Order and stability of generalized Padé approximations

Given a sequence of integers [ n 0 , n 1 , … , n r ] , where n 0 , n r ⩾ 0 and n i ⩾ − 1 , i = 1 , 2 , … , r − 1 , a sequence of r polynomials ( P 0 , P 1 , … , P r ) is a generalized Padé approximation to the exponential function if ∑ i = 0 r exp ( ( r − i ) z ) P i ( z ) = O ( z p + 1 ) , where the order of the approximation p is given by p = ∑ i = 0 r ( n i + 1 ) − 1 . The main result of this paper is that if 2 n 0 > p + 2 , then ∑ i = 0 r w r − i P i ( z ) is not the stability polynomial of an A-stable numerical method. This result, known as the Butcher–Chipman conjecture, generalizes the corresponding result for rational Padé approximations. The special case, formerly known as the Ehle conjecture [B.L. Ehle, A-stable methods and Padé approximations to the exponential, SIAM J. Math. Anal. 4 (1973) 671–680], was subsequently proved by Hairer, Nørsett and Wanner [G. Wanner, E. Hairer, S.P. Nørsett, Order stars and stability theorems, BIT 18 (1978) 475–489].