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