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A few words about convergence We have been looking at e a as our measure of convergence A more technical means of differentiating the speed of convergence looks at asymptotic convergence

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Definition: Rate of convergence if we say that the method converges to x-true with order p>0. Higher p is faster convergence. p=1 is linear p=2 is quadratic

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Lambda is asymptotic error constant Bisection: p=1 Regula falsi: p=1.4 to 1.6

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Another open method is fixed point iteration Idea: rewrite original equation f(x)=0 into form x=g(x). Use iteration x i+1 =g(x i ) to find a value that reaches convergence Example:

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For our Manning’s equation problem becomes

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Fortran program performing fixed-point iteration for Manning’s eq. example

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Fixed point iteration doesn’t always work. Basically, if |g’(x)| is <1 near the intersection with the x line, it will work. (See your book for derivation). Example where it doesn’t work

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King of the root-finding methods Newton-Raphson method Based on Taylor series expansion

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Truncate to get At the root, f(x i+1 )=0, so and

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Note that an evaluation of the derivative is required. You may have to do this numerically. However, can converge very quickly.

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Example using our Manning’s equation problem The derivative of this w.r.t h is

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Spreadsheet example

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Error analysis and convergence of Newton- Raphson The error of the Newton-Raphson method can be estimated from Because the error at time i+1 is proportional to the square of the previous error, the number of correct decimal places doubles each iteration

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Although Newton-Raphson converges very rapidly, it can diverge, and fail to find roots. 1) if an inflection point is near the root 2) if there is a local minimum or maximum 3) if there are multiple roots 4) if a zero slope is reached

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Secant method continued There is an alternate secant method that uses a perturbation method to approximate derivative. Start with

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Now plug this approximation for the derivative into the Taylor series approximation used in Newton-Raphson: becomes

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No derivative evaluation - like the secant method Only one initial guess is needed - like Newton-Raphson method Matlab example

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