*DECK CORTB
SUBROUTINE CORTB (NM, LOW, IGH, AR, AI, ORTR, ORTI, M, ZR, ZI)
C***BEGIN PROLOGUE CORTB
C***PURPOSE Form the eigenvectors of a complex general matrix from
C eigenvectors of upper Hessenberg matrix output from
C CORTH.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C4
C***TYPE COMPLEX (ORTBAK-S, CORTB-C)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is a translation of a complex analogue of
C the ALGOL procedure ORTBAK, NUM. MATH. 12, 349-368(1968)
C 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 CORTH.
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 information about the unitary trans-
C formations used in the reduction by CORTH in their
C strict lower triangles. AR and AI are two-dimensional
C REAL arrays, dimensioned AR(NM,IGH) and AI(NM,IGH).
C
C ORTR and ORTI contain further information about the unitary
C transformations used in the reduction by CORTH. Only
C elements LOW through IGH are used. ORTR and ORTI are
C one-dimensional REAL arrays, dimensioned ORTR(IGH) and
C ORTI(IGH).
C
C M is the number of columns of Z=(ZR,ZI) 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
C M 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 ORTR and ORTI have been altered.
C
C Note that CORTB preserves vector Euclidean norms.
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 CORTB
C
INTEGER I,J,M,LA,MM,MP,NM,IGH,KP1,LOW,MP1
REAL AR(NM,*),AI(NM,*),ORTR(*),ORTI(*)
REAL ZR(NM,*),ZI(NM,*)
REAL H,GI,GR
C
C***FIRST EXECUTABLE STATEMENT CORTB
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
IF (AR(MP,MP-1) .EQ. 0.0E0 .AND. AI(MP,MP-1) .EQ. 0.0E0)
1 GO TO 140
C .......... H BELOW IS NEGATIVE OF H FORMED IN CORTH ..........
H = AR(MP,MP-1) * ORTR(MP) + AI(MP,MP-1) * ORTI(MP)
MP1 = MP + 1
C
DO 100 I = MP1, IGH
ORTR(I) = AR(I,MP-1)
ORTI(I) = AI(I,MP-1)
100 CONTINUE
C
DO 130 J = 1, M
GR = 0.0E0
GI = 0.0E0
C
DO 110 I = MP, IGH
GR = GR + ORTR(I) * ZR(I,J) + ORTI(I) * ZI(I,J)
GI = GI + ORTR(I) * ZI(I,J) - ORTI(I) * ZR(I,J)
110 CONTINUE
C
GR = GR / H
GI = GI / H
C
DO 120 I = MP, IGH
ZR(I,J) = ZR(I,J) + GR * ORTR(I) - GI * ORTI(I)
ZI(I,J) = ZI(I,J) + GR * ORTI(I) + GI * ORTR(I)
120 CONTINUE
C
130 CONTINUE
C
140 CONTINUE
C
200 RETURN
END