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Unformatted text preview: Solutions for Homework 9 Foundations of Computational Math 2 Spring 2011 Problem 9.1 In this problem we consider the numerical approximation of the integral I = integraldisplay 1 1 f ( x ) dx with f ( x ) = e x . In particular, we use a priori error estimation to choose a step size h for Newton Cotes or a number of points for a Gaussian integration method. 9.1.a Consider the use of the composite Trapezoidal rule to approximate the integral I . Use the fact that we have an analytical form of f ( x ) to estimate the error using the composite trapezoidal rule and to determine a stepsize h so that the error will be less than or equal to the tolerance 10 2 . Approximately how many points does your h require? Solution: We must use the error expression for the composite trapezoidal rule, the expression for the appropriate derivative bound, and the required tolerance to get an a priori estimate for h . f ( x ) = e x = f ( x ) f ( x ) f (1) 2 . 72 E =- h 2 12 ( b- a ) f ( ) h 2 12 10 2 ( b- a ) f ( ) = . 06 f ( ) . 06 2 . 72 . 02 h . 14 n = ( b- a ) h = 2 . 14 14 steps Using the composite trapezoidal rule yields the following results: n I n I- I n 2 2 . 5308- . 1926 8 2 . 3623- . 01223 14 2 . 3544- . 00399 16 2 . 35346- . 00305 1 9.1.b Consider the use of the Gauss-Legendre method to approximate the integral I . Use n = 1, i.e., two points x and x 1 with weights and 1 . Use the fact that we have an analytical form of f ( x ) to estimate the error that will result from using the two-point Gauss-Legendre method to approximate the integral. How does your estimate compare to the tolerance 10 2 used in the first part of the question? Recall that for n = 1 we have the Gauss Legendre nodes x - . 5774 and x 1 . 5774. Apply the method to approximate I and compare its error to your prediction. The true value is I = integraldisplay 1 1 e x dx 2 . 3504 Solution: Recall the error formula for Gauss Legendre E = 2 2 n +3 [( n + 1)!] 4 (2 n + 3) [(2 n + 2)!] 3 f (2 n +2) ( ) We have f (2 n +2) ( x ) = e x e 1 2 . 72 and for n = 1 E = 2 5 [2!] 4 5 [4!] 3 (2...
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This note was uploaded on 11/10/2011 for the course MAD 5404 taught by Professor Gallivan during the Spring '11 term at FSU.
- Spring '11