*DECK HTRIBK
SUBROUTINE HTRIBK (NM, N, AR, AI, TAU, M, ZR, ZI)
C***BEGIN PROLOGUE HTRIBK
C***PURPOSE Form the eigenvectors of a complex Hermitian matrix from
C the eigenvectors of a real symmetric tridiagonal matrix
C output from HTRIDI.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C4
C***TYPE SINGLE PRECISION (HTRIBK-S)
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 TRBAK1, NUM. MATH. 11, 181-195(1968)
C by Martin, Reinsch, and Wilkinson.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
C
C This subroutine forms the eigenvectors of a COMPLEX HERMITIAN
C matrix by back transforming those of the corresponding
C real symmetric tridiagonal matrix determined by HTRIDI.
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 N is the order of the matrix. N is an INTEGER variable.
C N must be less than or equal to NM.
C
C AR and AI contain some information about the unitary
C transformations used in the reduction by HTRIDI in the
C strict lower triangle of AR and the full lower triangle of
C AI. The remaining upper parts of the matrices are arbitrary.
C AR and AI are two-dimensional REAL arrays, dimensioned
C AR(NM,N) and AI(NM,N).
C
C TAU contains further information about the transformations.
C TAU is a one-dimensional REAL array, dimensioned TAU(2,N).
C
C M is the number of eigenvectors to be back transformed.
C M is an INTEGER variable.
C
C ZR contains the eigenvectors to be back transformed in its first
C M columns. The contents of ZI are immaterial. ZR and ZI are
C two-dimensional REAL arrays, dimensioned ZR(NM,M) and
C 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 Note that the last component of each returned vector
C is real and that vector Euclidean norms are preserved.
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 HTRIBK
C
INTEGER I,J,K,L,M,N,NM
REAL AR(NM,*),AI(NM,*),TAU(2,*),ZR(NM,*),ZI(NM,*)
REAL H,S,SI
C
C***FIRST EXECUTABLE STATEMENT HTRIBK
IF (M .EQ. 0) GO TO 200
C .......... TRANSFORM THE EIGENVECTORS OF THE REAL SYMMETRIC
C TRIDIAGONAL MATRIX TO THOSE OF THE HERMITIAN
C TRIDIAGONAL MATRIX. ..........
DO 50 K = 1, N
C
DO 50 J = 1, M
ZI(K,J) = -ZR(K,J) * TAU(2,K)
ZR(K,J) = ZR(K,J) * TAU(1,K)
50 CONTINUE
C
IF (N .EQ. 1) GO TO 200
C .......... RECOVER AND APPLY THE HOUSEHOLDER MATRICES ..........
DO 140 I = 2, N
L = I - 1
H = AI(I,I)
IF (H .EQ. 0.0E0) GO TO 140
C
DO 130 J = 1, M
S = 0.0E0
SI = 0.0E0
C
DO 110 K = 1, L
S = S + AR(I,K) * ZR(K,J) - AI(I,K) * ZI(K,J)
SI = SI + AR(I,K) * ZI(K,J) + AI(I,K) * ZR(K,J)
110 CONTINUE
C .......... DOUBLE DIVISIONS AVOID POSSIBLE UNDERFLOW ..........
S = (S / H) / H
SI = (SI / H) / H
C
DO 120 K = 1, L
ZR(K,J) = ZR(K,J) - S * AR(I,K) - SI * AI(I,K)
ZI(K,J) = ZI(K,J) - SI * AR(I,K) + S * AI(I,K)
120 CONTINUE
C
130 CONTINUE
C
140 CONTINUE
C
200 RETURN
END