Linear multistep methods and root conditions

For a constant positive step h, write a k-step formula with fixed coefficients $\sum_{j=0}^k\alpha_jY_{n+j}=h\sum_{j=0}^k\beta_j f_{n+j}$ and $\alpha_k\ne0$. Its polynomials are $\rho(\zeta)=\sum\alpha_j\zeta^j$ and $\sigma(\zeta)=\sum\beta_j\zeta^j$. The coefficient at the newest derivative decides whether the method is explicit or implicit. These polynomials organize the history recurrence; they are different objects from a one-step Runge–Kutta stability function.

Substituting a constant exact solution tests that rho vanishes at one. Substituting a linear exact solution tests that rho prime at one equals sigma at one. Together these give first-order consistency for a regular multistep formula. Higher-order moment equations come from further Taylor terms. Consistency compares a single exact history with the formula; it does not yet control what happens when earlier computed values contain small errors. That propagation issue is precisely why a separate stability condition is necessary.

Apply the method to the zero differential equation. The history then follows the homogeneous recurrence defined by rho. A root outside the unit disk gives exponential growth of a perturbation with step index. A repeated root on the unit circle gives polynomial growth through its associated recurrence solutions. To keep perturbations uniformly controlled while the number of steps increases like one divided by h, all roots must have modulus at most one and every root on the unit circle must be simple.

For fixed coefficients and constant steps, that root condition is equivalent to zero-stability. Consistency together with zero-stability yields convergence for sufficiently regular Lipschitz initial-value problems with convergent starting data. To obtain a particular order p, starting values must be accurate to order h to the p as well as the formula having the appropriate local order. A parasitic root strictly inside the disk decays, but a root near the boundary can still make startup transients conspicuous.

The two-step Adams–Bashforth formula uses current and previous derivatives, so an initial-value problem must supply an additional starting value by another method or an exact expression. Poor startup can dominate an otherwise accurate formula. Startup accuracy is assessed at the starting mesh points, not only by the formal order label of the startup routine. For example, one Euler step from exact data has order-h-squared starting error and can be sufficient to start a second-order two-step method.

Variable-step multistep methods change coefficients according to recent step ratios. A constant-step root calculation does not automatically establish stability for arbitrary sequences of ratios. Large step jumps can amplify the history representation, and practical codes restrict or manage them while adjusting order and error estimates. It is therefore important to distinguish a proof about a fixed formula from a claim about a complete adaptive implementation. Both consistency and stability need to match the actual history update being used.