subroutine dgeco(a,lda,n,ipvt,rcond,z) integer lda,n,ipvt(1) double precision a(lda,1),z(1) double precision rcond c c dgeco factors a double precision matrix by gaussian elimination c and estimates the condition of the matrix. c c if rcond is not needed, dgefa is slightly faster. c to solve a*x = b , follow dgeco by dgesl. c to compute inverse(a)*c , follow dgeco by dgesl. c to compute determinant(a) , follow dgeco by dgedi. c to compute inverse(a) , follow dgeco by dgedi. c c on entry c c a double precision(lda, n) c the matrix to be factored. 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 and the multipliers c which were used to obtain it. c the factorization can be written a = l*u where c l is a product of permutation and unit lower c triangular matrices and u is upper triangular. c c ipvt 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 precision(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 dgefa c blas daxpy,ddot,dscal,dasum c fortran dabs,dmax1,dsign c c internal variables c double precision ddot,ek,t,wk,wkm double precision anorm,s,dasum,sm,ynorm integer info,j,k,kb,kp1,l c c c compute 1-norm of a c anorm = 0.0d0 do 10 j = 1, n anorm = dmax1(anorm,dasum(n,a(1,j),1)) 10 continue c c factor c call dgefa(a,lda,n,ipvt,info) c c rcond = 1/(norm(a)*(estimate of norm(inverse(a)))) . c estimate = norm(z)/norm(y) where a*z = y and trans(a)*y = e . c trans(a) is the transpose of a . the components of e are c chosen to cause maximum local growth in the elements of w where c trans(u)*w = e . the vectors are frequently rescaled to avoid c overflow. c c solve trans(u)*w = e c ek = 1.0d0 do 20 j = 1, n z(j) = 0.0d0 20 continue do 100 k = 1, n if (z(k) .ne. 0.0d0) ek = dsign(ek,-z(k)) if (dabs(ek-z(k)) .le. dabs(a(k,k))) go to 30 s = dabs(a(k,k))/dabs(ek-z(k)) call dscal(n,s,z,1) ek = s*ek 30 continue wk = ek - z(k) wkm = -ek - z(k) s = dabs(wk) sm = dabs(wkm) if (a(k,k) .eq. 0.0d0) go to 40 wk = wk/a(k,k) wkm = wkm/a(k,k) go to 50 40 continue wk = 1.0d0 wkm = 1.0d0 50 continue kp1 = k + 1 if (kp1 .gt. n) go to 90 do 60 j = kp1, n sm = sm + dabs(z(j)+wkm*a(k,j)) z(j) = z(j) + wk*a(k,j) s = s + dabs(z(j)) 60 continue if (s .ge. sm) go to 80 t = wkm - wk wk = wkm do 70 j = kp1, n z(j) = z(j) + t*a(k,j) 70 continue 80 continue 90 continue z(k) = wk 100 continue s = 1.0d0/dasum(n,z,1) call dscal(n,s,z,1) c c solve trans(l)*y = w c do 120 kb = 1, n k = n + 1 - kb if (k .lt. n) z(k) = z(k) + ddot(n-k,a(k+1,k),1,z(k+1),1) if (dabs(z(k)) .le. 1.0d0) go to 110 s = 1.0d0/dabs(z(k)) call dscal(n,s,z,1) 110 continue l = ipvt(k) t = z(l) z(l) = z(k) z(k) = t 120 continue s = 1.0d0/dasum(n,z,1) call dscal(n,s,z,1) c ynorm = 1.0d0 c c solve l*v = y c do 140 k = 1, n l = ipvt(k) t = z(l) z(l) = z(k) z(k) = t if (k .lt. n) call daxpy(n-k,t,a(k+1,k),1,z(k+1),1) if (dabs(z(k)) .le. 1.0d0) go to 130 s = 1.0d0/dabs(z(k)) call dscal(n,s,z,1) ynorm = s*ynorm 130 continue 140 continue s = 1.0d0/dasum(n,z,1) call dscal(n,s,z,1) ynorm = s*ynorm c c solve u*z = v c do 160 kb = 1, n k = n + 1 - kb if (dabs(z(k)) .le. dabs(a(k,k))) go to 150 s = dabs(a(k,k))/dabs(z(k)) call dscal(n,s,z,1) ynorm = s*ynorm 150 continue if (a(k,k) .ne. 0.0d0) z(k) = z(k)/a(k,k) if (a(k,k) .eq. 0.0d0) z(k) = 1.0d0 t = -z(k) call daxpy(k-1,t,a(1,k),1,z(1),1) 160 continue c make znorm = 1.0 s = 1.0d0/dasum(n,z,1) call dscal(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