MTH510_II Non-Linear Equations

MTH510_II Non-Linear Equations - 1 II Non-Linear Equations...

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Unformatted text preview: 1 II Non-Linear Equations 2 2.1 Introduction • Common problem in Applied Mathematics f(x)=0 • Algebraic equations • Transcendental functions – Including Trig., log, exp to function x of polynomial order th i f f x f y f y f i n n n n- = + + +-- 1 1 1 L 10 ) ln( 5 ) cos( 2 = + + = + + x x x x 3 Solution Methods • Using formula- for simple cases (quadratic eqn.) • Graphical method-gives rough estimate • Trial & Error – Tedious and Inadequate • Numerical Method 4 2.2 Graphical method • Simple method – for one variable eqn. • Limited practical value • Plot the function and observe the crossing points @ x= x r , f(x r )=0 f(x) Root x y x r 5 Ways Roots may Occur x l x l x u 1-root 0-root 2-root 3-root x u (a) (b) (c) (d) x x x x f(x) f(x) f(x) f(x) 6 2.3 Bracketing Method • Two methods: Bisection & False Position • Need initial guess for the bracket • Globally convergent • Parallel usage of Graph- reduces computation • Cannot identify multiple roots CHE 338 7 2.3.1 Bisection Method • Incremental search method/Graphical - to identify the root location interval ; • Interval always divided in half • Method systematically move the end points closer until we obtain a small bracket x u x l x r 1 L. Interval U. Interval x r 2 x x l r = 1 x x u r = 2 f (x) x 8 Algorithm 1 Choose lower & upper guesses ( x l , x u ) such that the function changes signs in the interval 2 Determine an estimate of the root x r by 3 Make evaluations a) If f(x l ). f(x r )<0, x u =x r and return to step 2 b) If f(x l ). f(x r )>0, x l =x r and return to step 2 c) If f(x l ). f(x r )=0 , root equals x r , terminate the computation 2 u l r x x x + = 9...
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This note was uploaded on 09/29/2010 for the course COMPUTER S cps615 taught by Professor Pro during the Spring '10 term at Randolph College.

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MTH510_II Non-Linear Equations - 1 II Non-Linear Equations...

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