Description

27.

If FACT = 'N', a diagonal pivoting method is used to factor the matrix $A$ as

$$A = U\, D\, U^T \mbox{ if {\bf UPLO} = 'U', or } A = L\, D\,... ...box{ if {\bf UPLO} = 'L'}\hspace{1.50 cm}(\mbox{\bf LA\_SYSVX})$$

or

$$A = U\, D\, U^H \mbox{ if {\bf UPLO} = 'U', or } A = L\, D\,... ...box{ if {\bf UPLO} = 'L'}\hspace{1.50 cm}(\mbox{\bf LA\_HESVX})$$

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric (LA_SYSVX) or Hermitian (LA_HESVX) and block diagonal with $1 \times 1$ and $2 \times 2$ diagonal blocks.

28.

If some $D_{i,i}=0$, so that $D$ is singular, then the routine returns with ${\bf INFO} = i$. Otherwise, the factored form of $A$ is used to estimate the condition number of the matrix $A$. If the reciprocal of the condition number is less than machine precision, ${\bf INFO} = n+1$, where $n$ is the order of $A$, is returned as a warning. However, the routine still goes on to solve for $X$. Iterative refinement is applied to improve the computed solution.

29.

LA_SYSVX and LA_HESVX also optionally compute, for each solution vector $X_j$, the estimated forward error bound and the componentwise relative backward error.