lizfcm/notes/Sep-25.org

• Steps
  • Calculus: defference quotient
  • Test.
  • Calculus: Taylor Serioes w/ Remainder
• Pseudo Code
• Error

ex: erfc(x) = ∫₀^x (\frac{2}{\sqrt{pi}})e^{-t^2} dt ex: IVP \frac{dP}{dt} = α P - β 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 → T P(T) ≈ n steps

Steps

Calculus: defference quotient

$f'(a) \approx \frac{f(a+h) - f(a)}{h}$

Test.

Roundoff for h ≈ 0

Calculus: Taylor Serioes w/ Remainder

$e_{abs}(h) \leq Ch^r$

(see Sep-20 . Taylor Series)

Pseudo Code

  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)

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