Introduction

These are a com­pi­la­tion of search­able com­po­nents which are typ­i­cally asked. (top level head­ings are cur­rently lower [FIX LATER])

Ordinary Differential Equations

Common: \[ f(x_0,y_0)=y' \]

• \(h\) is the step size
• \(x_0\) and \(y_0\) are start­ing val­ues

Runge Kutta Methods

The gen­eral form is

\[ y_1=y_0+k \]

Runge Kutta I

\[ k=hf(x_0,y_0) \]

Runge Kutta II

\[ k_1=hf(x_0,y_0) \]

\[ k_2=hf(x_0,y_0+k_1) \]

\[ k=\frac{1}{2}(k_1+k_2) \]

Runge Kutta III

\[ k_1=hf(x_0,y_0) \]

\[ k_2=hf(x_0+\frac{h}{2},y_0+\frac{k_1}{2}) \]

\[ k'=hf(x_0+h,y_0+k_1) \]

\[ k_3=hf(x_0+h,y_0+k') \]

\[ k=\frac{1}{6}(k_1+4k_2+k_3) \]

Runge Kutta IV

\[ k_1=hf(x_0,y_0) \]

\[ k_2=hf(x_0+\frac{h}{2},y_0+\frac{k_1}{2}) \]

\[ k_3=hf(x_0+\frac{h}{2},y_0+\frac{k_2}{2}) \]

\[ k_4=hf(x_0+h,y_0+k_3) \]

\[ k=\frac{1}{6}(k_1+2k_2+2k_3+k_4) \]

Milne’s Predictor Corrector

\[ y_4^P=y_0+4\frac{h}{3}(2f_1-f_2+2f_3) \]

\[ y_4^C=y_2+\frac{h}{3}(f_2+4f_3+f_4) \]

Root Finding Methods

Newton Raphson

\[ x_{i+1}=x_i-\frac{f(x_i)}{f'(x_i)} \]

Regula Falsi

With \(a\) and \(b\) as the in­ter­val points.

Evaluate \(f(a)\) and \(f(b)\).

\[ x_r=\frac{af(b)-bf(a)}{f(b)-f(a)} \]

Check

\[f(a)f(x_r)<0\implies{b}=x_r\] \[f(a)f(x_r)>0\implies{a}=x_r\]

Secant

\[ x_{i+1}=\frac{x_{i-1}f_i-x_i{f_{i-1}}}{f_i-f_{i-1}} \]

Bisection

With \(a\) and \(b\) as the in­ter­val points such that \(f(a)f(b)<0\).

\[ x_r=\frac{a+b}{2} \]

Check

\[f(a)f(x_r)<0\implies{b}=x_r\] \[f(a)f(x_r)>0\implies{a}=x_r\]

SOR

\[ x_i^k=x_i^{k-1}+w[x_{gs}^k-x_i^{k-1}] \]

Where:

• \(x_{gs}^k\) is the \(k^{th}\) it­er­a­tion Gauss-Seidel vec­tor
• \(w\in[1,2]\) for speed and is usu­ally taken as \(1.2\)

\[ x_{gs}^{k+1}=C+{B}X^k \]