This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Closed NewtonCotes Integration James Keesling This document will discuss NewtonCotes 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 NewtonCotes Integration We only cover the NewtonCotes 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 NewtonCotes 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?...
View
Full
Document
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.
 Spring '07
 JURY
 Calculus

Click to edit the document details