subroutine spoco(a,lda,n,rcond,z,info)
integer lda,n,info
real a(lda,1),z(1)
real rcond
c
c spoco factors a real symmetric positive definite matrix
c and estimates the condition of the matrix.
c
c if rcond is not needed, spofa is slightly faster.
c to solve a*x = b , follow spoco by sposl.
c to compute inverse(a)*c , follow spoco by sposl.
c to compute determinant(a) , follow spoco by spodi.
c to compute inverse(a) , follow spoco by spodi.
c
c on entry
c
c a real(lda, n)
c the symmetric matrix to be factored. only the
c diagonal and upper triangle are used.
c
c lda integer
c the leading dimension of the array a .
c
c n integer
c the order of the matrix a .
c
c on return
c
c a an upper triangular matrix r so that a = trans(r)*r
c where trans(r) is the transpose.
c the strict lower triangle is unaltered.
c if info .ne. 0 , the factorization is not complete.
c
c rcond real
c an estimate of the reciprocal condition of a .
c for the system a*x = b , relative perturbations
c in a and b of size epsilon may cause
c relative perturbations in x of size epsilon/rcond .
c if rcond is so small that the logical expression
c 1.0 + rcond .eq. 1.0
c is true, then a may be singular to working
c precision. in particular, rcond is zero if
c exact singularity is detected or the estimate
c underflows. if info .ne. 0 , rcond is unchanged.
c
c z real(n)
c a work vector whose contents are usually unimportant.
c if a is close to a singular matrix, then z is
c an approximate null vector in the sense that
c norm(a*z) = rcond*norm(a)*norm(z) .
c if info .ne. 0 , z is unchanged.
c
c info integer
c = 0 for normal return.
c = k signals an error condition. the leading minor
c of order k is not positive definite.
c
c linpack. this version dated 08/14/78 .
c cleve moler, university of new mexico, argonne national lab.
c
c subroutines and functions
c
c linpack spofa
c blas saxpy,sdot,sscal,sasum
c fortran abs,amax1,real,sign
c
c internal variables
c
real sdot,ek,t,wk,wkm
real anorm,s,sasum,sm,ynorm
integer i,j,jm1,k,kb,kp1
c
c
c find norm of a using only upper half
c
do 30 j = 1, n
z(j) = sasum(j,a(1,j),1)
jm1 = j - 1
if (jm1 .lt. 1) go to 20
do 10 i = 1, jm1
z(i) = z(i) + abs(a(i,j))
10 continue
20 continue
30 continue
anorm = 0.0e0
do 40 j = 1, n
anorm = amax1(anorm,z(j))
40 continue
c
c factor
c
call spofa(a,lda,n,info)
if (info .ne. 0) go to 180
c
c rcond = 1/(norm(a)*(estimate of norm(inverse(a)))) .
c estimate = norm(z)/norm(y) where a*z = y and a*y = e .
c the components of e are chosen to cause maximum local
c growth in the elements of w where trans(r)*w = e .
c the vectors are frequently rescaled to avoid overflow.
c
c solve trans(r)*w = e
c
ek = 1.0e0
do 50 j = 1, n
z(j) = 0.0e0
50 continue
do 110 k = 1, n
if (z(k) .ne. 0.0e0) ek = sign(ek,-z(k))
if (abs(ek-z(k)) .le. a(k,k)) go to 60
s = a(k,k)/abs(ek-z(k))
call sscal(n,s,z,1)
ek = s*ek
60 continue
wk = ek - z(k)
wkm = -ek - z(k)
s = abs(wk)
sm = abs(wkm)
wk = wk/a(k,k)
wkm = wkm/a(k,k)
kp1 = k + 1
if (kp1 .gt. n) go to 100
do 70 j = kp1, n
sm = sm + abs(z(j)+wkm*a(k,j))
z(j) = z(j) + wk*a(k,j)
s = s + abs(z(j))
70 continue
if (s .ge. sm) go to 90
t = wkm - wk
wk = wkm
do 80 j = kp1, n
z(j) = z(j) + t*a(k,j)
80 continue
90 continue
100 continue
z(k) = wk
110 continue
s = 1.0e0/sasum(n,z,1)
call sscal(n,s,z,1)
c
c solve r*y = w
c
do 130 kb = 1, n
k = n + 1 - kb
if (abs(z(k)) .le. a(k,k)) go to 120
s = a(k,k)/abs(z(k))
call sscal(n,s,z,1)
120 continue
z(k) = z(k)/a(k,k)
t = -z(k)
call saxpy(k-1,t,a(1,k),1,z(1),1)
130 continue
s = 1.0e0/sasum(n,z,1)
call sscal(n,s,z,1)
c
ynorm = 1.0e0
c
c solve trans(r)*v = y
c
do 150 k = 1, n
z(k) = z(k) - sdot(k-1,a(1,k),1,z(1),1)
if (abs(z(k)) .le. a(k,k)) go to 140
s = a(k,k)/abs(z(k))
call sscal(n,s,z,1)
ynorm = s*ynorm
140 continue
z(k) = z(k)/a(k,k)
150 continue
s = 1.0e0/sasum(n,z,1)
call sscal(n,s,z,1)
ynorm = s*ynorm
c
c solve r*z = v
c
do 170 kb = 1, n
k = n + 1 - kb
if (abs(z(k)) .le. a(k,k)) go to 160
s = a(k,k)/abs(z(k))
call sscal(n,s,z,1)
ynorm = s*ynorm
160 continue
z(k) = z(k)/a(k,k)
t = -z(k)
call saxpy(k-1,t,a(1,k),1,z(1),1)
170 continue
c make znorm = 1.0
s = 1.0e0/sasum(n,z,1)
call sscal(n,s,z,1)
ynorm = s*ynorm
c
if (anorm .ne. 0.0e0) rcond = ynorm/anorm
if (anorm .eq. 0.0e0) rcond = 0.0e0
180 continue
return
end