Module numerical_methods 

Numerical methods: root finding, ODE solvers, BVP, finite differences, numerical differentiation, quadrature, matrix decompositions, eigensolvers, continuation methods, and bifurcation detection.

Structs

BifurcationPoint

Detected bifurcation point.

Rk45State

Dormand-Prince RK45 adaptive step integrator.

RootResult

Result of a root-finding computation.

Enums

BifurcationType

Bifurcation type.

Functions

adaptive_simpson

Adaptive Simpson’s rule.

arnoldi_eigenvalues

Arnoldi iteration for finding a few eigenvalues of a large matrix.

bisect

Bisection method (guaranteed convergence on bracket).

brentq

Brent’s method (Brentq) for robust bracketed root finding.

bvp_shoot

Solve a second-order BVP y’‘=f(x,y,y’) with y(a)=ya, y(b)=yb.

cholesky

Cholesky decomposition for symmetric positive definite matrices.

deriv_central

Central difference derivative.

deriv_complex_step

Complex-step derivative (machine precision accuracy).

deriv_forward

Forward difference derivative.

detect_bifurcations

Detect bifurcation points on a solution branch.

euler_ode

Euler’s method for ODEs: dy/dt = f(t, y).

eye

Build identity matrix.

fd1d_laplacian

1D finite difference Laplacian matrix (Dirichlet BCs).

fd2d_laplacian

2D finite difference Laplacian (5-point stencil).

fd3d_laplacian

3D finite difference Laplacian (7-point stencil).

frobenius_norm

Frobenius norm of a matrix.

gauss_hermite_integrate

Integrate using Gauss-Hermite quadrature.

gauss_hermite_nodes

Gauss-Hermite quadrature nodes and weights (n ≤ 5).

gauss_legendre_integrate

Compute ∫_a^b f(x)dx using n-point Gauss-Legendre quadrature.

gauss_legendre_nodes

Gauss-Legendre quadrature nodes and weights for n-point rule (n ≤ 5).

hessian

Hessian via central differences (n×n matrix for n-dim function).

infinity_norm

Infinity norm (max row sum).

inverse_power_iteration

Inverse power iteration to find smallest eigenvalue.

jacobian

Jacobian via central differences (n_out × n_in).

lu_decomp

LU decomposition with partial pivoting.

lu_solve

Solve Ax = b using LU decomposition.

mat_mul

Matrix-matrix multiply.

mat_vec

Matrix-vector multiply.

newton_raphson

Newton-Raphson root finding.

power_iteration

Power iteration to find the largest eigenvalue and eigenvector.

pseudo_arclength_continuation

Pseudo-arclength continuation for parameter-dependent systems.

qr_decomp_gs

QR decomposition via Gram-Schmidt (returns Q, R).

qr_eigenvalues

QR algorithm for computing all eigenvalues of a symmetric matrix.

rk2

Runge-Kutta 2nd order (Heun’s method).

rk4

Classic 4th-order Runge-Kutta.

rk45_adaptive

Adaptive RK45 integrator.

rk45_step

Perform one RK45 step. Returns (y_new, error_estimate).

secant_method

Secant method for root finding (no derivative required).

solve_poisson_1d

Solve the 1D Poisson equation -u’’ = f on [0,1] with u(0)=u(1)=0.

svd_singular_values

Compute SVD via Golub-Reinsch (simplified: returns singular values only).

transpose

Transpose an m×n matrix.