Abstract
A complete method and expressions for computing Mathieu characteristic numbers (MCNs) of integer orders are given. The paper reviews available algorithms and their shortcomings, and then suitable methods are selected on the basis of simplicity and accuracy. For small orders$(n < 4)$, new approximate formulas for MCNs are developed; these have much larger range than available expressions. A new normalization technique for the MCNs is proposed. With the aid of this normalization a new approximate recurrence relation for MCNs is obtained. Using the same normalization, general formulas are obtained for regional limits where the approximate and asymptotic expressions can be used to compute MCNs. With these expressions, the algorithms for computing MCNs can handle very high orders, up to n = 1000 and beyond.