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Unformatted text preview: j = O(h ): 0 J i N i i 22 (9.3.9) We can get more detailed information about errors at the collocation points by subtracting (9.2.7b) from (9.3.2a) while using (9.3.1a) X
J e( ) = e(x 1 ) + h
ij i; i
k Taking a matrix norm =1 a A( )e( ) + h E
jk ik ik i j je( )j (1 + C1h )je(x 1 )j + C2h +1:
J ij i i; i In obtaining this relationship, we bounded the Runge-Kutta coe cients and jAj by their
maximal values and used consistency to infer that e( ) does not di er from e(x 1 ) by
more than O(h ). The last term above was bounded using (9.3.6).
Since (9.3.9) implies that je(x )j = O(h ), j = 0 1 : : : J , we have
ik i; i J j 0 i max
1 je( )j = O(h ): (9.3.10) J N j ij J Furthermore, since the exact and numerical solutions satisfy the di erential equation
(9.3.1) at the collocation points, we have e ( ) = A( )e( ):
0 ij ij ij Taking a vector norm and using (9.3.10)
0 i max
N je ( )j = O(h ): J (9.3.11) J 0 j ij Finally, applying the interpolation formula (3) to y and using (9.2.3a), we have
0 e (x) =
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
- Spring '14
- The Land