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subroutine tred2(nm,n,a,d,e,z)
c
integer i,j,k,l,n,ii,nm,jp1
double precision a(nm,n),d(n),e(n),z(nm,n)
double precision f,g,h,hh,scale
c
c this subroutine is a translation of the algol procedure tred2,
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 reduces a real symmetric matrix to a
c symmetric tridiagonal matrix using and accumulating
c orthogonal similarity transformations.
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 the real symmetric input matrix. only the
c lower triangle of the matrix need be supplied.
c
c on output
c
c d contains the diagonal elements of the tridiagonal matrix.
c
c e contains the subdiagonal elements of the tridiagonal
c matrix in its last n-1 positions. e(1) is set to zero.
c
c z contains the orthogonal transformation matrix
c produced in the reduction.
c
c a and z may coincide. if distinct, a is unaltered.
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
do 100 i = 1, n
c
do 80 j = i, n
80 z(j,i) = a(j,i)
c
d(i) = a(n,i)
100 continue
c
if (n .eq. 1) go to 510
c .......... for i=n step -1 until 2 do -- ..........
do 300 ii = 2, n
i = n + 2 - ii
l = i - 1
h = 0.0d0
scale = 0.0d0
if (l .lt. 2) go to 130
c .......... scale row (algol tol then not needed) ..........
do 120 k = 1, l
120 scale = scale + abs(d(k))
c
if (scale .ne. 0.0d0) go to 140
130 e(i) = d(l)
c
do 135 j = 1, l
d(j) = z(l,j)
z(i,j) = 0.0d0
z(j,i) = 0.0d0
135 continue
c
go to 290
c
140 do 150 k = 1, l
d(k) = d(k) / scale
h = h + d(k) * d(k)
150 continue
c
f = d(l)
g = -sign(sqrt(h),f)
e(i) = scale * g
h = h - f * g
d(l) = f - g
c .......... form a*u ..........
do 170 j = 1, l
170 e(j) = 0.0d0
c
do 240 j = 1, l
f = d(j)
z(j,i) = f
g = e(j) + z(j,j) * f
jp1 = j + 1
if (l .lt. jp1) go to 220
c
do 200 k = jp1, l
g = g + z(k,j) * d(k)
e(k) = e(k) + z(k,j) * f
200 continue
c
220 e(j) = g
240 continue
c .......... form p ..........
f = 0.0d0
c
do 245 j = 1, l
e(j) = e(j) / h
f = f + e(j) * d(j)
245 continue
c
hh = f / (h + h)
c .......... form q ..........
do 250 j = 1, l
250 e(j) = e(j) - hh * d(j)
c .......... form reduced a ..........
do 280 j = 1, l
f = d(j)
g = e(j)
c
do 260 k = j, l
260 z(k,j) = z(k,j) - f * e(k) - g * d(k)
c
d(j) = z(l,j)
z(i,j) = 0.0d0
280 continue
c
290 d(i) = h
300 continue
c .......... accumulation of transformation matrices ..........
do 500 i = 2, n
l = i - 1
z(n,l) = z(l,l)
z(l,l) = 1.0d0
h = d(i)
if (h .eq. 0.0d0) go to 380
c
do 330 k = 1, l
330 d(k) = z(k,i) / h
c
do 360 j = 1, l
g = 0.0d0
c
do 340 k = 1, l
340 g = g + z(k,i) * z(k,j)
c
do 360 k = 1, l
z(k,j) = z(k,j) - g * d(k)
360 continue
c
380 do 400 k = 1, l
400 z(k,i) = 0.0d0
c
500 continue
c
510 do 520 i = 1, n
d(i) = z(n,i)
z(n,i) = 0.0d0
520 continue
c
z(n,n) = 1.0d0
e(1) = 0.0d0
return
end
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