*DECK COMHES SUBROUTINE COMHES (NM, N, LOW, IGH, AR, AI, INT) C***BEGIN PROLOGUE COMHES C***PURPOSE Reduce a complex general matrix to complex upper Hessenberg C form using stabilized elementary similarity C transformations. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4C1B2 C***TYPE COMPLEX (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 COMHES, 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 COMPLEX 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 parameters, AR and AI, as declared in the calling C program dimension statement. NM is an INTEGER variable. C C N is the order of the matrix A=(AR,AI). N is an INTEGER C variable. N must be less than or equal to NM. C C LOW and IGH are two INTEGER variables determined by the C balancing subroutine CBAL. If CBAL has not been used, C set LOW=1 and IGH equal to the order of the matrix, N. C C AR and AI contain the real and imaginary parts, respectively, C of the complex input matrix. AR and AI are two-dimensional C REAL arrays, dimensioned AR(NM,N) and AI(NM,N). C C On OUTPUT C C AR and AI contain the real and imaginary parts, respectively, C of the upper Hessenberg matrix. The multipliers which C were used in the reduction are stored in the remaining C triangles under the Hessenberg matrix. C C INT contains information on the rows and columns C interchanged in the reduction. Only elements LOW through C IGH are used. INT is a one-dimensional INTEGER array, C dimensioned INT(IGH). C C Calls CDIV for complex division. 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 CDIV 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 COMHES C INTEGER I,J,M,N,LA,NM,IGH,KP1,LOW,MM1,MP1 REAL AR(NM,*),AI(NM,*) REAL XR,XI,YR,YI INTEGER INT(*) C C***FIRST EXECUTABLE STATEMENT COMHES LA = IGH - 1 KP1 = LOW + 1 IF (LA .LT. KP1) GO TO 200 C DO 180 M = KP1, LA MM1 = M - 1 XR = 0.0E0 XI = 0.0E0 I = M C DO 100 J = M, IGH IF (ABS(AR(J,MM1)) + ABS(AI(J,MM1)) 1 .LE. ABS(XR) + ABS(XI)) GO TO 100 XR = AR(J,MM1) XI = AI(J,MM1) I = J 100 CONTINUE C INT(M) = I IF (I .EQ. M) GO TO 130 C .......... INTERCHANGE ROWS AND COLUMNS OF AR AND AI .......... DO 110 J = MM1, N YR = AR(I,J) AR(I,J) = AR(M,J) AR(M,J) = YR YI = AI(I,J) AI(I,J) = AI(M,J) AI(M,J) = YI 110 CONTINUE C DO 120 J = 1, IGH YR = AR(J,I) AR(J,I) = AR(J,M) AR(J,M) = YR YI = AI(J,I) AI(J,I) = AI(J,M) AI(J,M) = YI 120 CONTINUE C .......... END INTERCHANGE .......... 130 IF (XR .EQ. 0.0E0 .AND. XI .EQ. 0.0E0) GO TO 180 MP1 = M + 1 C DO 160 I = MP1, IGH YR = AR(I,MM1) YI = AI(I,MM1) IF (YR .EQ. 0.0E0 .AND. YI .EQ. 0.0E0) GO TO 160 CALL CDIV(YR,YI,XR,XI,YR,YI) AR(I,MM1) = YR AI(I,MM1) = YI C DO 140 J = M, N AR(I,J) = AR(I,J) - YR * AR(M,J) + YI * AI(M,J) AI(I,J) = AI(I,J) - YR * AI(M,J) - YI * AR(M,J) 140 CONTINUE C DO 150 J = 1, IGH AR(J,M) = AR(J,M) + YR * AR(J,I) - YI * AI(J,I) AI(J,M) = AI(J,M) + YR * AI(J,I) + YI * AR(J,I) 150 CONTINUE C 160 CONTINUE C 180 CONTINUE C 200 RETURN END