As shown in section 32 a method of order p will

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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 k+j X z i Pk=0 pi z i k+j +1) i (3.3.6) i! = Pj=0 qizi + O(z i i=0 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 (3.3.6) gives 2 (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...
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