This guide helps you quickly recognize which numerical method to use for typical computational and exam problems. It summarizes the main clues and recommended routines from the KI-NUM toolkit.
Clues:
- Integral sign (∫), area under a curve, estimate a definite integral, analyze error vs. n.
Examples:
- “Compute ∫₁² (1/x) dx using Newton-Cotes, plot error.”
Recommended Methods:
SimpsonRuleMidpointRuleTrapezoidalRule
Clues:
- Matrix equation Ax = b, "solve for x", matrix entries defined by integrals.
Examples:
- “Solve Ax = b where A_ij = ∫₀¹ t^{i+j-2} dt.”
Recommended Methods:
GaussGaussPivotGaussSeidelJacobi
Clues:
- "Find a polynomial through all data points", "interpolate", curve must pass through all given points.
Examples:
- “Fit a polynomial exactly through given data, plot result.”
Recommended Methods:
LagrangeNewtonInterpolationVandermonde
Clues:
- “Best fit”, “approximate”, “regression”, “trend”, “minimize sum of squared errors”, noisy data.
Examples:
- “Fit a polynomial of degree 2 to experimental data.”
Recommended Methods:
LSA(Least Squares Approximation)LSS(Least Squares Solution)
Clues:
- “Fit a nonlinear model”, e.g. a(p) = am * bp / (1+bp), “estimate model parameters”.
Examples:
- “Fit a(p) model to data, estimate am and b.”
Recommended Methods:
NonlinearFit
Clues:
- “Find where f(x)=0”, “find intersection”, “where do two curves cross”.
Examples:
- “Find x such that f(x) = 0”, or intersection of two fitted curves.
Recommended Methods:
Bisection- (Newton/Secant if implemented)
| Problem Type | Keywords / Signs | Use These Methods |
|---|---|---|
| Definite Integral | ∫, area under curve, “estimate” | SimpsonRule, MidpointRule, TrapezoidalRule |
| Linear System | Ax = b, solve for x | Gauss, GaussPivot, GaussSeidel, Jacobi |
| Interpolation | through all points, “interpolate” | Lagrange, NewtonInterpolation, Vandermonde |
| Approximation/Trend | “Best fit”, regression, trend | LSA, LSS |
| Nonlinear Fit | Model with parameters, a(p), etc. | NonlinearFit |
| Root/Intersection | f(x)=0, intersection, “crosses” | Bisection |
- Exact fit through all points? → Use Interpolation methods.
- Best fit/trend (not exact)? → Use Regression/Approximation.
- System of equations? → Use Linear Solvers.
- Integral/area under curve? → Use Numerical Integration.
- Root/intersection/zero? → Use Root Finding.
- Estimate parameters in a nonlinear formula? → Use NonlinearFit.