*DECK BALANC
SUBROUTINE BALANC (NM, N, A, LOW, IGH, SCALE)
C***BEGIN PROLOGUE BALANC
C***PURPOSE Balance a real general matrix and isolate eigenvalues
C whenever possible.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C1A
C***TYPE SINGLE PRECISION (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 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 REAL 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 parameter, A, as declared in the calling program
C dimension statement. NM is an INTEGER variable.
C
C N is the order of the matrix A. N is an INTEGER variable.
C N must be less than or equal to NM.
C
C A contains the input matrix to be balanced. A is a
C two-dimensional REAL array, dimensioned A(NM,N).
C
C On OUTPUT
C
C A contains the balanced matrix.
C
C LOW and IGH are two INTEGER variables such that A(I,J)
C is 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 BALANCE appears in
C BALANC 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 BALANC
C
INTEGER I,J,K,L,M,N,JJ,NM,IGH,LOW,IEXC
REAL A(NM,*),SCALE(*)
REAL C,F,G,R,S,B2,RADIX
LOGICAL NOCONV
C
C***FIRST EXECUTABLE STATEMENT BALANC
RADIX = 16
C
B2 = RADIX * RADIX
K = 1
L = N
GO TO 100
C .......... IN-LINE PROCEDURE FOR ROW AND
C COLUMN EXCHANGE ..........
20 SCALE(M) = J
IF (J .EQ. M) GO TO 50
C
DO 30 I = 1, L
F = A(I,J)
A(I,J) = A(I,M)
A(I,M) = F
30 CONTINUE
C
DO 40 I = K, N
F = A(J,I)
A(J,I) = A(M,I)
A(M,I) = F
40 CONTINUE
C
50 GO TO (80,130), IEXC
C .......... SEARCH FOR ROWS ISOLATING AN EIGENVALUE
C AND PUSH THEM DOWN ..........
80 IF (L .EQ. 1) GO TO 280
L = L - 1
C .......... FOR J=L STEP -1 UNTIL 1 DO -- ..........
100 DO 120 JJ = 1, L
J = L + 1 - JJ
C
DO 110 I = 1, L
IF (I .EQ. J) GO TO 110
IF (A(J,I) .NE. 0.0E0) GO TO 120
110 CONTINUE
C
M = L
IEXC = 1
GO TO 20
120 CONTINUE
C
GO TO 140
C .......... SEARCH FOR COLUMNS ISOLATING AN EIGENVALUE
C AND PUSH THEM LEFT ..........
130 K = K + 1
C
140 DO 170 J = K, L
C
DO 150 I = K, L
IF (I .EQ. J) GO TO 150
IF (A(I,J) .NE. 0.0E0) GO TO 170
150 CONTINUE
C
M = K
IEXC = 2
GO TO 20
170 CONTINUE
C .......... NOW BALANCE THE SUBMATRIX IN ROWS K TO L ..........
DO 180 I = K, L
180 SCALE(I) = 1.0E0
C .......... ITERATIVE LOOP FOR NORM REDUCTION ..........
190 NOCONV = .FALSE.
C
DO 270 I = K, L
C = 0.0E0
R = 0.0E0
C
DO 200 J = K, L
IF (J .EQ. I) GO TO 200
C = C + ABS(A(J,I))
R = R + ABS(A(I,J))
200 CONTINUE
C .......... GUARD AGAINST ZERO C OR R DUE TO UNDERFLOW ..........
IF (C .EQ. 0.0E0 .OR. R .EQ. 0.0E0) GO TO 270
G = R / RADIX
F = 1.0E0
S = C + R
210 IF (C .GE. G) GO TO 220
F = F * RADIX
C = C * B2
GO TO 210
220 G = R * RADIX
230 IF (C .LT. G) GO TO 240
F = F / RADIX
C = C / B2
GO TO 230
C .......... NOW BALANCE ..........
240 IF ((C + R) / F .GE. 0.95E0 * S) GO TO 270
G = 1.0E0 / F
SCALE(I) = SCALE(I) * F
NOCONV = .TRUE.
C
DO 250 J = K, N
250 A(I,J) = A(I,J) * G
C
DO 260 J = 1, L
260 A(J,I) = A(J,I) * F
C
270 CONTINUE
C
IF (NOCONV) GO TO 190
C
280 LOW = K
IGH = L
RETURN
END