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The HCubature module is a pure-Julia implementation of multidimensional "h-adaptive" integration. then hcubature(f, a, b) computes the integral, adaptively subdividing the integration volume into smaller and smaller pieces until convergence is achieved to the desired tolerance (specified by optional rtol and atol keyword arguments. Because hcubature is written purely in Julia, the integrand f(x) can return any vector-like object (technically, any type supporting +, -, * real, and norm: a Banach...

ForwardDiff implements methods to take derivatives, gradients, Jacobians, Hessians, and higher-order derivatives of native Julia functions (or any callable object, really) using forward mode automatic differentiation (AD). While performance can vary depending on the functions you evaluate, the algorithms implemented by ForwardDiff generally outperform non-AD algorithms (such as finite-differencing) in both speed and accuracy. Functions like f which map a vector to a scalar are the best case...

This package provides guaranteed methods for finding roots of functions, i.e. solutions to the equation f(x) == 0 for a function f. To do so, it uses methods from interval analysis, using interval arithmetic from the IntervalArithmetic.jl package by the same authors. The basic function is roots. A standard Julia function and an interval is provided and the roots function return a list of intervals containing all roots of the function located in the starting interval.