# cbal.f

          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