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- *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
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