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Numerical Root-Finding & Quadrature Methods

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Abstractions about numerical methods for solving equations and evaluating integrals, including interval enclosure (bisection method, interval arithmetic), integral-equation recovery like Carleman's equation, weighted quadrature and special functions (Gauss–Jacobi quadrature, multivariate gamma function), and density-based sampling such as slice sampling and uniform distributions.

7 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

  • Bisection Method — Locate a zero of a continuous real function by preserving an opposite-sign interval and repeatedly replacing it with the sign-changing half, obtaining a deterministic enclosure whose width halves at every step.
  • Carleman's equation — Recover an unknown density on a finite interval from a first-kind integral equation with logarithmic kernel, retaining the endpoint weight and the exceptional interval-length solvability condition.
  • Continuous Uniform Distribution — The bounded continuous probability law whose density is constant, assigning probability in direct proportion to interval length.
  • Gauss–Jacobi Quadrature — An n-node Gaussian rule for the Jacobi weight (1-x)alpha(1+x)beta on [-1,1], using roots of the degree-n Jacobi polynomial and integrating weighted polynomials through degree 2n-1 exactly.
  • Interval Arithmetic — Arithmetic on interval enclosures that preserves inclusion of every admissible exact result, with outward rounding in finite-precision implementations.
  • Multivariate Gamma Function — A dimension-indexed special function that evaluates a gamma-type integral over the cone of real symmetric positive-definite matrices, factors into shifted ordinary gamma terms, and normalizes Wishart-family matrix distributions.
  • Slice Sampling — An MCMC method that adds a height beneath an unnormalized density and updates within its level-set slice using a transition that preserves the target distribution.