subroutine ortbak(nm,low,igh,a,ort,m,z)
c
integer i,j,m,la,mm,mp,nm,igh,kp1,low,mp1
double precision a(nm,igh),ort(igh),z(nm,m)
double precision g
c
c this subroutine is a translation of the algol procedure ortbak,
c num. math. 12, 349-368(1968) by martin and wilkinson.
c handbook for auto. comp., vol.ii-linear algebra, 339-358(1971).
c
c this subroutine forms the eigenvectors of a real general
c matrix by back transforming those of the corresponding
c upper hessenberg matrix determined by orthes.
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 low and igh are integers determined by the balancing
c subroutine balanc. if balanc has not been used,
c set low=1 and igh equal to the order of the matrix.
c
c a contains information about the orthogonal trans-
c formations used in the reduction by orthes
c in its strict lower triangle.
c
c ort contains further information about the trans-
c formations used in the reduction by orthes.
c only elements low through igh are used.
c
c m is the number of columns of z to be back transformed.
c
c z contains the real and imaginary parts of the eigen-
c vectors to be back transformed in its first m columns.
c
c on output
c
c z contains the real and imaginary parts of the
c transformed eigenvectors in its first m columns.
c
c ort has been altered.
c
c note that ortbak 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
la = igh - 1
kp1 = low + 1
if (la .lt. kp1) go to 200
c .......... for mp=igh-1 step -1 until low+1 do -- ..........
do 140 mm = kp1, la
mp = low + igh - mm
if (a(mp,mp-1) .eq. 0.0d0) go to 140
mp1 = mp + 1
c
do 100 i = mp1, igh
100 ort(i) = a(i,mp-1)
c
do 130 j = 1, m
g = 0.0d0
c
do 110 i = mp, igh
110 g = g + ort(i) * z(i,j)
c .......... divisor below is negative of h formed in orthes.
c double division avoids possible underflow ..........
g = (g / ort(mp)) / a(mp,mp-1)
c
do 120 i = mp, igh
120 z(i,j) = z(i,j) + g * ort(i)
c
130 continue
c
140 continue
c
200 return
end