### 2.3.2 Stiff Problems: Backward Differentiation Formulas     continued...

Note that the smaller the solution component becomes, the harder RKSUITE works. Using a code designed specifically for stiff problems such as the code VODE presented in the next section, the number of function evaluations would have been approximately 120 for the range of -values considered.

A class of multi-step formulas which are highly effective in solving stiff problems are based on numerical differentiation. Again, we start by interpolating the previously computed solution values as well as the new one by a polynomial . The derivative of the solution at is then approximated by . The approximation is related to the differential equation by insisting that it satisfy the differential equation at :

Substituting for in this equation, we obtain the family of backward differentiation formulas, the BDFs:

These formulas were popularized by Gear, and are sometimes known as Gear's formulas. They are implicit like the Adams-Moulton formulas, but not as accurate for formulas of the same order and not stable for orders 7 and up. However, at the orders for which the formulas are stable, they are much more stable than the Adams-Moulton formulas. The formulas (52) cannot be evaluated by simple iteration because this restricts the step size just as much as stability does for much less stable formulas. In practice, a modified Newton iteration (Linear Algebra Chapter) is used to solve the nonlinear algebraic equations for ; this requires approximating partial derivatives and solving systems of linear equations.