NewtonCotes - Closed Newton-Cotes Integration James...

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Unformatted text preview: Closed Newton-Cotes Integration James Keesling This document will discuss Newton-Cotes Integration. Other methods of numerical integration will be discussed in other posts. The other methods will include the Trapezoidal Rule, Romberg Integration, and Gaussian Integration. 1 Principle of Newton-Cotes Integration We only cover the Newton-Cotes closed formulas. The interval of integration [ a,b ] is partitioned by the points. a,a + b- a n ,a + 2 b- a n ,...,b We estimate the integral of f ( x ) on this interval by using the Lagrange interpolating polynomial through the following points. ( a,f ( a )) , a + b- a n ,f a + b- a n ,..., ( b,f ( b )) The formula for the integral of this Lagrange polynomial simplifies to a linear combi- nation of the values of f ( x ) at the points x i = a + i b- a n i = 0 , 1 , 2 ,...,n . In the next section we give a method for calculating the coefficients for this linear combination. 2 The Newton-Cotes Closed Formula We wish to estimate the following integral. Z b a f ( x ) dx 1 We use the value of the function at the following points { a + i b- a n i = 0 , 1 , 2 ,...,n } . Our estimate will have the following form. Z b a f ( x ) dx = A f ( a ) + A 1 f a + b- a n + A 2 f a + 2 b- a n + + A n f ( b ) So, what values should we use for the coefficients and how can we calculate them?...
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This note was uploaded on 07/14/2011 for the course MAC 3472 taught by Professor Jury during the Spring '07 term at University of Florida.

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NewtonCotes - Closed Newton-Cotes Integration James...

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