**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.

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,*)

**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.

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.

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).

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.

INFO INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

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.

[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.

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.

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).

**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

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

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