(20) Algorithms Root Finding

(20) Algorithms Root Finding - Root Finding Root Finding...

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11/5/2008 1 Root Finding Nathan Friedman Fall, 2008 Root Finding ± Many applications involve finding the roots of a function f(x). ± That is we want to find a value or 2008 Root Finding 2 That is, we want to find a value or values for x such that f(x)=0 Roots of a Quadratic We have already seen an algorithm for finding the roots of a quadratic We had a closed form for the solution 2008 Root Finding 3 We had a closed form for the solution, given by an explicit formula There are a limited number of problems for which we have such explicit solutions Root Finding ± What if we don’t have a closed form for the roots? 2008 Root Finding 4 Root Finding ± What if we don’t have a closed form for the roots? ± We try to generate a sequence of 2008 Root Finding 5 We try to generate a sequence of approximations x 1 , x 2 , …,x n until we (hopefully) obtain a value very close to the root Example: Firing a Projectile Find the angle at which to fire a projectile at a target Given: 2008 Root Finding 6 ± the velocity, v ± the distance to the base of the target, x ± the height of the target, h Find: the angle at which to aim, a
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11/5/2008 2 Example: Firing a Projectile The physics of the problem tells us that h = v sin a t - ½ g t 2 2008 Root Finding 7 t = x / (v cos a) where g is the gravitational constant. Example: Firing a Projectile Taking the equations: h = v*sin(a)*t - ½ g*t 2 t =x/(v*cos(a)) 2008 Root Finding 8 By substituting, we have h = x*tan(a) - 0.5*g*(x 2 /(v 2 cos 2 (a)) The angle a is a root of f(a) = x*tan(a)-0.5*g*(x 2 /(v 2 *cos 2 (a))-h The Bisection Method ± Suppose we have an interval that we know contains a root ± The bisection method uses this 2008 Root Finding 9 The bisection method uses this information to obtain better and better approximations The Bisection Method ± We start with an interval that contains exactly one root of the function ± The function must change signs on that 2008 Root Finding 10 The function must change signs on that interval. ± (If the function changes signs, in fact there must be an odd number of roots in the interval) The Bisection Method ± To get started, we must bracket a root ± How do we bracket the root? From our knowledge of the function 2008 Root Finding 11 ± ± By searching along axis at fixed increments until we find that the sign of f(x) changes The Bisection Method ± Once we have an interval containing the root(s), we narrow down the search ± Similar to binary search: divide the interval in half and look for a sign change in one of the
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(20) Algorithms Root Finding - Root Finding Root Finding...

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