JOB = 'N' or JOB = 'P', or N = 0;
LDWORK = 6*N, if (JOB = 'S' or JOB = 'B') and THRESH >= 0;
LDWORK = 8*N, if (JOB = 'S' or JOB = 'B') and THRESH < 0.
On exit, if JOB = 'S' or JOB = 'B', DWORK(1) and DWORK(2)
contain the initial 1-norms of S(s,s) and H(s,s), and
DWORK(3) and DWORK(4) contain their final 1-norms,
respectively. Moreover, DWORK(5) contains the THRESH value
used (irrelevant if IWARN = 1 or ILO = N).
Warning Indicator
IWARN INTEGER
= 0: no warning;
= 1: scaling has been requested, for THRESH = -2 or
THRESH = -4, but it most probably would not improve
the accuracy of the computed solution for a related
eigenproblem (since maximum norm increased
significantly compared to the original pencil
matrices and (very) high and/or small scaling
factors occurred). The returned scaling factors have
been reset to 1, but information about permutations,
if requested, has been preserved.
Error Indicator
INFO INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Balancing consists of applying a (symplectic) equivalence
transformation to isolate eigenvalues and/or to make the 1-norms
of each pair of rows and columns indexed by s of S and H nearly
equal. If THRESH < 0, a search is performed to find those scaling
factors giving the smallest norm ratio or product defined above
(see the description of the parameter THRESH).
Assuming JOB = 'S', let Dl and Dr be diagonal matrices containing
the vectors LSCALE and RSCALE, respectively. The returned matrices
are obtained using the equivalence transformation
( Dl 0 ) ( A D ) ( Dr 0 ) ( Dl 0 ) ( C V ) ( Dr 0 )
( ) ( ) ( ), ( ) ( ) ( ).
( 0 Dr ) ( E A' ) ( 0 Dl ) ( 0 Dr ) ( W -C' ) ( 0 Dl )
For THRESH = 0, the routine returns essentially the same results
as the LAPACK subroutine ZGGBAL [1]. Setting THRESH < 0, usually
gives better results than ZGGBAL for badly scaled matrix pencils.
References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
[2] Benner, P.
Symplectic balancing of Hamiltonian matrices.
SIAM J. Sci. Comput., 22 (5), pp. 1885-1904, 2001.
Numerical Aspects
The transformations used preserve the skew-Hamiltonian/Hamiltonian structure and do not introduce significant rounding errors. No rounding errors appear if JOB = 'P'. If T is the global transformation matrix applied to the right, then J'*T*J is the global transformation matrix applied to the left, where J = [ 0 I; -I 0 ], with blocks of order N.Further Comments
If THRESH = -2, the increase of the maximum norm of the scaled submatrices, compared to the maximum norm of the initial submatrices, is bounded by MXGAIN = 100. If THRESH = -2, or THRESH = -4, the maximum condition number of the scaling transformations is bounded by MXCOND = 1/SQRT(EPS), where EPS is the machine precision (see LAPACK Library routine DLAMCH).Example
Program Text
* MB4DPZ EXAMPLE PROGRAM TEXT
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 10 )
INTEGER LDA, LDC, LDDE, LDVW
PARAMETER ( LDA = NMAX, LDC = NMAX, LDDE = NMAX,
$ LDVW = NMAX )
* .. Local Scalars ..
CHARACTER*1 JOB
INTEGER I, ILO, INFO, IWARN, J, N
DOUBLE PRECISION THRESH
* .. Local Arrays ..
COMPLEX*16 A(LDA, NMAX ), C( LDC, NMAX ), DE(LDDE, NMAX),
$ VW(LDVW, NMAX)
DOUBLE PRECISION DWORK(8*NMAX), LSCALE(NMAX), RSCALE(NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL MB4DPZ
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, THRESH
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99985 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( DE(I,J), J = 1,N+1 ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( VW(I,J), J = 1,N+1 ), I = 1,N )
CALL MB4DPZ( JOB, N, THRESH, A, LDA, DE, LDDE, C, LDC, VW,
$ LDVW, ILO, LSCALE, RSCALE, DWORK, IWARN, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( DE(I,J), J = 1,N+1 )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( C(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99993 ) ( VW(I,J), J = 1,N+1 )
40 CONTINUE
WRITE ( NOUT, FMT = 99992 ) ILO
WRITE ( NOUT, FMT = 99991 )
WRITE ( NOUT, FMT = 99984 ) ( LSCALE(I), I = 1,N )
WRITE ( NOUT, FMT = 99990 )
WRITE ( NOUT, FMT = 99984 ) ( RSCALE(I), I = 1,N )
IF ( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' ) ) THEN
IF ( .NOT.( THRESH.EQ.-2 .OR. THRESH.EQ.-4 ) ) THEN
WRITE ( NOUT, FMT = 99989 )
WRITE ( NOUT, FMT = 99984 ) ( DWORK(I), I = 1,2 )
WRITE ( NOUT, FMT = 99988 )
WRITE ( NOUT, FMT = 99984 ) ( DWORK(I), I = 3,4 )
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99984 ) ( DWORK(5) )
ELSE
WRITE ( NOUT, FMT = 99986 ) IWARN
END IF
END IF
END IF
END IF
*
99999 FORMAT (' MB4DPZ EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB4DPZ = ',I2)
99997 FORMAT (' The balanced matrix A is ')
99996 FORMAT (/' The balanced matrix DE is ')
99995 FORMAT (' The balanced matrix C is ')
99994 FORMAT (/' The balanced matrix VW is ')
99993 FORMAT (20( 1X, G11.4, SP, F9.3, S, 'i ') )
99992 FORMAT (/' ILO = ',I4)
99991 FORMAT (/' The permutations and left scaling factors are ')
99990 FORMAT (/' The permutations and right scaling factors are ')
99989 FORMAT (/' The initial 1-norms of the (sub)matrices are ')
99988 FORMAT (/' The final 1-norms of the (sub)matrices are ')
99987 FORMAT (/' The threshold value finally used is ')
99986 FORMAT (/' IWARN on exit from MB4DPZ = ',I2)
99985 FORMAT (/' N is out of range.',/' N = ',I5)
99984 FORMAT (20(1X,G11.4))
END
Program Data
MB4DPZ EXAMPLE PROGRAM DATA
2 B -3
(1,0.5) 0
0 (1,0.5)
0 0 0
0 0 0
(1,0.5) 0
0 (-2,-1)
1 -1.e-12 0
(-1,0.5) -1 0
Program Results
MB4DPZ EXAMPLE PROGRAM RESULTS The balanced matrix A is 1.000 -0.500i 0.000 0.000i 0.000 0.000i 1.000 +0.500i The balanced matrix DE is 0.000 0.000i 0.000 +0.000i 0.000 +0.000i 0.000 +0.000i 0.000 +0.000i 0.000 +0.000i The balanced matrix C is 2.000 -1.000i 1.000 -0.500i 0.000 0.000i 1.000 +0.500i The balanced matrix VW is 0.000 0.000i 1.000 +0.000i 0.000 0.000i 0.000 0.000i 1.000 +0.000i -0.1000E-11 +0.000i ILO = 2 The permutations and left scaling factors are 4.000 1.000 The permutations and right scaling factors are 4.000 1.000 The initial 1-norms of the (sub)matrices are 1.118 2.118 The final 1-norms of the (sub)matrices are 1.118 2.118 The threshold value finally used is -3.000