Cheatsheet

Floating Point Numbers

System defined by $β, t, L, U$ where:
- $β =$ base (usually 2)
- $t =$ mantissa digits (precision)
- $L, U =$ exponent bounds
Normalized form: $\pm 0. d_{1} d_{2} d_{3} . . . d_{t} \times β^{p}$ where $d_{1} \neq 0$ and $L \leq p \leq U$
IEEE Single Precision (32 bits): $β = 2, t = 24, L = - 126, U = 127$
IEEE Double Precision (64 bits): $β = 2, t = 53, L = - 1022, U = 1023$

Absolute error: $E_{a b s} = | x_{e x a c t} - x_{a p p r o x} |$
Relative error: $E_{r e l} = \frac{| x_{e x a c t} - x_{a p p r o x} |}{| x_{e x a c t} |}$
$s$ significant digits if: $0.5 \times 10^{- s} \leq E_{r e l} \leq 5 \times 10^{- s}$
Machine epsilon (round to nearest): $E = \frac{1}{2} β^{1 - t}$
Machine epsilon (truncation): $E = β^{1 - t}$

Lagrange Form: $p (x) = \sum_{i = 1}^{n} y_{i} L_{i} (x)$ where:
- $L_{i} (x) = \prod_{j = 1, j \neq i}^{n} \frac{x - x_{j}}{x_{i} - x_{j}}$
Warning: High-degree polynomials may oscillate (Runge's phenomenon)

Cubic polynomial on each interval: $S_{i} (x) = a_{i} + b_{i} (x - x_{i}) + c_{i} (x - x_{i})^{2} + d_{i} (x - x_{i})^{3}$
Interpolation conditions: $S_{i} (x_{i}) = y_{i}$ , $S_{i} (x_{i + 1}) = y_{i + 1}$
Continuity conditions: $S_{i}^{'} (x_{i + 1}) = S_{i + 1}^{'} (x_{i + 1})$ , $S_{i}^{″} (x_{i + 1}) = S_{i + 1}^{″} (x_{i + 1})$
Boundary conditions:
- Clamped: $S^{'} (x_{1})$ and $S^{'} (x_{n})$ specified
- Natural/free: $S^{″} (x_{1}) = S^{″} (x_{n}) = 0$
- Not-a-knot: Match 3rd derivatives between end segments

Method	Formula	Error	Stability
Forward Euler	$y_{n + 1} = y_{n} + h f (t_{n}, y_{n})$	$O (h^{2})$	Conditional: $h < \frac{2}{λ}$
Backward Euler	$y_{n + 1} = y_{n} + h f (t_{n + 1}, y_{n + 1})$	$O (h^{2})$	Unconditional
Improved Euler	$y_{n + 1} = y_{n} + h f (t_{n}, y_{n})$ $y_{n + 1} = y_{n} + \frac{h}{2} [f (t_{n}, y_{n}) + f (t_{n + 1}, y_{n + 1})]$	$O (h^{3})$	Conditional: $h < \frac{2}{λ}$
Midpoint	$y_{n + \frac{1}{2}} = y_{n} + \frac{h}{2} f (t_{n}, y_{n})$ $y_{n + 1} = y_{n} + h f (t_{n} + \frac{h}{2}, y_{n + \frac{1}{2}})$	$O (h^{3})$	Conditional
RK4	$k_{1} = h f (t_{n}, y_{n})$ $k_{2} = h f (t_{n} + \frac{h}{2}, y_{n} + \frac{k_{1}}{2})$ $k_{3} = h f (t_{n} + \frac{h}{2}, y_{n} + \frac{k_{2}}{2})$ $k_{4} = h f (t_{n} + h, y_{n} + k_{3})$ $y_{n + 1} = y_{n} + \frac{1}{6} (k_{1} + 2 k_{2} + 2 k_{3} + k_{4})$	$O (h^{5})$	Conditional

Convert to system of first-order ODEs using $y_{i} = y^{(i - 1)}$ for $i = 1, 2, . . ., n$

Use two methods of different order
Estimate error: $e r r = | y_{n + 1}^{A} - y_{n + 1}^{B} |$
Adjust step size: $h_{n e w} = h_{o l d} (\frac{t o l}{| y_{n + 1}^{A} - y_{n + 1}^{B} |})^{1 / p}$