Description

1.

If ${\bf FACT}$ = 'E', then real row scaling factors $R_i$ and/or real column scaling factors $C_i$ are computed to equilibrate the system. The form of the equilibrated system depends on the value of TRANS:

$$\begin{array}{c\vert c} {\bf TRANS} & \mbox{ The equilibrate... ...hit{diag}(R)^{-1}X] =\mathit{diag}(C)\:B \\ \hline \end{array}$$

Depending on the value of EQUED determined during the equilibration, the matrices $\mathit{diag}(R)$ and/or $\mathit{diag}(C)$ may be implicitly the identity matrix:

$$\begin{array}{c\vert c\vert c} {\bf EQUED} & \mathit{diag}(R... ...'} & \mathit{diag}(R) & \mathit{diag}(C) \\ \hline \end{array}$$

2.

If ${\bf FACT}$ = 'N', matrix $A$ is factored as $A=P L U$, where $P$ is a permutation matrix, $L$ is unit lower triangular, and $U$ is upper triangular. If ${\bf FACT}$ = 'E', the equilibrated matrix is factored as $PLU$.

3.

If some $U_{i,i}=0$, so that $U$ is singular, 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.

4.

LA_GESVX also optionally computes the reciprocal pivot growth factor and, for each solution vector $X_j$, the estimated forward error bound and the componentwise relative backward error.