c
c zsico factors a double complex symmetric matrix by elimination with
c symmetric pivoting and estimates the condition of the matrix.
c
c if rcond is not needed, zsifa is slightly faster.
c to solve a*x = b , follow zsico by zsisl.
c to compute inverse(a)*c , follow zsico by zsisl.
c to compute inverse(a) , follow zsico by zsidi.
c to compute determinant(a) , follow zsico by zsidi.
c
c on entry
c
c a double complex(lda, n)
c the symmetric matrix to be factored.
c only the 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 a block diagonal matrix and the multipliers which
c were used to obtain it.
c the factorization can be written a = u*d*trans(u)
c where u is a product of permutation and unit
c upper triangular matrices , trans(u) is the
c transpose of u , and d is block diagonal
c with 1 by 1 and 2 by 2 blocks.
c
c kpvt integer(n)
c an integer vector of pivot indices.
c
c rcond double precision
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.
c
c z double complex(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
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 zsifa
c blas zaxpy,zdotu,zdscal,dzasum
c fortran dabs,dmax1,dcmplx,iabs
c
subroutine zsico(a,lda,n,kpvt,rcond,z)
integer lda,n,kpvt(*)
double complex a(lda,*),z(*)
double precision rcond
c
c internal variables
c
double complex ak,akm1,bk,bkm1,zdotu,denom,ek,t
double precision anorm,s,dzasum,ynorm
integer i,info,j,jm1,k,kp,kps,ks
c
double complex zdum,zdum2,csign1
double precision cabs1
double precision dreal,dimag
double complex zdumr,zdumi
dreal(zdumr) = zdumr
dimag(zdumi) = (0.0d0,-1.0d0)*zdumi
cabs1(zdum) = dabs(dreal(zdum)) + dabs(dimag(zdum))
csign1(zdum,zdum2) = cabs1(zdum)*(zdum2/cabs1(zdum2))
c
c find norm of a using only upper half
c
do 30 j = 1, n
z(j) = dcmplx(dzasum(j,a(1,j),1),0.0d0)
jm1 = j - 1
if (jm1 .lt. 1) go to 20
do 10 i = 1, jm1
z(i) = dcmplx(dreal(z(i))+cabs1(a(i,j)),0.0d0)
10 continue
20 continue
30 continue
anorm = 0.0d0
do 40 j = 1, n
anorm = dmax1(anorm,dreal(z(j)))
40 continue
c
c factor
c
call zsifa(a,lda,n,kpvt,info)
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 u*d*w = e .
c the vectors are frequently rescaled to avoid overflow.
c
c solve u*d*w = e
c
ek = (1.0d0,0.0d0)
do 50 j = 1, n
z(j) = (0.0d0,0.0d0)
50 continue
k = n
60 if (k .eq. 0) go to 120
ks = 1
if (kpvt(k) .lt. 0) ks = 2
kp = iabs(kpvt(k))
kps = k + 1 - ks
if (kp .eq. kps) go to 70
t = z(kps)
z(kps) = z(kp)
z(kp) = t
70 continue
if (cabs1(z(k)) .ne. 0.0d0) ek = csign1(ek,z(k))
z(k) = z(k) + ek
call zaxpy(k-ks,z(k),a(1,k),1,z(1),1)
if (ks .eq. 1) go to 80
if (cabs1(z(k-1)) .ne. 0.0d0) ek = csign1(ek,z(k-1))
z(k-1) = z(k-1) + ek
call zaxpy(k-ks,z(k-1),a(1,k-1),1,z(1),1)
80 continue
if (ks .eq. 2) go to 100
if (cabs1(z(k)) .le. cabs1(a(k,k))) go to 90
s = cabs1(a(k,k))/cabs1(z(k))
call zdscal(n,s,z,1)
ek = dcmplx(s,0.0d0)*ek
90 continue
if (cabs1(a(k,k)) .ne. 0.0d0) z(k) = z(k)/a(k,k)
if (cabs1(a(k,k)) .eq. 0.0d0) z(k) = (1.0d0,0.0d0)
go to 110
100 continue
ak = a(k,k)/a(k-1,k)
akm1 = a(k-1,k-1)/a(k-1,k)
bk = z(k)/a(k-1,k)
bkm1 = z(k-1)/a(k-1,k)
denom = ak*akm1 - 1.0d0
z(k) = (akm1*bk - bkm1)/denom
z(k-1) = (ak*bkm1 - bk)/denom
110 continue
k = k - ks
go to 60
120 continue
s = 1.0d0/dzasum(n,z,1)
call zdscal(n,s,z,1)
c
c solve trans(u)*y = w
c
k = 1
130 if (k .gt. n) go to 160
ks = 1
if (kpvt(k) .lt. 0) ks = 2
if (k .eq. 1) go to 150
z(k) = z(k) + zdotu(k-1,a(1,k),1,z(1),1)
if (ks .eq. 2)
* z(k+1) = z(k+1) + zdotu(k-1,a(1,k+1),1,z(1),1)
kp = iabs(kpvt(k))
if (kp .eq. k) go to 140
t = z(k)
z(k) = z(kp)
z(kp) = t
140 continue
150 continue
k = k + ks
go to 130
160 continue
s = 1.0d0/dzasum(n,z,1)
call zdscal(n,s,z,1)
c
ynorm = 1.0d0
c
c solve u*d*v = y
c
k = n
170 if (k .eq. 0) go to 230
ks = 1
if (kpvt(k) .lt. 0) ks = 2
if (k .eq. ks) go to 190
kp = iabs(kpvt(k))
kps = k + 1 - ks
if (kp .eq. kps) go to 180
t = z(kps)
z(kps) = z(kp)
z(kp) = t
180 continue
call zaxpy(k-ks,z(k),a(1,k),1,z(1),1)
if (ks .eq. 2) call zaxpy(k-ks,z(k-1),a(1,k-1),1,z(1),1)
190 continue
if (ks .eq. 2) go to 210
if (cabs1(z(k)) .le. cabs1(a(k,k))) go to 200
s = cabs1(a(k,k))/cabs1(z(k))
call zdscal(n,s,z,1)
ynorm = s*ynorm
200 continue
if (cabs1(a(k,k)) .ne. 0.0d0) z(k) = z(k)/a(k,k)
if (cabs1(a(k,k)) .eq. 0.0d0) z(k) = (1.0d0,0.0d0)
go to 220
210 continue
ak = a(k,k)/a(k-1,k)
akm1 = a(k-1,k-1)/a(k-1,k)
bk = z(k)/a(k-1,k)
bkm1 = z(k-1)/a(k-1,k)
denom = ak*akm1 - 1.0d0
z(k) = (akm1*bk - bkm1)/denom
z(k-1) = (ak*bkm1 - bk)/denom
220 continue
k = k - ks
go to 170
230 continue
s = 1.0d0/dzasum(n,z,1)
call zdscal(n,s,z,1)
ynorm = s*ynorm
c
c solve trans(u)*z = v
c
k = 1
240 if (k .gt. n) go to 270
ks = 1
if (kpvt(k) .lt. 0) ks = 2
if (k .eq. 1) go to 260
z(k) = z(k) + zdotu(k-1,a(1,k),1,z(1),1)
if (ks .eq. 2)
* z(k+1) = z(k+1) + zdotu(k-1,a(1,k+1),1,z(1),1)
kp = iabs(kpvt(k))
if (kp .eq. k) go to 250
t = z(k)
z(k) = z(kp)
z(kp) = t
250 continue
260 continue
k = k + ks
go to 240
270 continue
c make znorm = 1.0
s = 1.0d0/dzasum(n,z,1)
call zdscal(n,s,z,1)
ynorm = s*ynorm
c
if (anorm .ne. 0.0d0) rcond = ynorm/anorm
if (anorm .eq. 0.0d0) rcond = 0.0d0
return
end