*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