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@@ -0,0 +1,41 @@
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+-- Exercise 4.2
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+-- George C. Privon
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+-- 2023-12-22
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+type R = Double
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+type Derivative = (R -> R) -> R -> R
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+
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+-- a function to take the numerical derivative, approximated for some dt
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+derivative :: R -> Derivative
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+derivative dt x t = (x (t + dt/2) - x (t - dt/2)) / dt
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+
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+-- the function we want to take the derivative of
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+myfunc :: R -> R
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+myfunc x = x**3
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+
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+-- exact derivative of our function of interest
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+dmyfunc :: R -> R
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+dmyfunc x = 3 * x**2
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+
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+-- take the derivative with dt=1
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+dOne :: R -> R
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+dOne x = derivative 1 myfunc x
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+
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+-- compare the exact derivative with the numerical derivative using dt=1
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+abserr :: R -> R
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+abserr x = abs (dmyfunc x - dOne x)
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+
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+-- compare the exact derivative with the numerical derivative, for a fixed
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+-- value, varying the dt
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+abserrdt :: R -> R
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+abserrdt dt = abs (dmyfunc 0.1 - derivative dt myfunc 0.1)
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+
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+
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+-- Solution discussion
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+-- a) The error introduced by the numerical derivative is a constant value of
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+-- 0.25.
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+-- b) The error in terms of `dt` (a in the variables of the example) follows
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+-- 0.25*dt**2
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+-- c) When x=4, a fractional error of 1% can be achieved with a
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+-- dt=sqrt(0.01*48/0.25)~1.3856406460551018.
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+-- When x=0.1, the same fractional error can be achieved with:
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+-- dt=sqrt(0.01*0.03/0.25)~3.4641016151377546e-2
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