09 RootsOfEquations

09 RootsOfEquations - Roots of Equations CEE 3804 Computer...

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Roots of Equations CEE 3804 Computer Applications for Civil and Environmental Engineers
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CEE 3804 - Rojiani 2 Root Finding z A root of f(x) is the value of x that makes f(x)=0 z Most root finding problems fall into 2 categories: z 1) find the real roots of algebraic and transcendental functions Usually finds 1 root based on preliminary estimate of its location (initial guess) z 2) find all real and complex roots of a polynomial z We’ll focus on 1) above
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CEE 3804 - Rojiani 3 Closed versus Open Methods z Root finding algorithms are sometimes divided into closed or open methods. z Closed methods are generally slow but very reliable. z Open methods are much faster but can diverge. z Open methods generally require a continuous function with continuous derivatives. Direct methods do not. z Example Open Method: Newton’s Method z Example Closed Method: Bisection
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CEE 3804 - Rojiani 4 Newton's method for root finding z Based on Taylor series z for root: f (x 1 ) = 0, solving for x 1 z approximate because dropped 2 nd order and higher terms z x 0 serves as the initial starting guess for the root () ( ) ( ) ( ) ... ' 0 0 1 0 1 + + = x f x x x f x f ( ) () 0 0 0 1 ' x f x f x x
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CEE 3804 - Rojiani 5 Newton Iteration z apply iteratively ( ) () i i i i x f x f x x ' 1 +
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This note was uploaded on 11/14/2008 for the course CEE 3804 taught by Professor Aatrani during the Fall '07 term at Virginia Tech.

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09 RootsOfEquations - Roots of Equations CEE 3804 Computer...

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