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### cos323_s06_lecture02_rootfinding

Course: LECTURE 323, Fall 2009
School: Princeton
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Word Count: 663

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Finding COS Root 323 1D Root Finding Given some function, find location where f(x)=0 Need: Starting position x0, hopefully close to solution Ideally, points that bracket the root f(x+) &gt; 0 f(x) &lt; 0 1D Root Finding Given some function, find location where f(x)=0 Need: Starting position x0, hopefully close to solution Ideally, points that bracket the root Well-behaved function What Goes...

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Finding COS Root 323 1D Root Finding Given some function, find location where f(x)=0 Need: Starting position x0, hopefully close to solution Ideally, points that bracket the root f(x+) > 0 f(x) < 0 1D Root Finding Given some function, find location where f(x)=0 Need: Starting position x0, hopefully close to solution Ideally, points that bracket the root Well-behaved function What Goes Wrong? Tangent point: very difficult to find Singularity: brackets don't surround root Pathological case: infinite number of roots e.g. sin(1/x) Bisection Method Given points x+ and x that bracket a root, find xhalf = (x++ x) and evaluate f(xhalf) If positive, x+ xhalf else x xhalf Stop when x+ and x close enough If function is continuous, this will succeed in finding some root Bisection Very robust method Convergence rate: Error bounded by size of [x+... x] interval Interval shrinks in half at each iteration Therefore, error cut in half at each iteration: | n+1| = | n| This is called "linear convergence" One extra bit of accuracy in x at each iteration Faster RootFinding Fancier methods get super-linear convergence Typical approach: model function locally by something whose root you can find exactly Model didn't match function exactly, so iterate In many cases, these are less safe than bisection Secant Method Simple extension to bisection: interpolate or extrapolate through two most recent points 3 2 4 1 Secant Method Faster than bisection: | n+1| = const. | n|1.6 Faster than linear: number of correct bits multiplied by 1.6 Drawback: the above only true if sufficiently close to a root of a sufficiently smooth function Does not guarantee that root remains bracketed False Position Method Similar to secant, but guarantee bracketing 2 3 4 1 Stable, but linear in bad cases Other Interpolation Strategies Ridders's method: fit exponential to f(x+), f(x), and f(xhalf) Van Wijngaarden-Dekker-Brent method: inverse quadratic fit to 3 most recent points if within bracket, else bisection Both of these safe if function is nasty, but fast (super-linear) function if is nice NewtonRaphson Best-known algorithm for getting quadratic convergence when derivative is easy to evaluate Another variant on the extrapolation theme 2 3 4 1 Slope = derivative at 1 f ( xn ) xn +1 = xn - f ( xn ) NewtonRaphson Begin with Taylor series f ( xn + ) = f ( xn ) + f ( xn ) + 2 want f ( xn ) + ... = 0 2 Divide by derivative (can't be zero!) f ( xn ) 2 f ( xn ) + + =0 f ( xn ) 2 f ( xn ) 2 f ( xn ) - Newton + + =0 2 f ( xn ) f ( xn ) 2 2 - Newton = n +1 ~ n 2 f ( xn ) NewtonRaphson Method fragile: can easily get confused Good starting point critical Newton popular for "polishing off" a root found approximately using a more robust method NewtonRaphson Convergence Can talk about "basin of convergence": range of x0 for which method finds a root Can be extremely complex: here's an example in 2-D with 4 roots Popular Example of Newton: Square Root Let f(x) = x2 a: zero of this is square root of a f'(x) = 2x, so Newton2 iteration is x -a xn +1 = xn - n 2 xn = 1 2 (x n + a xn ) "Divide and average" method Reciprocal via Newton Division is slowest of basic operations On some computers, hardware divide not available (!): simulate in software a b 1 =...

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