subroutine imtqlv(n,d,e,e2,w,ind,ierr,rv1)
c
integer i,j,k,l,m,n,ii,mml,tag,ierr
real d(n),e(n),e2(n),w(n),rv1(n)
real b,c,f,g,p,r,s,tst1,tst2,pythag
integer ind(n)
c
c this subroutine is a variant of imtql1 which is a translation of
c algol procedure imtql1, num. math. 12, 377-383(1968) by martin and
c wilkinson, as modified in num. math. 15, 450(1970) by dubrulle.
c handbook for auto. comp., vol.ii-linear algebra, 241-248(1971).
c
c this subroutine finds the eigenvalues of a symmetric tridiagonal
c matrix by the implicit ql method and associates with them
c their corresponding submatrix indices.
c
c on input
c
c n is the order of the matrix.
c
c d contains the diagonal elements of the input matrix.
c
c e contains the subdiagonal elements of the input matrix
c in its last n-1 positions. e(1) is arbitrary.
c
c e2 contains the squares of the corresponding elements of e.
c e2(1) is arbitrary.
c
c on output
c
c d and e are unaltered.
c
c elements of e2, corresponding to elements of e regarded
c as negligible, have been replaced by zero causing the
c matrix to split into a direct sum of submatrices.
c e2(1) is also set to zero.
c
c w contains the eigenvalues in ascending order. if an
c error exit is made, the eigenvalues are correct and
c ordered for indices 1,2,...ierr-1, but may not be
c the smallest eigenvalues.
c
c ind contains the submatrix indices associated with the
c corresponding eigenvalues in w -- 1 for eigenvalues
c belonging to the first submatrix from the top,
c 2 for those belonging to the second submatrix, etc..
c
c ierr is set to
c zero for normal return,
c j if the j-th eigenvalue has not been
c determined after 30 iterations.
c
c rv1 is a temporary storage array.
c
c calls pythag for sqrt(a*a + b*b) .
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
ierr = 0
k = 0
tag = 0
c
do 100 i = 1, n
w(i) = d(i)
if (i .ne. 1) rv1(i-1) = e(i)
100 continue
c
e2(1) = 0.0e0
rv1(n) = 0.0e0
c
do 290 l = 1, n
j = 0
c .......... look for small sub-diagonal element ..........
105 do 110 m = l, n
if (m .eq. n) go to 120
tst1 = abs(w(m)) + abs(w(m+1))
tst2 = tst1 + abs(rv1(m))
if (tst2 .eq. tst1) go to 120
c .......... guard against underflowed element of e2 ..........
if (e2(m+1) .eq. 0.0e0) go to 125
110 continue
c
120 if (m .le. k) go to 130
if (m .ne. n) e2(m+1) = 0.0e0
125 k = m
tag = tag + 1
130 p = w(l)
if (m .eq. l) go to 215
if (j .eq. 30) go to 1000
j = j + 1
c .......... form shift ..........
g = (w(l+1) - p) / (2.0e0 * rv1(l))
r = pythag(g,1.0e0)
g = w(m) - p + rv1(l) / (g + sign(r,g))
s = 1.0e0
c = 1.0e0
p = 0.0e0
mml = m - l
c .......... for i=m-1 step -1 until l do -- ..........
do 200 ii = 1, mml
i = m - ii
f = s * rv1(i)
b = c * rv1(i)
r = pythag(f,g)
rv1(i+1) = r
if (r .eq. 0.0e0) go to 210
s = f / r
c = g / r
g = w(i+1) - p
r = (w(i) - g) * s + 2.0e0 * c * b
p = s * r
w(i+1) = g + p
g = c * r - b
200 continue
c
w(l) = w(l) - p
rv1(l) = g
rv1(m) = 0.0e0
go to 105
c .......... recover from underflow ..........
210 w(i+1) = w(i+1) - p
rv1(m) = 0.0e0
go to 105
c .......... order eigenvalues ..........
215 if (l .eq. 1) go to 250
c .......... for i=l step -1 until 2 do -- ..........
do 230 ii = 2, l
i = l + 2 - ii
if (p .ge. w(i-1)) go to 270
w(i) = w(i-1)
ind(i) = ind(i-1)
230 continue
c
250 i = 1
270 w(i) = p
ind(i) = tag
290 continue
c
go to 1001
c .......... set error -- no convergence to an
c eigenvalue after 30 iterations ..........
1000 ierr = l
1001 return
end