*DECK TRBAK3
SUBROUTINE TRBAK3 (NM, N, NV, A, M, Z)
C***BEGIN PROLOGUE TRBAK3
C***PURPOSE Form the eigenvectors of a real symmetric matrix from the
C eigenvectors of a symmetric tridiagonal matrix formed
C by TRED3.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C4
C***TYPE SINGLE PRECISION (TRBAK3-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 TRBAK3,
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 TRED3.
C
C On Input
C
C NM must be set to the row dimension of the two-dimensional
C array parameter, Z, as declared in the calling program
C 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 NV is an INTEGER variable set equal to the dimension of the
C array A as specified in the calling program. NV must not
C be less than N*(N+1)/2.
C
C A contains information about the orthogonal transformations
C used in the reduction by TRED3 in its first N*(N+1)/2
C positions. A is a one-dimensional REAL array, dimensioned
C A(NV).
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 TRBAK3 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 TRBAK3
C
INTEGER I,J,K,L,M,N,IK,IZ,NM,NV
REAL A(*),Z(NM,*)
REAL H,S
C
C***FIRST EXECUTABLE STATEMENT TRBAK3
IF (M .EQ. 0) GO TO 200
IF (N .EQ. 1) GO TO 200
C
DO 140 I = 2, N
L = I - 1
IZ = (I * L) / 2
IK = IZ + I
H = A(IK)
IF (H .EQ. 0.0E0) GO TO 140
C
DO 130 J = 1, M
S = 0.0E0
IK = IZ
C
DO 110 K = 1, L
IK = IK + 1
S = S + A(IK) * Z(K,J)
110 CONTINUE
C .......... DOUBLE DIVISION AVOIDS POSSIBLE UNDERFLOW ..........
S = (S / H) / H
IK = IZ
C
DO 120 K = 1, L
IK = IK + 1
Z(K,J) = Z(K,J) - S * A(IK)
120 CONTINUE
C
130 CONTINUE
C
140 CONTINUE
C
200 RETURN
END