49 lines
989 B
Org Mode
49 lines
989 B
Org Mode
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ex: erfc(x) = \int_{0}^x (\frac{2}{\sqrt{pi}})e^{-t^2 }dt
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ex: IVP \frac{dP}{dt} = \alpha P - \beta P^2
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P(0) = P_0
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Explicit Euler Method
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$\frac{P(t + \Delta t) - P(t)}{\Delta t} \approx \alpha P(t) - \beta P^2(t)$
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From 0 \rightarrow T
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P(T) \approx n steps
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* Steps
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** Calculus: defference quotient
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$f'(a) \approx \frac{f(a+h) - f(a)}{h}$
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** Test.
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Roundoff for h \approx 0
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** Calculus: Taylor Serioes w/ Remainder
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$e_{abs}(h) \leq Ch^r$
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(see Sep-20 . Taylor Series)
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* Pseudo Code
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#+BEGIN_SRC python
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for i in range(n):
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a12 = a12 + x[i+1]
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a22 = a22 + x[i+1]**2
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a21 = a12
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b1 = y[0]
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b2 = y[0] * x[0]
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for i in range(n):
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b1 = b1 + y[i+1]
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b2 = b2 + y[i+1]*x[i+1]
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detA = a22*a11 - a12*a21
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c = (a22*b1 - a12*b2) / detA
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d = (-a21 * b1 + a11 * b2) / detA
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return (c, d)
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#+END_SRC
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* Error
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We want
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$e_k = |df(h_kk) - f'(a)|$
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$= |df(h_k) - df(h_m) + df(h_m) - f'(a)|$
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$\leq |df(h_k) - df(h_m)| + |df(h_m) - f'(a)|$ and $|df(h_m) - f'(a)|$ is negligible
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