SLATEC Routines --- CBAL ---

*DECK CBAL
      SUBROUTINE CBAL (NM, N, AR, AI, LOW, IGH, SCALE)
C***BEGIN PROLOGUE  CBAL
C***PURPOSE  Balance a complex general matrix and isolate eigenvalues
C            whenever possible.
C***LIBRARY   SLATEC (EISPACK)
C***CATEGORY  D4C1A
C***TYPE      COMPLEX (BALANC-S, CBAL-C)
C***KEYWORDS  EIGENVECTORS, EISPACK
C***AUTHOR  Smith, B. T., et al.
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 the two-dimensional
C          array parameters, AR and AI, as declared in the calling
C          program dimension statement.  NM is an INTEGER variable.
C
C        N is the order of the matrix A=(AR,AI).  N is an INTEGER
C          variable.  N must be less than or equal to NM.
C
C        AR and AI contain the real and imaginary parts,
C          respectively, of the complex matrix to be balanced.
C          AR and AI are two-dimensional REAL arrays, dimensioned
C          AR(NM,N) and AI(NM,N).
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 INTEGER variables such that AR(I,J)
C          and AI(I,J) 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 permutations and
C          scaling factors used.  SCALE is a one-dimensional REAL array,
C          dimensioned SCALE(N).
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
C***REFERENCES  B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
C                 Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
C                 system Routines - EISPACK Guide, Springer-Verlag,
C                 1976.
C***ROUTINES CALLED  (NONE)
C***REVISION HISTORY  (YYMMDD)
C   760101  DATE WRITTEN
C   890831  Modified array declarations.  (WRB)
C   890831  REVISION DATE from Version 3.2
C   891214  Prologue converted to Version 4.0 format.  (BAB)
C   920501  Reformatted the REFERENCES section.  (WRB)
C***END PROLOGUE  CBAL