An extension of Chebfun to two dimensions. Chebfun is an object-oriented system that was designed on MATLAB for efficient handling of univariate scalar functions, and was first presented by Z. Battles and the second author [SIAM J. Sci. Comput. 25, No. 5, 1743–1770 (2004; Zbl 1057.65003)]. In this paper the reader is introduced to Chebfun2, an extension of Chebfun to represent and manipulate scalar-valued functions of two variables (Chebfun2 objects) and vector-valued functions with two components (Chebfun2v objects), all of them defined on rectangles. An essential point of this system is that the scalar functions are represented in terms of sums of functions of the form u(y)v(x) (low rank approximants), where u(y) and v(x) are univariate functions which in turn are represented as Chebfun objects. The so-called low rank approximations are constructed using an iterative algorithm that is, in a sense, equivalent to the Gaussian elimination with full pivoting. According to the authors, the applicability of this approach relies on a practical fact: an important number of functions of two variables is of low rank or can be approximated by one of this type. For example, the global minimum of a complicated function that can be written as one of rank 4 is found. The paper also discusses the role of the Chebfun2 technology in some relevant issues as global optimization, singular value decomposition, root finding, and vector calculus.

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  1. Massei, Stefano; Palitta, Davide; Robol, Leonardo: Solving rank-structured Sylvester and Lyapunov equations (2018)
  2. Piazzon, Federico; Vianello, Marco: A note on total degree polynomial optimization by Chebyshev grids (2018)
  3. Després, Bruno: Polynomials with bounds and numerical approximation (2017)
  4. Georgieva, I.; Hofreither, C.: An algorithm for low-rank approximation of bivariate functions using splines (2017)
  5. Hashemi, Behnam; Trefethen, Lloyd N.: Chebfun in three dimensions (2017)
  6. Nakatsukasa, Yuji; Noferini, Vanni; Townsend, Alex: Vector spaces of linearizations for matrix polynomials: a bivariate polynomial approach (2017)
  7. Slevinsky, Richard Mikael; Olver, Sheehan: A fast and well-conditioned spectral method for singular integral equations (2017)
  8. Trefethen, Lloyd N.: Cubature, approximation, and isotropy in the hypercube (2017)
  9. Wilber, Heather; Townsend, Alex; Wright, Grady B.: Computing with functions in spherical and polar geometries. II: The disk (2017)
  10. Du, Kui: On well-conditioned spectral collocation and spectral methods by the integral reformulation (2016)
  11. Noferini, Vanni; Townsend, Alex: Numerical instability of resultant methods for multidimensional rootfinding (2016)
  12. Plestenjak, Bor; Hochstenbach, Michiel E.: Roots of bivariate polynomial systems via determinantal representations (2016)
  13. Townsend, Alex; Wilber, Heather; Wright, Grady B.: Computing with functions in spherical and polar geometries. I. The sphere (2016)
  14. Gräf, Manuel; Hielscher, Ralf: Fast global optimization on the torus, the sphere, and the rotation group (2015)
  15. Hochstenbach, Michiel E.; Muhič, Andrej; Plestenjak, Bor: Jacobi-Davidson methods for polynomial two-parameter eigenvalue problems (2015)
  16. Mitrano, Arthur A.; Platte, Rodrigo B.: A numerical study of divergence-free kernel approximations (2015)
  17. Nakatsukasa, Yuji; Noferini, Vanni; Townsend, Alex: Computing the common zeros of two bivariate functions via Bézout resultants (2015)
  18. Townsend, Alex; Olver, Sheehan: The automatic solution of partial differential equations using a global spectral method (2015)
  19. Boyd, John P.: Solving transcendental equations. The Chebyshev polynomial proxy and other numerical rootfinders, perturbation series, and oracles (2014)
  20. Sorber, Laurent; van Barel, Marc; De Lathauwer, Lieven: Numerical solution of bivariate and polyanalytic polynomial systems (2014)

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