If obtaining the analytic derivatives is out of the question, viable alternatives remain. The gradient can be approximated by finite-differences of function values, such as
for suitably chosen intervals . Alternatively, automatic differentiation, essentially a new algebraic construct [19,35,53], may be used. In any case, these calculated derivatives may then be used in a gradient or quasi-Newton method. Such alternatives will generally provide significant improvement in computational cost and reliability, as will be discussed in the following sections.