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. Mourrain, Bernard; Telen, Simon; Van Barel, Marc: Truncated normal forms for solving polynomial systems: generalized and efficient algorithms (2021)
  2. Bu, Ling-Ze; Zhao, Wei; Wang, Wei: Tensor train-Karhunen-Loève expansion: new theoretical and algorithmic frameworks for representing general non-Gaussian random fields (2020)
  3. Aiton, Kevin W.; Driscoll, Tobin A.: An adaptive partition of unity method for multivariate Chebyshev polynomial approximations (2019)
  4. Gorodetsky, Alex; Karaman, Sertac; Marzouk, Youssef: A continuous analogue of the tensor-train decomposition (2019)
  5. Krause, Andrew L.; Ellis, Meredith A.; Van Gorder, Robert A.: Influence of curvature, growth, and anisotropy on the evolution of Turing patterns on growing manifolds (2019)
  6. Massei, Stefano; Mazza, Mariarosa; Robol, Leonardo: Fast solvers for two-dimensional fractional diffusion equations using rank structured matrices (2019)
  7. Piazzon, Federico: Pluripotential numerics (2019)
  8. Sánchez-Garduño, Faustino; Krause, Andrew L.; Castillo, Jorge A.; Padilla, Pablo: Turing-Hopf patterns on growing domains: the torus and the sphere (2019)
  9. Massei, Stefano; Palitta, Davide; Robol, Leonardo: Solving rank-structured Sylvester and Lyapunov equations (2018)
  10. Piazzon, Federico; Vianello, Marco: A note on total degree polynomial optimization by Chebyshev grids (2018)
  11. Després, Bruno: Polynomials with bounds and numerical approximation (2017)
  12. Georgieva, I.; Hofreither, C.: An algorithm for low-rank approximation of bivariate functions using splines (2017)
  13. Hashemi, Behnam; Trefethen, Lloyd N.: Chebfun in three dimensions (2017)
  14. Nakatsukasa, Yuji; Noferini, Vanni; Townsend, Alex: Vector spaces of linearizations for matrix polynomials: a bivariate polynomial approach (2017)
  15. Rajyaguru, Jai; Villanueva, Mario E.; Houska, Boris; Chachuat, Benoît: Chebyshev model arithmetic for factorable functions (2017)
  16. Slevinsky, Richard Mikael; Olver, Sheehan: A fast and well-conditioned spectral method for singular integral equations (2017)
  17. Trefethen, Lloyd N.: Cubature, approximation, and isotropy in the hypercube (2017)
  18. Wilber, Heather; Townsend, Alex; Wright, Grady B.: Computing with functions in spherical and polar geometries. II: The disk (2017)
  19. Bornemann, Folkmar: The SIAM 100-Digit Challenge: a decade later. Inspirations, ramifications, and other eddies left in its wake (2016)
  20. Du, Kui: On well-conditioned spectral collocation and spectral methods by the integral reformulation (2016)

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