*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