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SUBROUTINE CBAL(NM,N,AR,AI,LOW,IGH,SCALE)
C
INTEGER I,J,K,L,M,N,JJ,NM,IGH,LOW,IEXC
DOUBLE PRECISION AR(NM,N),AI(NM,N),SCALE(N)
DOUBLE PRECISION C,F,G,R,S,B2,RADIX
LOGICAL NOCONV
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 ARITHMETIC IS REAL THROUGHOUT.
C
C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C THIS VERSION DATED AUGUST 1983.
C
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