Singular or Ill-Conditioned Jacobian

The Jacobian is factorized in the solution of the system (7.38). The original solvers used Gauss factorization (or its PLR variant) since it is the least computationally onerous later on, however, many authors recommended QR factorization because 

Whichever factorization is adopted, it is mandatory that the algorithm be able to correctly identify whether the matrix is ill-conditioned and, if it is, the solution must have a reasonably small norm (see Section 8.2). 

Two further differences from multidimensional optimization should also be highlighted  

1) It is always necessary to evaluate the condition number in the solution of nonlinear systems. Conversely, in optimization problems, this is not required as simply checking that the diagonal elements are kept positive during the factorization, suffices. 

2) If PLR factorization (the most efficient method) is adopted, the Jacobian cannot be modified when it is singular, analogous to the singular or nonpositive definite Hessian. 

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