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The numerical solution of a one-step method depends on the initial condition <math> y_0 </math>, but the numerical solution of an ''s''-step method depend on all the ''s'' starting values, <math> y_0, y_1, \ldots, y_{s-1} </math>. It is thus of interest whether the numerical solution is stable with respect to perturbations in the starting values. A linear multistep method is ''zero-stable'' for a certain differential equation on a given time interval, if a perturbation in the starting values of size ε causes the numerical solution over that time interval to change by no more than ''K''ε for some value of ''K'' which does not depend on the step size ''h''. This is called "zero-stability" because it is enough to check the condition for the differential equation <math> y' = 0 </math> {{harv|Süli|Mayers|2003|p=332}}.
If the roots of the [[Characteristic equation (calculus)|characteristic polynomial]] ρ all have modulus less than or equal to 1 and the roots of modulus 1 are of multiplicity 1, we say that the ''root condition'' is satisfied. A linear multistep method is zero-stable if and only if the root condition is satisfied {{harv|Süli|Mayers|2003|p=335}}.
Now suppose that a consistent linear multistep method is applied to a sufficiently smooth differential equation and that the starting values <math> y_1, \ldots, y_{s-1}</math> all converge to the initial value <math> y_0 </math> as <math> h \to 0 </math>. Then, the numerical solution converges to the exact solution as <math> h \to 0 </math> if and only if the method is zero-stable. This result is known as the ''Dahlquist equivalence theorem'', named after [[Germund Dahlquist]]; this theorem is similar in spirit to the [[Lax equivalence theorem]] for [[finite difference method]]s. Furthermore, if the method has order ''p'', then the [[global truncation error|global error]] (the difference between the numerical solution and the exact solution at a fixed time) is <math> O(h^p) </math> {{harv|Süli|Mayers|2003|p=340}}.
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