Numerial Analysis

• Newton-Raphson Method

The root \(x^{*}\) of the function \(f\) can be found by iterating with the following update rule: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}. \]

The initial value \(x_0\) needs to be at sensible location for this to converge. Different initial value may converge to different roots.

Similarly, the root of the derivative of a function can also be found. \[ x_{n+1} = x_n - \frac{f'(x_n)}{f''(x_n) }. \]

For multivariate function \(f\colon V \to W\): \[ x_{n+1} = x_n - J_f(x_n)^{-1}f(x_n) \] where \(J_f\) is the Jacobian

The critical points for multivariate functions, such as local extrema or constrained extrema, can be found like: \[ x_{n+1} = x_n - H_f(x_n)^{-1}J_f(x_n) \] where \(H_f\) is the Hessian.

For \(f\in C^4[a, b]\), there exists \(\xi\in (a,b)\) such that \[ \int_a^bf(x) = \frac{b-a}{6} \]

Attracting fixed pint is the points in the neighborhood of the fixed points converges to the fixed point. It is a kind of attractor.

It is often numerically unstable, especially for stiff equations, meaning that it tends to diverge, gaining or losing fictitious energy.

• RK4

\[ x_{i+1} = x_i + \frac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4)\\ t_{i+1} = t_i + h \] where

\begin{align*} k_1 &= f(t_i, x_i) \\ k_2 &= f\left(t_i + \frac{h}{2}, x_i + \frac{h}{2}k_1\right) \\ k_3 &= f\left(t_i + \frac{h}{2}, x_i + \frac{h}{2}k_2\right) \\ k_4 &= f(t_i + h, x_i + hk_3) \\ \end{align*}

• It is averaging over the explicit Euler, implicit Euler, and other two, in order to reduce the error.
• It performs really well for extensive amount of time.

Numerical techniques for solving differential equations by approximating derivatives with finite difference.