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

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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...
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