ODE solvers

Euler’s method (first order): $y_{n+1} = y_n + hf(t_n, y_n)$, error $O(h)$

Runge–Kutta 4 (RK4): computes four slope estimates per step, error $O(h^4)$:

\[y_{n+1} = y_n + \tfrac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4)\]

Numerical linear algebra

• LU decomposition: $A = LU$ — solves $Ax = b$ in $O(n^3)$
• Conjugate gradient: iterative solver for symmetric positive-definite systems — $O(n\sqrt{\kappa})$
• QR algorithm: computes eigenvalues iteratively

Numerical integration (quadrature)

Trapezoidal rule: $\int_a^b f\,dx \approx \tfrac{h}{2}[f(a) + 2\sum_{i=1}^{n-1}f(x_i) + f(b)]$, error $O(h^2)$

Gaussian quadrature: $n$ optimally chosen points achieve $O(h^{2n})$ accuracy.