PGE 310 - HW5 - Solution

# PGE 310 - HW5 - Solution - The University of Texas at...

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1 The University of Texas at Austin PGE 310:Formulation and Solution in Geosystems Engineering Homework #5: Truncation Error (Taylor Series) and Root Finding in MATLAB By HosseinRoodi 1. Truncation Error: Taylor Series (HAND CALCULATION) Given below equation for ) ( x f consider that you want to calculate the value of the function at 4 x : 2 sin ) ( 2 4 x x e x f x x e x f x e x f x e x f x x e x f x x e x f x x x x x sin 256 1 ) ( ' ' ' ' cos 64 1 ) ( 1 sin 16 1 ) ( cos 4 1 ) ( 2 sin ) ( 4 4 4 4 2 4 You know the true solution is 12468 . 9 2 / 4 4 sin ) 4 ( 2 4 4 e f . But you are requested to estimate ) 4 ( f using Taylor series. So, take 6972 . 1 ) 2 ( & 2 0 0 x f x as your starting point and estimate ) 4 ( f four times using 1term, 2 terms, 3terms and 4 terms of Taylor series. Calculate the true relative percent error relative true and complete the below table: | | | | : .... ) ( ! 3 ) ( ) ( ! 2 ) ( ) ( ) ( ) ( ) ( 0 3 0 1 0 2 0 1 0 0 1 0 1 value true value estimated value true error percent relative True x f x x x f x x x f x x x f x f relative t Using 1 term: ) ( ) ( 0 1 x f x f 6972 . 1 ) 2 ( ) 4 ( f f % 81 81 . 0 | 12468 . 9 | | 6972 . 1 12468 . 9 | or relative t Using 2 terms: ) ( ) ( ) ( ) ( 0 0 1 0 1 x f x x x f x f 2263 . 6 ) 2645 . 2 ( * 2 6972 . 1 ) 2 ( ) 2 4 ( ) 2 ( ) 4 ( f f f % 32 32 . 0 | 12468 . 9 | | 2263 . 6 12468 . 9 | or relative t Using 3 terms: ) ( ! 2 ) ( ) ( ) ( ) ( ) ( 0 2 0 1 0 0 1 0 1 x f x x x f x x x f x f

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2 1207 . 10 9472 . 1 2 2263 . 6 ) 2 ( 2 ) 2 4 ( 2263 . 6 ) 4 ( 2 f f % 11 11 . 0 | 12468 . 9 | | 1207 . 10 12468 . 9 | or relative t Using 4 terms: ) ( ! 3 ) ( ) ( ! 2 ) ( ) ( ) ( ) ( ) ( 0 3 0 1 0 2 0 1 0 0 1 0 1 x f x x x f x x x f x x x f x f 5532 . 9 ) 4256 . 0 ( 6 8 1207 . 10 ) 2 ( ! 3 ) 2 4 ( 1207 . 10 ) 4 ( 3 f f % 5 05 . 0 | 12468 . 9 | | 5532 . 9 12468 . 9 | or relative t ) 4 ( f relative true True Solution 1247 . 9 1 term of Taylor series 6972 . 1 % 81 2 term of Taylor series 2263 . 6 % 32 3 term of Taylor series 1207 . 10 % 11 4 term of Taylor series 5532 . 9 % 5 SHOW CLEARLY ALL YOUR WORKS AND FORMULATIONS. (30 %)
3 2. Root Finding Method: (HAND CALCULATION& MATLAB PLOTTING) Consider that you want to find the root of the following equation: 4 8 2 x e x 3 4 32 2 ) ( 8 2 ) ( x e x f x e x f x x Using two iterations, solve 0 8 2 4 x e x for x using the following methods: a) Bisection Method with 0 L x and 1

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## This note was uploaded on 02/13/2012 for the course PGE 310 taught by Professor Klaus during the Spring '06 term at University of Texas at Austin.

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PGE 310 - HW5 - Solution - The University of Texas at...

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