Numerical differentiation

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Numerical differentiation Recall finite
differences from first week
Derived from Taylor series
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Neglecting all tersms higher than first order
Thats the forward difference - also backwards
and centered difference
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Why is centered finite difference O(h2)?
Subtract second equation from first
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We can combine Taylor series expansions in many
different ways to get estimates of
derivatives Example backwards second derivative,
O(h2) Start with
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Multiply first equation by -5, second equation by
4 and add together

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Multiply third equation by -1 and add to above
result

Rearrange
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Where did I get -5, 4,-1?
We multiply 1st equation by a, second by b, third
by c
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Now sum all equations and collect terms
Decide what derivatives we want to make disappear
- want a second derivative only - eliminate first
and third
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Three unknowns - 2 equations - make an assumption
Let c-1
Can solve by hand
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If we have more derivatives to get rid of, use
matrix methods - always one assumption
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More Richardson extrapolation Recall
Can do the same thing with derivatives
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Use same approach as Romberg integration with
halving the step size Example Formula for active
lateral pressure coefficient Ka with internal
angle of friction f and wall with slope b and
flat top is
Use Richardson/Romberg approach to estimate
at b10 degrees and f15 degrees
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Use O(h2) estimates to get O(h6) estimate
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Now do Richardson/Romberg trick
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Derivatives of unequally spaced data Can use
matrix approach with different amounts of
h Example given values of f at x(1,2,5.5,9)
determine f at 2
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Let h1, x2 (values at 1,2,5.5,9)
Equations to get rid of f and f are
and assume a value for c
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Let c1, then a-22.8667, b-8.5333 then
or
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Derivatives of unequally spaced data Another way
is to take derivative of interpolating
polynomial Lagrange polynomial - second order in
this case
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Derivatives and integrals with errors in
data Errors in data points can cause
problems esp. with differentiation Example
with and without noise True
derivative is 2x-6
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Look at ratio of noise in y to noise in dy/dx
For differentiation, fit a smooth line to the
data, then take derivative