Algorithms

Straight line fitting — closed-form fit of y = a + b*x, with parameter variances, covariance, and a goodness-of-fit probability. See Straight Line Fitting Algorithm for a detailed description.

Linear least-squares fitters — fit a linear combination of user-supplied basis functions by solving the normal equations directly (no iteration required). See Linear Least-Squares Fitting Algorithm for a detailed description.

Linear fitting by singular value decomposition — the same linear least-squares problem, solved via SVD instead of the normal equations for robustness to (near-)degenerate basis sets. See Linear Fitting by Singular Value Decomposition for a detailed description.

Levenberg-Marquardt fitters — iterative nonlinear least-squares fitting. See Levenberg-Marquardt Fitting Algorithm for a detailed description.

Trapezoidal rule — the simplest quadrature rule, sampling the integrand at evenly spaced points over a finite, non-singular interval; the basic building block that the other integration methods refine. See Trapezoidal Rule Integration for a detailed description.

Simpson’s rule — combines two successive trapezoidal-rule refinements with a fixed Richardson-extrapolation weighting, converging faster on smooth integrands at essentially no extra cost. See Simpson’s Rule Integration for a detailed description.

Romberg integration — generalizes Simpson’s rule to a full polynomial extrapolation of several successive refinements to the continuum limit; typically the fastest-converging option for sufficiently smooth, non-singular integrands. See Romberg Integration for a detailed description.

Midpoint rule — an open quadrature formula (never evaluates the integrand at the interval’s endpoints), with four variants that handle improper integrals: semi-infinite/infinite ranges, an inverse square-root singularity at either bound, and exponential decay to infinity. See Midpoint Rule Integration for Improper Integrals for a detailed description.

Double exponential rule — a variable transformation that suppresses arbitrary endpoint singularities and converges exponentially fast, without needing to tailor the substitution to the specific singularity. See Double Exponential Rule Integration for a detailed description.

Polynomial interpolation — draws the unique low-order polynomial through the nearest few tabulated points, via Neville’s algorithm; also covers plain linear interpolation and the shared table-search machinery every 1D interpolator below builds on. See Polynomial Interpolation for a detailed description.

Cubic spline interpolation — a piecewise cubic curve with continuous first and second derivatives across tabulated points. See Cubic Spline Interpolation for a detailed description.

Rational function interpolation — quotients of polynomials, better suited than plain polynomials to functions with poles; includes a pole-free barycentric variant. See Rational Function Interpolation for a detailed description.

Coefficients of the interpolating polynomial — the narrower problem of recovering the polynomial’s actual coefficients rather than just evaluating it. See Coefficients of the Interpolating Polynomial for a detailed description.

Two-dimensional grid interpolation — bilinear, tensor-product polynomial, and bicubic-spline interpolation on a complete rectangular grid. See Two-Dimensional Grid Interpolation for a detailed description.

Scattered-data interpolation — radial basis function interpolation, Shepard interpolation, and kriging, for data with no grid structure. See Scattered-Data Interpolation for a detailed description.

Curve interpolation in multiple dimensions — cubic-spline interpolation along an ordered, arc-length parameterized curve through scattered points. See Curve Interpolation in Multiple Dimensions for a detailed description.

Golden section search — the simplest reliable minimizer, needing only function evaluations; also covers the downhill bracketing routine shared by every optimizer below. See Golden Section Search for a detailed description.

Brent’s method — parabolic interpolation with a golden-section fallback, superlinearly convergent. See Brent’s Method for a detailed description.

Brent’s method with derivatives — the same method, using the sign of the derivative to pick better trial points. See Brent’s Method with Derivatives for a detailed description.

Downhill simplex method (Nelder-Mead) — a self-contained, derivative-free method that crawls downhill using a simplex of N+1 points; slow but robust. See Downhill Simplex Method (Nelder-Mead) for a detailed description.

Powell’s method — direction-set method built on repeated 1D line minimizations, requiring no gradient. See Powell’s Method for a detailed description.

Conjugate gradient methods — Fletcher-Reeves and Polak-Ribière variants, building conjugate directions from gradient evaluations in O(N) storage. See Conjugate Gradient Methods for a detailed description.

Quasi-Newton method (BFGS) — builds up an approximate inverse Hessian from gradient evaluations in O(N²) storage. See Quasi-Newton Method (BFGS) for a detailed description.

Bisection method — halves the bracket every iteration; slow but cannot fail. See Bisection Method for a detailed description.

Secant and false position methods — linear interpolation between two points, differing in which point is kept from one iteration to the next. See Secant and False Position Methods for a detailed description.

Ridder’s method — exponential interpolation, quadratically convergent while always staying bracketed. See Ridder’s Method for a detailed description.

Brent’s method — combines bisection, secant interpolation, and inverse quadratic interpolation for reliability and speed. See Brent’s Method (Root Finding) for a detailed description.

Newton-Raphson method — quadratically convergent tangent-line iteration requiring the function’s derivative, plus a "safe" variant with a bisection fallback. See Newton-Raphson Method for a detailed description.

Closed-form polynomial roots — exact solutions for degree 1–3 polynomials, requiring no iteration. See Closed-Form Polynomial Roots for a detailed description.

Laguerre’s method — finds all (real and complex) roots of a polynomial of arbitrary degree, via deflation and polishing. See Laguerre’s Method for Polynomial Roots for a detailed description.

Polynomial — core value type: arithmetic, evaluation, roots, derivatives, and integrals.

LMSEPolynomialEstimator / WeightedPolynomialEstimator — least mean square error fit of a polynomial to (possibly weighted) value/derivative/integral evaluations.

Robust polynomial fitting, built on the robust framework below: RANSACPolynomialRobustEstimator, LMedSPolynomialRobustEstimator, MSACPolynomialRobustEstimator, PROSACPolynomialRobustEstimator, PROMedSPolynomialRobustEstimator.

RANSAC — random sampling and consensus, requires a fixed inlier threshold. See RANSAC (Random Sample Consensus) for a detailed description.

LMedS — least median of squares, no threshold required, computed dynamically instead. See LMedS (Least Median of Squares) for a detailed description.

MSAC — M-estimator sample consensus, a bounded-loss/median hybrid of RANSAC and LMedS. See MSAC (M-estimator Sample Consensus) for a detailed description.

PROSAC — progressive sample consensus, exploits a quality ordering of the samples for large speedups over RANSAC. See PROSAC (Progressive Sample Consensus) for a detailed description.

PROMedS — combines PROSAC’s progressive sampling with LMedS’s threshold-free median criterion. See PROMedS (Progressive Least Median of Squares) for a detailed description.

FastRandomSubsetSelector (and the SubsetSelector framework) — minimal-subset sampling shared by all of the above.

1D convolution — discrete convolution of a signal with a kernel, with configurable edge-handling (zero, constant, repeat, mirror). See 1D Discrete Convolution for a detailed description.

Kalman filter — recursive linear state estimator (predict/correct) for discrete-time linear systems. See Kalman Filter for a detailed description.

Measurement noise covariance estimation — running estimate of a sensor’s measurement noise covariance from samples taken while the underlying state is constant. See Measurement Noise Covariance Estimation for a detailed description.