where .
Order of the method is the largest such that:
By Taylor expanding, we find
for .
Or rewriting:
This is necessary and sufficient condition.
By defining polynomials and we can write this concisely as
Dahlquist Equivalence Theorem
We say a polynomial obeys the root condition if all its roots are and all the roots of unit modulus are simple.
Then the multi-step method is convergent if and only if and obeys the root condition.
Constructing multi-step methods
Find a which obeys the root condition and .
Then find a s.t.
To see where this comes from, sub in and note that .