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hw8_solutions - Homework 8 Solutions Math 471 Fall 2007...

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Homework 8 Solutions Math 471, Fall 2007 Assigned: Friday, November 2, 2007 Due: Friday, November 9, 2007 Include a cover page Clearly label all plots using title , xlabel , ylabel , legend Use the subplot command to compare multiple plots Include printouts of all Matlab code, labeled with your name, date, section, etc. 1. (Interpolating Polynomials) The function f ( x ) = 1 / ( x + 1) is given at the four points x 0 = 1, x 1 = 2, x 2 = 3, x 3 = 4. (a) Write the interpolating polynomial in Lagrange form. ( x - 2)( x - 3)( x - 4) (1 - 2)(1 - 3)(1 - 4) 1 2 + ( x - 1)( x - 3)( x - 4) (2 - 1)(2 - 3)(2 - 4) 1 3 + ( x - 1)( x - 2)( x - 4) (3 - 1)(3 - 2)(3 - 4) 1 4 + ( x - 1)( x - 2)( x - 3) (4 - 1)(4 - 2)(4 - 3) 1 5 . (b) Write the interpolating polynomial in Newton form. 1 2 - 1 6 ( x - 1) + 1 24 ( x - 1)( x - 2) - 1 120 ( x - 1)( x - 2)( x - 3) (c) Evaluate f (1 . 5) and f (5) using the interpolating polynomial. Which approximate value is more accurate? f (1 . 5) 0 . 403125 , f (5) 0 . 133333 (d) Use the error formula to find an upper bound on the maximum error || f - p 3 || = max 1 x 4 | f ( x ) - p 3 ( x ) | . max 1 ξ 4 f (4) ( ξ ) 4!
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