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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: balances a pair of general complex matrices (A,B)

```NAME
LAPACK-3 - balances a pair of general complex matrices (A,B)

SYNOPSIS
SUBROUTINE ZGGBAL( JOB,  N,  A,  LDA, B, LDB, ILO, IHI, LSCALE, RSCALE,
WORK, INFO )

CHARACTER      JOB

INTEGER        IHI, ILO, INFO, LDA, LDB, N

DOUBLE         PRECISION LSCALE( * ), RSCALE( * ), WORK( * )

COMPLEX*16     A( LDA, * ), B( LDB, * )

PURPOSE
ZGGBAL balances a pair of general complex matrices (A,B).  This
involves, first, permuting A and B by similarity transformations to
isolate eigenvalues in the first 1 to ILO\$-\$1 and last IHI+1 to N
elements on the diagonal; and second, applying a diagonal similarity
transformation to rows and columns ILO to IHI to make the rows
and columns as close in norm as possible. Both steps are optional.
Balancing may reduce the 1-norm of the matrices, and improve the
accuracy of the computed eigenvalues and/or eigenvectors in the
generalized eigenvalue problem A*x = lambda*B*x.

ARGUMENTS
JOB     (input) CHARACTER*1
Specifies the operations to be performed on A and B:
= 'N':  none:  simply set ILO = 1, IHI = N, 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 the matrices A and B.  N >= 0.

A       (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the input matrix A.
On exit, A is overwritten by the balanced matrix.
If JOB = 'N', A is not referenced.

LDA     (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).

B       (input/output) COMPLEX*16 array, dimension (LDB,N)
On entry, the input matrix B.
On exit, B is overwritten by the balanced matrix.
If JOB = 'N', B is not referenced.

LDB     (input) 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.

WORK    (workspace) REAL array, dimension (lwork)
lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
at least 1 when JOB = 'N' or 'P'.

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

FURTHER DETAILS
See R.C. WARD, Balancing the generalized eigenvalue problem,
SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

LAPACK routine (version 3.2)     March 2011                       LAPACK-3(3)
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