**Purpose**

To balance a pair of N-by-N real matrices (A,B). This involves, first, permuting A and B by equivalence transformations to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal of A and B; and second, applying a diagonal equivalence transformation to rows and columns ILO to IHI to make the rows and columns as close in 1-norm as possible. Both steps are optional. Balancing may reduce the 1-norms of the matrices, and improve the accuracy of the computed eigenvalues and/or eigenvectors in the generalized eigenvalue problem A*x = lambda*B*x. This routine may optionally improve the conditioning of the scaling transformation compared to the LAPACK routine DGGBAL.

SUBROUTINE MB04DL( JOB, N, THRESH, A, LDA, B, LDB, ILO, IHI, $ LSCALE, RSCALE, DWORK, IWARN, INFO ) C .. Scalar Arguments .. CHARACTER JOB INTEGER IHI, ILO, INFO, IWARN, LDA, LDB, N DOUBLE PRECISION THRESH C .. Array Arguments .. DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), LSCALE(*), $ RSCALE(*)

**Mode Parameters**

JOB CHARACTER*1 Specifies the operations to be performed on A and B: = '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 and B. 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 A(s,s) and B(s,s), with s = ILO:IHI. 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(A(s,s),1)/norm(B(s,s),1), norm(B(s,s),1)/norm(S(s,s),1) ) (1) has the smallest value, for the threshold values used, where A(s,s) and B(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(A(s,s),1)*norm(B(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(s) and RSCALE(s), 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) DOUBLE PRECISION 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 balanced matrix A. In particular, the strictly lower triangular part of the first ILO-1 columns and the last N-IHI rows of A is zero. LDA INTEGER The leading dimension of the array A. LDA >= MAX(1,N). B (input/output) DOUBLE PRECISION array, dimension (LDB, N) On entry, the leading N-by-N part of this array must contain the matrix B. On exit, the leading N-by-N part of this array contains the balanced matrix B. In particular, the strictly lower triangular part of the first ILO-1 columns and the last N-IHI rows of B is zero. If JOB = 'N', the arrays A and B are not referenced. LDB INTEGER The leading dimension of the array B. LDB >= MAX(1, N). ILO (output) INTEGER IHI (output) INTEGER ILO and IHI are set to integers such that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j = 1,...,ILO-1 or i = IHI+1,...,N. If JOB = 'N' or 'S', ILO = 1 and IHI = N. LSCALE (output) DOUBLE PRECISION array, dimension (N) Details of the permutations and scaling factors applied to the left side of A and B. If P(j) is the index of the row interchanged with row j, and D(j) is the scaling factor applied to row j, then LSCALE(j) = P(j) for j = 1,...,ILO-1 = D(j) for j = ILO,...,IHI = P(j) for j = IHI+1,...,N. The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1. RSCALE (output) DOUBLE PRECISION array, dimension (N) Details of the permutations and scaling factors applied to the right side of A and B. If P(j) is the index of the column interchanged with column j, and D(j) is the scaling factor applied to column j, then RSCALE(j) = P(j) for j = 1,...,ILO-1 = D(j) for j = ILO,...,IHI = P(j) for j = IHI+1,...,N. The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1.

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 A(s,s) and B(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 = IHI).

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 an equivalence transformation to isolate eigenvalues and/or to make the 1-norms of the rows and columns ILO,...,IHI of A and B 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*A*Dr and Dl*B*Dr. For THRESH = 0, the routine returns essentially the same results as the LAPACK subroutine DGGBAL [1]. Setting THRESH < 0, usually gives better results than DGGBAL 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.

No rounding errors appear if JOB = 'P'.

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**

* MB04DL 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, LDB PARAMETER ( LDA = NMAX, LDB = NMAX ) * .. Local Scalars .. CHARACTER*1 JOB INTEGER I, ILO, INFO, IWARN, J, N DOUBLE PRECISION THRESH * .. Local Arrays .. DOUBLE PRECISION A(LDA, NMAX), B(LDB, NMAX), DWORK(8*NMAX), $ LSCALE(NMAX), RSCALE(NMAX) * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. External Subroutines .. EXTERNAL MB04DL * .. 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 = * ) ( ( B(I,J), J = 1,N ), I = 1,N ) CALL MB04DL( JOB, N, THRESH, A, LDA, B, LDB, ILO, IHI, 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 = 99995 ) ( A(I,J), J = 1,N ) 10 CONTINUE WRITE ( NOUT, FMT = 99996 ) DO 40 I = 1, N WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,N ) 40 CONTINUE WRITE ( NOUT, FMT = 99994 ) ILO WRITE ( NOUT, FMT = 99993 ) IHI WRITE ( NOUT, FMT = 99991 ) WRITE ( NOUT, FMT = 99995 ) ( LSCALE(I), I = 1,N ) WRITE ( NOUT, FMT = 99990 ) WRITE ( NOUT, FMT = 99995 ) ( 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 = 99995 ) ( DWORK(I), I = 1,2 ) WRITE ( NOUT, FMT = 99988 ) WRITE ( NOUT, FMT = 99995 ) ( DWORK(I), I = 3,4 ) WRITE ( NOUT, FMT = 99987 ) WRITE ( NOUT, FMT = 99995 ) ( DWORK(5) ) ELSE WRITE ( NOUT, FMT = 99986 ) IWARN END IF END IF END IF END IF * 99999 FORMAT (' MB04DL EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from MB04DL = ',I2) 99997 FORMAT (' The balanced matrix A is ') 99996 FORMAT (/' The balanced matrix B is ') 99995 FORMAT (20(1X,G12.4)) 99994 FORMAT (/' ILO = ',I4) 99993 FORMAT (/' IHI = ',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 MB04DL = ',I2) 99985 FORMAT (/' N is out of range.',/' N = ',I5) END

MB04DL EXAMPLE PROGRAM DATA 4 B -3 1 0 -1e-12 0 0 -2 0 0 -1 -1 -1 0 -1 -1 0 2 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

MB04DL EXAMPLE PROGRAM RESULTS The balanced matrix A is 2.000 -1.000 0.000 -1.000 0.000 1.000 -0.1000E-11 0.000 0.000 -1.000 -1.000 -1.000 0.000 0.000 0.000 -2.000 The balanced matrix B is 1.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 1.000 ILO = 2 IHI = 3 The permutations and left scaling factors are 2.000 1.000 1.000 2.000 The permutations and right scaling factors are 2.000 1.000 1.000 2.000 The initial 1-norms of the (sub)matrices are 2.000 1.000 The final 1-norms of the (sub)matrices are 2.000 1.000 The threshold value finally used is 0.2500E-12