A few words about convergence

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A few words about convergence We have been
looking at ea 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 pgt0. Higher p is faster convergence. p1 is
linear p2 is quadratic
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Lambda is asymptotic error constant Bisection
p1 Regula falsi p1.4 to 1.6
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Another open method is fixed point iteration
Idea rewrite original equation f(x)0 into form
xg(x). Use iteration xi1g(xi) to find a value
that reaches convergence Example
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For our Mannings equation problem
becomes
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Fortran program performing fixed-point iteration
for Mannings eq. example
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Fixed point iteration doesnt always
work. Basically, if g(x) is lt1 near the
intersection with the x line, it will work. (See
your book for derivation).
Example where it doesnt 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(xi1)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 Mannings 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 i1 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