CSC_349_HANDOUT#26

CSC_349_HANDOUT#26 - COMPUTER SCIENCE 349A Handout Number...

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1 COMPUTER SCIENCE 349A Handout Number 26 NEWTON-COTES CLOSED QUADRATURE FORMULAS The case n = 1 (Section 21.1 of the textbook): f(x) P(x) h a = x 0 b = x 1 x Here a b h = . The (linear) interpolating polynomial is P ( x ) = x x 1 x 0 x 1 f ( x 0 ) + x x 0 x 1 x 0 f ( x 1 ) . The quadrature formula for approximating f ( x ) dx a b is obtained by integrating P ( x ) : [] . since , ) ( ) ( 2 ) ( 2 ) ( 2 2 ) ( 2 ) ( ) ( ) ( ) ( ) ( 0 1 1 0 1 0 1 0 0 1 0 2 0 1 1 1 2 1 0 0 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 1 0 x x h x f x f h x f x x x f x x x x x x x x f x x x x x x f dx x f x x x x dx x f x x x x dx x P dx x f x x x x x x x x b a x x = + = + = + = + = ∫∫
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2 This is the trapezoidal rule . Its error term can be obtained by integrating the error term of the Lagrange form of the interpolating polynomial, which for n = 1 is ). )( ( 2 )) ( ( ) ( ) ( 1 0 x x x x x f x P x f = ξ Note that does depend on x ; for each value of x in [ a , b ], there is a different value
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This note was uploaded on 12/10/2011 for the course CSC 349 taught by Professor Oadje during the Spring '11 term at University of Victoria.

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CSC_349_HANDOUT#26 - COMPUTER SCIENCE 349A Handout Number...

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