# 318 to obtain s x utn ytn h bi xutn tn 1 cih utn

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ). We'll review some of the details describing the derivation of (3.3.18). Additional material appears in most elementary ^ numerical analysis texts 4]. Let F ( ) = F (tn 1 + h) and approximate it by a Lagrange interpolating polynomial of degree s ; 1 to obtain ; ^ F( ) = s X j =1 ^ ^ F (cj )Lj ( ) + Ms( ) F (s)( ) s! 2 (0 1) (3.3.20a) where Ms ( ) = s Y i=1 ( ; ci): (3.3.20b) (Di erentiation in (3.3.20a) is with respect to , not t.) Integrate (3.3.20a) and use (3.3.15b) to obtain Z1 s X ^ ^ ^ F ( )d = bj F (cj ) + Es j =1 0 where Z1 Z 1Y s ^s = 1 Ms( )F (s) ( ( ))d = 1 ^ ^ ( s) E s! s! 0 i=1 ( ; ci)F ( ( ))d : 0 (3.3.21a) (3.3.21b) In Newton-Cotes quadrature rules, such as the trapezoidal and Simpson's rules, the evaluation points ci, i = 1 2 : : : s, are speci ed a priori. With Gaussian quadrature, however, the points are selected to maximize the order of the rule. This can be done by ^ expanding F (s)( ( )) in a Taylor's series and selecting the ci, i = 1 2 : : : s, to annihilate as many terms as possible. Alternatively, and equivalent...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online