*DECK COMBAK SUBROUTINE COMBAK (NM, LOW, IGH, AR, AI, INT, M, ZR, ZI) C***BEGIN PROLOGUE COMBAK C***PURPOSE Form the eigenvectors of a complex general matrix from the C eigenvectors of a upper Hessenberg matrix output from C COMHES. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4C4 C***TYPE COMPLEX (ELMBAK-S, COMBAK-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 COMBAK, 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 This subroutine forms the eigenvectors of a COMPLEX GENERAL C matrix by back transforming those of the corresponding C upper Hessenberg matrix determined by COMHES. C C On INPUT C C NM must be set to the row dimension of the two-dimensional C array parameters, AR, AI, ZR and ZI, as declared in the C calling program dimension statement. NM is an INTEGER C variable. 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. C C AR and AI contain the multipliers which were used in the C reduction by COMHES in their lower triangles below C the subdiagonal. AR and AI are two-dimensional REAL C arrays, dimensioned AR(NM,IGH) and AI(NM,IGH). C C INT contains information on the rows and columns C interchanged in the reduction by COMHES. Only C elements LOW through IGH are used. INT is a C one-dimensional INTEGER array, dimensioned INT(IGH). C C M is the number of eigenvectors to be back transformed. C M is an INTEGER variable. C C ZR and ZI contain the real and imaginary parts, respectively, C of the eigenvectors to be back transformed in their first M C columns. ZR and ZI are two-dimensional REAL arrays, C dimensioned ZR(NM,M) and ZI(NM,M). C C On OUTPUT C C ZR and ZI contain the real and imaginary parts, respectively, C of the transformed eigenvectors in their first M columns. 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 COMBAK C INTEGER I,J,M,LA,MM,MP,NM,IGH,KP1,LOW,MP1 REAL AR(NM,*),AI(NM,*),ZR(NM,*),ZI(NM,*) REAL XR,XI INTEGER INT(*) C C***FIRST EXECUTABLE STATEMENT COMBAK IF (M .EQ. 0) GO TO 200 LA = IGH - 1 KP1 = LOW + 1 IF (LA .LT. KP1) GO TO 200 C .......... FOR MP=IGH-1 STEP -1 UNTIL LOW+1 DO -- .......... DO 140 MM = KP1, LA MP = LOW + IGH - MM MP1 = MP + 1 C DO 110 I = MP1, IGH XR = AR(I,MP-1) XI = AI(I,MP-1) IF (XR .EQ. 0.0E0 .AND. XI .EQ. 0.0E0) GO TO 110 C DO 100 J = 1, M ZR(I,J) = ZR(I,J) + XR * ZR(MP,J) - XI * ZI(MP,J) ZI(I,J) = ZI(I,J) + XR * ZI(MP,J) + XI * ZR(MP,J) 100 CONTINUE C 110 CONTINUE C I = INT(MP) IF (I .EQ. MP) GO TO 140 C DO 130 J = 1, M XR = ZR(I,J) ZR(I,J) = ZR(MP,J) ZR(MP,J) = XR XI = ZI(I,J) ZI(I,J) = ZI(MP,J) ZI(MP,J) = XI 130 CONTINUE C 140 CONTINUE C 200 RETURN END