Description

12.

If FACT = 'E', then real scaling factors $S_i$ are computed to equilibrate the system:

$$[diag(S) A\: diag(S)][diag(S)^{-1} X] = diag(S) B$$

Depending on the value of EQUED determined during the equilibration, the matrix $diag(S)$ may be implicitly the identity matrix:

$$\begin{array}{c\vert c} {\bf EQUED} & diag({\bf S}) \\ \hlin... ...ty \\ \hline \mbox{'Y'} & diag({\bf S}) \\ \hline \end{array}$$

13.

If FACT = 'N', the Cholesky decomposition is used to factor the matrix $A$ as

$$A = U^H U \mbox{ if {\bf UPLO} = 'U', or } A = L\:L^H \mbox{ if {\bf UPLO} = 'L'}$$

where $U$ is an upper triangular matrix and $L$ is a lower triangular matrix ($L=U^H$). If FACT = 'E', the equilibrated matrix is factored as $U^HU$ or $LL^H$.

14.

If the leading minor of order $i$ of (the equilibrated) $A$ is not positive definite, then the routine returns with ${\bf INFO} = i$. Otherwise, an estimate of the condition number of (the equilibrated) $A$ is found using the above factorization. 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.

15.

LA_POSVX also optionally computes, for each solution vector $X_j$, the estimated forward error bound and the componentwise relative backward error.