- 5.
- If = 'E', then real row scaling factors
and/or real column scaling factors are computed to equilibrate
the system. The form of the equilibrated system depends on the
value of
**TRANS**:

Depending on the value of**EQUED**determined during the equilibration, the matrices and/or may be implicitly the identity matrix:

- 6.
- If
**FACT**= 'N', the decomposition with row interchanges is used to factor as , where is a product of permutation and unit lower triangular matrices with subdiagonals, and is upper triangular with superdiagonals. If = 'E', the equilibrated matrix is factored as . - 7.
- If some , so that is singular, then the routine returns with . Otherwise, an estimate of the condition number of (the equilibrated) is found using the above factorization. If the reciprocal of the condition number is less than machine precision, , where is the order of , is returned as a warning. However, the routine still goes on to solve for . 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 , the estimated forward error bound and the componentwise relative backward error.