*DECK TRBAK1 SUBROUTINE TRBAK1 (NM, N, A, E, M, Z) C***BEGIN PROLOGUE TRBAK1 C***PURPOSE Form the eigenvectors of real symmetric matrix from C the eigenvectors of a symmetric tridiagonal matrix formed C by TRED1. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4C4 C***TYPE SINGLE PRECISION (TRBAK1-S) C***KEYWORDS EIGENVECTORS OF A REAL SYMMETRIC MATRIX, EISPACK C***AUTHOR Smith, B. T., et al. C***DESCRIPTION C C This subroutine is a translation of the ALGOL procedure TRBAK1, C NUM. MATH. 11, 181-195(1968) 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 REAL SYMMETRIC C matrix by back transforming those of the corresponding C symmetric tridiagonal matrix determined by TRED1. C C On Input C C NM must be set to the row dimension of the two-dimensional C array parameters, A and Z, as declared in the calling C program dimension statement. NM is an INTEGER 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 A contains information about the orthogonal transformations C used in the reduction by TRED1 in its strict lower C triangle. A is a two-dimensional REAL array, dimensioned C A(NM,N). C C E contains the subdiagonal elements of the tridiagonal matrix C in its last N-1 positions. E(1) is arbitrary. These C elements provide the remaining information about the C orthogonal transformations. E is a one-dimensional REAL C array, dimensioned E(N). C C M is the number of columns of Z to be back transformed. C M is an INTEGER variable. C C Z contains the eigenvectors to be back transformed in its C first M columns. Z is a two-dimensional REAL array, C dimensioned Z(NM,M). C C On Output C C Z contains the transformed eigenvectors in its first M columns. C C Note that TRBAK1 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 TRBAK1 C INTEGER I,J,K,L,M,N,NM REAL A(NM,*),E(*),Z(NM,*) REAL S C C***FIRST EXECUTABLE STATEMENT TRBAK1 IF (M .EQ. 0) GO TO 200 IF (N .EQ. 1) GO TO 200 C DO 140 I = 2, N L = I - 1 IF (E(I) .EQ. 0.0E0) GO TO 140 C DO 130 J = 1, M S = 0.0E0 C DO 110 K = 1, L 110 S = S + A(I,K) * Z(K,J) C .......... DIVISOR BELOW IS NEGATIVE OF H FORMED IN TRED1. C DOUBLE DIVISION AVOIDS POSSIBLE UNDERFLOW .......... S = (S / A(I,L)) / E(I) C DO 120 K = 1, L 120 Z(K,J) = Z(K,J) + S * A(I,K) C 130 CONTINUE C 140 CONTINUE C 200 RETURN END