Chebfun in three dimensions

We present an algorithm for numerical computations involving trivariate functions in a 3D rectangular parallelepiped in the context of Chebfun. Our scheme is based on low-rank representation through multivariate adaptive cross approximation (MACA). The component 1D functions are represented by finite Chebyshev expansions, or trigonometric expansions in the periodic case. Numerical experiments show the power and convenience of Chebfun3 for problems such as function manipulation, differentiation, optimization, and integration, as well as for exploration of fundamental issues of multivariate approximation and low-rank compression.