Unformatted text preview: ) and With Rjk normalized by (3.3.5b), there are k + j + 1 undetermined parameters in
(3.3.5a) that can be determined by matching the rst k + j + 1 terms in the Taylor's
series expansion of ez . Thus, the error of the approximation should be O(zk+j+1). Using
(3.3.5c), we have
X z i Pk=0 pi z i
i! = Pj=0 qizi + O(z
Equating the coe cients of like powers of z determines the parameters pi, i = 0 1 : : : k
and qi, i = 1 2 : : : j .
Example 3.3.4. Find the (2,0) Pade approximation of ez . Setting j = 2 and k = 0 in
(1 + z + z2 )(1 + q1z + q2 z2 ) = p0 :
Equating the coe cients of zi , i = 0 1 2, gives p0 = 1 1 + q1 = 0
22 1 +q +q =0
212 Thus, q1 = ;1 p0 = 1 q2 = 1=2: Using (3.3.5), the (2,0) Pade approximation is R20 (z) = 1 ; z 1 z2 =2 :
Additionally, ez = R20 (z) + O(z3): Some other Pade approximations are presented in Table 3.3.1. We recognize that
the (0,1) approximation corresponds to Euler's method, the (1,0) method corresponds
to the backward Euler method, an...
View Full Document
This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
- Spring '14