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Root Finding COS 323

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1-D Root Finding Given some function, find location where f ( x )=0Given some function, find location where f ( x )=0 Need:Need: – Starting position x 0, hopefully close to solution – Ideally, points that bracket the root f ( x + ) > 0 f ( x – ) < 0

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1-D Root Finding Given some function, find location where f ( x )=0Given some function, find location where f ( x )=0 Need:Need: – Starting position x 0, hopefully close to solution – Ideally, points that bracket the root – Well-behaved function

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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)

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Example: Press et al., Numerical Recipes in C Equation (3.0.1), p. 105 2 2 ln((Pi - x) ) 2 ln((Pi - x) ) f := x -> 3 x + ------------- + 1 f := x -> 3 x + ------------- + 1 4 Pi Pi > evalf(f(Pi-1e-10)); 30.1360472472915692... > evalf(f(Pi-1e-100)); 25.8811536623525653... > evalf(f(Pi-1e-600)); 2.24285595777501258... > evalf(f(Pi-1e-700)); -2.4848035831404979... > evalf(f(Pi+1e-700)); -2.4848035831404979... > evalf(f(Pi+1e-600)); 2.24285595777501258... > evalf(f(Pi+1e-100)); 25.8811536623525653... > evalf(f(Pi+1e-10)); 30.1360472510614803...

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Bisection Method Given points x + and x – that bracket a root, find x half = ½ ( x + + x – ) and evaluate f ( x half )Given points x + and x – that bracket a root, find x half = ½ ( x + + x – ) and evaluate f ( x half ) If positive, x + x half else x – x halfIf positive, x + x half else x – x half Stop when x + and x – close enoughStop when x + and x – close enough If function is continuous, this will succeed in finding some rootIf function is continuous, this will succeed in finding some root

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Bisection Very robust methodVery robust method Convergence rate: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 | = ½ | n | – This is called “linear convergence” – One extra bit of accuracy in x at each iteration

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Faster Root-Finding Fancier methods get super-linear convergenceFancier 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

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Secant Method Simple extension to bisection: interpolate or extrapolate through two most recent pointsSimple extension to bisection: interpolate or extrapolate through two most recent points 1 2 3 4

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Secant Method Faster than bisection: | n +1 | = const. | n | 1.6Faster than bisection: | n +1 | = const. | n | 1.6 Faster than linear: number of correct bits multiplied by 1.6Faster 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 functionDrawback: the above only true if sufficiently close to a root of a sufficiently smooth function – Does not guarantee that root remains bracketed

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False Position Method Similar to secant, but guarantee bracketingSimilar to secant, but guarantee bracketing Stable, but linear in bad casesStable, but linear in bad cases 1 2 3 4

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Other Interpolation Strategies Ridders’s method: fit exponential to f ( x + ), f ( x – ), and f ( x half )Ridders’s method: fit exponential to f ( x + ), f ( x – ), and f ( x half ) Van Wijngaarden-Dekker-Brent method: inverse quadratic fit to 3 most recent points if within bracket, else bisectionVan 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) if function is niceBoth of these safe if function is nasty, but fast (super-linear) if function is nice

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Newton-Raphson Best-known algorithm for getting quadratic convergence when derivative is easy to evaluateBest-known algorithm for getting quadratic convergence when derivative is easy to evaluate Another variant on the extrapolation themeAnother variant on the extrapolation theme 1 2 3 4 Slope = derivative at 1

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Newton-Raphson Begin with Taylor seriesBegin with Taylor series Divide by derivative (can’t be zero!)Divide by derivative (can’t be zero!)

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Newton-Raphson Method fragile: can easily get confusedMethod fragile: can easily get confused Good starting point criticalGood starting point critical – Newton popular for “polishing off” a root found approximately using a more robust method

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Newton-Raphson Convergence Can talk about “basin of convergence”: range of x 0 for which method finds a rootCan talk about “basin of convergence”: range of x 0 for which method finds a root Can be extremely complex: here’s an example in 2-D with 4 rootsCan be extremely complex: here’s an example in 2-D with 4 roots Yale site Yale site D.W. Hyatt D.W. Hyatt

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Popular Example of Newton: Square Root Let f ( x ) = x 2 – a : zero of this is square root of aLet f ( x ) = x 2 – a : zero of this is square root of a f ’( x ) = 2 x, so Newton iteration is f ’( x ) = 2 x, so Newton iteration is “Divide and average” method“Divide and average” method

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Reciprocal via Newton Division is slowest of basic operationsDivision is slowest of basic operations On some computers, hardware divide not available (!): simulate in softwareOn some computers, hardware divide not available (!): simulate in software Need only subtract and multiplyNeed only subtract and multiply

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Rootfinding in >1D Behavior can be complex: e.g. in 2DBehavior can be complex: e.g. in 2D

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Rootfinding in >1D Can’t bracket and bisectCan’t bracket and bisect Result: few general methodsResult: few general methods

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Newton in Higher Dimensions Start withStart with Write as vector-valued functionWrite as vector-valued function

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Newton in Higher Dimensions Expand in terms of Taylor seriesExpand in terms of Taylor series f ’ is a Jacobian f ’ is a Jacobian

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Newton in Higher Dimensions Solve for Solve for Requires matrix inversion (we’ll see this later)Requires matrix inversion (we’ll see this later) Often fragile, must be carefulOften fragile, must be careful – Keep track of whether error decreases – If not, try a smaller step in direction

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