Module numerical 

Numerical methods: root finding, quadrature, ODE solvers, linear algebra, interpolation. All implementations are from scratch — no external crates.

Structs

CubicSpline

Natural cubic spline interpolant.

Matrix

Dense matrix stored in row-major order.

Functions

adams_bashforth4

Adams-Bashforth 4-step method. Seeds first 4 steps with RK4, then applies the multi-step formula.

adaptive_simpson

Adaptive Simpson’s rule with recursive subdivision.

barycentric

Barycentric coordinates of point p w.r.t. triangle (a, b, c). Returns (u, v, w) such that p = ua + vb + w*c.

bilinear

Bilinear interpolation on a unit square. tl=top-left, tr=top-right, bl=bottom-left, br=bottom-right. tx, ty in [0,1].

bisect

Bisection method. Requires f(a)*f(b) < 0. Returns the root within tolerance tol, or None if bracket invalid or no convergence.

brent

Brent’s method — superlinear convergence without derivative. Requires f(a)*f(b) <= 0.

cholesky

Cholesky decomposition for symmetric positive-definite matrices. Returns lower triangular L such that A = L * L^T. Returns None if not SPD.

determinant

Determinant via LU decomposition.

eigenvalues_2x2

Analytic eigenvalues of a 2x2 matrix.

euler

Forward Euler method. Returns list of state vectors at each step.

fixed_point

Fixed-point iteration: x_{n+1} = g(x_n).

gauss_legendre

Gauss-Legendre quadrature. Supports n = 1..=5 nodes (pre-computed). Maps from [-1,1] to [a,b].

gram_schmidt

Gram-Schmidt orthonormalization.

illinois

Illinois method — a regula falsi variant with superlinear convergence.

inverse

Matrix inverse via LU. Returns None if singular.

lagrange_interp

Lagrange polynomial interpolation at x.

leapfrog

Leapfrog (Störmer-Verlet) symplectic integrator for N-body-style systems. positions and velocities are flat arrays of length 3*N. forces_fn takes positions and returns force vectors (acceleration). Returns steps of (positions, velocities).

lerp

Linear interpolation between a and b.

lu_decompose

LU decomposition with partial pivoting. Returns (L, U, pivot) or None if singular.

lu_solve

Solve LUx = Pb using forward/back substitution.

matmul

Matrix multiplication. Panics if dimensions mismatch.

monte_carlo_integrate

Multi-dimensional Monte Carlo integration. bounds is a slice of (low, high) per dimension. Uses a simple LCG for reproducible sampling.

muller

Muller’s method — quadratic interpolation, can find complex roots (returns real part here).

natural_cubic_spline

Build a natural cubic spline through (xs, ys).

newton_raphson

Newton-Raphson method.

power_iteration

Power iteration for the dominant eigenvalue/eigenvector.

qr_decompose

Thin QR decomposition via Gram-Schmidt.

rbf_interpolate

2D Radial Basis Function interpolation using multiquadric RBF. centers: list of (x, y) center points, values: function value at each center.

rk4

Classical 4th-order Runge-Kutta.

rk45

Dormand-Prince RK45 adaptive step integrator. Returns (time_points, state_vectors).

romberg

Romberg integration — Richardson extrapolation on the trapezoidal rule.

secant

Secant method — Newton without explicit derivative.

simpsons

Simpson’s 1/3 rule. n must be even; if odd, n is incremented by 1.

simpsons38

Simpson’s 3/8 rule. n must be a multiple of 3; adjusted upward if not.

solve_linear

Solve Ax = b via LU decomposition. Returns None if singular.

svd_2x2

Analytic 2x2 SVD: A = U * diag(sigma) * V^T.

transpose

Matrix transpose.

trapezoid

Trapezoidal rule with n sub-intervals (n must be >= 1).

verlet

Störmer-Verlet integrator for second-order ODE x’’ = a(x). Returns vec of (t, x, v).