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 %)
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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 ! R x Iteration 1: Negative x f x Positive x f x u u l l 5634 . 2 ) ( 1 2 ) ( 0 # ! !
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PGE 310 - HW5 - Solution - The University of Texas at...

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