*DECK ELMHES
SUBROUTINE ELMHES (NM, N, LOW, IGH, A, INT)
C***BEGIN PROLOGUE ELMHES
C***PURPOSE Reduce a real general matrix to upper Hessenberg form
C using stabilized elementary similarity transformations.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C1B2
C***TYPE SINGLE PRECISION (ELMHES-S, COMHES-C)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is a translation of the ALGOL procedure ELMHES,
C NUM. MATH. 12, 349-368(1968) by Martin and Wilkinson.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971).
C
C Given a REAL GENERAL matrix, this subroutine
C reduces a submatrix situated in rows and columns
C LOW through IGH to upper Hessenberg form by
C stabilized elementary similarity transformations.
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 LOW and IGH are two INTEGER variables determined by the
C balancing subroutine BALANC. If BALANC has not been
C used, set LOW=1 and IGH equal to the order of the matrix, N.
C
C A contains the input matrix. A is a two-dimensional REAL
C array, dimensioned A(NM,N).
C
C On OUTPUT
C
C A contains the upper Hessenberg matrix. The multipliers which
C were used in the reduction are stored in the remaining
C triangle under the Hessenberg matrix.
C
C INT contains information on the rows and columns interchanged
C in the reduction. Only elements LOW through IGH are used.
C INT is a one-dimensional INTEGER array, dimensioned INT(IGH).
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 ELMHES
C
INTEGER I,J,M,N,LA,NM,IGH,KP1,LOW,MM1,MP1
REAL A(NM,*)
REAL X,Y
INTEGER INT(*)
C
C***FIRST EXECUTABLE STATEMENT ELMHES
LA = IGH - 1
KP1 = LOW + 1
IF (LA .LT. KP1) GO TO 200
C
DO 180 M = KP1, LA
MM1 = M - 1
X = 0.0E0
I = M
C
DO 100 J = M, IGH
IF (ABS(A(J,MM1)) .LE. ABS(X)) GO TO 100
X = A(J,MM1)
I = J
100 CONTINUE
C
INT(M) = I
IF (I .EQ. M) GO TO 130
C .......... INTERCHANGE ROWS AND COLUMNS OF A ..........
DO 110 J = MM1, N
Y = A(I,J)
A(I,J) = A(M,J)
A(M,J) = Y
110 CONTINUE
C
DO 120 J = 1, IGH
Y = A(J,I)
A(J,I) = A(J,M)
A(J,M) = Y
120 CONTINUE
C .......... END INTERCHANGE ..........
130 IF (X .EQ. 0.0E0) GO TO 180
MP1 = M + 1
C
DO 160 I = MP1, IGH
Y = A(I,MM1)
IF (Y .EQ. 0.0E0) GO TO 160
Y = Y / X
A(I,MM1) = Y
C
DO 140 J = M, N
140 A(I,J) = A(I,J) - Y * A(M,J)
C
DO 150 J = 1, IGH
150 A(J,M) = A(J,M) + Y * A(J,I)
C
160 CONTINUE
C
180 CONTINUE
C
200 RETURN
END