- 12.
- If
**FACT**= 'E', then real scaling factors are computed to equilibrate the system:

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

- 13.
- If
**FACT**= 'N', the Cholesky decomposition is used to factor the matrix as

where is an upper triangular matrix and is a lower triangular matrix (). If**FACT**= 'E', the equilibrated matrix is factored as or . - 14.
- If the leading minor of order of (the equilibrated) is not positive definite, 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.
- 15.
**LA_POSVX**also optionally computes, for each solution vector , the estimated forward error bound and the componentwise relative backward error.