| c |
| c dpoco factors a double precision symmetric positive definite |
| c matrix and estimates the condition of the matrix. |
| c |
| c if rcond is not needed, dpofa is slightly faster. |
| c to solve a*x = b , follow dpoco by dposl. |
| c to compute inverse(a)*c , follow dpoco by dposl. |
| c to compute determinant(a) , follow dpoco by dpodi. |
| c to compute inverse(a) , follow dpoco by dpodi. |
| c |
| c on entry |
| c |
| c a double precision(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 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. if info .ne. 0 , rcond is unchanged. |
| 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 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 dpofa |
| c blas daxpy,ddot,dscal,dasum |
| c fortran dabs,dmax1,dreal,dsign |
| c |
| subroutine dpoco(a,lda,n,rcond,z,info) |
| integer lda,n,info |
| double precision a(lda,n),z(n) |
| double precision rcond |
| c |
| c internal variables |
| c |
| double precision ddot,ek,t,wk,wkm |
| double precision anorm,s,dasum,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) = dasum(j,a(1,j),1) |
| jm1 = j - 1 |
| if (jm1 .lt. 1) go to 20 |
| do 10 i = 1, jm1 |
| z(i) = z(i) + dabs(a(i,j)) |
| 10 continue |
| 20 continue |
| 30 continue |
| anorm = 0.0d0 |
| do 40 j = 1, n |
| anorm = dmax1(anorm,z(j)) |
| 40 continue |
| c |
| c factor |
| c |
| call dpofa(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 = 1.0d0 |
| do 50 j = 1, n |
| z(j) = 0.0d0 |
| 50 continue |
| do 110 k = 1, n |
| if (z(k) .ne. 0.0d0) ek = dsign(ek,-z(k)) |
| if (dabs(ek-z(k)) .le. a(k,k)) go to 60 |
| s = a(k,k)/dabs(ek-z(k)) |
| call dscal(n,s,z,1) |
| ek = s*ek |
| 60 continue |
| wk = ek - z(k) |
| wkm = -ek - z(k) |
| s = dabs(wk) |
| sm = dabs(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 + dabs(z(j)+wkm*a(k,j)) |
| z(j) = z(j) + wk*a(k,j) |
| s = s + dabs(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 = 1.0d0/dasum(n,z,1) |
| call dscal(n,s,z,1) |
| c |
| c solve r*y = w |
| c |
| do 130 kb = 1, n |
| k = n + 1 - kb |
| if (dabs(z(k)) .le. a(k,k)) go to 120 |
| s = a(k,k)/dabs(z(k)) |
| call dscal(n,s,z,1) |
| 120 continue |
| z(k) = z(k)/a(k,k) |
| t = -z(k) |
| call daxpy(k-1,t,a(1,k),1,z(1),1) |
| 130 continue |
| s = 1.0d0/dasum(n,z,1) |
| call dscal(n,s,z,1) |
| c |
| ynorm = 1.0d0 |
| c |
| c solve trans(r)*v = y |
| c |
| do 150 k = 1, n |
| z(k) = z(k) - ddot(k-1,a(1,k),1,z(1),1) |
| if (dabs(z(k)) .le. a(k,k)) go to 140 |
| s = a(k,k)/dabs(z(k)) |
| call dscal(n,s,z,1) |
| ynorm = s*ynorm |
| 140 continue |
| z(k) = z(k)/a(k,k) |
| 150 continue |
| s = 1.0d0/dasum(n,z,1) |
| call dscal(n,s,z,1) |
| ynorm = s*ynorm |
| c |
| c solve r*z = v |
| c |
| do 170 kb = 1, n |
| k = n + 1 - kb |
| if (dabs(z(k)) .le. a(k,k)) go to 160 |
| s = a(k,k)/dabs(z(k)) |
| call dscal(n,s,z,1) |
| ynorm = s*ynorm |
| 160 continue |
| z(k) = z(k)/a(k,k) |
| t = -z(k) |
| call daxpy(k-1,t,a(1,k),1,z(1),1) |
| 170 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 |
| 180 continue |
| return |
| end |
