Question

According to one theorem, one of two conditions for this property is that the unit disk contains all zeros of a certain polynomial, with the unit circle containing only simple zeros. That “root condition” must hold in order for a method to have this property, as proven by Germund Dahlquist. For consistent methods, this property is equivalent to stability, as stated by the Lax equivalence (10[2])theorem. Practical methods (10[1])of (-5[1])solving ODEs (“O-D-E’s”) have a form of this property defined by the relation between (-5[1])the step size and the local error at a given point. (10[2])For the Euler (“OY-ler”) method, this property has a relatively low namesake order and rate. (10[1])The (10[1])root (10[1])test and the (10[1])ratio (10[1])test (10[1])can be applied to a (10[1])Taylor (10[2])expansion (10[1])to find this property’s (10[1])namesake (-5[2])radius. (10[3])For (10[1])10 points, name this property of methods or series that approach a finite limit. ■END■ (10[3]0[1])

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