List of Numerical Analysis Topics - Numerical Quadrature (integration)

Numerical Quadrature (integration)

Numerical integration — the numerical evaluation of an integral

• Rectangle method — first-order method, based on (piecewise) constant approximation
• Trapezoidal rule — second-order method, based on (piecewise) linear approximation
• Simpson's rule — fourth-order method, based on (piecewise) quadratic approximation
  • Adaptive Simpson's method
• Boole's rule — sixth-order method, based on the values at five equidistant points
• Newton–Cotes formulas — generalizes the above methods
• Romberg's method — Richardson extrapolation applied to trapezium rule
• Gaussian quadrature — highest possible degree with given number of points
  • Chebyshev–Gauss quadrature — extension of Gaussian quadrature for integrals with weight (1 − x2)±1/2 on
  • Gauss–Hermite quadrature — extension of Gaussian quadrature for integrals with weight exp(−x2) on
  • Gauss–Jacobi quadrature — extension of Gaussian quadrature for integrals with weight (1 − x)α (1 + x)β on
  • Gauss–Laguerre quadrature — extension of Gaussian quadrature for integrals with weight exp(−x2) on
  • Gauss–Kronrod quadrature formula — nested rule based on Gaussian quadrature
  • Gauss–Kronrod rules
• Tanh-sinh quadrature — variant of Gaussian quadrature which works well with singularities at the end points
• Clenshaw–Curtis quadrature — based on expanding the integrand in terms of Chebyshev polynomials
• Adaptive quadrature — adapting the subintervals in which the integration interval is divided depending on the integrand
• Monte Carlo integration — takes random samples of the integrand
  • See also #Monte Carlo method
• Lebedev quadrature — uses a grid on a sphere with octahedral symmetry
• Sparse grid
• Coopmans approximation
• Numerical differentiation — for fractional-order integrals
  • Numerical smoothing and differentiation
  • Adjoint state method — approximates gradient of a function in an optimization problem
• Euler–Maclaurin formula