*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