subroutine trbak1(nm,n,a,e,m,z)
c
      integer i,j,k,l,m,n,nm
      double precision a(nm,n),e(n),z(nm,m)
      double precision s
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 two-dimensional
c          array parameters as declared in the calling program
c          dimension statement.
c
c        n is the order of the matrix.
c
c        a contains information about the orthogonal trans-
c          formations used in the reduction by  tred1
c          in its strict lower triangle.
c
c        e contains the subdiagonal elements of the tridiagonal
c          matrix in its last n-1 positions.  e(1) is arbitrary.
c
c        m is the number of eigenvectors to be back transformed.
c
c        z contains the eigenvectors to be back transformed
c          in its first m columns.
c
c     on output
c
c        z contains the transformed eigenvectors
c          in its first m columns.
c
c     note that trbak1 preserves vector euclidean norms.
c
c     questions and comments should be directed to burton s. garbow,
c     mathematics and computer science div, argonne national laboratory
c
c     this version dated august 1983.
c
c     ------------------------------------------------------------------
c
      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. 0.0d0) go to 140
c
         do 130 j = 1, m
            s = 0.0d0
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