 Backward differentiation formula

A backward differentiation formula is an implicit linear multistep method obtained by interpolating recent solution values and matching the interpolant's derivative to the ordinary differential equation at the newest time. At constant step $h$, the first-order formula is Backward Euler method; the second-order formula isThe new value appears inside $f$, so a nonlinear system may have to be solved at every step. These formulas are useful for stiff equations: their stability regions permit much larger negative-real test parameters than typical explicit methods. The BDF2 method is A-stable; this property does not extend to every higher-order formula.^[ref]