Description

5.

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:

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

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

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

6.

If FACT = 'N', the $L\,U$ decomposition with row interchanges is used to factor $A$ as $A = L\,U$, where $L$ is a product of permutation and unit lower triangular matrices with $kl$ subdiagonals, and $U$ is upper triangular with $kl + ku$ superdiagonals. If ${\bf FACT}$ = 'E', the equilibrated matrix is factored as $LU$.

7.

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.

8.

LA_GBSVX 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.