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

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1 II Non-Linear Equations
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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 - = + + + - - 0 0 1 1 1 L 0 10 ) ln( 0 5 ) cos( 2 = + + = + + x x x x
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3 Solution Methods Using formula- for simple cases (quadratic eqn.) Graphical method-gives rough estimate Trial & Error – Tedious and Inadequate Numerical Method
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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
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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)
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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
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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
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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 + =
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