subroutine cortb(nm,low,igh,ar,ai,ortr,orti,m,zr,zi)
c
integer i,j,m,la,mm,mp,nm,igh,kp1,low,mp1
double precision ar(nm,igh),ai(nm,igh),ortr(igh),orti(igh),
x zr(nm,m),zi(nm,m)
double precision h,gi,gr
c
c this subroutine is a translation of a complex analogue of
c the algol procedure ortbak, num. math. 12, 349-368(1968)
c 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 complex general
c matrix by back transforming those of the corresponding
c upper hessenberg matrix determined by corth.
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 cbal. if cbal has not been used,
c set low=1 and igh equal to the order of the matrix.
c
c ar and ai contain information about the unitary
c transformations used in the reduction by corth
c in their strict lower triangles.
c
c ortr and orti contain further information about the
c transformations used in the reduction by corth.
c only elements low through igh are used.
c
c m is the number of columns of zr and zi to be back transformed.
c
c zr and zi contain the real and imaginary parts,
c respectively, of the eigenvectors to be
c back transformed in their first m columns.
c
c on output
c
c zr and zi contain the real and imaginary parts,
c respectively, of the transformed eigenvectors
c in their first m columns.
c
c ortr and orti have been altered.
c
c note that cortb 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 (ar(mp,mp-1) .eq. 0.0d0 .and. ai(mp,mp-1) .eq. 0.0d0)
x go to 140
c .......... h below is negative of h formed in corth ..........
h = ar(mp,mp-1) * ortr(mp) + ai(mp,mp-1) * orti(mp)
mp1 = mp + 1
c
do 100 i = mp1, igh
ortr(i) = ar(i,mp-1)
orti(i) = ai(i,mp-1)
100 continue
c
do 130 j = 1, m
gr = 0.0d0
gi = 0.0d0
c
do 110 i = mp, igh
gr = gr + ortr(i) * zr(i,j) + orti(i) * zi(i,j)
gi = gi + ortr(i) * zi(i,j) - orti(i) * zr(i,j)
110 continue
c
gr = gr / h
gi = gi / h
c
do 120 i = mp, igh
zr(i,j) = zr(i,j) + gr * ortr(i) - gi * orti(i)
zi(i,j) = zi(i,j) + gr * orti(i) + gi * ortr(i)
120 continue
c
130 continue
c
140 continue
c
200 return
end