SUBROUTINE CBAL(NM,N,AR,AI,LOW,IGH,SCALE)
C***BEGIN PROLOGUE  CBAL
C***DATE WRITTEN   760101   (YYMMDD)
C***REVISION DATE  830518   (YYMMDD)
C***CATEGORY NO.  D4C1A
C***KEYWORDS  EIGENVALUES,EIGENVECTORS,EISPACK
C***AUTHOR  SMITH, B. T., ET AL.
C***PURPOSE  Balances a complex general matrix and isolates eigenvalues
C            whenever possible.
C***DESCRIPTION
C
C     This subroutine is a translation of the ALGOL procedure
C     CBALANCE, which is a complex version of BALANCE,
C     NUM. MATH. 13, 293-304(1969) by Parlett and Reinsch.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 315-326(1971).
C
C     This subroutine balances a COMPLEX matrix and isolates
C     eigenvalues whenever possible.
C
C     On INPUT
C
C        NM must be set to the row dimension of two-dimensional
C          array parameters as declared in the calling program
C          dimension statement.
C
C        N is the order of the matrix.
C
C        AR and AI contain the real and imaginary parts,
C          respectively, of the complex matrix to be balanced.
C
C     On OUTPUT
C
C        AR and AI contain the real and imaginary parts,
C          respectively, of the balanced matrix.
C
C        LOW and IGH are two integers such that AR(I,J) and AI(I,J)
C          are equal to zero if
C           (1) I is greater than J and
C           (2) J=1,...,LOW-1 or I=IGH+1,...,N.
C
C        SCALE contains information determining the
C           permutations and scaling factors used.
C
C     Suppose that the principal submatrix in rows LOW through IGH
C     has been balanced, that P(J) denotes the index interchanged
C     with J during the permutation step, and that the elements
C     of the diagonal matrix used are denoted by D(I,J).  Then
C        SCALE(J) = P(J),    for J = 1,...,LOW-1
C                 = D(J,J)       J = LOW,...,IGH
C                 = P(J)         J = IGH+1,...,N.
C     The order in which the interchanges are made is N to IGH+1,
C     then 1 to LOW-1.
C
C     Note that 1 is returned for IGH if IGH is zero formally.
C
C     The ALGOL procedure EXC contained in CBALANCE appears in
C     CBAL  in line.  (Note that the ALGOL roles of identifiers
C     K,L have been reversed.)
C
C     Questions and comments should be directed to B. S. Garbow,
C     APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
C     ------------------------------------------------------------------
C***REFERENCES  B. T. SMITH, J. M. BOYLE, J. J. DONGARRA, B. S. GARBOW,
C                 Y. IKEBE, V. C. KLEMA, C. B. MOLER, *MATRIX EIGEN-
C                 SYSTEM ROUTINES - EISPACK GUIDE*, SPRINGER-VERLAG,
C                 1976.
C***ROUTINES CALLED  (NONE)
C***END PROLOGUE  CBAL