Methods for solving single variable equations (polynomial roots finding)
- Bisection method
- Fixed point iteration method
- Newton's method
- Secant method
Methods for accelerating convergence
- Aitken's delta^2 method
- Steffensen's method
Methods for solving linear systems
- Standard Gaussian elimination
- Gaussian elimination with partial pivoting
- Gaussian elimination with scaled partial pivoting
- Gaussian elimination with complete (full) partial pivoting
- Jacobi iteration method
- Gauss-Seidel iteration method
- Successive Over Relaxation (SOR) method
Methods for computing eigenvalues (spectral radius)
- Standard power method
- Inverse power method
- Inverse power method + deflation method
- Symmetric power method
- Householder's method + QR factorization method
Methods for polynomial Interpolation
- Taylor expansion (single point, but can do derivatives)
- Lagrange interpolation (multiple points)
- Neville's method (easy updating)
- Newton's divided-difference formula (generalized Taylor form)
- Hermite interpolation (multiple points with derivatives)
- Cubic spline interpolation (piecewise, to avoid oscillation)
Methods for numerical differentiation
- Forward/backward difference formula (1st order accuracy)
- Three-point endpoint/midpoint formula (2nd order accuracy)
- Five-point endpoint/midpoint formula (4th order accuracy)
- 2nd derivative midpoint formula (2nd order accuracy)
- * balancing the truncation error and round-off error
- * Richardson's extrapolation (a general way to enhance the accuracy)
Methods for numerical integration
- Trapezoidal rule (3rd order accuracy)
- Simpson's rule (5th order accuracy)
- Closed Newton-Cotes formulas (generalization)
- Open Newton-Cotes formulas (generalization)
- Composite numerical integration
- Romberg integration (refinement)
- Adaptive quadrature methods
- Gaussian quadrature (with Legendre polynomials)
Methods for approximation
- Least squares approximation (discrete/continuous, Chebyshev polynomials)
- Rational function approximation
- Trigonometric polynomial approximation