subroutine zhico(a,lda,n,kpvt,rcond,z) integer lda,n,kpvt(1) complex*16 a(lda,1),z(1) double precision rcond c c zhico factors a complex*16 hermitian matrix by elimination with c symmetric pivoting and estimates the condition of the matrix. c c if rcond is not needed, zhifa is slightly faster. c to solve a*x = b , follow zhico by zhisl. c to compute inverse(a)*c , follow zhico by zhisl. c to compute inverse(a) , follow zhico by zhidi. c to compute determinant(a) , follow zhico by zhidi. c to compute inertia(a), follow zhico by zhidi. c c on entry c c a complex*16(lda, n) c the hermitian 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 output 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*ctrans(u) c where u is a product of permutation and unit c upper triangular matrices , ctrans(u) is the c conjugate 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 complex*16(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 zhifa c blas zaxpy,zdotc,zdscal,dzasum c fortran dabs,dmax1,dcmplx,dconjg,iabs c c internal variables c complex*16 ak,akm1,bk,bkm1,zdotc,denom,ek,t double precision anorm,s,dzasum,ynorm integer i,info,j,jm1,k,kp,kps,ks c complex*16 zdum,zdum2,csign1 double precision cabs1 double precision dreal,dimag complex*16 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 zhifa(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)/dconjg(a(k-1,k)) akm1 = a(k-1,k-1)/a(k-1,k) bk = z(k)/dconjg(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 ctrans(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) + zdotc(k-1,a(1,k),1,z(1),1) if (ks .eq. 2) * z(k+1) = z(k+1) + zdotc(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)/dconjg(a(k-1,k)) akm1 = a(k-1,k-1)/a(k-1,k) bk = z(k)/dconjg(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 ctrans(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) + zdotc(k-1,a(1,k),1,z(1),1) if (ks .eq. 2) * z(k+1) = z(k+1) + zdotc(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