Unformatted text preview: tability. Thus, we apply the
general method (3.2.3) to the test equation (3.2.8) to obtain yn = yn 1 + zbT Y (3.3.3a) (I ; zA)Y = yn 1l (3.3.3b) ; ; where Y, l, A, and b and are de ned by (3.2.6) and z = h .
Eliminating Y in (3.3.3a) by using (3.3.3b) we nd yn = R(z)yn ; (3.3.4a) 1 where R(z) = 1 + zbT (I ; zA) 1 l:
; (3.3.4b) The region of absolute stability is the set of all complex z where jR(z)j 1. While R(z)
is a polynomial for an explicit method, it is a rational function for an implicit method.
21 Hence, the region of absolute stability can be unbounded. As shown in Section 3.2, a
method of order p will satisfy R(z) = ez + O(zp+1):
Rationalfunction approximations of the exponential are called Pade approximations. De nition 3.3.1. The (j k) Pade approximation Rjk (z) is the maximumorder approximation of ez having the form zk
P
Rjk (z) = Qk (z) = p0 + p1 z + :: :: :: + pkzj
(z) q + q z + + q
j 0 j 1 (3.3.5a) where Pk and Qj have no common factors, Qj (0) = q0 = 1 (3.3.5b) Rjk (z) = ez + O(zk+j+1): (3.3.5c...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, yn, Tn, Numerical ordinary differential equations

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