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Unformatted text preview: MA2213 Lecture 12 REVIEW 1.1 on page 10 11. Compute Compute quadratic Taylor polynomials for , 1 ) ( lim  x x e x g x x ) ( lim ) ( f ; ), ( ) ( f x g x x g x x 12. Compute , ) 1 log( ) ( lim + x x x x g x where g is the function in problems 11 and 12. 1.2 on page 18 1. Bound the error in using on the interval x e ]. 1 , 1 [ to approximate Use the remainder ) ( p 3 x x a c c n a x x R n n n and between ), ( f )! 1 ( ) ( ) ( ) 1 ( 1 + + + = on page 11 and take ) ( p 3 x to be the third degree Taylor polynomial about . = a Suggestion Study bounding the error on page 15 Review power series and formuli on page 17 1.2 on page 19 14. (a) Obtain the Taylor polynomial for 2 1 1 ) ( f t t + about ). ( tan ) ( 1 x x g obtain a Taylor series with remainder for . = a 14. (b) Use the method developed on page 14 to Suggestion: Use part (a) + = x t dt x 2 1 1 ) ( tan and 2.2 on page 54 Review the elementary concepts in Chapter 2 about errors, particularly the sources of error on pages 4547, and lossofsignificance errors (and how to reduce such losses) on pages 4750 5. (a) 5. (d) 6. (c) 2 cos 1 x x 1 1 3 + x x e e x x 2 3.2 on page 88 Review Newtons method for finding roots, in particular Example 3.2.2 on pages 8183 that explains the importance of choosing a sufficiently close initial estimate, and the error analysis on pages 8386 4. m a a m , , a positive integer Read about order of convergence on page 101 : A sequence n x converges to with order of convergence 1 p if there exists a constant c such that . ,     1   + n x c x p n n 3 , 2 , 1 = p linear, quadratic, cubic convergence. 3.4 on page 96 4. Experimentally confirm the error estimate 13 . 1 where and is a sequence that converges n x to obtained by the secant method. ,     62 . 1 1   + n x c x n n Question What is the order of convergence of the secant method for this problem ? is the unique positive root of 1 6 x x Question What is the order of convergence of Newtons method, the secant method, for finding the root of the function 2 ) ( f x x ? 4.1 on pages 134135 16. As a generalized interpolation problem, find the quadratic polynomial for which . 4 ) 1 ( , 1...
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This note was uploaded on 01/06/2012 for the course MA 2213 taught by Professor Michael during the Fall '07 term at National University of Singapore.
 Fall '07
 Michael
 Polynomials, Numerical Analysis

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